An Invitation to Formal Reasoning The logic of terms
FRED SOMMERS Harry A. Wolfson Professor of Philosophy, Emeritus, Brandeis University GEORGE ENGLEBRETSEN Bishop's University
Ash gate Aldershot • Burlington USA • Singapore • Sydney
© Fred Sommers and George Englebretsen 2000 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the publisher. Published by Ashgate Publishing Ltd Gower House Croft Road Aldershot Hants GUll 3HR England Ashgate Publishing Company 131 Main Street Burlington Vermont 05401 USA Ashgate website: http://www.ashgate.com
British Library Cataloguing in Publication Data Sommers, Fred, 1923An invitation to formal reasoning : the logic of terms 1. Logic 2. Reasoning 3. Language and logic I. Title II. Englebretsen, George 160 Library of Congress Control Number: 00132808 ISBN 0 7546 1366 6
Printed and bound by Athenaeum Press, Ltd .. Gateshead, Tyne & Wear.
Contents Preface
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Chapter 1 Reasoning 1. Introduction 2. The Form of an Argument 3. A Word About the Form of Statements 4. The Form of Singular Statements 5. Terms and Statements 6. Symbolizing Compound Statements 7. A Word About Validity 8. How Material Expressions are Meaningful 9. Terms 10. Some Terms are 'Vacuous' 11. Statement Meaning 12. Truth and Correspondence to Facts 13. Propositions 14. 'States of Affairs' 15. The facts and the FACTS 16. What Statements Denote 17. Summary and Discussion on the Meaning of Statements
1 4 4 5 7 9 11 13 13 14 17 19 20 21 22 22 23
Chapter 2 Picturing Propositions 1. State Diagrams 2. Representing Singular Propositions 3. Entailments 4. Negative Entailments 5. STATES and states 6. Positive and Negative 'Valence' 7. The Limitations of State Diagrams 8. The Statement Use of Sentences 9. Truth Relations 10. Logical Syntax
25 28 30 33 35 36 36 38 39 40
v
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An Invitation to Formal Reasoning 11. Term Way vs. Predicate Way 12. Some Useful Terminology 13. Subjects and Predicates
Chapter 3 The Language of Logic (I) 1. Introduction 2. Writing 'Y some X' as an Algebraic Expression 3. Affirmation (+) and Denial () 4. Binary and Unary Uses of a Sign 5. Positive and Negative Valence 6. Contrary Terms and Sentences 7. 'Every' 8. Why Some Equal Sentences are not Logically Equivalent 9. Eforms and Aforms 10. Transcribing Affirmative Statements 11. How to Tell the Valence ofEform Statements 12. Negative Valence= Universal Quantity 13. The Law of Commutation in Eform 14. 'Every' in Eform Transcriptions 15. 'Isn't' 16. The General Conditions ofEquivalence 17. The General Form of Statements 18. The Logical Law of Commutation Applied to Compound Terms 19. The Logical Law of Association 20. Derivations 21. More on Regimenting Sentences 22. Uniquely Denoting Terms and Singular Statements 23. Identities Chapter 4 The Language of Logic (II) 1. Compound Statements 2. 'If... then' 3. More on Transcription
4. 'Or' 5. Representing Internal Structures 6. The General Form of Compound Statements 7. Direct Transcriptions
Contents
8. Relational Statements 9. A Word About Pairing 10. Subject/Predicate; Predicate/Subject 11. 'Dyadic Normal Fonns' 12. Commuting Relational Terms 13. Immediate Inferences from Relational Statements 14. Obversion 15. The Passive Transformation 16. Simplification 17. Pronouns and Proterms Appendix to Chapter 4 18. Bounded Denotation 19. Terms in their Contexts 20. Rules for Using Markers
vn 88 89 91 92 93 95 96 97 98 99 102 103 106
Chapter 5 Syllogistic 1. Validity 2. Inference 3. Enthymemes 4. Why REGAL Works 5. Inconsistent Conjunctions: The Telltale Characteristics 6. Equivalent Conjunctions 7. How This is Related to REGAL 8. Syllogisms with Singular Statements 9. The Laws ofldentity 10. Proofs ofThese Laws 11. The Matrix Method for Drawing Conclusions 12. Venn Diagrams
109 114 118 122 124 127 128 129 130 131 133 135
Chapter 6 Relational Syllogisms 1. Introduction 2. Applying the Dictum to Relational Arguments 3. Distributed Terms 4. Applying DDO 5. Indirect Proofs for Relational Arguments 6. Transforming Arguments 7. Annotating a ProofofValidity 8. Arguing with Pronominal Sentences
139 140 141 143 147 148 150 151
viii An Invitation to Formal Reasoning 9. Distributed Proterms
158
Chapter 7 Statement Logic 163 1. Introduction 165 2. Contradictions 166 3. Tautology 167 4. Inconsistent Statements 5. Contingent Statements 167 6. Direct Proofs 168 7. Rules of Statement Logic Used in Proofs 169 8. Disjunctive Normal Forms (DNF) 175 9. Inconsistency and Validity 176 10. Graphic Representation of Compound Statements 178 11. Regimenting Statements for Treeing ·183 12. Large Trees 185 13. Drawing Conclusions 192 14. Partial Disjunctions 193 15. Using the Tree Method for Annotated Proofs 199 16. Statement Logic as a Special Branch of Syllogistic Logic 201 17. Venn Diagrams for the Singleton Universe of Propositional Logic 209 Chapter 8 Modem Predicate Logic 1. Syntax 2. MPL: The Predicate Way 3. General Sentences in MPL 4. The Logical Language of MPL 5. Singular Sentences in MPL 6. How the Logical Syntax ofMPL is 'Ontologically Explicit' 7. Dyadic Normal Forms 8. Translating Pronominalizations 9. Preparing the TFL Bridge 10. Identity in MPL 11. Logical Reckoning in MPL 12. Transformation Rules 13. Rules oflnference 14. Literal Formulas 15. Reckoning in MPL
213 214 215 216 220 222 224 227 229 230 232 233 235 237 240
Contents
16. Canonical Normal Forms (CNF) 17. Indirect Proofs in MPL 18. Relational MPL Arguments 19. Identity Arguments in MPL
IX
241 242 244 249
Rules, Laws and Principles
253
A Note on Further Reading
259
Preface
It seems to be a fairly widely held belief among contemporary teachers of logic that one must introduce logic via the propositional, and then predicate, calculus. In particular, one would not, even if he or she believed otherwise, properly or fairly serve novice students by offering them instead something like syllogistic logic. Nonetheless, we intend to do just that here: introduce the subject of formal logic by way of a system that is 'like syllogistic logic'. Our system, like oldfashioned, traditional syllogistic, is a term logic. Our version of logic ('termfunctor logic', TFL) shares with Aristotle's syllogistic the insight that the logical forms of statements that are involved in inferences as premises or conclusions can be construed as the result of connecting pairs of terms by means of a logical copula (functor). This insight contrasts markedly with that which informs today's standard formal logic ('modern predicate logic', MPL). That version of logic is due to the work of the great nineteenth century innovator in logic, Gottlob Frege. His insight concerning the logical form of statements was inspired by the language of mathematics. It construes the logical form of statements as the result of functions (incomplete expressions like 'the square root of. .. ' or ' .. .loves ... ') being completed by the insertion of the appropriate arguments (namelike expressions such as '2'or 'Romeo' and 'Juliet'). This difference between TFL and MPL is important because formal logic takes the validity or invalidity of inferences to depend completely on the forms of the statements making up those inferences. Formal logic rests on a theory of logical form (syntax). A second important difference between TFL and MPL is this. Most of the time when inferences are made we need to pay attention to the forms of the statements involved. But sometimes, especially when most or all of those statements are compounds of simpler statements, we can ignore the particular forms of the statements and concentrate instead on the arrangements of simple statements used to form the compounds and, ultimately, the inference itself. The calculus ofMPL which accounts for these kinds of inferences is called 'propositional'. Modern logicians take the logic of unanalyzed statements, the propositional calculus, to be the foundation of all of MPL. This is why
X
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teachers today begin the introduction of fonnal logic with the propositional calculus. When Aristotle invented syllogistic, indeed the whole field offormal logic, in the fourth century B.C. he dealt first and foremost with the logic of terms, the logic of inferences that depend for their validity on the arrangement ofterms within their statements. The logic of propositions was only developed later by Stoic logicians, and then, many centuries later, again by Frege. Following an inspiration by the great seventeenth century polymath Leibniz, TFL incorporates the logic of propositions into the logic of terms by construing entire statements, or propositions, as themselves nothing more than complex terms. So, where MPL sees propositional logic as 'foundational', or primary logic, TFL takes the logic of terms as primary. As it happens, some version ofTFL, either Aristotle's syllogistic or, later, the Scholastic logicians' revised traditional syllogistic, dominated the field offonnallogic until the end of the nineteenth century. Yet even by the beginning of that century logicians had come to agree that traditional syllogistic logic was inadequate for the analysis of a wide variety of inferences. When Frege built MPL he offered logicians a system of logic far more powerful than any system that had gone before it. The power of MPL (its ability to offer analyses of a wide variety of kinds of inference) coupled with Frege's claim that the logic could serve as the foundation of mathematics (by the late nineteenth century mathematicians had become quite worried about the foundations of their field), insured that it would displace the old logic in short order. Today the hegemony ofMPL is almost complete. Still, there is a price to be paid. MPL is indeed powerful, but it is not simple and the logical forms which it ascribes to statements are remote from their natural language forms. Traditional formal logic lacked the scope enjoyed by MPL by not being able to analyze a number of types of inference. Yet it did at least enjoy the double advantage of (i) being simple to learn and use and (ii) construing the logical forms of statements as close to their natural language forms. Clearly a system of fonnal logic which has the power of MPL and the simplicity and naturalness of traditional logic would provide the best ofboth logical worlds. Beginning in the late 1960s Fred Sommers set himself the task of developing a system offormallogic (viz., TFL) that was powerful, natural and simple. The challenge faced by Sommers in accomplishing this was threefold. The first was to extend the power of term logic by incorporating into it the kinds of inferences beyond the powers oftraditionallogic. Those inferences were of three types: inferences involving statements with relational expressions, inferences involving statements with singular terms, and
xii An Invitation to Formal Reasoning inferences involving unanalyzed statements. The second challenge was to offer a theory oflogical form, or syntax, that was natural in the way that the syntax ofMPL was not. The third challenge was to provide a symbolic algorithm (a system of symbols along with rules for manipulating them) much simpler than the one employed by MPL (viz., 'the firstorder predicate calculus with identity'). During the past three decades Sommers has perfected just such a system of formal logic. TFL is at least as powerful as MPL, and it is far simpler and more natural. The most important factor accounting for the difficulty in learning and using MPL is its theory of logical form. By requiring statements to be analyzed as functions completed by arguments it achieves its great power, construing singular, general, relational, and compound sentences in a uniform manner. Predicates (like 'is wise' or 'runs'), quantifiers (like 'some' and 'every'), relational expressions (like 'loves' or 'taught'), and 'sentential connectives' (like 'and', 'only if, or 'not') are all taken to be function expressions. Proper names ('Socrates', 'Romeo'), personal pronouns ('it', 'they', 'he', 'her', etc.), and entire sentences are all taken to be arguments. Thus the following sentences can be given a uniform function/argument(s) analysis. (1) (2) (3) (4)
Socrates is wise Some philosopher is wise Romeo loves Juliet It is cold and it is wet
Symbolically, predicates are symbolized by uppercase letters, proper names by appropriate lowercase initials, pronouns by lowercase letters at the end of the alphabet, unanalyzed propositions by lowercase letters near the middle of the alphabet, and quantifiers by special symbols incorporating the pronouns for which those quantifiers serve as grammatical antecedents. Function expressions are written to the left of their arguments. Finally, parentheses are used as punctuations to ease the reading of formulas. The sentences above are usually formulated by 'translating' them into the standard symbolic notation. Thus:
(1.1) Ws (2.1) (Ex)(Px & Wx)
(3.1) Lrj
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(4.1) c & w
These formulas are unnatural, and the more complex a statement is the farther its logical form is from its natural language form. A sentence such as 'Every dog has a master' is first paraphrased as 'Each thing, call it x, is such that if it is a dog then there exists at least one thing, call it y, such that it is a master and x has it'. This is finally formulated as: (x)(Dx => (Ey)(My & Hxy)). Almost any teacher of MPL today will admit that the most difficult thing students must learn is this process oftranslation. Simple English sentences are paraphrased into sentences saturated with pronouns and sentential connectives, which had no place in the original. TFL requires only the minimum of'regimentation' (paraphrasing into a standard pattern) before symbolization. Translation is replaced by 'transcription'. This is because the TFL syntax of pairs of connected terms is close to the grammatical form of most natural language statements. The symbolic language of TFL is exceptionally easy to learn. All simple terms, singular, general, relational, are marked by uppercase letters. Unanalyzed propositions are symbolized by lowercase letters. All terms are either simple or complex. All complex terms are pairs of connected terms. All unanalyzed statements are complex terms. All terms are either positive or negative. All statements are affirmed or denied. All termpairs are connected by positive or negative functors. Plus and minus signs(+/) are used for all of these. As in arithmetic or algebra, positive signs are often suppressed (compare: '+3+(+4)=+7', read 'positive 3 added to positive 4 equal positive 7', which is normally written as '3+4=7', and read as '3 plus 4 equals 7'). Examples of positive/negative simple terms are 'wise/nonwise' (written: '+W/W'), 'happy/unhappy' ('+H/H'), and 'massive/massless' ('+M/M'). Connective . of pIuses and mmuses . (vtz., . ' +... + ' , ' +... ' , ' ...  ' , an d ' functors are patrs ... +'). The first ofthese indicates the quantity(+ for 'some', 'at least one', etc.;  for 'all', 'every', etc.). The second part of the connective functor indicates the copula (e.g., 'is', 'are', 'was', 'isn't', 'ain't'). Singular terms are marked with an asterisk,*. They have 'wild' quantity; they are indifferently + or, (written '±'). Parentheses are used to group pairs of connected terms. Our sample sentences above would be formulated in TFL as follows. (1.2) ±S*+W (2.2) +P+W (3.2) ±R*+(L±J*)
xiv An Invitation to Formal Reasoning (4.2) +c+w Note that unanalyzed statements (as in 4) are connected by the same functors as other term pairs. This is because such functors only represent relations with given formal features. Thus, for example, the+ ... + functor is symmetric, but not reflexive or transitive. These are just the features that guarantee the validity of such inferences as 'Some philosopher is wise, therefore some wise (person) is a philosopher' and 'It is cold and it is wet, so it is wet and it is cold'. From the point of view of 'formal' logic, only these formal features are of interest. Another source ofdifficulty for beginning students ofMPL is the large variety of rules required to adequately construct proofs of valid inferences. In addition to rules for the propositional calculus, there are rules for eliminating and for introducing each ofthe quantifiers and for manipulating identities. The relative naturalness of TFL' s syntax has already given it a degree of simplicity, which is now augmented by its algorithm for proofs. Since all formative expressions are plus or minus signs, it is easy to show that proof amounts to addition and subtraction (this turns out to be the 'cancelling of middle terms' familiar in traditional syllogistic). The present text book is intended as a tool for the introduction ofTFL to the beginning student of logic. It also includes a final chapter introducing standard MPL. One of the important advantages of coming to formal logic through TFL is that it makes the subsequent learning ofMPL so much easier. For TFL provides 'bridging formulas' that ease the usually difficult translation process that takes natural language statements into MPL formulas. The text contains several exercise sections and a summary of the main rules, laws and principles ofTFL. It is designed so that it could be used for selfteaching. But it is also designed to be used in classrooms as an introductory text for a onesemester course in formal logic. For those going on to do more mathematical logic it is an appropriate and (because of the bridging formulas) useful first text. For the more philosophically oriented it contains extensive discussions of important issues at the intersections of semantics, metaphysics, epistemology and logic. There has been much enthusiasm is recent years for either the replacement or supplementation of courses in formal logic with courses in informal logic. Much of this enthusiasm is due to disenchantment with MPL, which is seen as remote from the ways in which we naturally and ordinarily use our reason and language. In addition, as students arrive at colleges and universities in larger numbers, with a greater variety of
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educational backgrounds and abilities, more of them are enrolled in introductory logic courses. The rigours of standard mathematical logic are often beyond the capabilities or interests of many such students. Thus there is often pressure to 'soften' the blow. We are convinced that mathematical logic ought to be taught, and taught with the appropriate high degree of formal rigour. We also believe that those who seek an account of reason which is more natural and simpler than the one embodied in MPL are right to do so. But one need not abandon formal logic to achieve this end. TFL represents a system of logic which is at once formal, rigorous, powerful, effective, natural and simple. Just as Sommers is the author of TFL, he is the true author of this text. It is a sign not only of his innate generosity but of his deep conviction that logic is more important than any logician that he freely shares his ideas with no concern for personal renown and little sense of proprietorship. Sommers began work on the text in the early 1980s and has thoroughly revised it several times in light of critical suggestions by myself and others, and as the result of his use of the material in the teaching of introductory logic courses over several years at Brandeis University. With less patience and judgment, I have urged completion and publication from the beginning. I have taught MPL for three decades. I have also used much of the material here in the teaching of introductory logic courses during the past few years. I have made use of that experience as well as the results of my own research in logic over the past quarter century to make some minor modifications and additions to the text. There are several people who have been instrumental over the past several years in helping to clarify the ideas in this book, offering critical commentaries, providing useful suggestions, or patiently listening to one or both of the authors go on and on about terms. In addition to the many students who have served as guinea pigs through those years, particular mention must be made of Michael Pakaluk, Graeme Hunter, Thomas Hood, Aris Noah, Philip Peterson, Lome Szabolcsi, George Kennard, William Purdy, Wallace Murphree and David Kelley. Note for Instructors: Some of the material presented in this text deals either with semantic issues or philosophical issues often deemed beyond the scope of a purely technical course in symbolic logic. The instructor who wishes to present a streamlined
xvi An Invitation to Formal Reasoning approach can safely ignore several sections of the text devoted to those less technical topics. For such a course, we would recommend the omission of the following sections: Chapter 1, sections 8 through 17; Chapter 2, sections 1 through 7; all of Chapter 4; Chapter 6, sections 8 and 9 are optional; Chapter 7, sections 16 and 17.
George Englebretsen Lennoxville, Quebec
1 Reasoning
1. Introduction A normal adult possesses information stored in memory in the form of statements like 'Socrates taught Plato', 'Frenchmen eat frog legs', 'my brother is taller than I am' and so forth. Some of the statements in our memory are false but most are true. In any case our ability to retrieve information from the stock of statements we believe to be true is useful to us in countless ways. A good memory is a distinct advantage in life. But just as important is our ability to reason with the information we have. We reason by using one or more of the stored statements as premises to derive another statement, a conclusion, which may not previously have been thought of but which may now be added to the store of information in our possession. Logic is the science that studies reasoning. It shows how to reason well and how to distinguish bad reasoning from good reasoning. A unit of reasoning is called an argument or inference. An argument consists of one or more premises together with the conclusion that has been drawn from them. Any argument is either valid or invalid. When an argument is valid, its conclusion is said to follow from or to be entailed by its premises. As an example of arguing from a single premise to a conclusion, suppose that, knowing of your interest in women's achievements, a friend asks you whether any woman has been a British Prime Minister. Let us say that your memory contains the statement (S 1) Some Prime Minister was a woman. Applying your reasoning capability you will take S 1 as a premise and immediately (perhaps even automatically and unconsciously) derive the conclusion: (S2) Some woman was a Prime Minister.
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An Invitation to Formal Reasoning
Let us call your argument 'A 1': AI:
S 1. Some Prime Minister was a woman. I S2. some woman was a Prime Minister.
(The forward stroke sign should be read as 'therefore' or 'hence'.) You offer S2 to your friend as the answer to his question. Note that S2 may not actually have been in your memory. However, since S2 is entailed by Sl, you can now add it to your store of information. Al is an example of immediate inference. In immediate inference the conclusion is drawn from a single premise. Reasoning like this takes place very quickly, and usually without the conscious application of a technique for deriving conclusions from premises. But a great deal of reasoning is done carefully and reflectively and in many cases by deliberately using a method that has to be learned. Consider an example taken from a book on logic written by Lewis Carroll, the author of Alice in Wonderland. Carroll asks the reader to draw a conclusion from the following premises: (1) Babies are illogical.
(2) Nobody is despised who can manage a crocodile. (3) Illogical persons are despised. Here one must reflect a bit before coming up with the conclusion Carroll has in mind: (4) No baby can manage a crocodile. The whole sequence of four statements is an argument. The first three statements are its premises. The fourth is its conclusion. The next Carroll example is more complicated; in solving it we are well advised not to rely on our unaided wits; it is the sort of problem that is best approached with a logical method or technique for solving just this sort of problem. (1) Everything not absolutely ugly, may be kept in a drawingroom. (2) Nothing that is encrusted with salt is ever quite dry. (3) Nothing should be kept in a drawing room unless it is free from damp. (4) Bathingmachines are always kept near the sea. (5) Nothing that is made of motherofpearl can be absolutely ugly.
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(6) Whatever is kept near the sea gets encrusted with salt. Here mere reflection may get one confused and we are better off not relying on our wits but on a mechanical procedure for drawing conclusions from premises. We will later learn how to do this example by representing the six premises as algebraic expressions that can be added like numbers to derive the conclusion in a mechanical way. By using this technique you will be able to do examples that look fairly complicated very quickly and surely. When one applies the algebraic method to the six premises given by Carroll, one derives the conclusion: (7) No bathingmachine is made of mother of pearl. (This too is Carroll's conclusion; he arrives at it by using a method quite similar to ours.) For the moment we have no method for drawing conclusions from premises. So we shall leave Carroll's droll arguments for later consideration. In learning logic, as in other fields of exact knowledge, we are better advised to begin by attending first to simple easytofollow examples. So let us look again at the simple example of reasoning, AI, where we moved from SI as premise to S2 as the conclusion. AI is a typical example of how we use a truth that we have stored in memory to derive a new truth that we have not (yet) stored. Now it may seem that the move from S I to S2 is trivial. In fact, even in so simple a case as AI, the practical value of being able to infer a new truth from the truths we have at our immediate disposal is enormous. Thus suppose you did not know whether either S I or S2 is true. To find out about S I one need only take a casual glance at the biographies of the British Prime Ministers. There are only nine of these and the biographies are publicly available. Anyone taking the trouble to do this will quickly discover that at least one of them (viz., Margaret Thatcher) is a woman. Having learned that S I is true we should now infer the truth of S2 as well. In this way we get to know about S2 indirectly. We get to S2 by deriving it from Sl. But suppose we were somehow incapal1le of reasoning in the manner of AI. We should then be forced to approach the question of the truth of S2 directly, in the same way we learned about the truth Sl. A direct approach would require us to examine the biographies of all woman to see whether any woman was a Prime Minister. Of course this is a practical impossibility. In effect, if we were unable to reason in the manner of AI, we could not arrive at the truth of S2 at all. Clearly the only sensible and practical way ofleaming
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that S2 is true is indirectly: by way of inferring S2 from S 1. And in fact that is how we do it: S 1 is our starting point and anyone who knows that S1 is true will unconsciously and immediately infer S2 from S 1.
2. The Form of an Argument
S 1 and S2 are converses of one another. Our confidence in the move from S 1 to its converse, S2, is due to the confidence we have in the general pattern of reasoning where we move from one statement taken as premise to its converse. Conversion is a valid form or pattern of reasoning. Let us call this pattern F 1.
F1:
someXisaY I some Y is an X
(Here again, the stroke sign is read as 'hence' or 'therefore.') F1 is also the pattern of the following argument: A2: S3 Some member of the Armed Services Committee is a Southerner. I S4 Some Southerner is a member of the Armed Services Committee. A2, like A1, is of form F1 and we have confidence in any argument of that form. F 1 is an abstract pattern of reasoning and any argument that fits this form is called an instance of this pattern. Thus A1 and A2 are instances ofF 1 and so is A3: A3
some farmer is a noncitizen I some noncitizen is a farmer
3. A Word About the Form of Statements
Every argument consists of two or more statements (a conclusion and one or more premises). Each statement within the argument has a form and the argument as a whole has a form. To reveal the form of a statement we simply replace its terms by 'place holder' letters like 'X' and 'Y'. A place holder letter does not stand for a term; it merely occupies the places that a term or
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letter that stands for a term occupies. For example, by putting 'X' in place of 'ape' and 'Y' in place of 'genius' we show that 'some ape is a genius' has the form 'some X is a Y'. If we do this systematically to each statement of an argument, the form of the whole argument stands revealed. For example, the arguments Al, A2, and A3 are then revealed as all being instances of the argument form F 1. The form of the following argument every cat is a feline no feline is a herbivore I no cat is a herbivore is: every X is a Y no Yis aZ I no X is aZ Another instance of an argument of this form is every Greek is a philosopher no philosopher is a vampire I no Greek is a vampire
4. The Form of Singular Statements
In a statement like 'A president of the United States slept here' the expression 'president of the United States' is being used as a general term. There are many presidents. And when 'president' is used as a general term, it may denote many individuals. But in some uses it denotes no more than one individual. So used 'president' is a uniquely denoting term (UDT). An example is 'President' as it occurs in 'The President is tired'. Proper names are almost always used in a uniquely denoting way. Thus 'Garbo' in 'Garbo was lonely' is a UDT. On the other hand, it is not a UDT in 'Roseanne Barr is no Garbo'. We refer to UDTs as singular terms. A statement whose subject term is singular (e.g., 'Garbo is beautiful', 'The President is tired') is called a
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singular statement. Statements with general terms in subject position (e.g., 'some presidents are funny') are general statements. In a statement like 'Mark Twain is Samuel Clemens' both terms are singular. Such statements are called 'identities'. The difference between a singular and a general statement is semantic; it lies solely in the difference in the terms and not in the forms of the statements. Consider the two sentences i) Garbo was laughing. ii) Children were laughing. Though (i) is singular and (ii) is general, they have the same form. 'Garbo was laughing' has the form 'Some X* is Y' while 'Children were laughing' has the form 'some X is Y'. (We mark UDT occurrence by affixing a star to ·the letter.) From a strictly logical point of view 'Garbo was laughing' should be 'some Garbo was laughing'. In practice that is not done. For we know that whenever 'some Garbo is P' is true, 'every Garbo is P' will also be true (there being only one person who is Garbo). Since 'some Garbo is P' entails 'every Garbo is P' we do not bother to use either 'some ' or 'every' before 'Garbo'. Generally, whenever N* is a proper name, we use the form 'N* is P' and not 'some N* is P'. Nevertheless, for the purpose of seeing how singular sentences function inside of arguments, their form must be made explicit. As speakers ofEnglish we are content to say 'Garbo is laughing'; as logicians we need to represent this as 'some Garbo* is laughing' a statement that entails 'every Garbo* is laughing'. ****************************************************************** Exercises:
I. What is the form of the following arguments? (use X,Y Z) 1. All geographers are patriots. /All patriots are geographers. 2. No geographers are logicians. /No logicians are geographers. 3. Some nonvoters are citizens. /Some citizens are nonvoters. 4. All citizens are patriots, Some natives are citizens. /Some natives are patriots. 5. Bill Clinton is the President. I The President is Bill Clinton. (hint: The premise has the form 'some X* is Y*'.)
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6. Only Garbo is laughing. (hint: Only X is Y can be construed as 'no nonX is Y'. Remember too that 'Garbo' is a UDT.) II. Give two instances for each of the following argument forms. 1. no X is Y I no Y is X 2. some X is Y I some Y is X 3. all X are Y, no Yare Z I no X are Z 4. every Y is nonX* I no nonX* is Y ******************************************************************
5. Terms and Statements Every statement consists of two kinds of components. Elementary statements consist of terms and expressions that join them, called term connectives. For example, 'some women are farmers' consists of the terms 'women' and 'farmer' and the term connective 'some ... are .. .'. Compound statements consist of component statements and expressions that join them called statement connectives. For example the compound statement 'if every person is mortal then some accidents will be fatal' has as its components the two elementary statements, 'every person is mortal' and 'some accidents will be fatal' joined by the statement connective 'if.. then .. '. The terms of an elementary statement are its 'material' components; they carry its matter or content. The term connective is the 'formative' component; it determines the form of the statement. An elementary· statement is either universal or particular in form. For example, 'every logician is a charmer', is universal, being ofthe form 'every X is Y'. 'Some logician is a charmer', which is ofthe form 'some X is Y', is a particular statement. In a compound statement, the component statements are the material elements and the statement connective that joins them is the formative element. Statements of the form 'ifx then y' are called 'conditionals'; those of the form 'x andy' are called 'conjunctions'. (The distinction between material and formative components was first drawn by medieval logicians; they called material expressions 'categorematic', contrasting them to the formative expressions, which they called 'syncategorematic'. Medieval logicians also used the vowels 'a' and 'i' to represent the term connectives in universal and particular statements. Thus a
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An Invitation to Formal Reasoning
statement of form 'every X is Y' was represented as 'YaX' while 'some X is Y' was represented as 'YiX' .) In the case of some arguments, called propositional arguments, the material elements are whole statements and the formative elements are 'statement connectives'. In dealing with propositional arguments we may ignore the internal form and content of the component statements. The following is an example of a propositional argument: A4: some roses are red and no violets are yellow I no violets are yellow and some roses are red Let 'p' stand for 'some roses are red' and let 'q' stand for 'no violets are yellow'. Herethestatements, 'p' and 'q', arethematerialelements. They are joined by the statement connective 'and'. Al may be represented as 'p and q lq and p'. We call 'p' and 'q' 'statement letters' . Unlike term letters, which are formulated using upper case letters, statement letters make use of lower case. The form of A4 is F4:
xandy ly andx
where x and y stand for any two statements. Clearly any propositional argument of the form F4 is valid no matter what statements we substitute for x andy. Thus the following instance ofF4 is valid: A5: some barbers are not Greeks and some Greeks are not barbers I some Greeks are not barbers and some barbers are not Greeks Let 'r' be the statement 'some barbers are not Greeks' and 's' be the statement 'some Greeks are not barbers'. Then we may represent A5 as 'r and s I s and r'. The statements in A5 are different in form and content from the statements in A4. But that does not matter since those differences play no part in the argument which is concerned simply with the move from a conjunction of the form 'x and y' to one of the form 'y and x'. Paying no attention to the internal form of the statements involved, we recognize that A4 and A5 are
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instances of the same general form: x andy I y and x. And any instance ofF4 is valid. Another example of an argument whose validity is not due to the internal form of its component statements is A6:
if there is smoke then there is fire
I if there is no fire then there is no smoke A6 is an instance of the general form: F2: ifx then y I if not y then not x Here too we may replace x and y by any two statements of whatever internal form and content and the result will be a valid argument. Statements of form 'x andy' and 'ifx then y' are called compound statements since they contain two or more component statements joined together by 'and' or 'if... then' or some other formative statement connective. Arguments involving compound statements are the subject of a special branch of logic called Statement Logic (also called Propositional Logic). When the components statements are joined by 'and', the compound statement is called a conjunction and the two components are called 'conjuncts'; an example is 'roses are red and violets are blue'. Suppose that 'p' and 'q' stand for the respective conjuncts. Then one common way to write 'roses are red and violets are blue' in logical language is to use a symbol to stand for the English word 'and'. Thus we may use '&' to represent 'and' and then represent the conjunction as 'p&q'. But another way is to use algebraic operators like '+' for the statement connective. We should then represent the conjunction as 'p+q'. The algebraic way is called a 'transcription' of the English sentence. We shall later find that the algebraic way oftranscribing English sentences makes it especially easy to 'reckon' with them logically.
6. Symbolizing Compound Statements We have been using upper case letters to stand for terms. We shall always use lower cases letters to stand for whole statement. For example, we may let 'p' stands for 'Socrates was executed', 'q' for 'Plato died in his sleep' and 'r' for
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An Invitation to Formal Reasoning
'Aristotle went into exile'. We may then form compound statements which have the elementary statements 'p', 'q' and 'r' as components. For example, we may use 'p', 'q' and 'r' to form such compound statements as 'p and q' (symbolically written 'p & q'), 'ifp then q' (which we symbolize as 'p => q'), 'p or q' (symbolized as 'p v q', 'We then read 'p & (q v r)' as 'p and q orr' and we write 'ifp then (q and r)' as 'p => (q & r)'. In symbolically representing any compound statement we adhere to the convention of using lower case letters to represent the component statements.
******************************************************************* Exercises: Using the symbols'' for 'not,'&' for 'and','=>' for 'ifthen", 'v' for 'or' we represent some common propositional statements thus: notp pandq ifp then q p orq rand (p orq) (randp)orq ifp and q then r if not p then q
p p&q p=>q pvq r&(pvq) (r&p)vq (p&q)=>r p ;:) q
Using the above symbols for the statement connectives, represent the following compound statements in the language of 'symbolic logic':
1. notp orr 2. ifp then notr 3. p ornotp 4. q and (ifr then s) 5. s or not(q and r) 6. if not either p or q then r 7. p or not (q and r) 8. not (p and (q or notr)) 9. (notp and q) or notr 10. p and (a and (rands))
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11. if neither p nor q then not r 12. not (if p then not q) and either p or r
****************************************************************** 7. A Word About Validity An argument whose conclusion follows from its premises is called valid. When an argument is valid and its premises are true its conclusion must also be true. Whether an argument is valid or not, depends on its form. Most forms of argument are invalid. When an argument form is valid, none ofits instances have true premises and a false conclusion. When an argument form is invalid, it will be possible to find an argument of that form that has true premises but a false conclusion. An argument whose premises are true and whose conclusion is false is clearly invalid. Producing one invalid instance shows that all instances of that form are invalid. F1, above, is an example of a valid argument form: it has no invalid instances. But consider F3:
some X is not a Y
I some Y is not an X Now it might seem to us that F3 is a valid argument form. But if it is a valid argument form then we should never be able to find an argument of form F3 whose premise is true but whose conclusion is false. If we can find but a single invalid instance ofF3, that would be conclusive evidence that all arguments of form F3 are invalid. So we look for an invalid instance and after some thought we could come up with the following argument: A7:
some horse isn't a colt
I some colt isn't a horse The conclusion of A7 is false even though its premise is true. So A7 is invalid. But A7 is an instance of F3. And this shows that F3 is an invalid argument form. We call A7 a 'refuting instance' ofF3. Once we find a single refuting instance of an argument form we immediately lose confidence in all of its instances. For example, the following argument, A8, is also invalid:
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AS:
An Invitation to Formal Reasoning some Greek is not a philosopher I some philosopher is not a Greek
Admittedly, AS looks like a good argument. But appearances are deceptive; the conclusion of AS does not follow from its premise. It is after all logically possible for its premise to be true and its conclusion false. For we may imagine a period in history when no one but a Greek is a philosopher, in which case 'some philosopher is not a Greek' is false even though 'some Greek is not a philosopher' is true. But we need not bother to imagine this. For we know that AS is invalid since it is an instance ofF3 and the validity ofF3 was refuted by A7. Thus F3 and Fl are different. Unlike F3, Fl is a valid form of argument and all of its instances are valid. And this means that given any two statements of form 'some X is a Y' and 'some Y is an X', we may be sure that if one of them is true so is the other. In other words we can never find an instance Fl whose premise is true and whose conclusion is false. Let us look also at a third argument form: F5:
no X is a Y /no Yis an X
F5, like Fl, is a valid argument form: no matter what terms we choose for X and Y, no matter what situations we imagine, we shall never find a refuting instance of form F3. In this respect F5 is like Fl and unlike F3. But now the question arises: what makes us so sure that Fl and F5 are valid forms? How do we know that we could 'never' find refuting instances for Fl or F5? After all, F3 also looked like a good way to reason yet we found it to be invalid. Why should we have more confidence in Fl and F5? We here touch on some fundamental issues in logic. Some statements entail one another, so that if one is true the other must also be true. What are statements, and how are they tied in this way? One approach to logic is by way of the concept of truth. 'What is truth?' When a statement is true, what is it about the world that makes it true? To answer such questions we must explain how the terms in a statement are related to things in the world and how the statement itself is related to the world as a whole.
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8. How Material Expressions are Meaningful A statement that has whole statements (represented by lower case letters) as its components is compound and these component statements are its material elements. The material elements of an elementary statement are its terms. Terms and statements are the two basic kinds of material expressions. A material expression (a term or a statement) is meaningful in three ways: 1. It expresses a sense or characterization. 2. It denotes something to which the characterization applies. 3. It signifies a characteristic, property or attribute.
9. Terms We first discuss terms. (1) The term 'farmer', for example, expresses the description or characterization, BEING A FARMER. We call BEING A FARMER the sense or expressive meaning of 'farmer'. {2) 'In a sentence like 'a farmer was going into the bam' the term 'farmer' denotes an individual to whom the characterization BEING A FARMER applies. (3) 'Farmer' signifies the characteristic or attribute, being a farmer, that any farmer possesses. The attribute signified by a term is called its 'significance'. For example, being wise or wisdom is the significance of 'wise'. Note the distinction between the characterization that describes a thing and the characteristic that the thing itself possesses when we correctly describe it. We adopt the practice of writing the characterization in upper case letters and the characteristic in lower case letters. We say that a term expresses a characterization and that it signifies a characteristic. For example, the term 'wise', will be said to express the characterization BEING WISE and to signify the corresponding characteristic of being wise or wisdom. The characterization BEING WISE is said to be 'true of any individual that possesses the characteristic ofwisdom. A term denotes a thing only if the characterization it expresses is true of that thing. In the statement 'someone wise advised me' the term 'wise', which expresses BEING WISE, denotes an individual that possesses the characteristic (being wise, wisdom) that 'wise' signifies. The characterization, BEING WISE, and the characteristic, being wise, correspond to one another.
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An Invitation to Formal Reasoning
Let '[wise]' represent the characterization BEING WISE and let '<wise>' represent the characteristic of wisdom that any wise person possesses. Then 'wise' expresses [wise] and signifies <wise> and what 'wise' expresses corresponds to what it signifies. Also [wise] correctly describes (is true of) whoever is wise. And generally, if'#' is a term, the characteristic ,<#>, that a #thing possesses is said to correspond to the characterization, [#], that characterizes (is true of) the #thing.
10. Some Terms are 'Vacuous' A term like 'mermaid', 'flying saucer' or 'woman who will love living on Pluto' does not fail to express a characterization. But it may fail to denote. Terms that fail to denote are called vacuous. Consider the term 'mermaid! in the statement 'a mermaid lives in the bay'. The sense or expressive meaning of 'mermaid' is BEING A MERMAID. The characterization expressed does not characterize anyone since no one possesses the characteristic of being a mermaid. Thus 'mermaid' fails to denote anyone or anything. Proper name terms are usually not vacuous. But some do express characterizations that characterize no one. There is an ancient tradition that a hero called Theseus founded the city of Athens. Suppose that Theseus was only a legendary figure and that no such person as Theseus ever existed. In that case no one ever possessed the characteristic of being Theseus and 'Theseus' fails to denote anyone. All the same, 'Theseus' has a sense; it expresses the characterization of BEING THESEUS, a characterization that is not true of anyone but which nevertheless is the expressive meaning of 'Theseus'. A vacuous term such as 'Theseus' or 'mermaid' is analogous to a statue of Theseus or a picture of a mermaid. The term 'mermaid' denotes no one. Similarly, the picture does not portray a mermaid (in the sense of 'portray' that a photo taken of a bridesmaid at my cousin's wedding is a portait of the bridesmaid). Nevertheless, the mermaid picture (like the term 'mermaid') is representational. We all understand what the picture means; as it were, the picture 'expresses'BEING A MERMAID. Similarly, the expressive meaning of the term 'mermaid' is understood by most speakers of English. In that sense what 'mermaid' means is something public and objective. We all 'grasp' it. (Considered as an object of understanding that we all grasp, the sense of a term is called a 'concept'.) We may put the matter
Reasoning
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this way: The characterization expressed by any term (even one that is vacuous) exists as a 'concept'. Thus every meaningful term expresses a sense (or concept) and no term is expressively vacuous. (This doctrine, that concepts exist, is called Conceptual Realism.) No meaningful term is expressively vacuous. What about signification? Can a term be vacuous by signifying nothing? Some persons are kind, others are cruel. But nobody is perfect. So there is kindness and there is cruelty but no perfection. If nobody is perfect, nothing possesses the characteristic of being perfect. What is the status of a characteristic like
that nobody possesses? There is an ancient dispute about characteristics that nothing possesses. Some philosophers and logicians, called Platonic Realists, follow Plato in holding that a characteristic <X> exists even if there are no X things so that nothing or no one possesses <X>. Others (among them, the authors of this text who are Conceptual Realists but not Platonic Realists) deny this. They hold that a term like 'mermaid' or 'perfect' expresses a characterization but it fails to signify any characteristic. For there is no such thing as perfection and no such characteristic as being a mermaid. (On the other hand, there are such things as BEING A MERMAID and BEING PERFECT. For example, these meanings are expressed by the two terms in 'no mermaid is perfect'.) According to this view (which we shall 'officially' adopt) a term, 'T', will always express a sense or characterization, [T], but ifthere are noTthings, then there will be no characteristic for 'T' to signify. Thus 'mermaid' is expressively meaningful; it expresses BEING A MERMAID but it lacks both denotation and significance. We noted earlier that when a term lacks denotation we call it 'vacuous'. The nonPlatonist philosopher believes that 'mermaid' is doubly vacuous: not only does it fail to denote, it also fails to signify. Using the square brackets for the characterizations and the angle brackets for the characteristics, we summarize the above account of the meaning of terms. If 'T' is a term, then 1. 'T' expresses [T] or BEING T. If there are T things then 2. 'T' denotes T things, 3. 'T' signifies , the attribute ofTness or being T, 4. [T] corresponds to . 5. [T] characterizes (is true of) some T thing If nothing is T, then 'T' is doubly vacuous because
16
An Invitation to Formal Reasoning 6. 'T' fails to denote aT thing. 7. 'T' fails to signify, (there being no such thing as ).
******************************************************************* Exercises: I. Give the sense and significance of the following terms: Examples: 'pious (person)' sense: BEING (A) PIOUS (PERSON), significance: being pious, piety 'logician' sense: BEING A LOGICIAN significance: being a logician 1. bridesmaid 2. red (thing) 3. unmarried ll. Repeat example I, using the bracket notation for the sense and significance of terms. Example: 'pious' sense: [pious] significance:
Ill. Which of the following terms lacks denotation and significance? bridesmaid mermaid sea cow sea squirrel
N. To what attributes, if any, do the following characterizations correpond? BEING A BACHELOR BEING A MARRIED BACHELOR BEING A MERMAID
******************************************************************
Reasoning
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11. Statement Meaning Statements and terms are the two basic kinds of material expressions. Terms are used for characterizing things in the world. Statements are used for characterizing the world itself. We use lower case letters to represent statements. Like a term, a statement, 's', has three modes of meaning: (1) it expresses a sense or characterization, [s]. (What a statement expresses is called a proposition); (2) it denotes the world characterized by the proposition it expresses and (3) it signifies a characteristic of the world. (The characteristic, <s>, signified by a statement is called afact.) By definition, a statement is an utterance that is being used for saying something. And 'what is said' or expressed is a proposition. The proposition expressed may or may not characterize the world. If it does, the proposition is called true. If it does not, the proposition is false. Calling a statement true is a convenient shorthand way of saying that it expresses a true proposition. Not all utterances are statements but no utterance that is a statement can fail to express a proposition. Since every statement expresses a proposition that is either true or false, every statement is itself said to be true or false. How is the world characterized? What are its characteristics? We may think of the world as a collection or totality of things. (Here 'thing' is used in its widest sense to apply to whatever may be said to be present in the world including such things as London, The President of the United States, snow, hurricanes, pollution, wisdom, democracies and friendship.) Any totality, be it large or small, finite or infinite, is basically characterized by what is present in it and by what is absent from it. We may speak of such characterizations as 'existential'. Consider the collection of things now lying on your desk. Assume that the constituents of this little totality include a pen, ink, a lamp, a notebook and nothing else. Among the infinity of things not in this totality are horses, mermaids, envelopes, screwdrivers, etc. Suppose we call a totality ' {Q} ish' if it has a Q thing as a constituent and 'un {Q} ish' if it has no Q constituent. We may then existentially characterize the totality of things on your desk by saying that it is {pen} ish, {ink} ish, {lamp} ish and {notebook} ish. But negative characterizations are also true of it: for example, the little totality can be characteritzed negatively by saying that it is un {horse} ish, un {screwdriver} ish, etc. Any statement is a truth claim. Looking into a drawer I say 'there is no screwdriver', thereby claiming that the little totality under consideration, whose constituents are the objects in the drawer, is un {srewdriver} ish. We call
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An Invitation to Formal Reasoning
the totality of things under consideration when a given statement is made or when a given argument is presented 'the domain of the claim' (DC). Logicians sometimes refer to the DC as 'the universe of discourse'. Very often the totality under consideration is the whole world. For example, in asserting 'some women are farmers' one expresses the characterization SOME WOMEN BEING FARMERS or BEING {WOMAN FARMER}ISH, claiming (correctly) that this characterizes the contemporaneous world. In what follows we shall assume that the DC of a statement is the world. Here are some existential characterizations (propositions) that are true of the world at this time: BEING {WOMAN FARMER} ISH (that some women are farmers) BEING {ELK} ISH (that there are elks) BEING UN{ELF}ISH (that there are no elves) BEING UN{MERMAID}ISH (that there are no mermaids)
If we use the bracket notation to represent what a statement expresses, then a statement, 's', expresses the proposition [s]. For example, 'some women are farmers' expresses the proposition [some women are farmers] and 'there are no mermaids' expresses [there are no mermaids]. Using the convention of upper case letters for representing characterizations, we can also write 'BEING {ELK}ISH' as 'THE EXISTENCE OF ELKS' and 'BEING {MERMAID} ISH' as 'THE EXISTENCE OF MERMAIDS'. As it happens that there are mermaids does not correctly characterize the world. Note that our upper case convention does not extend to the form 'that s'. Thus we say that 'there are elks' expresses the proposition that there are elks. Equivalently we could say it expresses the proposition BEING {ELK} ISH, THERE BEING ELKS, THE EXISTENCE OF ELKS. To every true characterization there corresponds a characteristic ofthe world. Consider the statement, 'there are elks'. This statement expresses the true proposition that there are elks, a proposition that is true of the world because of the presence of elks, an existential characteristic of the world. In general, a totality that has a Q constituent has the characteristic of {Q}ishness. That is to say, the existence (or presence) of a Q thing ({Q}ishness) is a (constitutive or 'existential') characteristic ofthe totality. If the world is Qish, then BEING {Q}ISH (the proposition expressed by 'there are Q things') corresponds to being Qish or {Q}ishness (a world characteristic). Consider 'there are no K things'. If the totality is un {K} ish, then it is characterized by the nonexistence ofK things (by un{K}ishness, by
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being un{K}ish). The proposition that there are no K things then corresponds to the un{K}ishness of the world. Un{elf}ishness (the nonexistence of elves) is a negative existential characteristic ofthe world; {elk}ishness (the existence of elks) is a positive existential characteristic of the world. A more familiar term for a positive or negative world characteristic is 'fact'. The existence of elks ( {elk}ishness) is a positive fact; the nonexistence of elves (un{elf}ishness) is a negative fact.
12. Truth and Correspondence to Facts The world's existential characteristics constitute the facts. Facts are what make true propositions true. A world characterization is true (or true of the world) if it corresponds to a characteristic of the world. Each of the above characterizations corresponds to an existential characteristic of the world. For example, the existence of women farmers ( {women farmer} ishness), is the fact that corresponds to and confers truth on such characterizations as SOME WOMEN BEING FARMERS, BEING {WOMAN FARMER}ISH, THE EXISTENCE OF WOMEN FARMERS, THERE BEING WOMEN FARMERS, {WOMEN FARMER}ISHNESS, and that there are women farmers, which are all different but equivalent expressions standing for [some women are farmers], the proposition expressed by 'some women are farmers'. A statement that expresses a true proposition is true. Since [some women are farmers] corresponds to <some women are farmers>, it is a true proposition and the statement expressing it is a true statement. A proposition (and the statement that expresses it) is false if the proposition does not correspond to any characteristic of the world. For example, SOME THING BEING AN ELF is a false characterization that does not correspond to any fact, there being no such fact as {elf}ishness. Among the world's existential characteristics (facts) are the following: the existence of horses; {horse} ishness, being {horse} ish the nonexistence of elves; un {elf} ishness, being un {elf} ish the existence of women farmers; {woman farmer}ishness the existence of elks; {elk} ishness the nonexistence of mermaids; un{mermaid}ishness Using the angle bracket notation for the facts that are characteristics of the world, we represent the fact signified by 'there are elks' by '
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An Invitation to Formal Reasoning
elks>' and the fact signified by 'there are no elves' by ''. This negative fact, corresponds to the proposition that there are no elves, making it true.
****************************************************************** Exercises: What fact makes the following true: 1. that there are no mermaids answer: the nonexistence of mermaids 2. THERE BEING RICH BACHELORS answer: ? 3. that some even number is prime answer: the existence of 4. [Bertrand Russell did not write Waverly] answer:< ? > 5. that no pope is female answer: the nonexistence of ?
****************************************************************** 13. Propositions Generally, any statement 's' expresses as its sense the proposition that s. (which we symbolically represent as '[s]'). The form 'that s' is so common a way oftalking about the sense of's' that we continue to use lower case letters for it. For example, in asserting 'some women are farmers' we claim that SOME WOMEN BEING FARMERS obtains, (is a true characterization of the world). Equivalently we are claiming truth for the proposition that some women are farmers. As it happens, the existence of women farmers is a fact; [some women are farmers] corresponds to the fact <some women are fanners>. So [some women are farmers] is a true proposition.
Reasoning
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Just as the proposition, [s], expressed by a statement 's' can be spoken of in different ways, so the fact, <s>, that 's' signifies, can be spoken of in different ways. Here are some of the ways we may speak of the fact that some women are farmers. some women being farmers the existence of women farmers {woman farmer} ishness the state of affairs in which some women are farmers All these phrases are equivalent ways of talking about one and the same fact: the existence of women farmers. We have now shown how to answer the question: What is there about the world that makes a true statement true? Consider any statement of form 'some thing is a Q thing'. Ifthe world is characterized by {Q}ishness, the statement signifies a fact that corresponds to its sense. That fact the existence of a Q thingmakes the statement true. If the world is un{Q}ish, the statement is false; in that case un {Q} ishness is a fact and the contradictory statement 'no thing is a Q thing' is true. Consider again the true statement 'there are no mermaids'. This statement expresses the true negative proposition that there are no mermaids. Equivalently, we may think of the proposition expressed as a STATE OF AFFAIRS: THE NONEXISTENCE OF MERMAIDS. The proposition is true (the STATE obtains) because un{mermaid}ishness, a negative existential characteristic of the world is a fact. This fact, which is signified by 'there are no mermaids', makes the proposition true. And that in tum means that the statement expressing it is true.
14. 'States of Affairs' We commonly speak of a statement as expressing a 'state of affairs'. Here one should distinguish between STATES that are expressed and the states that are signified. What a statement expresses is a proposition or STATE OF AFFAIRS, what it signifies, if anything, is a fact or state of affairs (lower case). False statements express STATES but they do not signify states. For example, THE EXISTENCE OF MERMAIDS is the STATE OF AFFAIRS expressed by 'there are mermaids'. But there is no such state of affairs as the existence of mermaids. So THE EXISTENCE OF MERMAIDS does not
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An Invitation to Formal Reasoning
obtain. (A STATE that corresponds to a state characterizes the world and is said to 'obtain'.)
15. The facts and the FACTS Here again we distinguish between an upper and lower case meaning of a key word. In its primary meaning the word 'fact' denotes an existential characteristic of the world but 'fact' is also often used as a synonym for 'true proposition' or STATE OF AFFAIRS that obtains. Taken as a synonym for 'true proposition' the word 'fact' should be written in upper case. For example, that some farmers are women is a true proposition; we may call this proposition a FACT. FACTS are true of the world. That there are mermaids (The EXISTENCE OF MERMAIDS) is not a FACT. On the other hand; the NONEXISTENCE of MERMAIDS, that there are no mermaids, is a FACT or true proposition. The nonexistence of mermaids is a fact (lower case). This fact is a negative existential characteristic of the world. FACTS correspond to, are made true by, facts.
16. What Statements Denote Suppose I am at the zoo and say 'that elk keeps staring at me'. The nonvacuous term 'elk' signifies the characteristic of being an elk and it denotes something that has the characteristic signified. Just as nonvacuous terms denote what they characterize so do true statements. A true statement such as 'there are elks' signifies a (positive, existential) characteristic of the world ( {elk} ishness) and it too denotes something that has the signified characteristic. Since it is the world that possesses the characteristic of {elk} ishness, the statement 'there are elks' denotes the world. A true negative statement such as 'there are no elves' signifies a (negative existential) characteristic of the world (its un{elf}ishness) and it too denotes what has the signified characteristic. Thus 'there are elks' and 'there are no elves' signify different facts but both denote one and the same world. The false statement 'there are elves' expresses a characterization ({ELF} ISHNESS) but, like the vacuous term, 'elf', the statement denotes nothing and signifies nothing. True statements signify positive or negative facts. Facts differ from one another. All true statements denote one and the same world. As for false statements, they have expressive meaning but apart from that they are vacuous.
Reasoning
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17. Summary and Discussion on the Meaning of Statements If 's' is a statement then 1. 's' expresses [s] (the proposition that s) If [s] is true of the world, then <s>, a fact, is an existential characteristic of the world and 2. 's' signifies <s>, 3. [s] corresponds to <s>, 4. [s] 'obtains', is true, is a FACT, 5. 's' denotes the world, 6. 's' is true. If there is no such fact as <s> then 1* [s] does not correspond to any fact, 2* [s] does not obtain, is false, is not a FACT, 3* 's' does not denote the world, 4 * 's' does not signify a fact, 5* 's' is false.
******************************************************************* Exercises: I. What proposition or STATE OF AFFAIRS is expressed by each of the following statements? What fact or state (if any) does it signify? What, if anything does it denote? (The answers to (1) and (2) are given by way of illustration.) 1. no senator is a citizen: expresses TilE NONEXISTENCE OF SENATORS WHO ARE CITIZENS claiming that it obtains. But it fails to signify a fact. (An equally good answer is: ( 1) expresses the proposition that there are no senators who are citizens but this proposition does not correspond to any state of affairs and so ( 1) is false and it does not denote the world. 2. some waiters are not friendly: expresses the proposition that some waiters are not friendly. (Equivalently it expresses SOME WAITERS NOT BEING FRIENDLY (a FACT) or THE EXISTENCE OF UNFRIENDLY WAlTERS.) It signifies a well known fact, the existence of unfriendly waiters. (Another technically correct answer is: (2)
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An Invitation to Formal Reasoning
expresses [some waiters are not friendly], a true proposition that corresponds to <some waiters are not friendly>, the fact signified by (2). THE PRESENCE OF UNFRIENDLY WAlTERS correctly characterizes (is true of) the world so the statement expressing this FACT denotes the world.) 3. 4. 5. 6.
not a creature was stirring some mammals lay eggs. some birds do not fly. no bird is immortal
II. Which of the following statements denote the world? 1. There are no elves. 2. Some citizens are not farmers. 3. All women are citizens. 4. Elvis lives. 5. The France is a republic. III. Which ofthe following is incorrect? 1. Any statement is a truth claim made with respect to a specific domain, called the domain of the claim (DC). 2. A statement's' signifies <s>, only if <s> is an existential characteristic of the DC of 's'. 3. If's' is a true statement, then [s] corresponds to the world. 4. A statement, 's' denotes its DC, if and only if <s> is a fact. 5. A statement, 's,' denotes its DC if and only if[s] is true of its DC. 6. A fact is a property of the world. 7. FACTS are true propositions. 8. FACTS correspond to facts. 9. Vacuous statements are meaningless. 10. That some dogs are not friendly is a negative FACT. 11. That no dogs are friendly is a negative FACT. 12. False statements are not vacuous.
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2 Picturing Propositions
1. State Diagrams The propositions expressed by statements are STATES OF AFFAIRS. In what follows we sometimes use 'STOA' or 'STATE' as an abbreviated way ofwriting 'STATE OF AFFAIRS'. A STOAmay or may not be a FACT. A STOA or proposition is true (obtains, is a FACT) if it corresponds to a fact and false if it does not. Calling a statement true (or false) is just a convenient way of saying that the proposition it expresses is true (or false). John Venn, a nineteenth century logician, introduced a way of depicting simple propositions or STOAs by means of diagrams. The two Venn Diagrams below show how to represent two STATES the world could be in with respect to the presence or absence of mermaids. Figure 1
Figure 2 mermaids
25
26
An Invitation to Formal Reasoning
A square figure represents all the things in the domain. Any things inside the circle are mermaids. Any things outside are nonmermaids. By placing a cross inside the circle labeled 'mermaids' we indicate that the circle is not empty. Thus the first diagram represents the positive STATE OF AFFAIRS: SOME THINGS BEING MERMAIDS or THE EXISTENCE OF MERMAIDS. In the second diagram the mermaid circle is shaded. By shading the circle we signify that it is empty. Figure 2 represents the negative STATE OF AFFAIRS: the NONEXISTENCE OF MERMAIDS. Figure 2 represents a STOA that is a FACT. Figure 1 represents a STOA that is not a FACT. The two statements that express the depicted STATES are: 1. 2.
there are mermaids (Figure 1) there are no mermaids (Figure 2)
(1) claims that the EXISTENCE OF MERMAIDS corresponds to a fact. (2) claims that the NONEXISTENCE OF MERMAIDS corresponds to a fact. What (1) claims is false; there is no such fact as the existence of mermaids. (2) is true; the nonexistence of mermaids is a fact. Logicians are particularly interested in the STATES expressed by certain basic statements. Let 'S' and 'P' be two terms. Then the STATE, expressed by 'someS is P' is THE EXISTENCE OF AN SP THING:
Figure 3
[someS is P)
The STATE expressed by 'noS is P' is THE NONEXISTENCE OF AN SP THING:
Picturing Propositions
Figure 4
27
[noS is P]
The STATE expressed by 'every Sis P' is THE NONEXISTENCE OF AN
S(P) TIUNG: Figure 5
[every S is P]
The STATE expressed by 'someS is not P' is THE EXISTENCE OF AN S(P)TIDNG:
Figure 6
[someS is not P]
Because 'no S is P' contradicts 'some S is P' Figure 4 is shaded (indicating absence) where Figure 3 shows occupancy or presence. Similarly the Figure
An Invitation to Formal Reasoning
28
for 'every S is P' shows absence where that for 'some S is not P' shows presence. Note that 'noS is notP' expresses the same STATE as 'every Sis P'. For example, 'no senator is not pragmatic' claims that the world is characterizedbyTHEREBEINGNONONPRAGMATICSENATORS. The same claim is made by 'every senator is pragmatic'; both statements express the STOA depicted by shading the S(P) segment (figure 5), depicting the absence of anything that is S and nonP.
2. Representing Singular Propositions A statement that has a proper name or other uniquely denoting term in subject position is called singular. Examples of singular statements are: Socrates is wise. Garbo is beautiful. The President is tired. Bigfoot is hairy.
S* is W G* is B P* is T B* isH
As we saw in the first chapter, a singular term letter is affixed with a star to indicate that the term it stands for is a uniquely denoting term (UDT), i.e., a term that applies to no more than one individual. Note that 'president' in the phrase 'The president' is represented by a starred letter. In the context of a phrase ofthe form 'the S', the term'S' is a UDT and so we star it. The singular statement 'Bigfoot exists' expresses the singular proposition: THE EXISTENCE OF BIGFOOT:
Figure 7
B*()
Picturing Propositions
29
All things other than Bigfoot are outside the circle. The circle represents a set of things which, if it is occupied, has only one thing in it, Bigfoot. The cross indicates that the set of things that are Bigfoot is not empty, thus the diagram represents the claim that some thing is Bigfoot, that Bigfoot exists. The corresponding negative proposition expressed by 'Bigfoot does not exist' would then be represented thus:
Figure 8
The shading indicates that nothing is Bigfoot. From a logical point of view, a singular statement, 'N* is P' has the form 'some X* is Y' since it claims existence. For example, 'Bigfoot is hairy' claims that the world is characterized by the presence of a hairy creature known as Bigfoot. However, since there is no more than one Bigfoot, 'some Bigfoot* is hairy' entails 'every Bigfoot* is hairy'. The STATE expressed by 'Bigfoot is hairy' is a STATE of absence as well as presence: Figure 9
[(some/every Bigfoot* is hairy]
H
30
An Invitation to Formal Reasoning
******************************************************************* Exercises: Represent the propositions expressed by the following statements by means of Venn diagrams: 1. There are Eskimo senators. 2. There are no women moonwalkers.
3. No one is perfect. 4. Socrates is wise. 5. All humans are mortal. 6. There is no such person as the present King of France. (hint use 'K*' for 'present King of France' and treat the statement as 'The K* does not exist' or as 'Nothing is K*'. · 7. Russell was a genius. 8. Some actors are not rich. 9. Some who are rich are not actors. 10. Every fool is unwise.
****************************************************************** 3. Entailments We noted earlier that the truth of any statement claiming THE EXISTENCE OF SOMETHING THAT IS BOTH X AND Y entails the truth of its converse. In other words if the proposition expressed by 'some X is Y' is true, then the proposition expressed by 'some Y is X' must also be true. And again, to say that one statement entails another is a convenient way of saying that the proposition expressed by the first statement entails the proposition expressed by the second. We now turn to the task of explaining how one true proposition can entail the truth of another proposition. Consider again the statement s3
some citizen is a farmer
We represent the STATE expressed by s3 (that is, the STATE [s3]) by:
Picturing Propositions
Figure 10
c
31
F
Figure 10 depicts the EXISTENCE OF A CITIZEN FARMER and s3 is true if [s3], the STATE depicted, characterizes the world. (As it happens, the existence of citizens who are also farmers is a fact. Thus [s3] is a true proposition and so the statement, s3, that expresses this proposition, is true.) Now consider s4
some farmer is a citizen
The STOA expressed by s4 is [s4], depicted in Figure 11
Figure 11
c
i.e., the EXISTENCE OF A FARMER WHO IS ALSO A CITIZEN. It is clear that the two Venn Diagrams are like two photos of the same state taken from different angles. It is clear, in other words, that the state that makes [s4] true is the very same state that makes [s3] true. Putting 'X' for 'citizen' and 'Y' for 'farmer' we see generally that any two statements of the form 'some X is a Y' and 'some Y is an X' will express one and the same STATE OF AFFAIRS. This being so, it will never be possible for one ofthese statements to be true and the other false. For if the STATE in question obtains, both will be true and ifthe STATE does not obtain both will be false. In this way the mutual entailment that holds between s3 and s4 is represented by the Venn diagrams that picture [s3] and [s4] as a single STATE.
32
An Invitation to Formal Reasoning Consider also the two statements:
s5 s6
no senator is an albino no albino is a senator
The following Venn Diagrams depict in different ways the single negative STATE expressed by s5 and s6.
Figure 12
In general, any two statements of the form 'no X is Y' and 'no Y is X' express one and the same (negative) STATE OF AFFAIRS, which makes it impossible for one to be true and the other false. When two statements express one and the same STATE OF AFFAIRS they mutually entail each other and we call them logically equivalent. Sometimes we have entailment one way but not the other way. Consider for example the following little argument: s7 s8
some farmer is a citizen and a poet /some farmer is a citizen
The two statements are not logically equivalent: [s7] entails [s8] but [s8] does not entail [s7]. We can see why by looking at the diagrams of the STATES each statement expresses.
Picturing Propositions
Figure 13 F
33
Figure 14
c
The diagram on the left represents [s7], the STOA expressed by s7: The EXISTENCE OF A FARMERCITIZENPOET. The diagram on the right represents [s8], the STOA expressed by s8: the EXISTENCE OF A FARMERCITIZEN. Note that the EXISTENCE OF A FARMERCITIZEN, represented in diagram 14 by an 'x' in the overlap of the two circles is already shown in the diagram for 13 as part of the STATE which is the EXISTENCE OF A FARMERCITIZENPOET. For in diagram 13 we already have an x in the overlap of the 'farmer' and 'citizen' circles. Thus, if the EXISTENCE OF A FARMER CITIZENPOET is a FACT, so is the EXISTENCE OF A FARMERCITIZEN. But the converse does not hold: diagram 14 could represent a FACT even if Diagram 13 did not. For it could be the case that there is no farmercitizen who is also a poet. The STATES depicted in the two diagrams are positive; the diagrams show that one of the STATES of EXISTENCE is included in the other, thereby grounding the entailment of one proposition by the other. More generally, if S 1 and S2 are any two statements and the STATE, [S 1] includes the STATE [S2], then Sl entails S2.
4. Negative Entailments A relation of entailment may hold between statements expressing negative STATES, STATES OF NONEXISTENCE. Ifthere are no mermaids, there are no fluteplaying mermaids: the NONEXISTENCE OF MERMAIDS excludes the EXISTENCE OF FLUTEPLAYING MERMAIDS ('includes' the NONEXISTENCE OF FLUTEPLAYING MERMAIDS). And that is why 'nothing is a mermaid' will entail 'nothing is a fluteplaying mermaid'.
34
An Invitation to Formal Reasoning
Note that the STATE diagram for the premise in which the mermaid area is empty already shows the smaller area for fluteplaying mermaids to be empty. Figure 15 [slO]
[s9]
Here, as in the case of s7 and s8, we have an entailment in one directi<>n. More often than not, we have no entailment either way. We noted above that statements of form 'some X isn't Y' and 'some Y isn't X' are not equivalent and that neither entails the other. For example, an argument using either sl1, 'some farmer isn't a citizen', or s12, 'some citizen isn't a farmer', as premise with the other as conclusion is invalid. These statements express different STATES OF AFFAIRS neither of which includes the other. Here are the respective diagrams:
figure 16
figure 17
[sll] F
[s12]
F
As the diagrams show, the STOAS expressed by s11 and s12 are distinct; neither is included in the other. So there is no entailment in either direction.
35
Picturing Propositions
******************************************************************** Exercises: Examine the following Venn diagrams. What proposition does each graphically represent? What proposition does it entail? (Explain how the entailed proposition, the conclusion, is depicted in the diagram that depicts the premise.)
c
F
c
F
****************************************************************** 5. STATES and states STATES that obtain correspond to states of affairs. For example the EXISTENCE OF A MILLIONAIRE WHO IS A FARMER AND A PHILOSOPHER obtains (is a FACT) because it corresponds to a fact: the existence of a millionaire who is a fanner and a philosopher. This fact includes another fact: the existence of a philosopher who is a millionaire. Thus, just as STATES include or exclude other STATES, so the corresponding states include or exclude other states. The nonexistence ofmennaids excludes the existence of unhappy mennaids so that 'there are no mermaids' entails the falsity of 'there are unhappy mennaids'. Equivalently we may say that the nonexistence ofmennaids includes the nonexistence of unhappy mennaids. so that 'there are no mennaids' entails 'there are no unhappy mennaids'. The existence of rich bridesmaids includes the existence of bridesmaids.
36
An Invitation to Formal Reasoning
6. Positive and Negative 'Valence' In a Venn diagram the STOA expressed by a statement claiming absence (nonexistence) is represented by shading; the STOA expressed by a statement that claims presence is depicted as unshaded and marked by a cross. Let us call any statement that claims presence 'positive in valence' and any statement that claims absence 'negative in valence'. Venn diagrams graphically show that two logically equivalent statements express one and the same STATE OF AFFAIRS. Now ifthat STATE is a STATE of presence, the two statements that express it will be positive in valence. If it is a STATE of absence the two statements expressing it will be negative in valence. In effect when two statements are logically equivalent, both statements must be positive in valence or else both must be negative; thus no statement claiming presence can possibly be equivalent to a statement that claims absence. To put this P
7. The Limitations of State Diagrams Venn Diagrams are useful for explaining how one statement may entail another by showing in a graphic way how one STATE OF AFFAIRS may include or exclude another. However, not all STATES can be represented graphically and the actual use of Venn diagrams for logical purposes is rather limited. As students of logic we want to learn how to infer conclusions from given premises. And we want techniques for checking the validity of a wide range of arguments. For example, we might wish to see whether an argument like 'every noncitizen is an alien, hence, anyone who arrests a nonalien arrests a citizen' is valid. (It is.) For arguments of this kind the method of checking validity by means of state diagrams will not work. Nor is it practical for solving long arguments like the second of the two Lewis Carroll examples given in Chapter 1. Indeed, the use of state diagrams quickly becomes impractical as soon as we leave the simplest sorts of cases. In the next chapter we shall introduce an algebraic way of representing statements that will prove very useful for the wider purposes of logic.
Picturing Propositions
37
******************************************************************** Exercises: Let F= farmer, G =gentleman, S =spy. I. Draw Venn diagrams depicting the STATE OF AFFAIRS EXPRESSED by each of the following statements: 1. no F is an S 2. some G is an F and an S 3. noS is G 4. some Fare not G 5. some nonF is a nonS 6. no nonF is both F and S 7. every Sis a G but not an F 8. whatever is an S is not a G 9. no nonG is an F 10. all S are both F and nonG II. Using the forms 'the existence of X' and the 'the nonexistence of X', what facts are signified by each of the following true statements? 1. no animal is a mermaid 2. some bridesmaid is a spy 3. no farmer is a patriot and a spy 4. some farmer is a gentleman and a patriot 5. some farmer is a noncitizen 6. no farmer is a mermaid 7. every gentleman is a nonspy 8. all spies are gentlemen 9. some farmer who is a spy is a noncitizen 10. no bridesmaid is a mermaid
What is the valence of each statement in exercises I and II?
*******************************************************************
38
An Invitation to Formal Reasoning
8. The Statement Use of Sentences A sentence being used to make a truth claim is a statement. But a sentence may be used in different ways. Consider 'The door will be shut'. Taken simply as a piece of language this sentence is a well formed grammatical part of English. But in the mouth of a speaker it takes on a special character. A speaker may use it to give an order to the hearer. So used the sentence is a command. This use of a sentence is typically marked by an exclamation sign or by the manner of its utterance by the speaker. But the same sentence may be used to make a promise to the hearer, giving assurance that the door will be shut. Or it may be used as a question. Here the tone may indicate that the speaker wants information from the hearer about the future position of the door. A fourth use is to convey information. The speaker is predicting that the door will be shut. So used, the sentence is a statement. In any of th~se different uses the sentence expresses the same STATE OF AFFAIRS: Spoken imperatively, 'the door will be shut' is an order to the hearer to act so that the STATE will obtain. Spoken as a promise, the speaker is assuring the hearer that he will so act. Spoken as a question, the speaker is seeking information about the STATE: will it or won't it obtain? Spoken assertively, the speaker is making a truth claim, a (true or false) prediction that the STATE of AFFAIRS will obtain. A statement, then, is an asserted sentence, an utterance claiming that the proposition or STATE OF AFFAIRS expressed corresponds to a fact, is true of the world. In none of the other uses do we characterize the success or failure ofthe speaker's utterance as either true or false. A command is obeyed or disobeyed but it is not either true or false. A question is answered or not answered, appropriate or inappropriate, but it makes no literal sense to speak of a true or a false question. A promise is kept or not kept but it is not literally correct to characterize a promise as either true or false (though we may say it was seriously or casually made; a totally insincere or 'false' promise is not really a promise at all, just as a false pearl is not really a pearl at all). Only a statement is a truth claim; only a statement can be characterized as succeeding or failing to be true or false. Logic is exclusively concerned with statements because only statements claim truth for the propositions they express and so only statements are evaluated as being true or false. Logicians speak of a statement's 'truth value'; it has the value True if it is true and the value False if it is false.
Picturing Propositions
39
9. Truth Relations One may accept or reject a truth claim. If you accept a statement as true, then you are rationally committed to accept certain other statements as true. For example, anyone who accepts 'some farmer is a gentleman and a scholar' as true is rationally committed to accept 'some scholar is a farmer and a gentleman' and 'some gentleman is a scholar' as true. In accepting the first statement you accept the claim that the EXISTENCE OF A SCHOLAR WHO IS A GENTLEMAN AND A FARMER is a FACT. But if that STATE obtains so does the EXISTENCE OF A SCHOLAR WHO IS A FARMER AND A GENTLEMAN and so does THE EXISTENCE OF A GENTLEMAN WHO IS A FARMER. You cannot rationally accept the first claim and reject the other two. Thus, truth has consequences. Logic studies the consequences of truth claims. The study of how statements are truth related to one another is central to logic. It can be approached in two ways. We have so far approached the question 'semantically'. Semantics studies the relation of statements to the propositions they express and to the states of affairs or facts that make them true or false, and it studies the relation of STATES OF AFFAIRS to one another. For example, in our study of Venn diagrams we noted how some STOAS include or exclude other STOAS. Such studies belong to semantics. A second approach to truth relations focuses on the formal, or 'syntactical', structure of a sentence. For example, we noted earlier that where S2 is the grammatical converse of S 1 they will always have the same truth value. (That is, we learned that any two sentences of the forms 'some X is Y' and some Y is X' are true together or false together.) This second approach is syntactical and in its way it is as fruitful an approach as the semantic one. lndeed the semantic and syntactic approaches complement each other and it is now time to focus attention on the form or syntax of the sentences that figure in arguments. Our topic for the remainder of this chapter and for the next two chapters is Syntax or more precisely, since our concern is with statements bearing truth values, Logical Syntax. Logical Syntax is the study of the form and the composition of the statementsentences that we evaluate as being true or false.
40
An Invitation to Formal Reasoning
10. Logical Syntax Syntax or sentence structure is of special concern to grammarians and to logicians. The logician's concern with sentence structure differs from the linguist's or grammarian's in two ways. In the first place the logician restricts himself to sentence forms that are appropriate to statements. In practice the exclusive concern with statementsentences restricts the logician to the consideration ofelementary sentences that are declarative or indicative in fonn. For example, a sentence of the form 'some X is a Y' is declarative and of logical interest while 'Is any X a Y?' is not declarative; it is neither true nor false and so of no interest to the logician. Thus the logician is interested in sentences that make statements, not in sentences used to express questions (or prayers or commands). In the second place the logician's interest in the structure of a sentep.ce is qualified by his concern with the role that the sentence plays in reasoning. This special concern forces the logician to analyze its form in a manner that is markedly different from the way a linguist analyzes the form of the sentence. Unlike the linguist, the logician must not only attend to the form of the individual sentence; he must also keep an eye on other sentences to which it is truth related (e.g., as premise to conclusion or as conclusion to premise). Analyzing a sentence syntactically is called 'parsing'. To appreciate the difference between the way a linguist approaches the task of syntactic analysis and the way a logician approaches it let us consider two simple sentences that are truth related and observe each at his job. The sentences are: some horse speaks French some speaker of French is a horse These sentences mutually entail one another. (An inference using either one as premise and the other as conclusion is valid.) The logician must bear this in mind when he comes to parse them. But the linguist need not take this into consideration and his way of parsing the sentences will not necessarily reflect a concern with their truth relations. The linguist parses a sentence by first dividing it into two parts: a Noun Phrase subject and Verb Phrase predicate. Thus 'some horse speaks French' has 'some horse' as its noun phrase (NP) and 'speaks French' as its verb phrase (VP). (The NPNP analysis is indicated by the stroke sign that is placed between the NP and the VP: some horse/speaks French.) A similar analysis is given to 'some speaker of French is a horse', whose Noun Phrase
Picturing Propositions
41
subject is 'some speaker of French' and whose Verb Phrase predicate is 'is a horse' ('some speaker of French/is a horse'). We do not here need to go into more detail about linguistic syntax. For right at the outset the logician will find this mode of analysis unsatisfactory. This is not to say that he denies that an NPIVP analysis may legitimately be given to any sentence. But for his purposes such an analysis is inadequate. For it is the job of Logic to explain why 'some horse speaks French, hence some speaker of French is a horse' is a valid inference. In any such explanation the logician will need to refer to expressions in the premise that are repeated in the conclusion. But ifwe adopt the NPIVP mode of analysis, the expressions in the premise of the inference are significantly different from the expressions that appear in the conclusion. Where the premise contains 'speaks French' and 'some horse', the conclusion has 'speaker of French' and 'is a horse'. Thus the linguist's NPIVP mode of analysis does not make clear the pattern of validity that is instanced by this inference. Now this complaint will not impress the linguist. For he will counter by pointing out that it is his job to parse each sentence separately to bring out its grammatical form; it is not his job to parse them in a manner that brings out the form of an argument in which one is a premise and the other is the conclusion. More generally the linguist may wish to deny that syntactic analysis must attend to the truth relations of the sentences being analyzed. If we accept his denial we may accept the view that the NPIVP analysis is a legitimate way of parsing sentences. If however we require that any syntactic analysis must also help us to see how sentences are logically related to each other, then we shall demand a parsing of a sentence that is sensitive to the way it is logically related to other sentences. In any case the logician cannot rest content with the NPIVP style of syntactic analysis; he must aim for a 'logical syntax' that parses sentences in a logically revealing way. The logician approaches the two sentences with the idea of parsing each one in a way that exposes the pattern of inference in which one is the premise and the other is the conclusion. Thus he is dealing with the form of the inference as well as the form of the individual sentence that enters into it as a premise or conclusion. Indeed the two tasks are one. A syntactic analysis of sentences that clarifies the inferences in which they play a role is an analysis of their 'logical syntax'. In dealing with the structure of any sentence, the logician is out to make its logical syntax explicit. How does the logician analyze the two sentences? To see the how and why of his analysis we note that the pattern of the inference in which we move from the premise 'some horse speaks French' to 'some speaker of French is a
42
An Invitation to Formal Reasoning
horse' involves a change of order in two expressions. In explaining why this inference is valid we want to say that the form of both sentences is the same, only that in the conclusion the order of two expressions contained in the premise has been switched. The trouble with the NPNP analysis was that it
did not bring to the fore the two material expressions that have been switched about in the conclusion. For example, the premise contains 'speaks French', which, speaking strictly, is not found in the conclusion (which has 'speaker of French'). Moreover, superficially at least, the forms of 'some horse speaks French' and 'some speaker of French is a horse' seem different. Thus the linguist's NPNP analysis does not meet the following two conditions that the logician here demands:
1. The two sentences must have the same structure. 2. We must be able to identify two expressions in the premises that ~re repeated in reverse order in the conclusion. An analysis of the structure of the two sentences that meets both of
these demands will reveal their logical syntax. It is, however, possible to meet these demands for a logical syntax in two different ways, one taking terms or nominals as the repeated expressions and the other taking predicates or verbs as the repeated expressions. It is perhaps not surprising that different logicians have shown decided preferences for one way or the other. In explaining the two alternatives we shall continue to use our two sentences by way of illustration. One mode of analysis, called the Term Way, identifies the repeated expressions as the two terms, 'horse' and 'speaker of French', and construes both sentences as having the form 'some.. .is a .. .' where the blanks indicate where the two terms (nouns or noun phrases) belong. Now 'some horse speaks French' is not explicitly of form 'some .. .is a .. .', nor does it contain the term 'speaker of French'. The terminist logicians (as we shall call them) deal with both of these points by construing 'some horse speaks French' as having the structure of 'some horse is a speaker of French'. In effect, the terminist rewrites, or logically regiments the inference in a way that brings out the logical form of both sentences. He does this by replacing the original premise by a paraphrase that explicitly has the favored two term structure. The conclusion is already of this form and the whole inference now reads: 'some horse is a speaker of French, hence some speaker of French is a horse'. In this form, both sentences are parsed in a manner that meets the requirements of a logically adequate syntactic analysis; i.e., one that attends to the form of the inference in which they figure as well as attending to the form of each as a
Picturing Propositions
43
separate sentence. According to the Term Way, when properly formulated, or regimented, the inference looks like this: ( 1)
some horse is a speaker of French /some speaker of French is a horse
The pattern of the inference is some X is a Y (hence) some Y is an X The terminist version of logical syntax was favored by Aristotle, the fourthcentury B.C. inventor offormallogic, and adopted by most logicians as the standard way of regimenting sentences that enter into deductive reasoning. It requires us to construe every simple sentence as having two nominal expressions, called terms connected by an expression such as 'some .. .is .. .' or 'every .. .is a .. .' or variants such as 'some ... are .. .', 'some ... will be .. .', 'a .. .is a ... ', 'all ... are ... ', and so forth. The NPNP analysis is not consistent with this. Thus any phrase of form 'some X' is a Noun Phrase and any phrase of form 'is a Y' is a Verb Phrase. But the terminist maintains that Noun and Verb Phrases are essentially of this special form each containing a term. Logically speaking, the Noun Phrase always consists of a word of quantity such as 'some' or 'all' followed by a nominal term and the Verb Phrase consists of a word of quality such as 'are' or 'were' followed by a second nominal term. In a proper paraphrase, the term in the 'Verb Phrase' must be interchangeable with the term in the 'Noun Phrase'. For example, 'children laughed' is construed by the terminist as 'some children were laughers' since this way of construing it gives us two nominal expressions as terms that can be interchanged (thereby allowing the inference to 'some laughers were children').
11. Term Way vs. Predicate Way In the nineteenth century another version of logical syntax was introduced by the great German logician, Gottlob Frege. This new version, which has gained wide acceptance among contemporary logicians, is called the 'Predicate Way'. According to the Predicate Way the two material expressions that are interchanged in our little inference are not the terms 'horse' and 'speaker of
44
An Invitation to Formal Reasoning
French' but the Verb Phrase predicates 'is a horse' and 'speaks French'. Logicians who favor the Predicate Way reconstrue the premise as 'something is such that it is a horse and it speaks French'. And they reconstrue the conclusion as 'something is such that it speaks French and it is a horse'. Using predicate verb phrases is an alternative way of satisfying the requirement that a logically adequate syntactic analysis of the premise and conclusion must construe them both as having the same form while identifying two expressions whose order in the premise is reversed in the conclusion. According to logicians who favor the Predicate Way, the expressions in question are not terms but predicates. And the pattern of inference is something is such that it X's andY's (hence) something is such that it Y's and X's where 'X's' and 'Y's' hold the place of predicates. Following the Predicate Way, (1) is an instance ofthat pattern: something is such that it is a horse and it speaks French
I something is such that it speaks French and it is a horse The present book develops the classical terminist approach to logical syntax. The style oflogic that adopts the term syntax is called Term Logic. In Chapter 8 we will be taking a closer look at the rival predicate approach. We will there compare Term Logic to Predicate Logic and show how they are related to one another. These two classical ways of doing logic are both legitimate and equally powerful from a formal standpoint. But the Term Way is closer to the syntax of a natural language like English, Spanish or French. Consider the fact that the Predicate Way construes an elementary sentence like 'some ape is black' as 'something is such that it is an ape and it is black', a sentence that contains the conjunctive form 'it is an ape and it is black' and the pronoun 'it'. These are complexities not found in Term Logic. In the next chapter we shall introduce a notation for representing sentence in Term Logic that is very much simpler than the notation currently used to represent sentences in Predicate Logic. Moreover, the actual working oflogical proofs in Term Logic is easier than it is in Predicate Logic. We have therefore chosen Term Logic for our working logic. Historically too, term logic came first. We shall not neglect Predicate Logic. But we shall treat it as an alternative system and defer its study to the time we have a thorough understanding of Term Logic. At that point Predicate Logic becomes much easier to understand.
Picturing Propositions
45
******************************************************************** Exercises:
I. Regiment the following sentences terrninistically. (The first two examples are for purposes of illustration.) 1. children were laughing (Acceptable answers: some children were laughing persons; some children were laughers.) 2. whales are mammalian (Acceptable answers: all whales are mammalian animals; every whale is a mammal.) 3. some Greeks are wise 4. all human beings die 5. some cats are tricolored 6. no horse speaks French 7. someone is talking 8. all fish swim 9. some mammals do not swim 10. whatever speaks French is a person II. Regiment examples 1, 3, and 5 in the 'predicate way'.
******************************************************************* 12. Some Useful Terminology The words 'some' and 'every' are called words of 'quantity'. Statements of form 'some X is Y' are called 'particular in quantity' because they speak of some X's but not of all X's. For example, 'some farmer is a citizen' is particular in quantity since, in it, the term 'farmer' denotes some farmer or farmers but not necessarily all farmers. By contrast, a statement of form 'every X is Y', which speaks of all the X's, is 'universal in quantity.' For example, 'every farmer is a citizen' is said to be universal in quantity. Sentences of from 'some X is Y' and 'every X is Y' are said to be affirmative. 'Some X is Y' is called a particular affirmation, 'every X is Y' is called a universal affirmation, despite the fact that its valence is negative. (Its Venn diagram is shaded.)
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An Invitation to Formal Reasoning
Denying a sentence changes its quantity. Thus 'no X is y' which is the denial of 'some X is Y' is a universal sentence and 'some X is not Y' (or 'not every x is Y'), which is the denial of 'every X is Y', is a particular sentence.
13. Subjects and Predicates In sentences ofthe form 'some/every X is a Y' the phrases 'some X' and 'every X' are called subjects. For example, 'some farmer' is a particular subject; 'every farmer' is a universal subject. The phrase 'is (a) Y' is called the predicate of the sentence. For example, 'is a citizen' and 'is a noncitizen' are predicates. In' some (every) X is a Y', 'X' is called the subject term andY is called the predicate term. Very often the terms of a sentence are not at the surface and we m~st rephrase the sentence to bring out its twoterm structure. For example, in 'some farmer drinks' the predicate 'drinks' is not a term and we regiment the sentence as 'some farmer is a drinker'. We then identify 'drinker' as the predicate term. We similarly regimented 'some horse speaks French' as 'some horse is a speaker of French' to bring out its logical form, which is 'some X is a Y'. This gave us the nominals 'horse' and 'speaker of French' as the two terms of the sentence.
***************************************************************** Exercises: Regiment the following sentences and identify a) their subject terms, and b) their predicate terms. i) only members were present ii) whoever errs is mortal iii) birds always have feathers iv) dogs sometimes bite v) Tom doesn't eat meat vi) dogs never lay eggs vii) a fish swims viii) a spider doesn't swim ix) one who listens learns x) the guests were all women
Picturing Propositions
47
Regiment and then identify the subjects and predicates of:
10 every citizen is a farmer 20 the whale is a manunal 3 the whale is hungry 4 children were laughing 50 frogs are amphibians 60 bats are not birds 7 no whale is a fish 8 no whale lays eggs 9 a frog leaps 100 a bird lays eggs 0
0
0
0
0
********************************************************************
3 The Language of Logic (I) 1. Introduction Any argument consists of statements. In this chapter and the next we will be learning how to represent statements in a way that makes it easy to manipulate them, 'adding them up' to arrive at conclusions. Our task here is similar to the task we once had of learning 'the language of mathematics'. As a first step we learned how to represent numbers like seven hundred and thirtyone and nine thousand six hundred and ninetyeight as 731 and 9,698. Using arabic notation made it easy for us to add these numbers in a mechanical way to get 10,429 as the sum. Analogously we will be learning a logical language, in which we represent statements in a notation that permits us to 'reckon' with them to derive conclusions from them in a mechanical way. Particular affirmations like 'some farmer is a citizen' and 'Tom is a citizen' have a very simple structure so we begin with them. The particular affirmation 'some farmer is a citizen' has the form 'some X is a Y' with 'X' as the subject term and 'Y' as the predicate term. The claim is that BEING A CITIZEN characterizes some farmer. We abbreviate 'BEING A Y characterizes (an) X' as 'Y some X'. We use term letters to stand for the terms. For example, the abbreviated form for 'BEING A CITIZEN characterizes some farmer' is 'C some F'; the abbreviated form for 'BEING A CITIZEN characterizes Tom' is 'C some T*'. The formula 'Y some X' will be called an 'Aform' to remind us that Aristotle was the first to introduce paraphrases that placed the predicate term on the left and the subject term on the right thereby reversing the natural English order of terms in a sentence. Aristotle's own paraphrase for 'some X is Y' was 'Y belongs to some X'. His paraphrase for 'every X is Y' was 'Y belongs to every X'. We shall speak of 'every' and 'some' as 'term connectives'. Thus 'every' is the term connective in the sentence 'Y every X' and 'some' is the term connective in 'Y some X'. Aform sentences are not natural to English speakers; in the next section we shall recapitulate our discussion for sentences that follow natural English order (Eform sentences). For the present, however, we find Aforms useful; by paraphrasing the Eform sentence 'some X is Y' as 'Y some X', 49
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An Invitation to Formal Reasoning
the Aform brings out the characterizing role of the predicate term, the denotative role of the subject term and the connecting role of the formative expression '(belongs to) some'. Moreover the Aform has the virtue of placing the term connective, '(belongs to) some', '(belongs to) every', entirely between the terms that it connects and this suggests that we could represent the sentence in an algebraic way.
2. Writing 'Y some X' as an Algebraic Expression
We will in fact represent 'Y some X' as 'Y+X', thereby transcribing 'some' as a plus sign. We noted earlier that 'some X is a Y' is equivalent to its converse 'some Y is an X'. The equivalence of 'Y some X' to 'X some Y' shows that 'some' does indeed behave in a pluslike manner. Using the Aform the equivalence of converses can be stated as 'Y some X = X some Y'. But now, representing 'some' as a plus sign, we can formulate the equivalence as a simple algebraic equation: Y+X=X+Y
3. Affirmation(+) and Denial()
Recognizing the pluslike character of 'some' is the first step in algebraic notation. We now proceed to add to our algebraic notation. Some statements are prefixed by a sign of negative judgment. For example, in 'not a creature was stirring' the word 'not' is a sign of negative judgment, or denial. The Aform of this sentence is 'not: stirring some creature', which algebraically transcribes as '(S+C)'. In this formula the external minus sign represents the word 'not' or the phrase 'it is not the case that'. Note that the affirmative statement 'a creature was stirring' lacks an explicit expression of positive judgment (affirmation); however we assume the sentence to be implicitly prefixed by 'it is the case that' or 'yes'. Transcribing the unspoken sign of affirmation by a plus sign we could represent 'yes: a creature was stirring' as '+(S+C)'. In general then, an affirmation has the form '+(Y+X)'; a denial has the form '(Y+X)'.
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4. Binary and Unary Uses of a Sign Note that in '+(Y+X)' the plus sign plays two quite different roles. Coming between the terms, it plays the 'binary' role of connecting, or 'adding', two elements. Coming before the whole sentence it plays a 'unary' role of qualifying that sentence as positive. The difference between the binary and unary uses of a sign can be seen if we attend to an arithmetical expression like '( 125)'. Coming between two numbers, the minus sign operates on them both simultaneously; its role in '125' being that of a binary subtraction operator. But in its initial occurrence in,' ( .... )', it is a unary operator; it is like the minus sign in '7', where the minus sign operates on a single number to transform it into a negative number. The following illustrates how we transcribe sentences in algebraic form. We give theEform (English Form) and then the Aform along with its algebraic transcription:
Eform some X is nonY nononX is Y no X is nonY some nonXis nonY
Aform transcription Aform +((Y)+X) nonY some X (Y+(X)) not: Y some nonX ((Y)+X) not: nonY some X +(( Y)+(X)) nonY some nonX
5. Positive and Negative Valence We speak of the positive or negative character of a statement as its 'valence'. A statement of the form '+(Y+X)' is positive in valence (or particular in quantity) and it expresses EXISTENCE or PRESENCE. A statement of the form '(Y+X)' is negative in valence (or universal in quantity) and it expresses NONEXISTENCE or ABSENCE. Statements of the same valence (or quantity) are called covalent. Statements that differ in valence are called divalent. (The reader should review section 6 of the preceding chapter.) When two statements are equivalent they express one and the same STATE OF AFFAIRS. The STOA they both express must either be positive or negative. So divalent statements cannot possibly be equivalent. For, of two divalent statements, one will signifY a positive STATE and the other a negative STATE. Thus covalence is a necessary condition of equivalence.
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Being able to represent logical equivalence algebraically is a distinct advantage. More generally, our use of algebraic transcription reveals the logical syntax of statements to us in a logically useful manner. It makes logical reckoning easy; indeed, using only very simple High School algebra, we shall soon be learning how to treat arguments as we treat sums, adding up premises to derive conclusions. Statements that differ only in their external signs are contradictory to one another. For exaniple, 'a creature was stirring' and 'not a creature was stirring' are contradictories and, generally, any two statements of the form '+( ... )'and( ... )' are contradictories. Contradictories are divalent. When two statements are equivalent so are their contradictories. Thus just as the equation +(Y+X)
= +(X+Y)
represents the equivalence yes: some X is Y
= yes: some Y is X
so the equation (Y+X)
= (X+Y)
represents the equivalence no X is Y
= no Y is X
6. Contrary Terms and Sentences
Terms that differ in sign are said to be logically contrary to one another. For example, the terms 'citizen' and 'noncitizen' are logically contraries. If 'C' represents 'citizen', 'C' represents 'noncitizen'. Two sentences that have contrary predicate terms are called contrary to one another. Thus the following pairs of sentences are contraries: every farmer is a citizen; every farmer is a noncitizen some farmer is a citizen; some farmer is a noncitizen
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no fanner is a citizen; no fanner is a noncitizen Generally, ' ... is a P' and ... is a nonP' are contraries.
7. 'Every' We now consider the fonn 'Y every X' which is 'Aristotelian' for 'every X is Y'. In dealing with statements of this fonn we take note of the following equivalence: El
Y every X= not: nonY some X
The statement on the right is 'Aristotelian' for 'it is not the case that some X is nonY' which transcribes as ' (( Y)+X)'. But what is the algebraic transcription of 'Y every X'? Note that both minus signs in '((Y)+X)' are unary signs. The external minus sign operates on the sentence '(Y)+X'. The internal minus sign operates on the tenn 'Y'. But now suppose we simplify ' (( Y)+X)' by driving in the external sign. The result is 'Y X', an expression that introduces a new binary use of'' standing for the tenn connective 'every'. We could now express the equivalence between 'every X is Y' and no X is nonY' as the equation E*l
YX =
(( Y)+X)
whose left side is the transcription of 'Y every X'. The equation suggests that 'every' in logic behaves like the binary minus sign of High School algebra. But we shall have to check this out; transcribing 'every' as a binary minus sign is justified only if 'every X is Y' does logically behave as 'Y X' in all cases where the word 'every' figures in a pair oflogically equivalent statements. Let us look therefore at some more equivalences involving 'every'. One such equivalence is: E2
every X is Y = every nonY is nonX
[The reader is invited to check the correctness of this and other equivalences by representing each side on Venn Diagrams. In the present instance the left side signifies the nonexistence of things that are X but not Y; the right side
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An Invitation to Formal Reasoning
signifies the nonexistence of things that are nonY and nonnonX. Since whatever is X is nonnonX, the Venn diagrams for both statements will be the same: shading out the area that is X but not Y.] Transcribing 'every' as a minus sign allows us to express E2 as an algebraic equation: E*2
YX = (X)(Y)
This gives us additional confidence in the minuslike character of 'every'. Another correct equivalence shows 'every' again behaving like a minus sign. E3 E*3
some X is Y = not every X is nonY Y+X = (( Y) X)
And yet another: E4 E*4
every nonXis Y =every nonY is X Y(X) = X(Y)
8. Why Some Equal Sentences are not Logically Equivalent Looking at a sentence like 'Y (X)' one may be moved to ask why we cannot simply equate it with 'Y+X' to give us the 'equivalence': Y(X) =Y +X every nonXis Y =some X is Y Though the two sentences in this algebraically correct equation are algebraically equal, their divalence prevents them from being logically equivalent. The sentence on the left is universal and negative in valence; it expresses a STATE of absence. The sentence on the right is particular and positive in valence; it expresses a STATE of presence. The covalence condition imposes a significant constraint on logical algebra. In ordinary arithmetic and algebra, 'two negatives make a positive' even when we are canceling a unary negative sign by a binary one, as we do in equating '12(5)' to '12+5'. But in logical reckoning, the cancellation of a unary minus sign by a binary one is prohibited. Binary cancellation illegitimately
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changes the valence of a sentence. In banning cancellation by a binary minus sign, logical algebra is strikingly unlike the High School algebra with which we are familiar. Students of logic who are using algebraic notation for their logical language should take note of this difference. Though a binary ('every') minus sign cannot cancel a unary ('not') minus sign, the unary minus can cancel a binary or a unary minus sign. Thus in E*3 '((Y)X)' (read: 'not every X is nonY') may be simplified by driving in the external unary sign to cancel both internal minus signs, thereby changing ' Y' to 'Y' and changing the binary minus sign for 'every' to the binary plus sign for 'some' to give us 'Y+X' (read: 'some X is Y').
9. Eforms and Aforms The Aforms 'Y some X' and 'Y every X' differ from the more natural Eforms 'some X is Y' and 'every X is Y' in three respects: (1) The order of the terms is different. In the Aform the subject term is on the right. (2) TheEform has the grammatical copula 'is' (or variants like 'are', 'will be', 'were', etc.). The Aform lacks the grammatical copula. (3) In theEform the words of quantity 'some' or 'every' come first and not between the two terms. All these differences are merely stylistic. It is obviously desirable to have a way of transcribing sentences that follows the natural order of English. With this in mind we introduce a convention of representing positive grammatical copulas such as 'is' and 'are' as plus signs. We now transcribe 'some X is Y' with two plus signs as '+X+Y', reading the .first plus sign as 'some' and the second as 'is'. Similarly 'every X is Y' is transcribed as 'X+Y'. This mode of transcription of'every X is Y' is formally less elegant than 'Y X';' X+Y' uses two signs for the expression that joins the terms where 'Y X' uses only one. Also' X+Y' obscures the binary character of 'every' as a minus sign. But these disadvantages are quite outweighed by the gain of being able to directly transcribe the Eform sentences into algebraic form: algebraic transcription now follows English order in a direct, natural and linear way. The following examples illustrate the way we transcribe Eforms in algebraic notation:
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An Invitation to Formal Reasoning
(Jo/e use the sign'=>' as an abbreviation for 'transcribes as'.) => every X is Y some X is Y ==> => some X is nonY not an X is a Y => no X isY => not every X is nonY =>
X+Y +X+Y +X+(Y) (+X+Y) (+X+Y) (X+(Y))
In the remainder of this chapter and for much of the next chapter, we will be learning how to represent all Eform statements as algebraic expressions. This will enable us to reckon with them in an algebraic manner. For example we will learn how set down the premises of a Lewis Carroll argument in algebraic form and to derive a conclusion from them by adding them. And we will be learning how to check a given piece of reasoning in much the way that we check the correctness of an example in a very simple kind of algebra whose only operative signs are '+' and ' '.
10. Transcribing Affirmative Statements If you say 'no farmer is a billionaire' and I disagree, I would probably express my disagreement by saying 'some farmer is a billionaire'. I might, for emphasis, say, 'it is the case that some farmer is a billionaire', but in most normal contexts affirmation is not explicitly expressed. Nevertheless we shall temporarily adopt a method of transcription that prefixes affirmations with an explicit sign of positive judgment corresponding to the external 'no' or 'not' of denial or negative judgment. Thus, just as we transcribe the sign of denial in 'not a creature was stirring', rendering this as '(+C+S)', so shall we transcribe a (tacit, unspoken) sign of affirmation 'yes' (or 'it is the case that') in representing 'a creature was stirring' as '+(+C + S)'. This method of transcribing will be adopted for most of this chapter. Later we shall be able to follow natural English usage by omitting signs of positive judgment, transcribing 'a creature was stirring' as '+C+S'.
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11. How to Tell the Valence ofEform Statements A statement of positive valence claims that something exists; a statement of negative valence claims that something does not exist. When two statements, S 1 and S2, are divalent one will express EXISTENCE, the other NONEXISTENCE. For S 1 and S2 to be equivalent, they would both have to express one and the same STATE OF AFFAIRS. It follows that divalent statements S 1 and S2 cannot be equivalent and that covalence is a necessary condition of equivalence. We are often in the position of wanting to know whether two statements are equivalent and one of the things we must know is whether they are covalent. It is therefore important for us to be able to inspect a statement and to determinepreferably at a glancewhat its valence is. Fortunately there is a simple way to do this:
Look at the judgment sign and then look at the sign of quantity: if these signs differ, the valence is negative; if they are the same, the valence is positive. For example, since the first two signs of'+{ F +C)' {the transcription of' every farmer is a citizen') differ, its valence is negative. By contrast, the valence of 'not every farmer is a citizen' is positive since the two initial signs of its transcription, ' ( F +C)', are the same. (The statement signifies the existence of an farmer that is a noncitizen.) Generally then, the valence of a universal statement of the form 'every X is a Y' is negative; such statements signify the nonexistence of anything that is both an X and a nonY. The valence of its contradictory 'not every X is a Y' (=> ( X+Y)) is positive; the initial signs are the same. Statements of this form are positive in valence, being equivalent to 'some X is nonY' and claiming the existence of something that is X and nonY.
12. Negative Valence= Universal Quantity Our criterion shows that affirmative statements beginning with 'every' are negative in valence. To see why consider the equivalence: every X is a Y = not: some X is a nonY
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An Invitation to Formal Reasoning +(X+Y) == (+X+(Y))
The equivalence is between two statements, an affirmation and a denial. The statement on the left is explicitly universal in being about every X. It is affirmative, but since it is equivalent to the statement on the right (which clearly claims the absence of any X that is nonY}, its valence must be negative. What is the 'quantity' of the negative statement on the right? A moment's reflection suggests that it is universal in quantity; it too speaks of all the Xs (saying of each and every X that it does not fail to be a Y). In effect the equation shows that 'no X is nonY' and 'every X is Y' have the same universal quantity and the same negative valence. To call a statement universal (or negative in valence) is thus tantamount to calling it negative in valence (or universal in quantity). Conversely, to call a statement particular in quantity (or positive in valence) is tantamount to calling it positive in valence (or particular in quantity).
13. The Law of Commutation in Eform The Aform equivalence '+(Y+X)=+(X+Y)' expresses the Logical Law of Commutation (LLC). The same law can be expressed using Eform transcriptions of 'some X is Y' and 'some Y is X':
LLC
+{+X+ Y)
=
+{+ Y+A}
LLC is a law of logic. It tells us that any two statements of form '+(+X+Y)' and '+(+Y+X)' signify one and the same state of affairs and it tells us this by stating their logical equivalence in the form of a simple algebraic equation in which the double plus functor (the term connective) represents 'some .. .is (a)'. If we change the sign on both sides ofthe equation we get another equivalence: (+X+Y} = (+Y+X) no X is Y = no Y is X
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******************************************************************* Exercises:
I. Transcribe each statement, inspect the first two signs and then say whether the statement is positive or negative in valence. 1. Not every ape is grey. 2. No ape is green. 3. Whales are mammals. 4. Children were laughing. 5. It is the case that every ape is unreasonable. 6. Some unbelievers are happy. 7. All unbelievers are curious. 8. Not all dogs are faithful. 9. Babies cry. 10. It is not the case that some cats are friendly. 11. No unbelievers are laughing. 12. It is not the case that no ape is green.
****************************************************************** 14. 'Every' in Eform Transcriptions We saw earlier that 'every' in the Afonn 'Y every X' transcribes as a minus sign. It is easy to show that our method of showing the minuslike character of 'every' carries over to the Efonn 'Every X is Y.' We first transcribe the equivalence of 'no X is nonY' to 'every X is Y' algebraically but without transcribing 'every': yes: every X is a Y = not: some X is a nonY +(every X+ Y) =(+X+( Y)) Since the two sides of the equation are equivalent they must be algebraically equal as well as covalent. Since the first two signs in the fonnula on the right differ, they must differ in the fonnula on the left. Putting '' for 'every' does just what we want: it assures covalence and it gives us a correct equation.
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An Invitation to Formal Reasoning
(1) yes: every X is a Y = not: some X is a nonY +(X+Y) = (+X+(Y)) Equation (1), which equates two Eform sentences, shows once again that 'every' has the character of a minus sign and should be transcribed as such. (We continue the practice of writing 'yes' for 'it is the case that', transcribing this expression for affirmative judgment by the '+' sign; we write 'not' for 'it is not the case that', transcribing these and others expressions for negative judgment by the'' sign.) Looking at equation ( 1) more carefully we note that the two (covalent and equal) sides differ in three ways: ( 1) They differ in the judgment sign; one signifies affirmation the other signifies denial. (2) They differ in the sign of quantity; one side has 'every' the other side has 'some'. (3) They differ in the sign of predicate quality; one has a positive term in predicate position, the other has a negative term. Any two statements that differ in just these three ways are called obverse to one another. And any two statements that are obverse to one another are equivalent. Thus we can form the obverse equivalent of any statement by ( 1) changing its judgment sign, (2) changing the sign of quantity and (3) by changing the quality of its predicate term. Here are two more examples of obverse equivalents: (2) not every A is nonB; some A is B (A+(B)) = +(+A+B) (3) no A is B; every A is nonB (+A+B) = +(A+(B)) Obversion further justifies our treating the formative words 'some' and 'is' as plus signs and 'every' and 'not' as minus signs. For, as (2) and (3) show, obversion means that we can drive an external minus sign inward algebraically, by changing the external judgment sign from'' to'+', by changing the sign of'quantity' from 'some'(+) to 'every' (),or from 'every' to 'some' and by changing the 'quality' of the predicate term from positive to negative or from negative to positive. More particularly, obversion shows that
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'every', in the Eform transcriptions, behaves like a minus sign and we so transcribe it. We shall formulate the Law ofObversion as:
Law ofObversion (Obv)
(+1X+I11
=
+(I+XI+Y)
15. 'Isn't' Any statement of the form 'some X is not a Y' is equivalent, by definition, to a statement ofthe form 'some X is a nonY'. The former sentence is in turn equivalent to 'some X isn't a Y'. We express these equivalences algebraically by equating '+(+X+(Y))', '+(+X+  Y)' and '+(+XY)'. In effect we contract 'is not' which transcribes as '+ 'to 'isn't' which transcribes as ' '. The use of a minus sign for 'isn't' allows us to express familiar logical relations in algebraic form: some X is a Y =some X isn't a nonY; +(+X+Y) =+(+X( Y)) some nonY is an X= some X isn't a Y; +(+( Y)+X) =+(+X Y) not every X is a Y =some X isn't a Y; (X+Y) =+(+X Y) Note: in logic, a binary minus sign has the meaning of'every'. But in all of the above equations, the minus sign is unary so cancellation is allowed.
16. The General Conditions of Equivalence Covalence does not suffice for equivalence. Thus 'some farmer isn't a citizen' and 'some citizen isn't a farmer' are covalent statements but they are not algebraically equal and so they are not equivalent. However, when two covalent statements are also algebraically equal their equivalence is guaranteed by the laws of commutation or obversion. Thus consider the pair: some nonsenator is a noncitizen; +(+(S)+(C)) some noncitizen is a nonsenator; +(+(C)+(S)) Both are positive in valence and in addition they are algebraically equal. Being positive in valence, they both signify a positive state of affairs, possibly the same state of affairs. But they are also algebraically equal so, by the law of
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An Invitation to Formal Reasoning
commutation, they are equivalent. Consider the pair every senator is a citizen; +( S+C) every noncitizen is a nonsenator; +( (C)+( S)) They too are covalent. And they are algebraically equal. We could show they are equivalent by appealing to obversion and to LLC. Thus +(S+C) = (+S+(C)) +((C)+(S)) = (+(C)+S)
byObv byObv
The two right sides are then shown to be equivalent by LLC. Since the two right sides are equivalent so are the two left sides. More generally, whenever two statements are covalent and equal, we can show they are equivalent by appeal to LLC and/or Obv. Equivalent statements are both equal and covalent.
So it is helpful that the word 'equivalent' connotes equality and covalence. We have then an easy test for equivalence. When two statements are transcribed in algebraic form we can see whether they are covalent and also see whether they are algebraically equal. Suppose we find they are both equal and covalent. Then we have determined that they are equivalent. If however we find that they are either divalent or unequal, then we judge them to be nonequivalent. Each condition, covalence and equality, is necessary and together they are sufficient for equivalence. We may state this as a fundamental general principle, called PEQ or the Principle of Equivalence: P EQ Two statements are logically equivalent if and only if they are covalent and equal.
Consider a pair of divalent statements that are algebraically equal: +(+S +C) some senators are Citizens no nonsenators are noncitizens  (+( S)+( C)). The transcriptions are algebraically equal but the two statements are not
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logically equivalent because they are divalent: the first statement is positive in valence, the second is negative. Consider whether 'no native is a noncombatant' and 'no noncombatant is a native' are equivalent. Looking at their transcriptions we find them both covalent and equal: (+N+(C)) = (+(C)+N) Since they satisfy both conditions specified by PEQ we conclude they are equivalent. PEQ illustrates how the algebraic notation is used for logical purposes, in this case as a test for equivalence. As a further example ofhow we use PEQ let us consider the sentences '(it is the case that) every noncitizen is an Asian' and '(it is the case that) every nonAsian is a citizen'. To determine whether they are equivalent we transcribe them algebraically and form the equation that asserts their equivalence: +((C)+A)
= +((A)+C)
Inspecting the first two signs of both statements we note that both statements are negative in valence since in both cases the first two signs are different. Thus the two statements meet the covalence condition for logical equivalence. We note also that the two statements are algebraically equal. Since they are equal as well as covalent, the statements are logically equivalent. Here are two more statements logically equivalent to '+( (C)+A)':  (+(C)+( A)) (+(A)C)
no noncitizen is a nonAsian not: some nonAsian isn't a citizen
******************************************************************** Exercises:
I. Give four more equivalents to 'every noncitizen is an Asian'. II. Regiment, transcribe algebraically and apply PEQ to determine which of the following are logically equivalent pairs. 1. every truthteller is a citizen; every noncitizen is a liar 2. only A's are B's; every B is an A
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An Invitation to Formal Reasoning 3. whales are mammals; no nonmammal is a whale
III. Give seven equivalents to 'some politicians aren't honest'.
N. Regiment and transcribe in both Aform and Eform: 1. Tom runs. (Answer: We regiment 'Tom runs' as '(some) Tom is a runner'. The Aform and Eform transcriptions are: R+T* and +T*+R. 2. The king is speaking. 3. The Earth is not flat. 4. Some birds do not fly. 5. It is not the case that all birds fly. ********************************************************************
17. The General Form of Statements The transcription of 'not every nonX isn't a nonY' is
((X)( Y)) In this transcription, all signs are minus. Note that each negative term has a minus sign. We could also introduce a convention transcribing positive terms in a way that gives each positive term an explicit sign to signify its positive quality. If we prefix each positive term with a positive sign, we should transcribe 'some X is a Y' thus: +(+(+X)+(+Y)) Now all signs are plus. In between these extremes we have many statements with mixed signs. Thus 'every X is a nonY' is +((+X)+( Y)) and 'some X isn't Y' is +(+(+X)(+Y))
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In these transcriptions all plus signs are explicit. The transcription overtly signifies the positive character of a term or a statement by prefixing it with a plus sign. This goes quite beyond the norms of usage. Thus the normal form is 'some X is a Y' which we could simply transcribe 'stenographically' as '+X+Y'. But the more explicit transcription, '+(+(+X)+(+Y))', brings out the fact that the statement is an affirmation and not a denial and it explicitly shows the positive character of both terms. Any fully explicit transcription has five signs. The first sign signifies affirmation ('yes') or denial ('no', 'not'). The next sign signifies 'some' or 'every' . The next sign qualifies the first term as positive or negative. The next sign signifies 'is' or 'isn't'. And the final sign qualifies the second term as positive or negative. Since there are 2 possibilities for each ofthe five terms, the total possible combinations is 2x2x2x2x2 or 32. The following omnibus formula is the general form that shows all the possible kinds of statement: ±(±(±X)±(±Y)) yes/no some/every X/nonX is/isn't Y/nonY
In transcribing an English sentence like 'every ape is a primate' we may choose to make all positive signs explicit, in which case every ape is a primate
==> +((+A)+(+P))
Or we may choose a semiexplicit transcription that suppresses the positive signs for terms but leaves the external judgment sign [=> +(A+P)]. Finally we may choose the most natural 'stenographic' style of transcription that also suppresses the positive judgment signs [=>  A+P]. The semiexplicit style of transcription has the advantage of making it easy to tell the valence of the statement by looking at the first two initial signs. But after a while one becomes adept at determining the valence of any sentence without using that technique and then the direct mode of transcription may be used. It is after all the most natural. For the time being however, the student is advised to transcribe sentences in the 'semiexplicit' mode, beginning each affirmative sentence with a positive plus sign of affirmative judgment.
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******************************************************************** Exercises: I. Give a fully explicit transcription for the following statements. 1. every noncitizen is eligible 2. some senator is not a farmer 3. not every farmer is a golfer 4. some nongolfers are noncitizens 5. every senator is a politician 6. some sen~tor isn't a noncitizen 7. no senator is a golfer 8. no farmer is ineligible 9. not every golfer is ineligible 10. it is not the case that every politician is a citizen IT. A semiexplicit transcription drops the positive signs for terms but keeps the external signs of affirmation and denial. For example, the semiexplicit transcription of 'some X is a Y' is '+(+X+Y)'. Give a semiexplicit transcription for each of the statements in exercise I.
m. A direct or 'no frills' transcription has no plus sign for affirmation as well as no plus signs for positive terms. For example '+X+Y' is a direct transcription for 'some X is a Y'. Give a direct transcription of each statement in Exercise I.
******************************************************************** In the remainder of the chapter we shall use either use direct transcriptions or 'semiexplicit' transcriptions that give algebraic expression to 'it is the case that'. In any case we shall omit all positive signs for positive terms. For example we shall transcribe 'every Greek is a philosopher' as '+(G+P)' or as 'G+P' but not as '+((+G)+(+P))'. (See previous exercise Part II.)
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******************************************************************** Exercises: Regiment where necessary, represent all sentences in semiexplicit transcriptions and apply PEQ to determine the equivalence or lack of equivalence of the following: 1.
every noncitizen speaks a foreign language everyone who doesn't speak a foreign language is a citizen 2.
not every ape can hop some that cannot hop are apes not every ape can hop not everything that can hop is an ape not every nonape can hop
******************************************************************** 18. The Logical Law of Commutation Applied to Compound Terms The law of commutation that applies to converse pairs of sentences also applies to expressions that we call compound terms. A compound term is formed by joining two terms by means of'and' or 'or' thereby forming a third term. For example taking the two terms 'gentleman' and 'scholar' we may form a single compound term, the conjunction, 'gentleman and scholar', that applies to any gentleman that is a scholar. Conjunctive compound terms obey the law of commutation. To see this let'+ ... + ... ' stand for 'both ... and ... '.Then the expression '<+G+S>' stands for 'both G and S' or 'G that is an S ',a term that denotes an individual that is both a G and an S. We adopt the practice of using angular brackets for representing compound terms. Now the law of commutation as it applies to 'both X andY' may be stated algebraically: <+X+Y> = <+Y+X> According to this law compound terms of form 'X andY', 'an X that is a Y'
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or 'an XY thing' denote the same thing as a term of form 'Y and X', 'a Y that is an X' or 'a YX thing'. For example, the statement 'some farmer is both a gentleman and a scholar' is algebraically transcribed as '+F+<+G+S>'. Note that by the law of commutation this statement is equivalent to '+F+<+S+G>' (read: 'some farmer is a scholar and gentleman'). Thus: +F+<+G+S> = +F+<+S+G>
19. The Logical Law of Association The reader may remember that the Law of Association is another fundamental law that governs the behavior of the plus sign in arithmetic. According to the law of association, adding y+z to x gives the same result as adding z to x+.y: Association:
x+(Y+z)
=
(x+y)+z
Association also governs the behavior of the plus signs in logic. Consider the following two statements: +F+<+G+S> +<+F+G>+S
some farmer is a gentleman and scholar some farmer and gentleman is a scholar
These statements are clearly equivalent: the existence of a farmer who is both a gentleman and a scholar is the very same state of affairs as the existence of a farmer and a gentleman who is a scholar. The Logical Law of Association may be stated thus: LLA
+X+<+Y+Z> = +<+X+Y>+Z some X is Y and Z = some X and Y is Z
For logical purposes such compound phrases as 'gentleman scholar', 'gentleman who is a scholar' and 'gentleman and scholar' are treated alike, all being transcribed as '<+G+S>' since all denote an individual who is both a gentleman and a scholar. For example, we transcribe the statement 'some citizen who is a farmer is a gentleman scholar' as '+<+C+F>+<+G+S>'. The phrase 'poor widow' denotes someone who is both a poor person and a widow. By the law of association the statement '+<+P+W>+S' ('some poor widow is starving') is equivalent to '+P+<+W+S>', which we read as 'some poor person
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is a widow who is starving'. Adjectives like 'poor' and 'starving' are represented by letters like 'P' and 'S', which we sometimes read as noun phrases ('poor person', 'starving person'). Thus in the above transcriptions 'P' is first read as the adjective 'poor' and then read as the noun phrase 'poor person'. The sentence 'some widow is poor' transcribes as '+W+P', which is equivalent by LLC to '+P+W' ('some poor person is a widow'). Here again the second occurrence of 'P' is given a noun phrase reading ('poor person').
20. Derivations Let us see how LLC and LLA may be applied to show that an argument is valid: The argument in question is: A1 (1) some woman who is British is a novelist +<+W+B>+N
I (2) some novelist who is British is a woman +<+N+B>+W To justify this inference we show how the conclusion can be derived from the premise by a series of steps called a derivation: 1. 2. 3. 4.
+<+W+B>+N +W+<+B+N> +<+B+N>+W +<+N+B>+W
prenuse 1,LLA 2,LLC 3,LLC
The sequence of four statements constitute a derivation (also called a 'deduction' or 'proof). The premise is the first line and needs no justification. Each subsequent line of the derivation is a step that is justified by applying a logical law to a previous line. The justification is stated on the right. For example, the third statement in the derivation is justified by applying LLC to the second statement. The expression on the right indicates that we got statement 3 by applying LLC to the statement on line 2. The derivation ends when we reach the desired conclusion. Given 'some citizen who is a farmer is a gentleman and a scholar' as premise we may want to show that 'some farmer is a scholar who is a citizen and a gentleman' follows. Algebraically transcribed this is the argument:
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A2 +<+C+F>+<+G+S> +F+<+S+<+C+G>>
preiDise conclusion
The following sequence of steps shows how we may deduce the conclusion from the premise. l.+<+C+F>+<+G+S> 2. +<+F+C>+<+G+S> 3 +F+<+C+<+G+S>> 4.+F+<<+G+S>+C> 5.+F+<<+S+G>+C> 6.+F+<+S+<+G+C>> 7.+F+<+S+<+C+G>>
premise l,LLC 2,LLA 3,LLC 4,LLC 5,LLA 6,LLC
The derivation that follows A3 shows it is a valid argument: A3 no poor widow is a Southern congresswoman (hence) no poor Southerner is a widow who is a congresswoman 1. 2. 3. 4. 5. 6.
(+<+P+W>+<+S+C>) (+P+<+W+<+S+C>>) (+P+<+<+S+C>+W>) (+P+<+S+<+C+W>>) (+<+P+S>+<+C+W>) (+<+P+S>+<+W+C>)
premise l,LLA 2,LLC 3,LLA 4,LLA 5,LLC
****************************************************************** Exercises: Provide a derivation for each of the following arguments:
1.
no B is a C that is a D /no D that is a C is a B
2.
some A is a B that is both a C and a D /some A that is a B is a C that is a D
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some A that is a B is a C that is an D /some D is an A that is both a B and a C
******************************************************************* 21. More on Regimenting Sentences An English sentence like 'every whale is a mammal' transcribes directly into
algebraic notation as ' W+M'. Similarly its equivalent, 'no nonrnammals are whales', transcribes directly as '(+( M)+W)'. Sentences that come ready made for direct transcription will be called 'canonical'. In reallife reasoning, however, canonical English sentences are the exceptions rather than the rule. We are just as likely to come across 'the whale is a mammal' and 'only mammals are whales' as 'every whale is a mammal' or 'no nonmammals are whales'. Consider the simple valid inference nothing but a bird flies I so whatever flies is a bird To expose the structure ofthis inference we must reformulate its two sentences, giving each one a canonical paraphrase. This process is called 'regimentation'. The first step in regimenting a sentence has already been discussed: one must isolate its terms, paraphrasing where that is necessary, and then assigning a term letter to each term. Once that is done it is often easy to reformulate it as a canonical sentence that can immediately be transcribed. In the present example we have expressions like 'nothing but' and 'whatever' for which we have no direct transcriptions. So we must rephrase the sentences canonically to make them fit for transcription. Thus we first assign B for 'bird' F for 'flier' or 'thing that flies' But the premise and conclusion need to be regimented a bit more. We paraphrase the premise as 'no non bird is a flier' [==> (+(B)+F)] and the conclusion as 'every flier is a bird' [=>  F+B]. Here are two sentences taken from an example in Lewis Carroll's book on logic that need regimentation.
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(1) Nothing that isn't free from damp should be kept in a drawing room. (2) Nothing that's encrusted with salt is ever quite dry. From these two sentences taken as premises Carroll draws the conclusion that nothing encrusted with salt should be kept in a drawing room. The terms of the argument are: thing free from damp: F thing that may be kept in a drawing room: D thing encrusted with salt: E We can now regiment the sentences to allow for direct transcription: no nonF is D => (+(F)+D) no E is F => (+E+F) I no E is D > (+E+D) Using PEQ we may give an argument that is equivalent to this one:  D+F; every thing that may be kept in a drawing room is free from damp F+( E); every thing that is free from damp is not encrusted with salt D+( E); /every thing that may be kept is a drawing room is not encrusted with salt Regimenting sentences to make them suitable for reckoning is an indispensable tool in practical reasoning. It is essential to expose the structure of an argument by giving each sentence its proper form. Consider for example the following simple argument that is clearly valid. A4 some sport cars that have no automatic transmissions are convertibles I some convertibles that lack automatic transmissions are sports cars Regimented and transcribed the argument looks like this: +<+S+( A)>+C
I <+C+( A)>+S
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******************************************************************* Exercises:
I. Give a formal derivation of the conclusion of the argument A4. II. Regiment and transcribe the following sentences. 1. None but the brave were undaunted. 2. Female acrobats are brave women who are strong and quick. 3. All but the undaunted are men who are unwilling and unable. *******************************************************************
22. Uniquely Denoting Terms and Singular Statements A term that denotes no more than one thing is called a uniquely denoting term and any statement that has a UDT in the subject position is called a singular statement. Proper names are an important kind ofUDT, appearing in singular statements such as 'London is foggy', 'Caruso was not a baritone' and 'Einstein is a genius'. In the singular sentence 'the Queen is gracious' the term 'Queen' may be used to uniquely denote Elizabeth II but in the general sentence 'every queen is an aristocrat' the same term is not a UDT. A singular statement makes a claim of existence and so its valence is positive. Thus 'Caruso is a tenor' claims that the world is characterized by the existence of a tenor who is Caruso. The term 'Caruso' is not an ordinary term since, by convention, it applies uniquely to a single individual: Enrico Caruso. (Of course, even a proper name like 'Caruso' can have a use that is not UDT: for example, praising someone's singing we might say 'He is a going to be another Caruso'. But in that second use, 'Caruso' is, strictly speaking, no longer a proper name.) UDTs, including proper names, have special logical characteristics and we mark the term letters that represent them with an asterisk. For example we use 'C*' for 'Caruso', 'E*' for Einstein' and 'Q*' for 'Queen'(in the phrase 'the Queen'). Since the valence of'Caruso is a tenor' is positive it is logically a particular statement and we so transcribe it: +(+C*+T) Literally this formula says that 'some Caruso' is a tenor. However 'Caruso'
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is a proper name that applies uniquely to no more than one person; if some Caruso is a tenor, then every Caruso is. Algebraically, '+(+C*+ entails '+(C*+T)'. Similarly '(some) Einstein is a genius'(=> +E*+G) entails 'every Einstein is a genius' ( E+G). In general, ifN* is a proper name conventionally used to denote some given individual, then any statement of the form 'some N* is P' will entail 'every N* is P'. This is so because there is no more than one N* so the difference between 'some N*' and 'every N*' is an idle difference. The fact that 'some N* .. .' entails 'every N* .. .' explains why the words 'some' and 'every' are not actually used before proper names: normally a subject of form 'some X' differs from 'every X', but in the case ofproper name subjects this difference makes no difference and so we do not mark it in actual discourse. Nevertheless, for logical purposes we bear in mind that the form of 'N* is P' is particular [> +(+N*+P)] claiming existence and that it entails a statement claiming nonexistence. For if 'N* is P' is true, then so is 'every N* is P' or 'NoN* is a nonP'. Thus the existentially positive statement 'N* is P' entails 'every N* is P ', which claims the nonexistence of anyone who is N and not P. For example, 'Caruso is a tenor' claims the existence of someone who is Caruso and who is a tenor. But since there is only one Caruso, this claim entails the negative claim that no one who is Caruso fails to be a tenor (the nonexistence of any nontenor who is Enrico Caruso). The sentence 'Caruso is a tenor' is transcribed as a particular sentence, '+C*+T'. But '+C*+T' entails 'C*+T' so we could think of 'Caruso is a tenor' as indifferently particular or universal. In a sense its quantity is 'wild'; we are free to give it whatever quantity we wish. To indicate wild quantity for a singular sentence, 'N* is P', we shall sometimes transcribe it with a double sign thus:
n'
±N*+P some/every N* is P The Venn diagram for a singular statement will depict two states of affairs, one positive, the other negative. Thus 'Caruso is a tenor' signifies the presence of a tenor who is Caruso and the absence of a nontenor who is Caruso:
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Figure 18 T
Many singular sentences have subjects of form 'the X' or 'that X'. These too are transcribed as particular sentences with a starred term in subject position. For example 'that white star is a dwarf is transcribed as '+W*+D' to indicate that the subject term is a UDT uniquely denoting the white star under consideration. Similarly 'the moon is setting' transcribes as '+M*+S'. Since any sentence with a UDT in subject position has 'wild' quantity we could transcribe 'the X is Y' as '±X*+Y'. For example, where 'Queen' denotes Elizabeth II' in 'the Queen is gracious', both 'some Queen is gracious' and 'every Queen is gracious' is true and we transcribe 'the Queen is gracious as '±Q*+G'. In some cases a singular statement will make only the negative claim. I may be unsure whether Homer existed but reasonably sure that ifhe did exist, he was blind. I should then wish to claim the nonexistence of anyone who was Homer but who was not blind. In that case my claim is represented by ' (+H*+( B)' or by ' (B)+( H*)'. The Venn diagram for this negatively existential claim will shade the H*( B) segment; it won't have a sign for the presence of a blind person who was Homer since my claim is merely negative:
Figure 19
B
A statement like 'Caruso is not a baritone' is not 'existentially' negative; it claims the existence of a nonbaritone who is Caruso. It is arguable that existentially negative singular statements are not really singular. In the
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case of a statement like 'Bigfoot does not exist', it is clear that what we are saying is quite general: namely that nothing is Bigfoot. To summarize: Normally, the uniquely denoting subject term of a definite singular statement denotes some given individual. Strictly speaking such a statement has the logical form of a particular (existential) statement. But since it entails the corresponding universal statement, a specification of quantity is pointless and neither sign of quantity is explicit in actual discourse. Definite singular statements have either particular or 'wild' quantity in transcription.
23. Identities
A singular statement whose predicate term is also singular is called an identity. Examples of identities are 'Twain is Clemens'[> +T*+C*] and 'The Queen of the United Kingdom is Elizabeth II' [=> +Q*+E*] and 'Twice seven is fourteen' [=> +T*+F*]. Any identity has the wild quantity of a singular statement and so does its converse. This makes identity arguments special. The logical laws governing identity statements are discussed below in section 6 of Chapter4. ******************************************************************* Exercises: Transcribe the following singular statements: 1. Santa Claus is kind. 2. Homer was a poet. 3. The Loch Ness Monster is shy. 4. Charles is the Prince ofWales. 5. The moon is made of green cheese. 6. Twain is not Shakespeare. 7. Twain is no Shakespeare. 8. The Bard is Shakespeare. 9. The United States Chief Executive is the President. 10. Logic is fun.
******************************************************************
4 The Language of Logic (II)
1. Compound Statements We have adopted the practice of representing terms by upper case letters. For example, we represent the compound term 'gentleman and scholar' as '<+G+S>'. By convention, we use lower case for statement letters. For example, we may let 'a' stand for 'roses are red'. We now show how to represent compound statements such as 'roses are red and violets are blue' in algebraic notation. Let 'a' represent the statement 'roses are red' and 'b' the statement 'violets are blue'. The word 'and' is used as a statement connective in forming the compound statement 'a and b'. A compound statement formed by using the word 'and' as the connecting expression is called a conjunction and each statement in it is called a conjunct. Thus, if 'p' is a statement and 'q' is a statement, then 'p and q' is a conjunction whose conjuncts are 'p' and 'q'. The algebraic transcription of 'p and q' is +p+q Here too we may preface the statement by some sign of affirmation ('yes', or 'it is the case that'), which we transcribe as an external plus sign. Our transcription then looks like this: +(+p+q) which reads: it is the case that: (both) p and q The denial of a statement is formed by prefixing it with an external minus sign. For example if'a' represents 'some farmer is a citizen' then 'a' (read 'not a') represents 'no farmer is a citizen'. In denying a conjunction we
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replace 'it is the case that' by 'it is not the case that' representing this negative phrase for statement denial by an external minus sign: (+p+q)
In explaining the plus/minus notation for elementary statements, we began with the one binary sign in 'P+S' and the one unary minus sign in such forms as ' (P+S)' and '( P)+S'. Using the unary minus and binary plus signs as a 'primitive' base, we introduced new signs by definition. In particular, we introduced the binary minus sign representing the term connective '(belongs to) every' and the plus sign for the positive copulas. If we had confined ourselves to the binary plus sign and the unary minus the range of inference of our logical system would not have been fundamentally affected. But we would have very limited expressive power and that would radically affect our ability to apply our logic to actual discourse. Since we would be limited to the basic formatives we should lack the copula and the binary minus sign for 'every'. Thus our logical language would require us to say 'not: nonprimate some ape'; we should have no way to transcribe and to deal logically with 'every ape is a primate'. This means we could infer 'not: A some (nonP)' from 'not:(nonP) some A' [==> (( P)+A) /(A+( P))] but we should have no way to express the inference 'every A is P /every nonP is nonA'. The same limitation in statement logic would confine us to 'and' and 'not'. We should then have no access to inferences involving such statement connectives as 'if and 'or'. In the next section we show again how to use the primitive base (binary '+', unary ' ') to define an algebraic representation for 'if. Later we extend our expressive power to cover the range of statement connectives that enter into the inferences we make in ordinary discourse.
2. 'If...then' Suppose you overhear someone saying 'there is smoke but there is no fire' and you disagree. Your denial is amounts to affirming the old saw 'if there is smoke then there is fire'. Statements of the form 'if p then q' are called conditionals (sometimes: hypotheticals). Of the two component statements in a conditional, the first is called the antecedent and the second is called the consequent. Thus, in 'if there is smoke then there is fire','there is smoke' is the antecedent and 'there is fire' is the consequent. Let 's' stand for the
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antecedent and 'f for the consequent. Then '+(+s+( t) )' transcribes the statement you overheard and ' (+s+( t))' transcribes your denial. The denial is equivalent to 'if s then f but we have as yet no way to transcribe 'if...then' algebraically. Nevertheless, we may assume that 'ifp then q' is defined as 'not both p and notq' and transcribe what we can: +(ifs then t) = (+s+(t)) The right side defines the left. The right is negative in valence so the left must also be negative in valence. But the left side is an affirmation implicitly prefaced by something like 'it is the case that'. It will therefore have an external plus sign. Thus 'if s then f will have the form of an assertion '+( )'. Since the valence is negative the next sign must be ' '. But the next sign represents 'if and this suggests that 'if (like 'every') is logically minus and that 'yes: ifs then f transcribes as '+(s+t)'. Once again we have used the equivalence of two statements to determine the plus/minus character of an important logical connective. We did this before when we defined 'every X is Y' by way of 'not some X isn't Y', revealing that 'every .. .is' is to be represented as ' ... +'. And now, by defining 'if x then y' in terms of 'not both x and noty', we reveal that the binary functor statement connective 'if ..then' is' ... +'. Algebraically our definitional equivalence looks like this: ifx then y =df. not both x and noty +(x+y) =df. (+x+(y)) The expression '=df should be read as 'is equivalent by definition to'.
******************************************************************* Exercises: Transcribe the following forms: 1. if x then both y and not z 2. neither y nor x (hint: treat it as a conjunction whose conjuncts are ' y ' and ' z ') . 3. y if not z 4. if z then y but not x 5. if not x then ify then z
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6. ifx andy then z 7. if not x, then if not y then z
******************************************************************* 3. More on Transcription
Our method of algebraic transcription has shown that the formative words, 'some', 'and' 'is' and 'then' are plus words. The formative words 'every', 'if, 'not' and 'isn't' are minus words. We have seen that 'isn't' is a contraction of 'is not'. In some languages there is a contractive word for 'and not'. If English had the word 'andn't' we should transcribe it as '' rendering 'p andn't q' as '+(+pq)'. Another possible contraction could be 'thenn't'.for 'then not'. If 'thenn 't' were English we could say things like 'if p thenn 't q', transcribing this as '+(pq)' The following formative words have a plus/minus representation: PLUS: 'yes', 'some', 'is', 'both', 'and', 'then' MINUS: 'not','every', 'if, 'isn't', 'andn't', 'thenn't' This master list is much larger than it looks. Thus 'all' is logically a variant of 'every' and it, too, is a minus word. All kinds of negative particles such as 'less', 'un' and 'non' are minus signs . 'Is', 'are' 'was' and other forms of positive copula are all transcribed as plus signs; the different forms and tenses of negative copulas such as 'isn't', 'aren't' 'won't be' are transcribed as minus signs. For example, 'all raindrops are colorless' transcribes'+( R+( C))' and 'children were shouting' transcribes as '+(+C+S)'. We have noted that some signs have no English equivalents. Consider '+(+p+( q))', the transcription of 'p and not q'. This is algebraically equivalent to '+pq', which we should have to read as 'p andn't q'. Suppose also that we could contract 'then not' into 'thenn't'. We could then say that 'if p then not q' is equivalent to 'if p thenn't q' and we could express the equivalence algebraically: p+( q) = pq The examples of 'andn't' and 'thenn't' show that algebraic transcription is in some respects richer than English. On the other hand, there
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are many formative words that are not variants of any words in the above list. These cannot be directly transcribed by a (single) plus or minus sign. 'Only' and 'unless' are examples. In transcribing a statement containing 'only' or 'unless' we must rephrase it as a statement that contains words from the above list or variants thereof 'Only' seems to mean something like 'no non', or 'not: some non'. For example, 'only citizens are voters' may be paraphrased as 'no noncitizens are voters' and then transcribed as ' (+(C)+V)'. 'Unless' amounts to 'if not' so we may paraphrase 'p, unless q' as 'if not q then p' and transcribe it as '+{ ( q)+p)'. No variant of'or' is on the list because 'or', like 'only' and 'unless', cannot be transcribed as a simple plus or minus sign. Statements of the form 'p or q' are common and we now turn to the question of how we may transcribe them.
4. 'Or' Among the logical words that cannot be directly transcribed, 'or' is the most important. Statements of the form 'p or q' are called disjunctions and their component statements are called disjuncts. A disjunction is equivalent to the denial of a conjunction. Consider the statement 'either Nellie is home or Tom is smoking'. This statement is equivalent to the denial: 'not: both Nellie is not home and Tom is not smoking'. More generally, since we can equate 'p or q' to a statement containing only the primitive plus/minus words 'and' and 'not', we may use the latter statement to define an algebraic form for 'p or q': p or q =df. {+(p)+{q)) whose right side is equivalent to '+{(p)(q))', which we could read as 'if not p thenn't not q'. Or more simply, we could avail ourselves of the word 'or' and simply read '(p)(q)' as 'p or q'. Algebraically the definition of'p or q' looks like this: +{(p)(q)) =df. {+{p)+(q)) Transcribing 'either Nellie is home or Tom is smoking' as (n)(t)
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we note that it is equivalent to ' ( n)+t' which is the transcription of the conditional: 'if Nellie is not home then Tom is smoking'. More generally: (p)(q) = (p)+q p or q = if not p then q This equivalence could have been used to define 'p or q' as 'if not p then q'. If English had a contraction for 'then not' we could say things like 'if not p thenn't not q', a form of compound statement that is equivalent to 'p or q' since it claims: ifp is false, q can't also be false. We may even think of'p or q' as an abbreviated way of saying just that. With that understanding we are free to transcribe 'p or q' as the formula: +((p)(q)) Another way of thinking about 'or' is to recognize the equivalence of 'neither p nor q' and 'not either p or q'. The former is equivalent to 'both not p and not q' (consider: 'neither Jan is going to the party nor Peter is crashing it', which is equivalent to 'both Jan is not going to the party and Peter is not crashing it'). So 'not either p or q' is equivalent to 'both not p and not q'. The latter is transcribed as '+(+(p)+{ q) )'. The negation of this is '(+{ p)+( q))', which is equivalentto '+( ( p)( q))', read: 'eitherp orq'. Here the first ' ( ' transcribes the word 'either' and the second transcribes the word 'or'. We shall adopt the four minus way oftranscribing 'p or q'. More often than not, we will drop the parentheses and transcribe 'p or q' as '  p  q'. The symmetry of this formula for disjunction is like the symmetry of '+p+q', the formula for conjunction. The transcription makes it clear that 'or' has itself no direct transcription as a plus or minus sign. In this respect, 'or' is like 'only' and unlike such logical words as 'if, 'and', and 'not', each of which is transcribed as a plus or as a minus sign. Statements of the form 'p or q' are disjunctions and negative in valence. And of course 'p or q' is equivalent to 'q or p'. Their equation shows this: +{(p)(q)) = +( ( q) ( q)). The two sides are covalent and equal. So they are equivalent. Note that while '+(+(+p)+{+q))' (the transcription of 'p and q') and '+{(p)(q))' (the transcription of 'p or q') are equal they are not equivalent since they are not covalent.
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5. Representing Internal Structures We said earlier that we can evaluate many arguments with compound statements merely by using statement letters. But it is sometimes the case that a transcription using only statement letters is inadequate. Consider the valid inference: every student is literate if Tim is not a student then he is ineligible I if Tim is illiterate then he is ineligible Using statement letters the inference would be represented thus: p [q]+[r] /[s] + [r]
But this way of representing the argument would make it impossible to show that it is valid. On the other hand, if we represent the internal structure of its component sentences, we should transcribe it thus: S+L  [+T*S]+(+T*+( E)] 1 [+T*+( L)]+[+T*+( E)] Now it is possible to prove the argument valid by methods that we shall learn in Chapter 6. (Note that the conclusion is equal to the sum of the premises). The point of this example is to illustrate that we must often transcribe compound sentences in a way that exposes the internal structure of the component sentences. Consider the conjunction, 'Paris is expensive but Madrid isn't'. Letting 'p' stand for 'Paris is expensive and 'q' for 'Madrid is expensive' we would transcribe the conjunction as '+p+( q)'. This however would give little clue to the real meaning of the sentence. Here again we are interested in the structure of the component sentences of a compound sentence. In the present case the following transcription will give us what we want: +[+P*+E]+[+M* E]
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Note, in this and the previous example, our use of square brackets to encase a sentence that occurs as a component in a compound sentence. Other examples of the use of square brackets are: some roses are white but every violet is blue: +[+R+W]+[ V+B] if any ape is a citizen then Ike is a citizen: [+A+C]+[+I*+C] all crows are black but not all are honest: +[C+B]+[(C+H)] ******************************************************************** Exercises: I. Transcribe the following statements algebraically. Use the underlined letters
to represent the component statements. 1. If there is no fire then there is no ~moke. 2. Either there is smoke or there is no fire. 3. That roses are red and violets are .Qlue isn't true. 4. It isn't true that if there is smoke then there is fire. 5. It isn't true that roses are red or violets are blue. II. Show that 'p unless q' and 'p or q' are equivalent. (hint: treat 'unless' as 'if not'.) III. If the internal structure of the components sentence in a compound are
made evident, we should transcribe 'if some citizen is a farmer, then some farmer is a citizen' thus:  [+F+C]+[+C+F] Transcribe the following sentences using the letters underlined as term letters (in upper case). Component sentences must now be encased in square brackets. 1. Some rose is nink and every violet is .Qlue. 2. If some farmer is a gentleman and ~cholar, then every farmer is literate. 3. If some farmer is a non£itizen and some farmer is a citizen then no scholar is a thief. IV. Any two statements that are equal and covalent are logically equivalent.
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Using this sufficient condition, the following statements are seen to be equivalent to '+(p+q)', i.e., 'ifpthenq': (+(q)+p) +( ( ( p))+q) +( (q)+( p))
not both not q and p either not p or q if not q then not p, either q or not p
Give three more equivalents to 'ifp then q'. V. 1. Give seven algebraic equivalents to '+p+q'. For each algebraic formula, give an English reading. If necessary, use contractions such as 'andn't' that are not found in English. 2. Give seven algebraic equivalents and their English counterparts to 'p+q'. VI. Show that the following pairs of compound statements are equivalent.
1. p and (q orr); neither not q and not r nor not p. 2. p or (q and r); not both not p and (not q or not r). VII. 'or' like 'and' is commutative and associative. Write out the laws of commutation and association for 'or'.
******************************************************************** 6. The General Form of Compound Statements
The contractive expressions 'andn't' and 'thenn't' are represented in the General Form of Compound Statement, in which all positive signs are fully explicit: ±(±(±p)±(±q))
yes/not; both/if; yes p/not p; and/andn't,thenlthenn't; yes q/not q This omnibus formula contains all of the possible ways that two statements may be joined to form a compound assertion or denial. For example, '((p)(q)' is the transcription of 'not: if not p thenn't not q' and '(+( p)(+q))' is the transcription of 'not: both not p andn't q'.
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In some languages, 'or' is repeated so that 'p or q' is expressed as 'or p or q'. Suppose that in some ofthose languages we have a word 'om't' so that 'or not p or not q' is expressed as 'om't p om't q'. This disjunction is equivalentto 'ifp then not q', which transcribes as '+((+p)+(q)'. But if we represented 'om't' as a minus sign we could transcribe 'om't p om't q' directly as '+( (+p) (+q))' (whose colloquial reading in normal English is 'not p or not q'). Note that 'om't' hasn't been represented as a contraction of 'or not' but as a new word in its own right that transcribes as a minus sign. (We may think of 'om't p om't q' as defined by the denial of 'p and q' via the equation: +(pq) =df. (+p+q).) Our discussion has shown that the possible ways of forming compound statements goes beyond the expressive powers of a given natural language like English or Greek, which lacks words such as 'thenn't' and 'om't'. All of the possible ways offorrning compounds implicit in the general form of compound statement are expressed in algebraic notation but not all can be expressed in English. We may nevertheless be content with the expressive powers of English. For anything that could be said in a language that contained the exotic contractions can be equivalently said in English. Even a contraction like 'isn't P' is no more than a convenience that could be dispensed with for all logical intents and purposes. After all 'x isn't P' [=> 'x P'] is simply short for 'x is nonP' [==> 'x+( P)'].
7. Direct Transcriptions In what follows we shall mainly stick to English words and we shall follow the practice of directly transcribing them. The logical English words for which we have direct transcriptions are:
PLUS some and is then
MINUS every not isn't if
We shall also drop the initial plus sign of assertion as well as the signs for positive terms. For example, we shall directly transcribe 'if p then q' as 'p+q' and 'p and q' as '+p+q'. Similarly 'every A is B' transcribes as 'A+B' and 'some A is B' as '+A+B'. The convenience of dispensing with
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all of the extraneous plus signs is obvious. But there is a price. By dropping the external sign of affirmation we lose the ability of mechanically determining the valence of a statement by comparing its first two signs to see whether they differ. For example, in determining the valence of'ifp then q' as negative, we compare the first two signs of '+( p+q)' and find them different. But if we drop the initial sign of affirmation and transcribe 'if p then q' simply as 'p+q' this test cannot be used. We could, however, look at 'p+q' and mentally supply an initial plus sign of affirmation. We could then 'see' that the 'next' sign is different and thus determine that the statement is negative in valence. We think you will agree that this is a small price to pay for the convenience of omitting the eternal plus sign and going over to direct transcription. One very quickly learns how to tell the valence of the simpler and more natural forms. ******************************************************************** Exercises: I. Give direct transcriptions of the following statements and say whether their valence is positive or negative (e.g., not p or not q: pq, negative). 1. not p andn't q 2. if not p then q 3. or p or q 4. orn't p orn't q 5. not: if not p thenn't q 6. if not p then both q and r 7 if both p and q then not r 8. both p and not q 9. if p then neither q nor r 10. if orn't p orn't q then not p II. Using English* (a language supplemented by 'andn't', 'orn't' and 'thenn't') read each ofthe following sentences. Where possible give alternative readings to a sentence. orn't p orn't q; ifp thenn't q Example: pq:
1. p(q) 2. +p(q)
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pq (pq){+(q)s) {p+q){p+r) {pq)(rs)
******************************************************************** 8. Relational Statements The statements we have so far considered have a subject of the form 'someS' or 'every S' and a predicate of the form 'is P' or 'is not P'. In a statement like 'some boy envies every astronaut' we find two subject expressions, 'some boy' and 'every astronaut' connected by the transitive verb 'envies'. A transitive term is a two faced expression turning one (active) face to one subject and the other (passive) face to a second subject; grammarians often call the subject expression that follows the transitive verb the 'object' of the verb. In the present example, the first subject term ('boy') denotes someone being characterized as envier (one who envies), the second subject term denotes someone being characterized as envied; the subject 'every astronaut' is the 'object' of the verb 'envies'. Although 'some boy admires every astronaut' contains two subjects, it can be parsed as a subject/predicate statement, and regimented as 'some boy is an envier of every astronaut' whose subject term is 'boy' and whose predicate term is 'envier of every astronaut'. We call 'envier of every astronaut' a complex term. A complex term consists of a relational term (in this case 'envier of') followed by a subject (in this case 'every astronaut'). The relational term 'envier (of)' is a third term that mediates between 'boy' and 'astronaut'. Logicians call it a 'two place' term since it pairs with two terms, 'boy' and 'astronaut'. The relational sentence 'some boy is an envier of every astronaut' thus has three terms which we may represent as 'B ', 'E' and 'A'. In this sentence 'B' pairs with the active side of 'E' since the claim is that 'boy' denotes something that 'envier' denotes. 'A' pairs with the passive side of 'E', the claim being that 'envied' denotes whatever 'astronaut' denotes.
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9. A Word About Pairing In a two term sentence such as 'some boy is an astronaut' the terms 'boy' and
'astronaut' are paired for codenoting some individual or individuals. Similarly, in 'every boy is an astronaut' the same terms are paired and the claim is that 'astronaut' denotes whatever 'boy' denotes. In a two term sentence there is no possibility of misunderstanding: the two terms pair with one another. Nevertheless, we could make the obvious explicit by supplying a pairing index for any pair of codenoting terms. If the index is numerical we should transcribe 'some boy is an astronaut' as '+B 1+A1' or as '+B5+A5 ' using any numeral we please as long as the two codenoting terms are given the same numeral. In practice, the terms in a two term sentences are not given pairing indices. But in the case of relational sentences where we have more than two terms we want a notation that keeps track of the term pairs. Using numerical pairing indices for transcribing 'some boy envies every astronaut' indicates how the active and passive sides of 'E' pair with 'B' and 'A' respectively:
In general, terms that pair with one another are used in a 'codenoting' way. Two terms that have a common numerical index form a 'proper (codenoting) pair'. The following (proper) pairs ofterms are implicit in 'some boy envies every astronaut': Bh E12 boy, envier A2, E12 astronaut, envied Bh (E 12, A2) 1 boy, envier of astronaut A2, (E 12, B1)2 astronaut, envied by boy In the following sentence, three subjects are related by a three place relational term: some sailor is giving every child a toy +S 1+(G 123 C2+T3) Some of the proper term pairs implicit in the sailor sentence are: sailor, giver child, getter
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T3, G123 toy, given C2, (G 123, T3h child, getter of toy T3, (G 123, C2h toy, given to child T3, (G 123, S1)3 toy, given by sailor Some sentences contain more than one relational term. Examples are every owner of a farm is an owner of an animal  (012+F2}1+(013+A3)1 someone who married a French woman is speaking to Tilly +(M12+<+F2+W2>2}1+(SI3+T*3)1 A subject whose term is complex is called complex. The last two sentences have complex subjects. In some sentences the object of a relation is a complex subject. Here are two examples: some boy envies every owner of a dog +BI+(E12(023+D3)2)1 every lobbyist is friendly with someone working for a member of the Senate  LI+(F12+(W23+(M34+S4)3)2)1 ******************************************************************** Exercises: I. Transcribe 'someone who is disdainful of everyone who believes in Voodoo
had been cursed by Otto'. II. Give five examples of term pairs in the above transcription.
III. Transcribe and give some term pairs for each of the following. 1. 2. 3. 4.
Anyone that loves a teller who is stealing from a bank is anxious. Any boy who owns a dog bathes it. Some man who gave a rose to Edith stole a flower from Sara. Every sailor who eats some fish is a man who thinks.
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5. No boy who loves all animals is unkind to Fido.
******************************************************************** 10. Subject/Predicate; Predicate/Subject In a sentence like 'some girl is loved', the subject 'some girl' is to the left of the predicate tenn 'loved'. But in a relational tenn like 'loves some girl' [=> L 12+G:z], the subject expression '+G2' is to the right of its predicate term, 'L 12 '. The predicate/subject fonn reminds us of sentences written in 'Afonn': no copula and the predicate tenn on the left. Indeed relational terms show that the Afonn is not altogether absent from English syntax. From a syntactical standpoint, the presence of an Afonn is just what distinguishes relational sentences. The basic overall sentence structure is subject/predicate (with implicit or explicit copula) but the relational tenns have the predicatesubject structure of 'Aristotelian' sentences: no copula, predicate tenn on the left and subject on the right. In English the basic nonnal fonn is subject/predicate (SIP). Any relational sentence could be 'nonnalized' by giving its relational tenns a subject/predicate fonn. To get the relational tenn 'loves some girl' in line with the basic SIP structure we could commute 'L 12+G2'. This would paraphrase ' B 1 + (L 12+G2)' to give us:
or 'every boy is what some girl is loved by' (or 'every boy some girl doth love'). In this paraphrase the relational tenn has the fonn of a nonnal 'NounphraseNerbphrase' (subject/predicate) English sentence whose algebraic fonn is '+S+P'. Let us call any expression offonn '±X±Y' a dyad. Note now that ' B 1+(+G2+L 12)' is fully dyadic in structure. The subsentence '+G2+L 12 ' is adyadofthefonn '+X+Y'. Themainsentenceisadyadofthefonn 'X+Y'. In the next section we show how to get at and make explicit the dyadic structure of any English statement/sentence.
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11. 'Dyadic Normal Forms' A dyad is an expression ofthe form '±X±Y'. Sentences, compound terms, relational terms all have dyadic structure. Any sentence is a dyad in which the first part, '±X', is the subject and the second part, '±Y', is the predicate. Some dyads are compound terms and others are relational terms. In 'some farmer is a gentleman and a scholar'[==> +F+<+G+S>] the whole sentence is a subjectpredicate dyad but its compound predicate term, '<+G+S>', is also a dyad. There are many English sentences that do not have a dyadic structure on the surface. But all can be given a dyadic paraphrase. Consider 'some farmer is a millionaire and a gentleman and a scholar' which we may freely transcribe as '+F+<+M+G+S>'. The predicate term ofthis sentence is not a dyad. But that may be remedied by rephrasing the sentence as 'some farmer is a millionaire who is a gentleman and a scholar[> +F+<+M+>]. The paraphrase, and its algebraic transcription, now has a dyad predicate term that 'nests' another dyad. In this way the whole sentence is formulated as fully dyadic: the sentence itself is a dyad and all nonsimple parts of it are dyads. Relational sentences also need some tinkering to get them into proper dyadic form. Thus in 'some boy is petting a dog'[=> +B 1+(P 12+D2)] the expression 'P 12+D2 ' has the subject 'a dog' on the right. To render this relational term as a proper dyad of form '+X+Y' we commute its material elements, putting the subject term 'D2 ' to the left. In effect we rewrite the sentence as '+B 1+(+D 2+P 12) 1' [read: 'some boy is what some dog is being petted by']. By putting each subject to the left of its own predicate the sentence has been reformulated as a dyad that nests another dyad. The whole sentence is then said to be in dyadic norma/form (DNF).
A sentence is in DNF if every one of its terms belongs to a dyad. To get a relational sentence into DNF we place each of its subject expressions to the left of its own predicate expression. Let us take the sailor sentence as another example. Its transcription is '+S 1+(G 123 C2+T3). This sentence has three subject expressions: 'some sailor', 'every child' and 'some toy'. The first of these subjects, '+S ', is already to the left, but the second and third subjects need to be relocated to the left of their own predicates. The resulting DNF formula is
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Here '+S 1' has the bracketed expression that follows it as its predicate. (The predicate says of some sailor that he gives every child a toy.) 'C2 ' has the bracketed expression that follows it as its predicate which says of every child that it gets a toy. And '+T 3 ' too has its own predicate which says of a toy that it is given. Each subject is now in its own dyad and the whole sentence is in 'dyadic normal form'. It is important to note that the numeral '1' pairs the terms of the outermost dyad, the numeral '2' pairs the terms of a nested dyad and the numeral '3' pairs the terms ofthe innermost nested dyad. Finally consider how we would get some boy who gave an old woman a flower was insincere into DNF. Its transcription is
Its DNF transform is
Each dyad has a pair of codenoting terms; the common numeral index keeps track of the term pairs.
12. Commuting Relational Terms Applying commutation to 'some A is a B that is a C' we get 'some A is a C that is a B': +A+<+B+C> = +A+<+C+B>. Applying Commutation to the relational term in '+A1+(R12+B 2)' gives us its DNF: '+A 1+(+B 2+Rn)'. For example, given 'some boy is petting a dog' we get 'some boy a dog is petting' or 'some boy is what some dog is petted by'.
The last example reminds us once again that any sentence can be formally rewritten in a way that brings out its dyadic structure, what we are calling its 'dyadic normal form'. When a sentence is in DNF each of its non
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elementary terms (relational or compound) will itself be a dyad of form '±X±Y' that may in turn 'nest' other dyads so that the whole sentence is a structure of dyads. As another example consider the difference between the English Normal Form and the Dyadic Normal form of some gambler who petted a dog is afraid of all black male cats ENF: +<+G1+(P 12+D 2)>+(A23 <+B3+M3+C3>3) DNF: +<+G1+(+D 2+P 12)>+(<+<+B3+M3>+C3>3+A23 ) (We begin now to suppress some of those numerical subscripts which are not essential.) The example illustrates that it is not always possible to get anything like a normal sounding DNF paraphrase of the original sentence. In this case the closest we could get was some gambler who a dog did pet is of every black male that is a cat afraid We do our reasoning in ordinary English and for that reason we shall not be making routine use ofDNF formulas. Nevertheless, sentences in DNF are theoretically important in revealing the basic twoterm structure of the sentences that enter into logical reasoning. One practical use of the DNF formulas comes into play when we wish to detach a subsentence from its context. For example, given 'some dog bit a man' we may want to infer 'a man was bitten'. We do this by first transcribing the premise in ENF as '+D 1+B 12+M2' and then transforming it into DNF:
We may then detach the 'subsentence', '+M2+B 12 '. The justification for detachment is discussed in section 16 below. G.W.F. Leibniz, one ofthe great logicians ofthe past five hundred years, noted that every relational sentence consists of subsentences, each containing two terms. Thus 'Paris loves Helen' is understood to say 'Paris loves and eo ipso Helen is loved'. Leibniz's idea was that these subsentences are implicit in 'Paris loves Helen'. Our DNF notation for relational sentences represents Leibniz's analysis in perspicuous fashion. The DNF of 'P* 1+(L 12+H* 2)' is 'P* 1+(H* 2+L 12)', from which we detach the subsentence '+H* 2+L 12 '. The idea that all ofthe component dyads are subsentences was
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also proposed by A. Arnaud, a contemporary ofLeibniz, who pointed out that 'a man who is wise is honest' contains the subsentence 'a man is wise'. This too is revealed in the algebraic notation '+<+M+W>+H', which contains the phrase 'man and wise' in the form of a sentence 'some man is wise'. DNF forms come into play when we come to compare term logic with modern predicate logic (see the discussion of the Term Way and the Predicate Way in Chapter 2, section 10, and Chapter 8). There DNF formulas serve as a bridge between the two approaches to logical reckoning. Indeed, using DNF we can show how to 'translate' any sentence of the term functor logic (TFL) into the logical language ofModern Predicate logic (MPL).
13. Immediate Inferences from Relational Statements Any inference in which a conclusion is deduced from a single premise is called an immediate inference. For example, given the premise 'all children are students' we can (immediately) infer 'no nonstudent is a child' by PEQ. We now discuss immediate inferences where the premise is a relational statement. In dealing with them we may need to apply laws like commutation, association and obversion to relational terms. Recall how the law of association enables us to infer 'some A that is B is C' from 'some A is a B that is C'. Algebraically this inference is: +A+<+B+C>/+<+A+B>+C Consider now the relational statement '+A 1+(R 12+B 2)' (read: 'some A is R to some B'). Applying association gives us '+(+A 1+R12)+B2' which may be read 'something that some A is R to is a B'. An instance of this form of argument is: some boy is afraid of some dog /something that some boy is afraid of is a dog
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14. Obversion We have seen (Chapter 3, section 14) that the two sentences 'not every A is a B' and 'some A is a nonB' are equivalent. We call either one the obverse of the other. One sentence is the obverse of another if and only if they differ in three ways: in external sign, in the quantity of their subjects and in the quality of their predicates. This is easily seen if we form the equation of the two obverses:  ( A+B) =+{+A+( B)) Similarly 'no A is B' and 'every A is nonB' are obverses: {+A+B) =+{A+{ B)) Obverses are equivalent by PEQ. Consider now the two sentences: some boy failed a test +B 1+({.P) 12+T2) some boy didn't pass every test +B 1(P 12  T 2) Note that the relational terms '+(( P) 12+T2)' and '{P 12  T 2)' are obverses of one another. That obversion applies to relational terms is further evidence that relational terms are sentential in nature. Transforming a sentence into its DNF brings this out. +B 1+{+T2+{ P) 12) some boy some test did fail +B 1 ( T 2+P d some boy not every test did pass Here the two obverse 'subsentences' are '(T+P)' and '+{+T+(P))' Applied to relational terms with more than two subject expressions, the same rule applies: drive in the minus sign so as to change the sign of the relational expression and the signs of quantity of each subject. For example from some sailor didn't give every child a toy we derive
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some sailor failedtogive some child every toy The equivalence shows that the right side is the obverse of the left since all quantities of the two subjects within the relational term have been changed:
15. The Passive Transformation A familiar type of relational inference is the move from the active to the passive form or from the passive to the active form of a relation. An example IS
some boy is petting a dog I some dog is petted by a boy +BI+(PI2+D2) I +D2+(PI2+BI) To show the validity of this inference we give an annotated proof. In an annotated proof, each step is justified by appeal to some relevant principle of logic.
1. +BI+(P12+D2) 2. +(PI2+02)+BI 3. +(+D2+P 12)+B 1 4. +D 2+(P 12+B 1)
premise 1, commutation 2, commutation 3, association
Line 4 is the conclusion we wanted to prove. As the annotation on the right indicates, line 4 is got from line 3 by applying the principle of association. In applying Commutation and Association to relational statements we must take care to deal only with genuine dyads. A genuine dyad consists of a pair of material elements that have a common subscript. To see the difference between a genuine and a nongenuine dyad suppose we are given the second step and try to move form there to 'P 12+(D 2+B 1)', justifying this move by Association. The move is wrong since the expression 'D2+B 1' is illformed: its two material elements have no common subscript.
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16. Simplification The following inference pattern exemplifies a principle called Simplification: p and (q and r) I q and r
In propositional logic, simplification consists of detaching one of the two conjuncts from a conjunctive premise. The detached conjunct is then asserted as the conclusion. For example, given 'Sam is tall and (Mary is pretty and Don is in love)' we may infer 'Mary is pretty and Don is in love' by simplification. Algebraically the move allows us to detach '(+y+z)' from '+x+(+y+z)' to conclude '+y+z'. More generally, simplification allows us to detach any wellformed dyad from an expression whose terms are connected by a binary commutative and associative functor (the'+ ... +' functor). We shall formulate the Law of Simplification as: Law of Simplification (LS) Any wellformed dyad may be detached from an expression whose terms are connected by a binary commutative and associative functor (viz., the '+ ... +'functor). Thus from 'some farmer is a gentleman and scholar' we may infer 'some gentleman is a scholar'; detaching '+G+S' from '+F+<+G+S>' is justified by the law of simplification. Simplification also applies to relational statements. Given the premise 'some boy is petting a dog'[=> +B 1+(P 12+D 2)] we may first derive its DNF, '+B 1+(+D 2+Pd', by commutation, from which, by simplification, we may detach '+D2+P 1/, 'some dog is petted', as the conclusion. The line of reasoning from 'some boy petted a dog to 'a dog was petted' can be made more explicit by giving an annotated proof.
1. +B1+(P12+D2) 2. +B 1+(+D2+P12) 3. +D 2+P 12
premise 1, commutation 2, simplification
The following sequence is a derivation of 'Brutus killed' from 'Brutus killed Caesar'. 1. +B1+(K12+C2) 2. +(+B1+K12)+C2
prenuse 1, association
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2, simplification To infer 'a truck is being sold' from 'some farmer is selling a truck to a neighbor' we first formulate the premise in DNF:
from which, by commutation and association, we derive
Next we detach 'some neighbor is buying a truck', i.e.,
Finally, we detach '+T2+S 123 ' as the conclusion. In a sentence like 'some poor man is reading a newspaper' we may detach two sentences 'some man is poor' and 'some newspaper is read'. The sentences 'some man is poor' and 'some newspaper is read' are called 'subsentences' of the larger sentence in which they are embedded. And generally, any compound or relational term within a sentence has the status of a 'subsentence'.
17. Pronouns and Proterms Very often we use a term and then use that term again to denote the very same thing as previously denoted. Consider the difference between the following two pairs of sentences: some men were shouting; some men were quiet +M+S; +M+Q In this pair, the recurring term, M, does not denote the same individuals. By contrast, in some men are shouting; they are alarmed +M'+S; ±M'+A
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the second occurrence ofM denotes 'the men in question'. Note the use of a common superscript to indicate that in both sentences the subject term denotes the same individuals. In the second sentence, the subject 'they' refers to the men in question already referred to. So we need a pair of terms that are designed to denote one and the same individuals in both sentences. Terms marked by the common superscript are calledproterms. Note also that the second sentence has 'wild quantity' since 'they' has the meaning of 'the men in question' and all, not merely some, of the men in question are denoted by this proterm. A context involving a pair of proterms is called a pronominalization. The first proterm of a pronominalization is called the antecedent; the second is called the pronoun. (The pronoun may be prefixed with wild quantity.) Two common forms of pronominalizations are some A is B and it is C if any A is a B then it is C
+[+A"+B]+[+A"+C] [+A'+B]+[+A'+C]
A third common form involves 'reflexive pronouns'. Here the antecedent and the pronoun occur in the same sentence. Examples are: every barber shaves himself some barber shaves himself
 B' l+Sl2+B' 2 +B' l+Sl2+B' 2
In these sentences both occurrences ofB denote the same barber, but as the numerical indices show, the first occurrence denotes the barber qua shaver, the second denote him qua shaved. The next example is a favorite of linguists: a boy who was fooling her kissed a girl who loved him
+<+B' I+(F12+G" 2)1>I+(K13+<+G" 3+(L34+B' 4)3>3)1 Here both occurrences of B and both occurrences of G are pronominal. But where B 1 denotes the boy qua footer, B4 denotes him as loved and where G 2 denotes the girl who was fooled, G3 denotes her as lover. Here are several other examples of reflexive pronominalizations: 1. some girl hates a lover of her mother
+G' 1+H12+(L23+(M34+G'4) 2. every poet understands every metaphor contained in his own work
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3. no barber shaves his uncle (+B' t+(S12+(U23+B' 3))) 4. some barber shaves himself and no one else +[+B' 1+S 12+B' 2]+[+B' 1(S 12+( B' 2))] Note that 'else' is not a pronoun but the contrary of a pronoun. We may apply a rule, called IPE (for Internal Pronoun Elimination) to the transcription of an internal pronominalization that allows us to remove the pronoun. Thus, given +B' 1+S 12+B' 2, the transcription of 'some barber shaves himself, we can eliminate the second proterm to give us +B 1+S 11 • Note that the elimination is accompanied by changing S 12 to S 11 . Applying IPE to +B' 1+S 12+B' 2is analogous to changing 'some barber shaves himself to 'some barber is a selfshaver'.
IPE IfP ',. ... P 'n is an internal pronominalization, remove P 'nand replace any remaining occurrence ofn by,. Applying IPE to sentence 1 above we have 1. some girl hates a lover of her mother +G' t+(H12+(L23+(M34+G' 4))) I 1.1 +G 1+(H12+(L23+M3t)) which is got by eliminating G' 4 and replacing 4 by 1 in M34 .
******************************************************************** Exercises:
I. Apply IPE to 2, 3 and 4 above to get 'pronounfree' transcriptions. II. Apply IPE also to 'some barber shaves himself but no one else'. (Hint: ( B ' 2), which transcribes 'else', is not a pronoun, so should not be eliminated.)
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Appendix to Chapter 4
18. Bounded Denotation The extension of a tenn comprises all the things that have the characteristic it signifies. For example whoever has the characteristic of being a snob is in the extension of the tenn 'snob'; thus all snobs are in the extension of 'snob' and all critics are in the extension of 'critic'. But a tenn in use need not denote everything in its extension. Thus in 'some critics are snobs; they admire only one another' the tenn 'critics' is being used with restricted denotation to denote certain critics (not necessary all who are snobs). Those who utter this sentence may indeed have certain critics in mind whom they could identify. But in any case they are restricting the denotation ofthe subject term to certain critics that are, in principle, identifiable. The denotation of 'critics' in this statement is said to be 'bounded'. A tenn that is being used with bounded denotation will be represented by a marked letter. Thus our sentence about the critics would be transcribed thus:
certain critics are snobs; anyone admired by some one of them is one of them. The case of bounded denotation is an object lesson in how the denotation of a tenn is context dependent. Consider 'critic' in 'every logician is a critic'. Here too 'critic' signifies the characteristic of being a critic (BEING A CRITIC) but in this use it has no denoting role at all. In 'some critic is a logician' 'critic' does have a denoting role; it denotes some critics. In 'every critic is unhappy', 'critic' denotes all the things that have the characteristic of being a critic, that is, it denotes all of its extension. A distributed tenn such as 'critic' in 'all critic like this play' denotes all of its extension. But an undistributed tenn (e.g. 'critic' in 'some critics dislike the play' does not denote all of its extension. [For more discussion of distribution see section 3 in Chapter 6.] In 'the critic liked the play' the term 'critic' uniquely denotes a particular critic whom the speaker could identify for us. In 'a critic is in the audience; the critic (in question) is taking notes' the term 'critic' is a 'protenn'. (Thus 'the critic' could be replaced by 'he' or by 'she' in its second occurrence.) The moral of these examples is that while the extension of a tenn is independent of context, what, if anything, it denotes can vary from one sentence to the next.
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19. Terms in their Contexts A term in a statement is meaningful in three distinct ways: (i) it expresses a sense, (ii) it signifies an attribute and (iii) it denotes (a) thing(s). The term 'wise person', for example, expresses the sense (description, characterization) of BEING A WISE PERSON and it signifies the attribute (property, characteristic) ofbeing a wise person (wisdom). What it denotes must satisfy the description by having the characteristic signified. The expressive and signifying meanings do not vary from context to context. But the denotation does. In 'some wise man gave Sally excellent advice' the term 'wise man' denotes a particular wise man. In 'every wise man is honest', the same term denotes all wise men. Terms are various. Some, like 'farmer', 'citizen' and 'lover(of)', are general descriptive expressions usually denoting more than one thing. Some, like 'Caruso' and 'fortysecond president of the United States', are uniquely denoting expressions (UDT's). Some, like 'he', and 'they', are 'proterms' that denote a thing or things previously denoted. Others have restricted denotation. For example, in 'Certain critics admire only one another' the term 'critics' occurs with restricted denotation, denoting persons in a restricted class within the larger class of critics. In many cases what determines the denotation of a term is the context of utterance. For example, in 'that ball is new' the term 'ball' uniquely denotes the ball in question. In most contexts the term 'moon' uniquely denotes the satellite of the Earth as contrasted with its occurrence in 'Jupiter has more than one moon' where it is not a UDT. Proper names by convention uniquely denote their bearers. For example, used as a proper name 'Caruso' denotes Enrico Caruso, the famous tenor. :r.t 'Elton is no Caruso' the term in predicate position is not a proper name but a general term. In evaluating arguments it is important that we know whether a term is a proper name since sentences containing proper names have logical properties that differ from those of sentences containing only general terms. For example, if the subject term of '(some) Sis P' is a proper name then we write it as '(some) S* is P' so we know that 'every S* is P' is entailed. To indicate how a term denotes in a context, we often make use of term markers. There are four kinds of term markers: 1. Pairing markers. This indicates how two terms are being paired for codenoting. 2. Pronominal markers. These indicate that a given recurrent term is a proterm,
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denoting the same thing again. 3. Uniquely denoting term markers (UDT markers). These show that a term has unique denotation in a given context. For example in 'the barber was asleep' [=> +B*+A] the term 'barber' has unique denotation. Proper names are UDTs in all contexts. 4. Restrictive markers. These indicate that a given term'S' is being used to denote the members of a restricted class of S things.
1. Pairing markers A pairing marker is a common numerical index affixed to a pair of terms to indicate codenotation. The numerals' 1' and '2' in some A is R to a B +A1+R12+B2 > > every A that is a B is R to a C <+AI+BI>I+RI2+C2
are pairing markers. In 'some dog was barking at a cat' [> +D 1+(B 12+C 2)] the index '2' pairs 'cat' and 'bark' for denoting a cat that is being barked at. Any twoterm sentence is implicitly pairmarked. For example, 'some fanner is a citizen' could be transcribed as '+F4+C/. In general we omit pair markers when transcribing simple nonrelational sentences.
2. Pronominal markers A pronominal marker is one or more prime superscripts that we affix to a recurrent term to indicate that its denotation is focused on some one thing or things. Superscripted terms are protenns. Examples are: some A is a B, that A is a C some A is R to itself every A is R to itself if any A is Bit is also C
+[+A'+B]+[+A'+C] +A' I+RI2+A' 2 A'I+RI2A'2 [+A'+B]+[A'+C]
The first subject in which the protenn occurs is the antecedent, the subsequent subjects are pronouns. The sequence of sentences containing antecedent and pronoun is a pronominalization. A single sentence can constitute a pronominalization. For example, in 'some barber shaved himself[=> +B' 1+S 12+B' 2] the common pronominal superscript indicates
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that the barber being shaved (denoted by 'B' 2 ') is the same as the barber that shaves (denoted by 'B' 1'). 3. Uniquely denoting term markers
Unlike the focus of a proterm like 'man' in 'that man', which differs from context to context depending on the antecedent, the focused denotation of some recurrent terms is conventionally and permanently fixed. Thus, the proper name 'Caruso' differs from 'he' in being always focused on the great Italian tenor who sang at the 'Met' in the twenties and thirties. Also, unlike proterms, which are usually descriptive ('an ape ...that ape'), proper names are nondescriptive. [But some proterms are nondescriptive. Thus you say 'a man is on the roof and I say 'it's not a man'. Here I focus on what you focused on but I deny the description 'man'. My pronoun 'it' may be understood to refer to 'the thing in question' i.e., that thing you took to be as man'. See F. Sommers, The Logic ofNatural Language, Oxford University Press, Oxford, 1982, Chapter 11, where it is argued that proper names are a special kind of nondescriptive proterms.] Proper names are one kind of UDT. We mark UDT term letters by a circumflex. Some UDTs are descriptive. Thus in 'the Sun is ninetythree million miles away' and 'The present monarch is a woman' the terms 'Sun' and 'present monarch' are descriptive UDTs. A descriptive UDT differs from a proterm in occurring without antecedent. Consider the difference between the above UDT occurrence of'Sun' and the occurrence of'Sun' as a proterm in a context like 'a sun was visible on the screen; it (the sun) was a white dwarf'. Descriptive UDTs are very much like proterms whose antecedent background is taken for granted. For example, in 'the present monarch is a woman' (said in 2000) the term 'present monarch' is a UDT. But we may also view it as a proterm whose contextual background is understood. If we made the background explicit we might have something like: A present monarch was crowned in 1953 . She (i.e., the monarch) was a woman. The transcription would then be +M'+C; +M'+W
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marker: the monarch is a woman [=> +MA+W] 4. Restrictive markers Some terms are used with restricted denotation. These are marked by two stroke signs. For example, 'critic' may denote a special group of critics in a context like the following: certain critics who are snobs admire only one another So used, the term 'critics' denotes the members of a subclass of critics and· we should transcribe the sentence in two parts, thus: +[+/C/+S]+[(A 12+/C/1)+/C/2] certain critics are snobs and anyone admired by one of them is one of them Restricted terms are not focused. They are like general terms only their range of denotation is bounded or restricted.
20. Rules for Using Markers The following rules govern the use of markers: 1. Pairing markers When a numerical (codenoting) index has been used as a marker on an undistributed term, 'X', another index may be used again in repeating the sentence. For example, we may transcribe 'some A is B' as '+A1+B 1' and then again as '+A2+~' and so on. Moreover, if an undistributed term occurs twice in different sentences it is generally advisable to use new numerical indices. For example, where 'some A is R to a B' has been transcribed as '+A1+R12+B2', weshouldnottranscribe 'some A is FtoaC' as '+A 1+F 12+C 2' but as '+A3+F34+C/. But where one or both occurrences are universally distributed, the same index may be used:
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some A is R to a B no A is F to any C
In general, an assignment of a codenoting index to a distributed term is always unrestricted. 2. Pronominal markers A similar rule holds for pronominal superscript indices. Here the antecedent determines the indexing. In a pronominalization whose antecedent proterm is undistributed we introduce a proterm superscript. But if we have a new pronominaliation, we cannot index new occurrences of the proterm by the same superscript. For example, having transcribed 'some A is P and that A is Q' as +[+A'+P]+[+A'+Q] we may not transcribe 'some A is Band it is C' as +[+A'+B]+[+A'+C] However, where context warrants an interpretation that continues to focus on the same Athing we have a single pronominalization and then we should continue to use the same superscript. For example: 'some A is B; it is C; it is D ... ' transcribes as +A'+B; +A'+C; +A'+D ... There is no restriction on the assignment of superscripts to proterms whose antecedents are distributed. For example, having transcribed 'some A is Band it is C' with a double prime superscript, we are free to transcribe 'if any A is D then it is E' using that same superscript: [+A"+D]+[ A"+E].
3. Uniquely denoting term markers Uniquely denoting terms, UDTs, are given asterisks. For example, a letter representing a proper name in transcription is given one. There is no restriction on the recurrent use of the same name over many different contexts. Other UDTs occur in phrases like 'the moon', 'the President'. However it is
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sometimes clear that the term 'S' in a phrase of form 'the S' is really a proterm subject. If so, we do not mark it as a UDT but as a proterm that is subject to the restrictions that govern how we assign pronominal superscripts. We should, for example, normally transcribe 'the moon is full' as '+M*+S'. On the other hand, we may be talking about one of Jupiter's moons and say 'the moon was first seen by Galileo'. Here 'the moon' is interchangeable with 'it' (that moon, the moon in question); we should then treat the expression 'the moon' as a pronoun and give 'moon' proterm status. 4. Restrictive markers There are no special rules for assigning the stroke markers to terms of restricted denotation.
5 Syllogistic
1. Validity
In this chapter we study arguments that have two or more premises. Any argument consists of n statements (a conclusion and n1 premises). An argument is called a syllogism if it has as many terms as it has statements, each term appearing twice in different statements. Here are two examples of syllogistic arguments: A1
all baboons are apes all apes are primates I all baboons are primates
A2
no primate is a reptile some reptiles are herbivores all baboons are primates /some herbivores are not baboons
Al is a syllogism with three recurrent terms and three statements. A2 is a syllogism with four recurrent terms and four statements. (Syllogisms with more than two premises are often referred to as polysyllogisms or sorites.) Logicians are especially interested in two kinds of syllogism: 1. Those containing only universal statements. (A1 is an example.) 2. Those containing exactly two particular statements one of which is the conclusion. (A2 is an example.) Syllogisms of type 1 are called Uregular Syllogisms of type 2 are called Pregular All other syllogisms are irregular. The reason for being interested in regular
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syllogisms is simple: only regular syllogisms are valid. (A1 is Uregular; A2 is Pregular.) Logicians speak of the mood of a syllogism. A syllogism that is of type 1 or type 2 is said to have a regular mood. If its mood is irregular, the syllogism is invalid. So the first question we ask when looking at a syllogism is: What is its mood? If its mood is regular, it may be valid. If its mood is irregular we may dismiss it forthwith as invalid. For example, looking at the argument: A*
Some cats are tigers some tigers are striped /some cats are striped
we know at once that, despite its plausible appearance, it does not have a valid form, being neither Uregular nor Pregular. The form of A* is: some X is Y some Yis Z /some X is Z (Since the form is invalid, there will be counterinstances. After a moments reflection we may think of a counterinstance like some cats are mangy animals some mangy animals are dogs /some cats are dogs) A second question we ask is: Do the premises add up to the conclusion? The answer to this question is again crucial. For only those syllogism that add up
are valid. These two conditionsbeing regular and adding upare the only conditions that a valid syllogism has to satisfy. We can state the criteria for syllogistic validity thus:
The REGAL Principle: A syllogism is valid ifand only ifits mood is regular and it adds up.
Syllogistic
Ill
We call this the Regal Principle because the word 'regal' connotes regular and ega/ (French for 'equal') thereby reminding us of the two conditions to look for in a syllogism. Each condition is necessary for validity. Taken together they are sufficient. REGAL is a decision procedure telling us how to proceed in deciding whether a given syllogistic inference is valid or not. In section 4 of this chapter we shall justify REGAL, explaining why Regularity and Equality do the trick. But for now we shall apply REGAL. In applying Regal to check the validity of a syllogism, we first look to see whether the syllogism is regular. If it is irregular we immediately judge the syllogism to be invalid. But if it is regular we go on to see whether the conclusion is equal to the sum of the premises. If it also adds up we judge the syllogism valid. Both conditions must hold; if either one fails to hold, we judge the syllogism invalid. If both do hold, we judge the syllogism valid. Let us first consider a U regular syllogism: Ul every Athenian is a Greek U2 every Greek is mortal (so) U3 every Athenian is mortal
A3
A+G
G+M 1A+M
A3 has three statements and three recurrent terms. It is Uregular and its conclusion is equal to the sum of its premises. So it is valid. That A3 is valid can also be seen if we represent the states of affairs signified by its premises in a single Venn diagram: Figure 20
The first premise is negative, signifying the nonexistence of nonGreek Athenians. This is represented by shading the A, nonG area. The second premise is also negative: it signifies the nonexistence of anyone who is both a
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Greek and a nonmortal. This is represented by the shaded G, nonM area. We now note that the A, nonM area is shaded. So the state of affairs signified by the conclusion, i.e., nonexistence of anyone that is both an Athenian and immortal is already represented in the diagram representing the states signified by the two premises. Thus the Venn diagram shows that AI is a valid syllogism. Consider the following Pregular argument: A4
U 1 every Greek is literate P2 some philosopher isn't a nonGreek P3 /some philosopher isn't illiterate
G+L +P(G) /+P(L)
G is the middle term (i.e., the one shared by the two premises but not the conclusion). Notice that the second occurrence ofG is algebraically positive since the two minus signs cancel out. Thus the conclusion results from adding the two premises in a way that cancels out the positive occurrence of G, replacing it by L. Since it is both regular and algebraically correct, A4 satisfies the Regal principle. Consider now the following syllogism, A4 *, that is equivalent to A4: A4*
U1 every Greek is literate P2 * some philosopher is a Greek P3* I some philosopher is literate
G+L +P+G /+P+L
Each statement in A4 * corresponds to an equivalent statement in A4. So A4 *, like A4, is valid. We have labeled the statements of A4 and A4* according to their quantity or valence, P for particular/positive and U for universal/negative. This gives the valence profile, or what we will call the mood, of the syllogism. Thus the mood of A4 and A4* is UP/P, a Pregular syllogism. This Venn diagram graphically shows the validity of A4*:
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Figure 21
******************************************************************** Exercises: Use the REGAL principle to check the validity of the following syllogisms.
1.
no man is an island every island is a land mass so, no man is a land mass
2.
some logicians are philosophers some philosophers are mathematicians so, some logicians are mathematicians
3.
some noncombatants were armed no soldiers were noncombatants so, some soldiers were unarmed
4.
every boy is a dog lover every dog lover is kind some kids are boys every kind person is gentle so, some kids are gentle
5.
no cheater is honest none who are dishonest are successful
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6.
some dogs are mean some dogs are vicious so, some dogs are mean and vicious
7.
all lovers are poets all poets are sentimental no biologists are sentimental some biologists are wise so, every lover is wise
8.
all lovers are poets all poets are sentimental no biologists are sentimental some biologists are wise so, some who are wise are not lovers
9.
some pilots are brave some cowards are not pilots so, no coward is brave
10.
some soldiers are brave and true whoever is brave and true will be honored by all citizens so, some soldiers will be honored by all citizens
******************************************************************** 2. Inference We have just learned a decision procedure for evaluating any syllogistic argument for validity. Evaluating arguments is a typical kind oflogical task. Another kind of logical task is inferential: to complete an incomplete argument by drawing a conclusion from a given set of premises. In solving inference problems we draw on our knowledge of the two characteristics that every valid syllogism must have: (i) it must add up and (ii) it must either consist entirely ofUstatements or else it must have a Pconclusion and exactly one Pstatement among the premises.
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Here is an inference problem taken from Lewis Carroll's logic book. Carroll gives us three premises and he asks the reader to supply a fourth statement as a conclusion to give us a complete syllogism: (1) No terriers wander among the signs of the Zodiac (2) Nothing that does not wander among the signs of the Zodiac is a comet. (3) Nothing but a terrier has a curly tail /(4) ? ? ? The tricky part of dealing with a problem of this kind is to sort out the terms and to paraphrase each sentence in a way that permits us to transcribe it algebraically (i.e., as a sentence that affirms or denies something of the form 'some/every X is/isn't Y'). This procedure is known as regimentation. [The reader may wish to look back at the discussion of regimentation in section 10 of Chapter 2 and sections 17 and 21 of Chapter 3.] The following transcriptions show how we may regiment the three premises of Carroll's example. Terms: T =terrier W = wanderer among the signs of the Zodiac C =comet S = curly tailed Using these term letters we may regiment the argument thus: (1) NoT is a W (2) no nonW is a C (3) no nonTis an S /(4) ? ? ?
(+T+W) (+(W)+C) (+(T)+S)
I???
We are being asked to supply the conclusion. Since all of the premises are Ustatements we know by the regularity requirement that the conclusion must also beaUstatement. So all we now need to do is add the premises. Driving minus signs inward and adding gives us TW(W)C(T)S = CS
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The expression on the right is the conclusion every comet isn't curly tailed. By PEQ, this is equivalent to '(+C+S)' or no comet is curly tailed. This conclusion is a Ustatement and it completes the syllogism as a valid argument. If a set of premises contains a single Pstatement they should be summed to a Pconclusion. Here is another inference problem adapted from Lewis Carroll (we supply the transcriptions using P, N, D and W as term letters). (1) all puddings are nice (2) some deserts are puddings (3) no nice things are wholesome /(4) ? ? ?
P+N +D+P (+N+W)
I???
Here we see a Pstatement among the premises so we know that the conclusion must be a Pstatement. We know that a conclusion may be drawn by adding up the premises and canceling the middle terms. Adding up we have  P+N+D+P N W ==  W+D or +D W Of the two algebraic alternatives for a conclusion, only +D W will give us a syllogism that is valid in mood. For we need a particular conclusion. So our answer to Lewis Carroll's problem is /(4)
+D W:
some deserts aren't wholesome
We are now ready to tackle the Lewis Carroll problem we cited in Chapter 1. Carroll asks the reader to draw a conclusion from the following premises: ( 1) Babies are illogical (2) Nobody is despised who can manage a crocodile (3) Illogical person are despised Regimenting and transcribing these algebraically we have:
1. B+(L) 2. (+D+M) 3. (L)+D
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All three premises are universal. So any valid syllogism with these three premises must be Uregular. This means that the conclusion must be universal. Adding up the premises gives us 'BM' (or, equivalently, '(+B+M)'). Thus the conclusion is: (4) No baby can manage a crocodile About Carroll's next example we said earlier that it is the sort of problem that is best approached with logical method or technique for solving just this sort of problem. We now possess that technique. ( 1) Everything not absolutely ugly may be kept in a drawing room. (2) Nothing that is encrusted with salt is ever quite dry. (3) Nothing should be kept in a drawing room unless it is free from damp. (4) Bathingmachines are always kept near the sea. (5) Nothing that is made of motherofpearl can be absolutely ugly. (6) Whatever is kept near the sea gets encrusted with salt. We transcribe it thus:
1. {U)+K 2.(+E+D) 3.(+KD) 4. B+S 5.(M+U) 6.S+E All six premises are universal. So again the conclusion must be universal. Adding up the premises gives us 'BM' or '{+B+M)', which is the transcription of 7. No bathing machine is made of mother of pearl.
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******************************************************************** Exercises: (Carroll examples) 1.
All cats understand French. Some chickens are cats. I? ? ?
2.
I have been out for a walk. I am feeling better.
I ? 3.
?
?
All my cousins are unjust. All judges are just.
I ?
5.
?
No bridecakes are wholesome. What is unwholesome should be avoided.
I ? 4.
?
?
?
Things sold in the street are of no great value. Nothing but rubbish can be had for a song. Eggs of the Great Auk are very valuable. It is only what is sold in the street that is really rubbish.
I ?
?
?
******************************************************************** 3. Enthymemes Some arguments don't have all their premises. This happens very often in conversation where the speaker may assume that a premise is so obvious that it doesn't have to be made explicit. An argument with a missing premise is called an enthymeme. One fairly common type of enthymeme is exemplified by the following argument:
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all cats are furry animals all cats are purry animals so, some purry animals are furry animals As it stands the argument is irregular and it does not add up. But both these defects are repaired if we added some cats are cats to the premises. Since this premise is obviously true (it is equivalent to some things are cats or there are cats) we can assume it as implicit part of the original argument. Other enthymemes omit commonly accepted truths which the argument does not bother to state. For example, you may overhear the following exchange between A and B:
A. Ted's not married B. How do you know? A. Well, I know he's a Catholic priest As argument may be stated thus: A5
Ted is a Catholic Priest
I Ted is not married
+T*+P /+T*M
As it stands A5 is incomplete. It has three terms but only two statements. So a tacit premise has been omitted. Assuming that A5 is valid we can complete it in a way that satisfies the requirements for syllogistic validity. Being valid, A5 must have two premises that add up to the conclusion. The conclusion of A5 is particular and so is the premise that is given us. So the missing premise must be universal. [Why?] This gives us the first bit of information as to the nature of the missing premise. To find out more about it we note that the term 'M' appears in the conclusion but not in the premises. So this term must be in the missing premise. Using this information we have, we may solve for the missing premise in As enthymeme. Let the missing premise be 'X+M'. Then, on the assumption that A5 is valid, the following equation must be correct. [+T*+C]+[ X+M]=[+T* M] This gives us:
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X+M = [+T*M][+T+C] = +T*MT*C = MC = (+M+C) = C+(M) A's enthymeme has now been reconstructed by making the tacit premise (starred) explicit: 1. * all Catholic priests are unmarried 2 Ted is a Catholic priest /3. Ted is not married
C+(M) +T*+C /+T*M
It is sometimes possible to reconstruct the argument of an enthymeme that has two missing premises. Consider: A6 every Catholic Qriest is an unmarried man I some hachelors are £lergymen which is formulated initially as: P+(M) /+B+C This enthymematic argument is informally valid and it can be made formally valid if we supply some premises that were too trivial to be expressed in ordinary discourse. It has four terms (P, M, B and C) but only two statements. So two premises are missing. Since the conclusion is a P statement, one of the premises must be a P statement. Since no valid argument can have two P premises, the other premise must be a U statement. The terms 'B' and 'C' must appear in these missing premises. So we let one be '+X+B' and the other be '+Y+C'. The equation ofthis syllogism is: [ P+( M)]+[±X+B]+[±Y +C]=+B+C This reduces to: ±X±Y=+P+M which, since one of the missing premises must be universal and the other particular, gives four possible pairs of equations for our two missing premises:
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1) +X+B = +P+B; Y+C = (M)+C 2) X+B = (P)+B; +Y+C=+M+C 3) +X+B = +M+B; Y+C = (P)+C 4) X+B = (M)+B; +Y+C=+P+C A tacit premise must be true. But only the fourth of these pairs has two true statements. Supplemented by the fourth pair we can reconstitute the enthymeme as a formally valid syllogism: A6*
every Catholic priest is an unmarried man every unmarried man is a bachelor some clergymen are priests I some bachelors are clergymen
now formulated as: P+(M) (M)+B +C+P /+B+C ******************************************************************** Exercises: Find the missing premise in each ofthe following: 1.
some birds swim so, some swimmers have wings
2.
some birds swim so, some swimmers are not fish
3.
all monks are bachelors some logicians are monks
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An Invitation to Formal Reasoning so, some logicians are unmarried 4.
all those who were uninvited were boys not one boy brought a gift every female brought a gift so, no girl was uninvited
5.
every politician is unprincipled all philosophers are principled so, no senator is a philosopher
******************************************************************** 4. Why REGAL Works
In this section we explain why regularity and (algebraic) correctness are the necessary and sufficient conditions for the validity of syllogistic arguments. Anyone who asserts an argument makes a claim that its conclusion cannot be false if its premises are true. Suppose that the argument has two premises, S1 and S2, and a conclusion, S3. Thus the argument is S1,S2/S3. The claim is that if S 1 and S2 are true, S3 must also be true. In effect the claim is that anyone who asserts the premises but denies the conclusion is inconsistent. Let us call the conjunction that consists of the premises and the denial of the conclusion the counterclaim of the argument. Thus the conjunction 'S1 and S2 and (S3)' is the counterclaim of 'S1,S2/S3' and, generally, the counterclaim of 'S1, ... Snl/Sn' is the conjunction 'Sl...and Sn 1 and ( Sn)'. The relation between any argument and its counterclaim is very intimate. The following principle, called the Principle ofValidity (PV), states this relation:
PV An argument, A, is valid if and only if its counterclaim, C(A), is inconsistent. Any conjunction can be thought of as the counterclaim of an argument. For example, 'p&q&r' is the counterclaim of 'p,q/ r'. For suppose someone had asserted the argument 'p,q/r'. Then its counterclaim is 'p and q and ( r)' or 'p and 'q and r'. A conjunction can serve as counterclaim to several
Syllogistic
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arguments. Thus, since 'p and q and r' is equivalent to 'p and rand q' and to 'q and rand p', it is also the counterclaim to two other arguments: 'p,r/q' and 'q,r/p'. A conjunction that is counterclaim to an argument will be said to reject that argument. For example, 'p and q and r' rejects the arguments 'p,q/r', 'p,r/q' and 'q.r/p'. More generally, any conjunction ofn conjuncts can be thought of as the counterclaim of n distinct arguments each having as conclusion the denial of one of the n conjuncts with the rest of the conjuncts serving as premises. Suppose now that C(A) is the counterclaim of some argument A. And suppose that we know that C(A) is inconsistent. Then, by the principle of validity, A is valid. From this we see that one way of finding out whether an argument is valid is to find out whether its counterclaim conjunction is inconsistent. If a conjunction, C, is inconsistent then all the arguments that C rejects are valid. IfC is consistent, then all the arguments it rejects are invalid. In what follows we shall learn how to examine a conjunction for tell tale signs of inconsistency. We call any argument ofn statements and n terms a syllogism. Let us call any conjunction that contains n conjuncts and n terms, a syllogistic conjunction. For example, some A is Band noB is C and every A is Cis a syllogistic conjunction with three terms and three statements. The question that concerns us is: How can we tell whether a syllogistic conjunction is inconsistent? In what follows conjunction will always mean syllogistic conjunction. Fortunately inconsistent conjunctions do have special telltale characteristics that can be detected by the use of certain procedures called decision procedures. Applying a decision procedure to a conjunction enables us to decide quickly whether it is consistent or not. With such a procedure in hand we can check the validity of any syllogistic argument, A. For suppose we want to know whether A is valid. By denying A's conclusion and conjoining it to the premises we form the counterclaim C(A). Now we apply our decision procedure to determine whether C(A) is inconsistent. Suppose the procedure delivers the verdict that C(A) is inconsistent. Then, by PV, we know that A is valid. Suppose that it delivers the verdict that C(A) is not inconsistent. Then, again by PV, we know that A is invalid. For if it were valid, its counterclaim would be inconsistent.
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5. Inconsistent Conjunctions: The Telltale Characteristics Consider the statement 'some ape is not an ape'. It is not a conjunction but let us call it that anyway, allowing for conjunctions that have only a single conjunct. Here then is an example of an inconsistent conjunction. Let us transcribe it and examine it to see what this inconsistent conjunction looks like:
cl
+AA
Here we have a syllogistic conjunction of n terms and n statements, for the case n= 1. c 1 has two quite distinctive characteristics, called P and Z respectively: (P) The conjunction contains exactly one particular statement. (Z) The algebraic value of the conjunction is zero. Clearly, all conjunctions of form '+X X' will be inconsistent. Let us keep these two criteria of inconsistency in mind. Despite the extreme simplicity of the example, we shall find that any conjunction of n statements and n recurrent terms that satisfies conditions (P) and (Z) is inconsistent. The P/Z conditions are necessary and sufficient conditions for the inconsistency of any syllogistic conjunction. Thus every conjunction that satisfies P/Z is inconsistent and no conjunction that fails to satisfy P/Z is inconsistent. Consider a conjunction of two conjuncts: c2
+[+X+Y]+[ X+( Y)] (some X is Y and every X is nonY)
Here too we have a conjunction that has exactly one particular statement (the other statement is universal) and it too adds up to zero. So, c2 is inconsistent. More generally, any two statements that are divalent and that sum to zero must be inconsistent. To see why, consider that two statements are equivalent if and only if they are covalent and equal. Now two statements S 1 and S2 are equivalent if and only ifthe conjunction, +S1+(S2), is inconsistent. Where S 1 and S2 are equivalent, S 1 and ( S2) will be divalent so that one will be particular and the other universal. Also, since Sl=S2, Sl+(S2)=0. Generally then, any conjunction of two statements that sums to zero and that has exactly one particular statement {the other being universal) will be inconsistent.
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We have now shown that P/Z holds for syllogistic conjunctions ofn conjuncts where n= 1 and n=2. Before going on to consider cases where n>2, we shall need to state a principle first stated by Aristotle.
The Principle ofTransitivity: If being B characterizes every A and being C characterizes every B then being C characterizes every A. According to the transitivity principle '+[ A+B]+[ B+C]' entails 'A+C'. Thus any argument of the following fonn is valid: (A1)
A+B B+C
1A+C If so, the counterclaim conjunction C(Al)
+[ A+B]+[ B+C]+[ ( A+C)]
is inconsistent. Note that the P/Z conditions for the inconsistency of a syllogistic conjunction are again satisfied: C(Al) has exactly one Pconjunct (the third) and its algebraic value is zero. The first conjuncts in C(Al) fonn a transitive chain of two links. Together they entail the transitive conclusion, 'A+C'. But the third conjunct, ' (A+C)', contradicts that conclusion. C(A1) is therefore inconsistent. Let us call any conjunctive chain of universal statements whose links areofthefonn '+[Xi+Xj]+[Xj+Xk]' a transitive chain. For example, the conjunctive chain of statements TC1
+[A+B]+[B+C]+[C+D]+[D+E]
is a transitive chain. By the principle oftransitivity, any conjunction ofn1 universal statements that fonns a transitive chain will entail a statement of the fonn 'X+Y' where X is the subject tenn of the first statement and Y is the predicate tenn of the last statement in the chain. If we now add to this transitive chain a particular nth statement ofthe fonn ' (X+Y) ',the completed conjunction will satisfy the P/Z condition for inconsistency. For now the whole conjunction will consist of n1 Ustatements of the transitive chain and the nth P statement that denies the conclusion they entail. Moreover the whole conjunction will now sum to zero. For example, suppose we added
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' (A+E)' to the transitive chain, TC 1. This would give us a conjunction consisting of five statements four of which are universal and one of which is particular, the whole summing to zero. We call a conjunction like this one, canonically inconsistent. A conjunction is canonically inconsistent if it has n1 universal statements that form a transitive chain and an nth statement that denies the transitive conclusion they entail. It is clear that any such canonically inconsistent conjunction will satisfy the P/Z conditions. The transitive chain can be as long as you please. For example, by the principle of transitivity a conjunction of 26 conjuncts of the form +[ A+B]+[ B+C]+[ C+D] ... +[ Y+Z]+[ ( A+Z)] will be inconsistent since the first n1 conjuncts form a transitive chain that entails  A+Z but the last conjunct is a particular statement that denies  A+Z. The whole conjunction is canonically inconsistent and it satisfies P/Z. In what follows we will show that any inconsistent conjunction is either itself canonically inconsistent or else is equivalent to one that is. We will also see that any conjunction equivalent to a canonically inconsistent conjunction will satisfy the P Z criterion for inconsistency. C(A1) is an example of a canonically inconsistent conjunction. Consider the following conjunction, C(A * 1), equivalent to C(A1). C(A*1)
+[(B)+{ A)]+[ {+B+(C))]+[ {+(B)+A)]
Each conjunct in C(A * 1) is equivalent to a corresponding conjunct in C(A1). Since equivalence preserves covalence C(A*1), like C(A1), has exactly one particular conjunct. Since equivalence preserves equality, the algebraic value ofC(A*l) will also sum to zero. And finally, since C(A*l) is equivalent to C{Al), it too, though not canonical, is an inconsistent conjunction. Our discussion has revealed the basic reason for the inconsistency of any P/Z syllogistic conjunction. We have shown this for the case of canonically inconsistent conjunctions. But what holds true for the canonical case also holds true for any equivalent conjunction that is equivalent to a canonically inconsistent conjunction. For suppose that a noncanonical conjunction Cj is equivalent to a canonical conjunction Ci. Since each conjunct of Cj is equivalent to a corresponding conjunct of Ci, the algebraic and mood properties of the two conjunctions will be exactly the same. Since Ci sums to zero so does Cj. Since Ci has exactly one particular statement so
Syllogistic
127
does Cj. Thus Cj too is an inconsistent conjunction that satisfies P/Z. The P/Z criteria characterizes any conjunction, canonical or not:
PIZ A syllogistic conjunction is inconsistent if and only if it is itself canonically inconsistent or else is equivalent to a canonically inconsistent conjunction.
6. Equivalent Conjunctions One way to form the equivalent of a conjunction is by reordering its conjuncts. For example, (S1 and S2 and S3) and (S2 and S3 and Sl) are equivalent. Another way is to form a conjunction that has equivalent conjuncts by PEQ. For example, 'some A is Band every B is C' is equivalent to 'some B is A and noB is nonC' because 'some A is B' and 'some B is A' are equivalent and so are 'every B is C' and 'noB is nonC'. When a conjunction is consistent all conjunctions equivalent to it are consistent. When a conjunction is inconsistent, all conjunctions equivalent to it are inconsistent. The set of equivalent conjunctions may be quite large. For example, c(A*1) is equivalent to c(Al). But many other conjunctions are too. Since each conjunct of a conjunction can be written in eight different equivalent ways, and since we can order the conjuncts in six different ways, we will have eight times eight times six conjunctions all equivalent to C(A1), all of which are inconsistent and each of which rejects a valid argument. Consider c(A1) and c(A*1) again: c(A1) c(A*1)
+[ A+B]+[ B+C]+[ ( A+C)] +[(B)+(A)] +[(+B+(C))] +[(+(B)+A)]
For all intents and purposes we may look upon these as the same. Both belong to the same equivalence class of conjunctions and both are inconsistent; c(A 1) is canonically inconsistent. But c(A * 1) is equivalent to it and it too is inconsistent. Since c(A*1) is inconsistent, any argument formed by taking two conjuncts from c(A* 1) as premises and the denial of the remaining conjunct as conclusion will be valid. For example, the syllogism  (+( B)+A), ( (B)+( A)/+B+( C) formed by taking the third and first conjuncts of c(A * 1) as premises and the
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denial of the second conjunct as the conclusion will be valid. Moreover c(A1) and c(A*1) are only two of8x8x6 equivalent conjunctions, any one ofwhich is a counterclaim that may be used to generate a valid syllogism in this manner.
7. How This is Related to REGAL We have learned why P/Z is a necessary and sufficient condition for inconsistency. We are now in position to say why any valid syllogism must be regular and have a conclusion equal to the sum of its premises. That is, we now show how the P/Z conditions for inconsistency are connected to the REGAL principle for validity. A syllogism is valid if and only if it has an inconsistent counterclaim. A counterclaim to a syllogism is a syllogistic conjunction. Now any inconsistent syllogistic conjunction ofn conjuncts satisfies the P/Z condition: (1)
U1+U2+ ... +Unl+Pn=O
Now (1) counterclaims syllogisms that satisfy one of the following two conditions: (2) (3)
Ul+U2 ... +Un1=Pn Pn+U2 ... +Un1= U1
where the statements on the left are the premises of the syllogism and the statement on the right is the conclusion. Equation (2) contains only Ustatements and corresponds to syllogisms that are Uregular. Equation (3) corresponds to syllogisms that are Pregular (the conclusion,' U1', is a Pstatement). No irregular syllogism has a counterclaim of the form (1). Now (1) is the general form of an inconsistent conjunction. Since irregular syllogisms have no counterclaims that have the form of ( 1), they are invalid. Thus regularity is a necessary condition of syllogistic validity. It is easy to see that equality is another necessary condition for validity. For ( 1) counterclaims syllogisms that deny one of its conjuncts. Thus (1) only counterclaims syllogisms that are formed as equations oftype (2) or type (3). In other words, the conclusion of a valid syllogism must be equal to the sum of the premises. Thus regularity and equality are the two conditions that any valid syllogism must satisfy.
Syllogistic
129
We have shown that only Regal syllogisms are valid. It remains to be shown that all Regal syllogisms are valid. That is, we now want to show that any syllogism that does satisfy regularity and equality is valid. To show that REGALITY is sufficient for validity we need only to reverse the reasoning: A REGAL syllogism must satisfy either (2) or (3). However, any syllogism that satisfies (2) or (3) equation has a counterclaim that satisfies the P/Z criteria of inconsistency embodied in (1): Ul+U2... +Un1+Pn=O. Thus the counterclaim of any REGAL syllogism is inconsistent. It follows that every REGAL syllogism is valid.
8. Syllogisms with Singular Statements Consider: A7
1. Socrates is an Athenian 2. Socrates is a genius /3. some Athenian is a genius.
A7 has two premises both of which are singular statements. And clearly A7 is a valid argument. To account for its validity we need to remind ourselves of the special characteristics of singular statements which we call their wild quality. A singular statement, 'S* is P', is particular ('+S*+P'). But 'S' applies only to one individual; so '+S*+P' entails 'S*+P'. Consider 'Socrates is an Athenian'. This statement makes a positive claitp: that the existence of an Athenian who is Socrates is a fact. Since 'Socrates' uniquely denotes a single individual, 'Socrates is an Athenian' entails 'No Socrates is a non Athenian' or 'Every Socrates is an Athenian'. Thus in transcribing 'Socrates is an Athenian' we have a choice. We may transcribe it as '+S*+A' ('some Socrates is an Athenian') or as 'S*+A' ('every Socrates is an Athenian'). Thus one way of transcribing A7 is: S*+A +S*+G /+A+G a valid Pregular syllogism in which U+P=P.
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******************************************************************** Exercises: The following syllogisms contain one or more singular premises. Determine their validity or invalidity. 1. All cads are untrustworthy. Sam is a cad. Therefore, Sam is untrustworthy. 2. Eve is tall and beautiful. Eve is also intelligent. So, someone tall and beautiful is intelligent. 3. Every country is aggressive. Nothing which is aggressive really prospers. Whatever fails to prosper will dissolve. Italy is a country. Hence, Italy will dissolve. 4. One who loved Sally was Max. Max was sincere. So, someone who was sincere loved Sally. 5. No experienced person is incompetent. Jenkins is always blundering. No competent person is always blundering. So, Jenkins is inexperienced.
******************************************************************** 9. The Laws ofldentity In some singular statements both terms are singular. We have seen that such singular statements are called 'identities'. Here are some examples of identities:
Syllogistic Two is the cube root of eight Sam Clemens is Mark Twain Mt. Everest is the tallest mountain
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+T*+C* +S*+T* +E*+T*
Let A*, B* and C* be singular terms. The following laws (called Laws of Identity) hold:
Laws ofIdentity: A* is B * IB * is A*. The Law ofSymmetry: The Law ofTransitivity: A* is B* and B* is C*/ A* is C* The Law of Reflexivity: A* is A*
10. Proofs of These Laws
An identity is a singular statement both of whose terms are singular, so its quantity is wild. Let 'X*' and 'Y*' be two singular terms. The laws of identity are special applications of the logical laws governing statements of the form 'some S is P' and 'every S is P'. Identity statements are special because of their wild quantity and because both terms are UDTs. Identity statements appear to be relational; 'twice three is six' seems like 'twice three is greater than five'. But that too is a common illusion: 'twice three is greater than five' is indeed relational but 'twice three is six' is an ordinary predication, no different from 'twice three is an even number'. Of course we can cast any predication in relational form. Aristotle's formula for a statement of the form 'some S is P' is 'P belongs to some S'. His formula for 'every Sis P' is 'P belongs to every S'. We may think of the phrases 'belongs to some' and 'belongs to every' as representing relations that tie the two terms in a predicative tie. Laws of immediate inference and syllogistic can then be reformulated as laws that govern the predicative relations 'belongs to some' and 'belongs to every'. These relations are then seen to have certain formal properties. 'Belongs to some' is symmetrical. Using Aforms, 'belongs to some' is represented by the plus sign: the symmetry of this relation is reflected in the equivalence of 'Y*+X*' to 'X*+Y*', and more generally of 'Y+X' to 'X+Y'. 'Belongs to every' is represented by the minus sign; its transitivity is reflected in the law that says 'Z* X*' follows syllogistically from 'Y* X* & Z* Y*'. That 'belongs to every' is reflexive can be shown by considering that '(X)*+X*' (read: 'non X* belongs to some X*') is selfcontradictory. Thus its denial,
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' ((X*)+X*)', is a tautology, or logical truism. But the denial is equivalent to 'X* X*', or '(being) X* belongs to every X*', which is the law of reflexivity for the identity relation. One principle of identity is known as the Principle oflndiscemibility. It says that whenever X* and Y* are identical, any property of X* is also a property ofY* so that there can be no discernible difference between X* and Y*. In effect, given any identity statement 'X* is Y* ', the principle asserts that whatever is true of X* is also true ofY*. The following syllogistic argument justifies this: (every) X* is Y* (some) X* is Q I (some) Y* is Q This tell us that where X* is (identical with) Y* and Qness is a property of X* it is also a property of Y*.
******************************************************************** Exercises: Using the identity laws, decide whether each of the following arguments are valid or invalid. 1.
Sam is Mark. Mark is funny. So, Sam is funny.
2.
The fortysecond President of the United States is Clinton. Clinton is the husband ofHillary. Thus, the fortysecond President of the United States is the husband of Hillary.
3.
3 is the squareroot of 9. 3 is the sum of 4 and  1. the squareroot of 9 is the sum of 2 and 1. so, the sum of 4 and  1 is the sum of 2 and 1.
Syllogistic
4.
Alan is Blake. Blake is not Cash. So, Alan is not Cash.
5.
The Queen is Elizabeth. Elizabeth is a Windsor. The Queen rules. So, a Windsor rules.
133
******************************************************************** 11. The Matrix Method for Drawing Conclusions Given any syllogism we check to see whether it is REGAL and in that way determine whether it is valid or invalid. But as often as not the problem is inferential: we are given premises and we want to draw a conclusion. In what follows we give a general technique for deriving a conclusion from two premises. The technique is justified by a fundamental Aristotelian principle called the 'Dictum de Omni' (or the 'Every Principle'). We shall accept Aristotle's Dictum as our starting point: Dictum de Omni (DDO) Whatever characterizes every X characterizes any X
For example, suppose we have the premise that every human being is mortal and also the premise that Socrates is a human being. According to the first premise, being mortal characterizes every human. According to the second premise, Socrates is a human being. Then, according to the Dictum, being mortal will characterize Socrates. DDO justifies drawing a conclusion from two premises both containing a common tenn 'X'. The premise ofthe fonn 'X+Y' says that being Y characterizes every X. In this premise, which we shall call the 'Omni' (or 'Donor') premise, 'X' has a negative occurrence algebraically. The other premise, called the 'Matrix' (or 'Host') premise, contains a positive occurrence of 'X'. We may represent this premise as ' ... X .. .' where the dots represent the matrix or environment of 'X'. Schematically the two premises look like this:
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An Invitation to Formal Reasoning Matrix premise Omni premise
... X ... X+Y
For example, the Matrix premise might be 'some X isn't aZ' [==> '+XZ'], in which 'X' is embedded in the matrix '+( ) Z'. The premises will then be: +XZ X+Y where 'X+Y' is the Omni premise and '+X Z' is the Matrix premise. The dictum allows us to replace the positive occurrence of 'X' in the host premise by the expression from the donor premise that characterizes every X. In effect we may here replace 'X' by 'Y' giving us the conclusion, '+Y Z' or 'some Y isn't Z'. Algebraically represented, the Dictum asserts the validity of all syllogisms that have the following pattern:
... X... every X is Y
I .. .Y ... Applying the Matrix Method to the premises 'every X is Y' and 'some X isn't Z', we have concluded 'some Y isn't Z' since this conclusion gives us a syllogism that follows the pattern: +XZ X+Y /+YZ
In effect, the Dictum tells us that any syllogism containing a matrix premise in which the middle term has positive occurrence and an omni premise in which the middle term has negative occurrence is valid if the conclusion is just like the matrix premise except that the middle term has been replaced by the term from the omni premise when the two premises have been added. It is not hard to show that the Matrix Method (MM) sanctions only REGAL syllogisms. Let us first consider regularity. MM requires that one premise must be a U statement and it further requires that the conclusion must be covalent with the matrix premise since it is exactly like the matrix premise except that 'X' has been replaced by 'Y'. So there are only two possibilities:
Syllogistic
135
either all three statements are U and the syllogism is Uregular, or the omni premise is U but the conclusion and matrix premise are Pstatements, in which case the syllogism is Pregular. That the conclusion will be equal to the sum of the two premises is also guaranteed: for the middle term 'X' is algebraically negative in the first premise and algebraically positive in the second premise, and it cancels out by addition to give us a conclusion exactly equal to the sum of the premises. Thus the matrix method yields only REGAL syllogisms.
******************************************************************** Exercises: Give an argument that the MM validates all threetermed Regal Syllogisms. (hint: Show that the matrix method will apply to validate all syllogisms generated by inconsistent sets ofthree statements that satisfy the P/Z conditions.)
******************************************************************** 12. Venn Diagrams [The reader may wish to review sections 1, 2 and 3 of Chapter 2 before reading this section.] Venn diagrams have the virtue of graphically showing how two states signify a third. Thus by jointly depicting the state of affairs signified by 'no P is M' and the state signified by 'someS isM' in a Venn diagram we find that the state of affairs signified by 'someS is non P' is already depicted, thereby showing us that this latter state characterizes any universe that is characterized by 'noM is P and someS is P'.
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Figure 22
s
p
Note that 'someS is P' is not depicted. This is evidence that the syllogism 'no M is P; some S is M /some S is P' is not valid. For if it were valid, the conclusion would be graphically present when we depict the premises. Though they have the merit of being graphic, Venn diagrams afe practically limited in application to arguments with no more than three premises. For that reason we are better off learning algebraic methods that have no such limitations. For example, using the REGAL we can check the validity of any nterms argument by seeing whether it is regular and adds up. Using P/Z we can check the validity of the argument by denying its conclusion and seeing whether its counterclaim is consistent or not. Though they are limited in scope, Venn Diagrams are aesthetically appealing and they give a good idea ofthe semantic conditions of validity. For both of these reasons they will rightly continue to be popular with students and professional logicians.
******************************************************************** Exercises:
I. Construct Venn diagrams to determine the validity or invalidity of the following.
1.
Every farmer is a landowner. Some women are farmers. Hence, some women are landowners.
2.
Morris is a fool. No fool ever succeeds.
Syllogistic
137
So, Morris will never succeed. 3.
Some boys are good students. Some good students are diligent. So, some boys are diligent.
4.
Some actors both sing and dance. So, some singers are dancers.
5.
No snakes are mammals. No mammals are insects. So, no snakes are insects.
II. Use the algebraic method for the above arguments.
********************************************************************
6 Relational Syllogisms
1. Introduction
In a statement like 'some Greek is wise' the term 'wise' is tied to the term 'Greek' in the subject 'some Greek'. In a statement like 'some Greek is wiser than every Barbarian' we have two subjects, 'some Greek' and 'every Barbarian'; the term 'wiser' is therefore tied to the two subject terms 'Greek' and 'Barbarian'. A term that is tied to one subject term is called monadic. Terms tied to two subject terms are called dyadic, or two place, terms. Dyadic terms are 'relational'. For example, in 'Tom is taller than Nancy' and 'Nancy is wiser than Tom' the terms 'taller (than)' and 'wiser (than)' are relational terms. Terms that relationally tie three subject terms are called triadic. For example, in 'some sailor is giving every child a toy' the term 'giving' is triadic, or three place, tying three subject terms: 'sailor ', child' and toy'. Statements containing transitive verbs like 'kisses' or 'gives' or relational terms like 'taller than' are called relational statements. Note that the transitive verb 'kisses' is implicitly relational: 'Tom kisses Nancy' amounts to 'Tom is a kisser of Nancy' which contains the dyadic relational term 'kisser of. For further discussion, including how to transcribe relational statements algebraically, the reader is advised to review sections 8 to 15 of Chapter 4 above. Arguments containing relational statements are called relational arguments. Now a standard syllogism has as many terms as it has statements but relational arguments have more terms than statements. So, strictly speaking, relational arguments are not syllogisms. Nevertheless, in deducing conclusions from relational premises we apply the basic syllogistic principle first formulated by Aristotle and known as the Dictum de Omni. For this reason we shall continue to speak of relational arguments as syllogistic arguments, calling them relational syllogisms.
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2. Applying the Dictum to Relational Arguments The most important innovation in dealing with relational arguments is the generalization of Aristotle's 'Dictum de Omni':
What is true ofevery Xis true ofwhatever is an X We want this dictum to apply to an argument like the following:
AI some sailor is giving every child a toy; +S 1+G123 C 2+T3 every woman kisses some child;  W 4+K42+C 2 /every woman kisses someone some sailor is giving a toy to
The first premise contains the subject phrase 'every child', which, however, is not the subject of the sentence. Instead, 'every child' is surrounded by another expression. Let us call this surrounding expression E*. Now the whole sentence could be viewed as an expression of form E*(C) in which E* represents 'some sailor gives a toy (to)' or '+S 1+G123+T3 ' as the environment of 'every child', or ' C'. In effect, the statement tells us that 'some sailor is giving a toy to' is true of every child. 'E*( C)' will be variously referred to as the 'Omni Premise', the 'Donor Premise' or the 'Estar Premise'. In the Omni Premise the middle term 'child' (represented by the term letter 'C') appears negatively in the phrase 'every child', whose environment is E*, or '+S 1+G 123+T3 '. 'C' also appears in the premise we call the 'Host' or 'Matrix Premise'. In the Host Premise 'child' has another environment, E, representing the expression 'every woman kisses some', or 'W4+K42+'. Thus our two premises are E*(C) and E(C) with the middle term 'C' having a negative occurrence in the Donor premise and a positive occurrence in the Host premise. The conclusion is formed by adding the Donor premise to the Host premise thereby canceling the middle term of the Host Premise, and replacing it by the environment of the Donor premise. In this process the Donor premise cancels into the Host premise, and then contributes to the Host, transforming it by replacing its canceled middle term byE*, resulting in a conclusion ofthe form 'E(E*)':
Relational Syllogisms
141
 W 4+K42+(+SI+Gm+T3h every woman kisses someone a sailor gives a toy to Here is the pattern of the argument: A1 M=C, E* = +S 1+G123+T3 E*(M) +S 1+G123 C2+T3 W4+K42+C 2 M=C, E = W4+K42+ E(M) IE(E*) / WI+KI2+(+SI+GI23+T3)2
Note that in the conclusion, 'what's true of every child', namely, that some sailor is giving a toy to it, is now true of what every women kisses, namely, a child. Note that numeral, '2', common to the middle term in both of its occurrences reappears in the conclusion. The rule of inference 'E(M),E*( M)IE(E*)' is called 'DDO' to remind us of the Latin phrase for Aristotle's syllogistic principle, the Dictum de Omni (the 'every principle').
DDO:
E(M), E*( M) I E(E*)
3. Distributed Terms The middle term of any valid syllogistic argument is positive in theEpremise and negative in theE* premise. In 'E*( M)', the minus sign means 'every' and the term 'M' is there said to be universally distributed. The idea of 4istribution is important for syllogistic reasoning and we pause to explain it further. (For an earlier discussion see section 18 of Chapter 4.) Syllogistic argument proceeds by canceling middles and it requires that one of the two opposing middles be universally distributed. Negative and positive occurrence is purely a matter of algebra. Thus 'M' has a negative occurrence and 'P' has positive occurrence in 'not every P isM' since its transcription is ' ( P+M)' which reduces to '+PM'. But a term can be negative without being distributed. If the argument is to be valid, the negative middle term in theE* premise must be 'universally distributed'; the minus sign must have the meaning 'every' or 'any'. For example, in '+PM' [=>'some P isn't M'], the minus sign does not have the meaning 'every' and '+PM' is not a proper E* statement. Since the application of DDO depends on our ability to recognize when a term that occurs negatively is universally
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distributed, we state the two conditions to look for in determining whether a given term is or is not universally distributed. A term 'M' is distributed in a sentence, S, if and only if (1) M is negative inS (2) Sis equivalent to a sentence ofform, 'E*(every M) '. According to condition (2) a term that is distributed in a sentence must have 'universal occurrence' in that sentence. According to condition ( 1) it must also be distributed in that sentence. An example where 'M' meets condition (2) without meeting condition (1) is 'Not every M is P' [=> ( M+P)] in which M is positive. An example where 'M' satisfies (1) without satisfying (2) is This sentence transcribes as 'Some A isn't either M or B'. '+A<(M)(B)>', which, as an algebraic expression, simplifies to AM B in which 'M' has a negative occurrence. However, despite being negative, 'M' does not have universal occurrence in '+A<( M)( B)>' since this sentence is not equivalent to any sentence of form 'E*(every M)'. Another example where 'M' is negative but not distributed is '+( M)+P' (Some non M is P). By contrast 'M', is distributed in 'no A gave an M to everyB', which transcribes as '(+A 1+G 123+M2 B 3)'. For when we drive the external minus sign inward the result is a sentence of the form E*(every M), in this case 'every A failedto give every M to some B'. The transcription ' A 1+( G 123) M 2+B3 ' reveals that 'M' occurs both negatively and universally in this statement. Generally then, a term 'M' is universally distributed in S if and only if S is equivalent to a sentence of the form E*( M), with 'M' having both negative and universal occurrence (reading' M' as 'every M'). In any valid syllogism the middle term is distributed in the E* premise; it is positive in the E premise. In the following transcription, the term 'G' is universally distributed: +B 1 (L 12+G2); (some boy doesn't love any girl) Here 'G' is distributed; to show it is we drive in the unary minus sign to form a phrase 'every G' thereby disclosing that 'G' has universal occurrence. Thus '+B 1(L 12+G2)' is equivalent to '+B 1+((L 12)G2)', in which 'G' has the meaning, 'every G'.
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Consider:
K. anything that's either A or M is D 'M' is distributed in K since (1) M occurs negatively in K and (2) K is equivalent to 'every M is either not A or D':
<AM>+D
= M+<AD>
Of course 'M' is distributed in any sentence that begins with 'every M'. It is also true that any sentence in which 'M' is distributed entails a sentence that begins with the phrase 'every M'. So we could also state the condition for the distribution of 'M' in S thus: 'M' is distributed in S 'everyM'.
if and only if S entails a sentence beginning with
******************************************************************** Exercises: Transcribe algebraically and determine whether the underlined term is universally distributed.
1. some American is neither a fool nor rich 2. Tom didn't hit any target 3. no owner of an animal is exempt 4. no snake is a mammal 5. it is not true that a sailor killed him
******************************************************************** 4. Applying DDO The terms in relational statements are normally transcribed with numerical indices. In applying DDO to relational arguments we assign the middle term the same numerals in both of the premises. Suppose that we are given the
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English sentences of AI and that we have transcribed them thus:
1. W6+~1+C1 2. +S 1+G123 C2+T3
every woman kisses some child some sailor is giving every child a toy
The second occurrence of 'C' is distributed. We have the right to assign any numeral we please to a distributed term so we shall want to assign it a numeral that will make it easy for us to deduce a conclusion from the two premises. Since we want the middles to cancel out, we want 'C/ in place of 'C/; so we replace '2' with '7' thereby rewriting the second premise as: +S 1+G173  C 7+T 3. We can now apply DDO to give us the conclusion 'W6+~7+(+S 1 +G 173 +T3 )/, which reads as before: 'every women kisses someone a sailor is giving a toy to'. The rule for assigning numerals to universally distributed occurrem:es of a term may be stated more formally as a Reassignment Rule:
RA .... T;.. .
/. .. 1j.. . RA tells us that we may replace the numeral index in a universally distributed occurrence of 'T' by any numeral we please, thereby replacing 'Ti' by 'Tj '. The numeral represented by 'i' is a pairing index and when we change it, we change its partners too. For example, we changed 'C/ to 'C/ but in so doing we also had to change the 'G 123 ' to 'G 173 '. We may avoid the use of RA by following the procedure of transcribing the E or 'host' premise first. Assume that the middle term in the E premise has the index 4 so that theE premise is of form E(M4). We take note of the numeral as we come to transcribe the E* premise. For now we simply use the same numeral for the middle of theE* premise, writing theE* premise as E*( M 4). Let us take a closer look at the form ofDDO inferences. E(M) E*(M) IE(E*)
DDO allows us to replace the middle term of the Host premise, replacing 'M' in 'E(M)' by the expression E* taken from the Donor premise. It does not
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allow any replacements in the Donor premise. Thus inferences of the form
E(M) E*(M) /E*(E) are invalid. For example, the following inference, A2, is invalid even though the two premises add up to the conclusion: A2 E(C) E*(C)
/E*(E)2
W4+K42+C2 +S 1+G 123 C2+T3 I +S1+Gm+(W4+K42)2+T3
The conclusion of A2 says that some sailor is giving someone kissed by every woman a toy. That is obviously unwarranted; this conclusion is the result of canceling illegitimately into the Donor premise and replacing its middle term by the environment of the Host premise. The opposite should be done. To ensure validity in all cases, one must eliminate the middle from the host replacing it by the environment of 'every M' taken from the Donor: in other words, the conclusion must be of the form E(E*) and not of the form E*(E). Here is another example ofthe proper application ofDDO. A3 E Premise: every Derby Winner is sired by a thoroughbred E* Premise: some book lists every thoroughbred Let 'L' stand for 'lists'. We transcribe and add E*( T) to E(T) to get E(E*). E(T) + E*(T)
==> E(E*) [W3+S23+T2]+[+Bt+L12T2] ==> W3+S23+(+Bt+L12)2
The conclusion of A3 is 'every Derby Winner is sired by something that some book lists'. Note that by Commutation, the conclusion is equivalent to
 W3+(St2+(L12+Bt)) every Derby Winner is sired by something listed in a book
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A3 is an argument of the form E*( M},E(M)/E(E*) and so it is valid by the Dictum de Omni. Now consider the following invalid argument: A4 E(T} + E*( T} I E*(E) +[+BI+LI2T2]+[W3+S32+T2] I +BI+LI2+(W3+S32) The invalid conclusion of A4 is the obviously false statement that some book lists something that has sired every Derby Winner. Here again the mistake is that we have a conclusion of form E*(E) in which contribution was made to, not by, the Donor premise.
******************************************************************** Exercises: What conclusion can be drawn from the following pairs of premises? 1.
every Pope prays for every sinner I am a sinner
2.
I pray for every sinner every priest prays for some sinner
3.
all animals are mortal some owners of an animal are cruel
4.
every lover sent a valentine to some girl every boy is a lover
5.
some dogs are owned by some fanner every farmer sells some produce
6.
some students fear all exams every teacher scares every student
7.
some dogs are timid
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no cat chases any dog 8.
9 is greater than 7 7 is a prime number
9.
some judges respect some barristers all barristers are shy
10.
no philosophers read all books all books are written for some good
********************************************************************
5. Indirect Proofs for Relational Arguments Given the premise 'some boy loves every girl' how can we show that 'every girl is loved by some boy' follows? Here is the argument:
1. +B 1+L 12 G2 I G2+LI2+BI We know that this argument is valid if and only if the conjunction of the premise with the denial of the conclusion is a contradiction. So we can show that the argument is valid by showingthat this counterclaim conjunction is a contradiction. The counterclaim has two conjuncts. Let us treat them as premises of an argument: (1) +B 1+L 12 G2 Premise Denial of concl. (2)  ( G2+L 12+B 1) 1+2= ((+BI+LI2)+LI2+BI) DDO
E*=+BI+LI22 E=( ( ... )+LI2+BI) E(E*)
The third statement, derived from the conjunction of the premise and the denial of the conclusion, is easily transformed into the overt contradiction: +(L 12+B 1)(L 12+B 1) or 'someone loved by a boy isn't loved by a boy'. Thus we have derived a contradiction from the assumption that the premise is true and the conclusion is false. This shows that the original argument is valid.
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The following famous argument is an instance of the general principle that from 'every A is B', 'whatever is R to an A is R to a B' follows. Premise: every horse is an animal Conclusion: every owner of a horse is an owner of an animal Here again we may show that this argument is valid by denying the conclusion and applying DDO to derive a contradiction. Counterclaim: ( 1) every horse is an animal
(2) some owner of horse doesn't own any animal H2+A2 +(012+H2) (012+A2) /+(0 12+A2)(0 12+A2); some owner of an animal doesn't own an animal A direct proof of the argument may also be given by treating it as an enthymeme (see section 3 of Chapter 5) with a missing premise that is 'tautological'.
A5 1.  H+A every horse is an animal premise 2.  (0 12+H2)+(0 12+H2) *every owner of a horse is an owner of a horse premise /3. (0 12+H2)+(0 12+A2) every owner of a horse is an owner of an animal The starred premise is a logical truism, as is any statement of the form 'X+X'. Any logical truism (other than the conditional statement which has the conjunction of the premises as its antecedent and the conclusion as its consequent) may be added as a premise to an argument.
6. Transforming Arguments A relational inference not given in DDO form can often be transformed into a DDO inference by showing that one of the premises is equivalent to an E*
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premise. For example, given the premise 'noM is P and someS isM' we may validly conclude 'some S is nonP'. To apply DDO we first use PEQ on the universal premise: E*(M) E(M) E(E*)
M+(P) = (+M+P) +S+M = +S+M +S+( P) = +S+( P)
noM is P someS isM I someS is non P
Consider the following inference: <+A+B>+C Every A that is B is a C SomeB is aD +B+D I something that's either not A or Cis aD I+< A+C>+D In this inference 'B' is the middle term. The inference is valid but it is not in DDO form since it lacks an Omni premise in which ' B' appears. It can however be put into the form of a DDO inference if we transform the first premise into its equivalent ' B +<A+C>'. Now the first premise is of the form E*( B). So we cancel 'B' in the Host premise, replacing it there by '< A+C>' to give us '+< A+C>+D'. Let us look at some more examples in which DDO figures.
1. every owner of a dog is kind 2. some dogs are beagles What conclusion may be derived from these two premises? We note that the middle term 'dog' occurs positively in (2). So (2) is the Host premise. Let us transcribe this as '+D2+B 2 '. Now 'dog', the middle term, is distributed in (1). But ( 1) needs to be transformed into an explicit E* premise in which we find 'every dog' surrounded by its environment E*. The following transformation shows that ( 1) is indeed equivalent to an E*( D) premise.
The third formula on the right shows that'D' is distributed in a statement of the form 'E*( D)'in whichE* = ( K) 1+(0 12). So we may applyDDOwith (2) as the donor orE premise to get'+(( K) 1+(0) 12)+B2 ', i.e. 'something every unkind person fails to own is a beagle', or more colloquially as '+((+( K) 1+(0) 12)) 2+B2 ', which says 'something no unkind person owns is
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a beagle'. Equivalently the conclusion is '+((0) 12 (K) 1)+B2 ' (read: 'something that is unowned by everyone who is unkind is a beagle').
7. Annotating a Proof of Validity The following argument is valid: All Greeks are philosophers All Athenians are rich I All Athenian Greeks are rich philosophers To show that it is valid we could use the indirect method of denying the conclusion and proving inconsistency.
1. G+P 2. A+R 3. (<+A+G>+<+R+P>) The third statement denies the conclusion and we must now show that 1, 2, and 3 are jointly inconsistent. We may do this in an annotated proof: 4.+<+A+G><+R+P> 5.+<+R+G><+R+P> 6 +<+R+P<+R+P>
3,PEQ 4+2, DDO 5+1, DDO
Thus we have derived a contradiction from 1, 2, and 3. We may also get the conclusion by a direct proof:
1. 2. 3. 4.
5
G+P A+R <+A+G>+<+A+G> <+A+G>+<+A+P> <+A+G>+<+R+P>
prermse prermse tautological premise 3+1, DDO 4+2, DDO
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******************************************************************** Exercises: Give indirect proofs to show that the following arguments are valid: 1.
all wise men are clearheaded some men are devious so, some who are devious are either not wise or clearheaded
2.
whoever tries to kiss every girl is rude Jack tries to kiss every girl so, Jack is rude
3.
whoever tries to kiss every girl is rude some girls are coy thus, someone who tries to kiss someone who is coy is rude
4.
every dog has some fleas all fleas cause some irritation so, all dogs have something that causes irritation
5.
John loves Mary Mary kissed Tom Tom fooled Lois so, John loves someone who kissed someone who fooled Lois
******************************************************************** 8. Arguing with Pronominal Sentences [The reader may wish to review sections 17 to 20 of Chapter 4 before proceeding.] I say that a man is at the door and then add that he (themaninquestion, that man) wants to speak to you. The form of the sentences is +M'+P; +M'+Q, in which the recurrent term has a common (prime) superscript to indicate that in both occurrences the term denotes one and the same thing: what the first 'proterm' denotes is again denoted by the second proterm. Taken together the two sentences constitute a pronominalization.
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Grammarians call the first subject in a pronominalization the 'antecedent subject' and the second subject the 'pronominal subject', or 'pronoun'. The antecedent and the pronoun differ in the vernacular ('some man ... the man ... '). But logically both subjects of a pronominalization are represented in the same way with a superscripted protenn: +M'. The prime superscript indicates that the same man is denoted by both occurrences of 'man' in 'a man' and 'the man'. Usually we write 'he' for 'the man in question' but logically what we have is a recurrent protenn focused on the same denotatum. In 'someS is a P~ that Sis a Q' the recurrent protenn 'S' codenotes a 'certainS' even though the S 'in question' is not identified for us by the speaker and may not even be known to the speaker. I may for example suspect that some students are cheating and say: Some students are cheating~ they will be found out +S'+C~+S'+F
In my second statement, I denote the 'students in question' whether or not my prediction that they will be exposed is true or false. I do not know who they are, and may never know, yet my protenns denote 'certain students' (they know who they are). Note that we may assign wild quantity to the pronominal subject since S' denotes all as well as some of the studentsinquestion. By assigning wild quantity to the pronoun we also mark the difference between antecedent and pronoun. The antecedent has particular quantity, the pronoun that 'refers back' to it is given wild quantity. Thus the pronominalization may be transcribed as
+[+S'+C]+[±S'+F] Suppose I add the following remark: However, some students were prepared~ they did not cheat Since the students now under consideration are not those previously denoted by the protenn S ', we transcribe this second pronominalization by giving different superscripts to the protenns: (1)
(2)
+[+S'+C)+[±S'+F] +[+S"+P]+[±S"C]
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Note that the proterms in (2) have been given double markings. In general, once we choose a superscript for the proterms in a given pronominalization whose antecedent is a particular subject of form 'some S ', the superscripted proterms denote a 'certain thing' and we may not use the same superscripted proterm again unless we intend to continue to denote the thing in question. Pronominalization can often be paraphrased by sentences that are pronoun free. Thus 'some A is a B; it is a C' can be paraphrased as 'some A is a B that is a C', which transcribes as '+A+<+B+C>'. The more serious pronominalizations are those that are not easily or naturally paraphrased away. Pronominalization that takes place in the same sentence often contains reflexive pronouns that cannot be naturally paraphrased in a pronoun free way. An example is 'some barber shaves himself: +B' 1+S 12+B' 2 • A favorite example of linguists is: 'a boy who was fooling her kissed a girl who loved him'. Here we have two pronominalizations, one of which linguists call cataphoric, or 'backward', since the pronoun 'her' appears before its antecedent. But again, while there is this difference between pronoun and antecedent in the vernacular, the difference is not represented in logical transcription, since all that logically matters is the fact that a single recurrent term is being used to denote one and the same thing. Thus the boyfooledgirl sentence transcribes as:
Note also that the proterms' numerals may differ: G' 2 is paired with F 12 as the girl who is being fooled, and G' 3 is paired with K 13 and L34 as the girl who is kissed and who loves, but the common superscript ensures that it's the same girl. Again, B' 1 is the footer and the kisser while B' 4 is the loved one but it's the same boy. The common superscripts signify the common focus. Similarly 'everybarbershaveshimselftranscribesas B' 1+S 12+B' 2, where each barberquapersonwhoshaves is the very same person as the barberquapersonwhoisshaved. In the next few paragraphs we discuss some special rules that we need for reckoning with sentences containing proterms. Consider again the case of simple pronominalization. I say that a man is P and follow this by saying that he is Q. My first sentence looks as if it should be transcribed as +M+P and indeed ifthis sentence were not part of a pronominalization, that would be the proper transcription. However, since we now have a pronominalization we cannot simply transcribe the whole as +M+P; +M+Q since there would then be no indication that the man in question
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of whom it is being said that he is Q is the one that has just been said to be P. Once the second sentence with its back reference to the first subject has entered the picture, we must view the two subject terms in tandem as denoting the same thing; in other words, both 'man' and 'he' now stand revealed as 'proterms' and their codenoting status must be marked by the pronominal superscript: +M'+P, ±M'+Q. More generally, any use of 'some X' is potentially a pronominal use and when 'X' is followed by a pronominal back reference to 'the X in question', it is actually so. Indeed, any sentence of form 'some X is Y' is trivially equivalent to a pronominalization: 'something is X and it is Y'. We may therefore introduce the following rule of inference called Pronominalization and referred to as P 1.
P1
some X is Y I so it (the X in question) is Y +X'+ Y I +X'+ Y
Notethattheproterm in +X'+Y shouldhavewildquantity. Given +X'+Y, we may infer' X'+Y' For if it is true that some students are cheating, then it is true that 'they' are cheating, i.e, that all of the students in question are cheating. This gives us another rule of inference: P2
+X'+Y I X'+Y
We will refer to P2 as the Rule of Wild Quantity (WQ). The following equivalences are selfexplanatory. In each of these cases the formula on the right is a Pronominal Expansion ofthe formula to its left: some A is B +A+B
= something is an A and it is a B = +[+T'+A]+[+T'+B]
some A is B = an A exists (is a thing) and it is a B +A+B = +[+A'+T]+[+A'+B] every A is B = if any thing is an A, it is a B  A+B =  [+T'+A]+[+T'+B] every A is B = if an A exists, it is a B A+B = [+A'+T]+[+A'+B]
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We shall refer to the statement on the right, the pronominal expansion, as the pronex of the statement on the left. The pronex formulas show that any particular statement or universal statement is equivalent to a pronominalization. Pronominal expansions may also be formulated as rules of inference: +X+Y I +[+T'+X]+[+T'+Y]  X+Y I [+T'+X]+[+T'+Y] One pronex of+X+Y is +[+X'+T]+[+X'+Y] ('some X is a thing and it is a Y'). Applying simplification, we may drop the first conjunct and get Pronominalization (PI) as a derived rule: PI
some X is a Y I so that X (or it) is a Y +X+Y /+X'+Y
PI allows us to replace +M+P by +M'+P, a sentence with a pronominal subject, thereby justifying transcribing 'some man is P and he is Q' as +[+M'+P]+[+M'+Q]. In effect, we understand 'some man is P and he is Q' as saying 'a certain man is P and that man is Q'. It is easy to see that PI allows for pronominalizing Y as well as X. Thus from+ X+Y we move by PEQ to+Y +X and thence by PI to+Y' +X, and again by PEQ to +X+Y' and by PI to +Y'+X'. This gives an important derived rule of inference:
Pia
+X+Y I+X'+Y'
The following simple argument shows how a pronominalization may figure in inference: · some Greek are Sophists they are philosophers they are Athenians /some Athenians are philosophers To show this argument is valid we transcribe the premises algebraically thus: 1. +G'+S 2. +G'+P 3.+G'+A
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The derivation proceeds: 4. G'+A 5. +A+P
3, WQ
2+4, DDO
In using DDO we leave the Host premise intact except for the middle term which is replaced by the environment of the Donor premise. When we apply DDO to inferences involving pronominalization we treat the superscript ofthe Host's middle term as part ofthe environment, E, leaving the superscript in place when we replace the middle term by E*. The following example illustrates this procedure: A6 1. some barber shaves himself 2. every barber is a philosopher I some philosopher shaves himself
+B'1+S12+B'2 B~+Pl
/+P' 1+S12+P' 2
E(B') E*(B') IE(E*)
Note again that the proterm 'barber' in premise ( 1) occurs twice with different numerical indices. In its first occurrence it denotes a barber that shaves; in its second occurrence it denotes a barber that is shaved. The common superscript indicates that it is the same barber. The environment E in the host expression is+ ... '+S 12+ .. .'. Note again that the middle term, 'barber' in the Host premise has superscripts. But these are part of its environment and they remain when 'barber' is replaced by 'philosopher'. Finally, we restate our rule IPE (from Chapter 4, section 17). This rule enables us to eliminate the pronouns from the transcription of any internal pronominalization. The Rule of Internal Pronoun Elimination is: IPE
Given any internal pronominalization, ... P'm .... P'n we may remove P'n and replace any occurrence of n by m.
For example, given +B' 1+S 12+B' 2 (the transcription of 'some barber shaves himself) we may remove B' 2 and replace '2' by' 1' in S12 thereby giving us a new transcription, B1+S 11 , which we might read as 'some barber is a self shaver'. Now we can deal with A6 as follows 1. +B' 1+S 12+B' 2 2. B 1+P 1
3. +B 1+S 11 4. +P 1+S 11
E(B) E*(B) 1, IPE 2+3, DDO
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Some things we talk about are impossible. Impossible things may be more or less obvious. That square circles and married bachelors are impossible is obvious. But it may not be so obvious that we are dealing with an impossible situation when someone asks us to think about what happens when an irresistible force meets up with an immovable object. It takes awhile to realize that a world containing a force that no object can resist cannot also be a world in which there is an object that no force can affect. One famous case of an impossibility is the barber who shaves all persons who don't shave themselves and refrains from shaving any that shave themselves. To prove that no such barber can possibly exist we put down the two conditions as premises. One condition is that the barber shaves all nonselfshavers. The other is that he refrains from shaving all selfshavers. 1. +B\+S12(S22) 2. ±B' 1+(SdS22
If such a barber exists, he will either be a selfshaver or a nonselfshaver. Assume the first alternative: 3. ±B' 2+S 22 4. ±B'2+(±B'1+(S12)) 5. ±B'2+(S22)
Assumption 2+3, DDO 4, IPE
Now 5 contradicts 3; so assume the other alternative, i.e. that the barber in question is a nonselfshaver: 6. ±B'2+(S22) 7. ±B' 2+(±B'1+S12)) 8. ±B2+S 22
Assumption 1+7, DDO 8, IPE
Here 8 contradicts 6. Since both possible assumptions lead to contradiction we conclude that the barber described in the premises cannot existhe is impossible.
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******************************************************************** Exercises: Give annotated proofs for each of the following valid inferences. 1.
all politicians admire themselves they are vain so, some who admire themselves are vain
2.
every critic is an author every critic reads every author so, every author reads himself
3.
some philosophers are wise they are also poor and they are humble thus, there are those who are wise and humble who are poor
4.
all politicians vote for themselves some senators are politicians they are crass so, some who are crass vote for themselves
5.
some girl was kissed by a boy he liked her so, she was kissed by one who liked her
******************************************************************** 9. Distributed Proterms We have said that a term, T, is distributed in ' ... T .. .' if and only if T is algebraically negative and ' ... T .. .' entails a statement of form ' ... every T .. .'. A proterm is distributed if its antecedent is distributed. Consider 'if some A is B then it is C' (the pronex of 'every A is Band C'). The pronex transcribes as '[+A'+B]+[±A'+C]', in which the proterm occurs twice as a distributed term.
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When a term is distributed we can assign to it any numeral we please. And if the term is a proterm, then the same rule holds for its superscripts. When a proterm is distributed we may assign to it any superscript we please, including superscripts already in use in other pronominalizations. For example, given the premises 'some barber loves himself [=> +B' 1+L 12+B' 2] and 'every barber shaves himself, we have distributed proterms in the E* premise. This means we can assign its numerals and superscripts to line them up with those of the Host premise, writing theE* premise as B' 1+S 1:itB' 2 to derive the conclusion: +B' 1+L 12+( B' 1+S 12) 2 [read 'some barber loves someone he shaves']. Here is the deduction:
1. +B' 1+L12+B' 2
prenuse prenuse
2.  B' 1+S 1:itB' 2 1+2 DDO 3. +B' 1+L 12+(Sl2+B'2)'2 /4. +B' 1+L 12+(±B\+S 12)' 2 Commutation The proterms ofpronominalizations ofthe form 'if any Tis a P then it is a Q' are distributed. (That the 'T' in the antecedent 'any T' is distributed is evident from the fact that the statement is equivalent to 'every Tis either not a P or a Q. ') Consider the statement: (9) if any students are cheating they will be found out Here I do not commit myself to saying that some students are cheating. In a sense the proterm does not denote at all; for there may not be any students of whom it could be said that they are 'the students in question'. In another sense we are inclined to say that whatever the antecedent term denotes is denoted by the proterm in the second sentence. So we have a kind of distributed codenoting. 'The' students that cheat (if any there be) are the students that will be exposed. In transcribing this pronominalization we again use superscripts: [+S'+C]+[ S'+F] Here again, since the proterms of (9) are distributed we can choose whatever superscripts we please; we need not worry about having used the same superscripts earlier. Thus suppose we had asserted (8) some students are late; they must wait to be seated +[+S"+L]+[+S"+W]
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and now wish to assert (9). We need not be concerned that the double prime has been used for (8). For (9) applies to any student, including the students denoted by (8). It is therefore legitimate to transcribe (8)and (9) as 8. +[+S"+L]+[+S"+W] 9. [+S"+C]+[S"+F]
As the last two examples illustrate, whenever we are given a pronominalization ' ... Ti .. .', where 'T' is a distributed proterm, we may change the superscript and form another pronominalization ' ... Tj .. .'. We state this as a new Reassignment Rule:
RA2
... 1' .. . I ... P .. .
Another example of a pronominalization whose proterms are distributed is every barber shaves himself which we transcribe as
Consider the following argument: some barber drinks every barber shaves his uncle so, someone who drinks shaves his uncle This argument is intuitively valid. To show that it is valid we shall give an annotated proof, moving step by step to the conclusion. 1. +B 1+D 1 2.  B' l+Sl2+(U2l+B' I) 3.  B1+S12+U21
4. +D 1+B 1 5. +DI+S12+U21
premtse premise 2, IPE 1, Commutation 3+4, DDO
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The next example has a pronoun whose antecedent proterm is a proper name. It also illustrates how to handle the 'exceptive' word 'else'. Show that 'Mark Twain is Sam Clemens' follows from the following two premises:
1. Mark Twain is funnier than anyone else. +T* 1+F 12 (T*h 2. He's not funnier than Samuel Clemens. +T* 1+( F 12)+C* 2 passive transformation 3. (T*h+F 12 T* 1 4. +C* 2+(F 12)+T* 1 2, passive transformation 5 +C* 2(F 12 T* 1) 4, obversion 6. C* 2(F 12T* 1) 5, WQ 7. (F 12 T* 1)C* 2 6, PEQ 8. (T*)2C* 2 3+7, DDO 9. C* 2+T* 2 8, PEQ The last line shows that 'Clemens is Twain' can be derived from the two premises in a series of justified steps. [For the use of passive transformation in steps 3 and 4 the reader should review section 15 of chapter 4.] ******************************************************************** Exercises: Give annotated proofs for the following: 1.
Bill admires everyone but Ned Ned is admired by everyone but Clint so, Bill is Clint
2.
some boy loves every girl if someone loves every girl he is a fool so, someone is a boy and he is a fool
3.
some actors love themselves all selflovers are vain hence, some who are vain are actors
4.
Tom is a citizen every citizen likes Tom Tom was mugged by a citizen
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5.
2 is smaller than any other prime no other prime is even 2 is not smaller than the square root of 4 the square root of 4 is even so, 2 is the square root of 4
********************************************************************
7 Statement Logic
1.
Introduction
Arguments consist of statements. But some arguments can be seen to be valid or invalid even if we know nothing about the internal structure or contents of the statements involved. The logic that deals with arguments of this kind is known as Statement or Propositional logic. In Statement Logic, we do not make use of term letters at all since terms are internal to the statements involved. Instead we use lower case statement letters. Each such letter represents a whole statement. Statement logic {also known as 'propositional logic') deals with arguments where we can ignore contents of the statements involved. Indeed when statement letters are used we often do know what statements they represent. But we do not care; the validity of the argument will not depend on the form or content of the statements involved. As an example of an argument in statement logic consider: If Winston Churchill is still alive then he is the oldest man alive. Churchill is not the oldest man alive. So Churchill is not still alive. This argument has the following pattern:
p+q =lL /p Any argument of this pattern is valid; e.g., this is valid and has that pattern. If no mammals lay eggs then the platypus is not a mammal. The platypus is a mammal. So it is not true that no mammals lay eggs. 163
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The plus and minus signs serve to operate on or to connect whole statements. Since we are not concerned with the internal composition of the statements we have no use for term connectives such as 'some is a' and 'every is a'. The basic connectives that figure in statement logic are the unary connective 'not' and the binary connective 'and' which are used to define a range of other formatives in the plus/minus notation. Here are some familiar transcriptions. notp bothp and q ifp then q p orq
p +p+q p+q (p)(q)
The general form of statement in statement logic is ±(±(±p)±(±q)) The external sign is the sign of judgment, plus for affirmation, minus for denial. We determine the valence of a statement in the usual way by comparing the external sign of judgment with the next sign: if these are the same the valence is positive; if they differ the valence is negative. For example ifp then q' whose fully explicittranscription is'+( (+p)+(+q))' is negative in valence. We shall however omit all the plus signs that signify affirmative judgment so that 'if p then q' will transcribe as 'p+q'. Statements of positive valence are equivalent to conjunctions. Statements ofnegative valence are equivalent to disjunctions. An affirmative statement of negative valence is either a conditional of form ifx then y or a disjunction of form either x or y. Strictly speaking we ought to transcribe 'p or q' as '(p)(q)' but we shall take the liberty of omitting the brackets thereby transcribing 'p or q' as  p q. Also we shall transcribe 'not p or q' as 'pq' or as'( p) q' which shows its equivalence to 'ifp then q' since the latter transcribes as the equal and covalent statement ' p+q'. Equality and covalence suffice for equivalence. For example, 'if p then q' is equivalent to 'not p or q' and also to 'if not q then not p since  p+q =  ( q)+( p). Now this means that we can replace any compound statement by a statement that is equal and covalent to it. However neither equality nor covalence is necessary for equivalence. For example any statement 'p' is equivalent to 'p or p' and also to 'p and p' but 'p' and 'p or p' are not equal
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algebraically since: p "'  p p. Nor is covalence necessary. Thus conjunctions and disjunctions are divalent. But sometimes a conjunction is equivalent to a disjunction. Thus 'p and p' is equivalent to 'p or p'. And, as we shall show in the next section, the conjunction '(p or p) and( q orr)' is equivalent to the disjunction 'either p and q or p and r'. Thus, in statement logic, the old 'principle of equivalence' serves only as a sufficient condition for equivalence but not as a necessary one. To remind ourselves of the difference we shall use lower case letters to represent the principle of equivalence as it applies to logic calling it: peq.
peq Two compound statements are equivalent if (but not 'only if) they are covalent and equal.
******************************************************************** Exercises. Apply peq to see whether any of the following are equivalent. (Remember, if two statements are not equivalent by peq, they may still be equivalent.)
1. if p then not r; not both r and p 2. ifp then r; notp orr orr 3. notp and notr; not: either p orr
******************************************************************** 2.
Contradictions
Some compound statements are contradictory others are tautological and still others are [_contingent. For example any statement of form 'p and not p' is a contradiction, any statement of form 'ifp then p' is a tautology, and any statement of form 'p or q' is contingent. We shall first discuss contradictions. It is easy to spot contradictions of form 'p and not p'. But not all contradictions are of this simple form. The following definition of contradiction builds up to a more general form.
1. Any statement of the form 'p & not p' is a contradiction.
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If c is a contradiction then any statement ofform
'p & c' is a contradiction. 3. Ifc1 and c2 are contradictions then any statement ofform 'cl or c2 ' is a contradiction.
2.
For example by clause 2. any statement ofthe form 'q and p and not pis a contradiction . And by clause 3 any statement of form 'q and p and not p or r and s and not s' is a contradiction. We shall speak of clause 3 type contradictions as 'disjunctive contradictions'. Each disjunct of a disjunctive contradiction is a conjunction of simple statements. And in each such conjunction we can find two conjuncts of opposite sign. So each such conjunction is a contradiction and, by clause 3, the whole disjunction of these contradictions is a contradiction. A disjunctive contradiction is the most general form of contradiction since we may consider contradictions oftype two and type one as special cases oftype three contradictions. For example, 'q and p and (not p)' can be considered a disjunctive contradiction consisting of a single contradictory 'disjunct'.
3. Tautology We use our definition of contradiction to define tautology. Any statement that is equivalent to the negation ofa contradiction is a tautology. For example 'if p then p' is a tautology since it is equivalent to ' (+p+( p))' which is the negation of a contradiction. A tautology is a logical truism and as such (as long as it is not merely the corresponding conditional of the argument) it may be inserted as a premise in any argument or as a step in any proof
The corresponding conditional of an argument is a conditional statement that has the conjunction of premises as its antecedent and the conclusion as its consequent. In an argument is valid, its corresponding conditional is a tautology.
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4.
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Inconsistent Statements
Any statement that is a contradiction or equivalent to a contradiction is said to be inconsistent. For example, '(p+p)' is an inconsistent statement since it is equivalent to '+p+(p)'. Strictly speaking only statements oftypes 1, 2 or 3 are contradictory. But since any inconsistent statement is equivalent to a contradiction, we shall loosely characterize any inconsistent statement as a 'contradiction' and any contradiction as an inconsistency. For example, we shall say that the inconsistent statement ' ( p+p)' is a contradiction since it is equivalent to one. Also, since any contradictory statement is inconsistent we shall often call it that. In effect we shall be using the terms 'inconsistent' and 'contradictory' interchangeably to characterize any statement that is equivalent to a contradiction of type 1, 2 or 3. The following statement is inconsistent: if(ifp then not both q and notq) then rand notr Proof:  [p+( (+q+( q)))]+[+r+( r)]  [+p+(+q+( q))] [+r+( r)]
peq
The second line is a type 3 contradiction. Derivatively, a series of statements is often characterized as inconsistent. Such a series is inconsistent if and only if the statement that is the conjunction of all of its members is itself inconsistent.
5.
Contingent Statements
A statement is 'contingent' if it is neither a contradiction nor a tautology. Most statements are contingent. For example, all simple statements are contingent. Compound forms such as 'p or q' and 'p and q' are contingent.
***************************************************************** Exercises Which of the following compound formulas is inconsistent? Tautologous?
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Contingent?
1. p or (q and notq) 2. (p and q and notp) or (neither p nor notp) 3. not: if(p and q) then (q orr) [hint: not: ifx then y = x and noty] 4. ifp and s then tors 5. neither q nor not(q and t) 6. ifp then notp 7. if not p then (p and q) 8. ifp then p and notp 9. (not both and not p) and (q or notq orr) 10. ifp then (q or p)
******************************************************************** 6.
Direct Proofs
An argument is valid only if its premises cannot be true if its conclusion is false. Let A be an argument. Call the conjunction of its premises with the denial of the conclusion of A 'the counterclaim of A'. One way to prove that A is valid is to prove that its counterclaim is inconsistent. A proof of validity by showing the inconsistency of the counterclaim is called an indirect proof (sometimes called a reductio). Another type of proof is direct. In a direct proof we assert the premises and move step by justified step to the conclusion. Such a proof is usually given in annotated form with the justifications for each step given on the right. For example, suppose, given the premise 'p and q' we should want to show that 'not if p then not q' follows. We could state the proof thus:
1. +p+q
/(p)+(q) 2. (p+(q)) 1, peq Step 2, which is also the conclusion, is derived from step 1 by peq. Another example of inference from a single premise is 1. +(+p+q)+r
/+(+r+q)+p
Statement Logic 2. +p+(+q+r) 3. +(+q+r)+p 4. +(+r+q)+p
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1, Association 2, Commutation 3, Commutation
The justifications for each step appear at the right. Again, the last step is the conclusion ofthe inference whose single premise is given as step 1.
7.
Rules of Statement Logic Used in Proofs
In this section we present some basic principles used in justifying the steps of proofs in statement logic.
Modus Ponens: Commutation and Association are familiar principles of great importance for statement logic. Another familiar rule of inference that applies to inferences from more than one premise is known as modus ponens. It tells us that given 'ifp then q' as one premise and 'p' as a second premise, we may infer 'q' as the conclusion.
1. p+q 2. p /q Note that the conclusion is got by adding the first premise to the second thereby using 'p' to cancel 'p' and to replace it by '+q'. We may think of Modus Ponens as a E/E* type of rule. Think of E* as the environment of 'if p'. Then in this case E* = +q. Think of E as the environment of 'p' in the second premise. In this case the environment is null. Then we can see the rule as justifying the replacing of'p' in E(p) byE*. Moreover this way oflooking at Modus Ponens is quite general since it applies to any two premises 'E*(if p)' and 'E(p)' where p has negative occurrence in theE* premise and positive occurrence in theE premise. Thus let###### be the environment of p and let ....... be the environment of 'ifp'. The rule permits us to derive'### ...... ###' from ' .. .ifp .. .' and '###p###':
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An Invitation to Formal Reasoning ... (ifp) ... ###p### I### ...... ###
E* (p) E(p) I E(E*)
This is our old friend, the donor host rule, in new guise. Only now the repeated ('middle') element 'p' represents a whole statement and 'p' has the meaning 'ifp'. In Syllogistic Logic the host/donor rule is a generalization of Aristotle's Dictum de Omni. In Statement Logic, the host/donor rule of inference is generalization ofthe rule popularly known as 'Modus Ponens' or MP. MP often figures in an annotated proof that a conclusion follows from given premises. In its most general form modus ponens may be formulated as: MP Given any premise ofform E(ifP) where 'p 'has negative occurrence and another premise M(p) where p has positive occurrence the conclusion M(E) follows.
Consider the following simple argument: If roses are red then daisies are white Roses are red and violets are blue I Daisies are white and violets are blue The argument is valid. Its annotated proof has the form: 1. ifp then q 2 p and r 3. q and r
premise premise 1,2, MP
p+q +p+r l+q+r
E*=+q E= ++r E(E*)
The pattern reveals nothing about the internal structure or content of 'p', 'q' or 'r'. But that is irrelevant: any argument of this pattern is valid. Sometimes we appeal both to MP and to peq. Consider: If every student reads some books then the classroom is a place of ideas The classroom is not a place of ideas I It is not the case that every student reads some books
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This argument has the following pattern: p+q
q I p Any argument of this pattern is valid. An annotated proof that  q follows from the premises 'p+q' and' q' is:
1. 2. 3.
p+q q (q)+(p) 4. I p
premise premise
1, peq 3,2,MP
Conjunction:
A principle called conjunction allows us to derive the conjunction of any premises as a conclusion from those premises. For example, given any two premises S1 and S2, 'S1 and S2' follows by the principle of conjunction (Conj):
1. S1 2. S2 3. I S1 and S2
prem prem Conj.
The rule of conjunction may be shown valid by an indirect proof. FQr suppose we denied the conclusion. This would give us
1. S1 2. S2 3. (+Sl+S2) We could then show that this leads to contradiction:
4. Sl+(S2)
3, peq
5. S2
4,1 MP
Step 5 and step 2 are contradictory.
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Conj, peq and MP are used to license the steps in many a proof. Applying these laws we can derive several other rules that are useful in proofs. Simplification:
One, called simplification (Simpl), allows us to infer a conjunct from a conjunction. The general pattern is 1. p and q I 2. p Simplification is another 'derived' rule that can be shown valid by applying rules already stated. We may indirectly prove that 'p and q I p' is a valid form of argument by asserting 'p and q' together with the denial ofp and showing that this leads to contradiction: 1. +p+q 2. p
3. +[+p+q] + (p) 4. +(p) + [+p+q] 5. +[+( p)+p]+q
premise negation of conclusion 1, Conj 3, Commutation 4, Association
Thus the conjunction of 1 and 2 leads to a contradiction (line 5). The proof of the laws of conjunction and simplification was indirect since it proceeded by showing the absurdity of denying the laws. Such proofs are often called 'reductio ad absurdum' proofs (proofs that reduce the opposite assumption to absurdity or contradiction). Here are several other inference patterns and a sketch of how they can themselves be justified by appeal to the laws already given. Conjunctive Iteration:
Conjunctive Iteration (CI) asserts that pis equivalent to 'p and p'. We may show this indirectly (that is, by 'reductio'). Assume p and deny 'p and p'. 1. p 2. (+p+p) 3. p+(p}
premise negation of conclusion 2, peq
Statement Logic 4. p 5. +p+(p)
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1, 3 MP 1,4 Conj
To show the reverse we assume 'p and p' and deny 'p' and showthatthis leads to contradiction.
1. 2. 3. 4.
+p+p p +(+p+p)+(p) +p+(+p+(p))
prenuse negation of conclusion 1,2 Conj 3, Association
4 is a contraction.
Disjunctive Iteration: According to Disjunctive Iteration (DI), given 'p', 'p orp' follows. We show this indirectly:
1. 2. 3. 4.
p (Pp) +(p)+(p) +p+(p)+ (p)
premise negation of conclusion 2, peq 3, Conj
4 is a contradiction. To show that 'p' follows from p p, we assume the opposite and show that ' p' and ' p p' are jointly inconsistent. We first use peq to replace' p p' by' (p)+p'. We then apply MP to get '+(p)+p' which (by peq, again) is equivalent to the overt contraction '+p+( p)'.
Disjunctive Addition: Disjunctive Addition (DA) allows us to derive 'p or q' from the premise 'p'. DAis valid. For suppose that 'p' is true and 'p or q' is false:
1. 2. 3. 4.
p (pq) +(p)+(q) +p+(p)+(q)
prenuse negation of conclusion 2,peq 1,3 Conj
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The last step is the contradiction derived from the assumption that 'p or q' is false in the case where 'p' is true.
The Law of 'And/ Or' Distribution: We said earlier that peq gives us a sufficient condition for equivalence. Iteration shows that two statements can be equivalent without being algebraically equal. The Law of' And/Or' Distribution (AOD) shows that two statements can be equivalent without being covalent. The law equates a conjunction with a disjunction: p or p and (q orr) +[pp] + [qr]
= =
p and q or p and r [+p+q][+p+r]
To prove that the right side follows from the left side we show that a contradiction follows from the assumption that the left side is true and the right side false (a socalled reductio argument). We should then have: 1. pp 2. qr 3.  [ [+p+q] [+p+r]] 4. +[ pq]+[ pr] 5. p 6. pq 7. pr 8. p+(q) 9. p+(r) 10. q 11. r 12. +(q)+(r) 13. (+(q)+(r)) 14. +[12]+[12]
premise premises premise 3, peq
1, DI 4, Simpl 4, Simpl 6,peq 7,peq
5, 8MP 5, 9MP 10, 11 Conj 2, peq 12, 13 Conj
By taking the denial of the right · side as a premise we have derived a contradiction. A similar contradiction can be derived by assuming the right side and denying the left. Thus the law of distribution holds. Our reductio argument has shown that we cannot consistently assume that it does not hold.
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The And/Or distribution law shows how a conjunction may be transformed into a disjunction whose disjuncts are conjunctions. We shall presently give a technique for doing this to any conjunction, however long or complicated. The Law of 'Or/And' Distribution: According to the Law of'Or/And' Distribution (OAD), 'p or (q and r) and '(p or q) and (p or r)' are equivalent. This too can be shown by the indirect method. By the laws of iteration 'p' and 'p or p' and 'p and p' are mutually equivalent and so one may replace the other. By the laws of distribution 'p&(q orr)' and '(p&q) or (p&r)' are equivalent (and mutually replaceable)~ also 'p or q&r' and '(p or q)&(p orr)' are equivalent and mutually replaceable. Thus we see that in addition to the equivalences that come under peq, we have others that do not conform. For we have equivalences like 'p =  p p' and '+p+p = p p' whose sides are divalent or unequal or both. In the case ofOAD and AOD both peq conditions are missing.
8. Disjunctive Normal Forms (DNF) A statement such as 'p' or 'p' does not have two or more statements as components. Such statements are not compound~ we call them simple. A disjunction is said to be in 'disjunctive normal form' when each of its disjuncts is (i) a simple statement or (ii) a conjunction of simple statements. For example 'either rands or p and q and m' is a statement in disjunctive normal form. On the other hand, 'either r or neither p nor q' which also has two disjuncts is not in DNF since the second disjunct is neither simple nor a conjunction of simple statements. Nevertheless, 'either r or neither p nor q' may be transformed into a DNF statement by applying peq to the second disjunct to give us the equivalent disjunction: either r or both not p and not q. Now the second disjunct is a conjunction of two simple statements, 'p' and 'q', and the whole disjunction is in 'normal form'. The importance of putting a disjunction into 'normal form' is this: When a disjunction is in normal form we can easily determine whether it is contradictory or not by a inspecting each of its disjuncts. In a DNF each disjunct that isn't simple is a conjunction of simple statements. We inspect each disjunct of the DNF looking to see whether it contains a contradictory
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pair of simple statements. If every disjunct has a contradictory pair, then the whole disjunction is inconsistent since it would then be a type 3 contradiction. But if even a single disjunct is contingent or tautological, the whole disjunction is consistent (not a contradiction). Thus, getting a DNF equivalent to a statement puts us in position to say whether the statement is consistent. As we shall see, that information is logically valuable for deciding the validity of arguments.
******************************************************************** Exercises: Transform the following statements into DNF disjunctions, then check for inconsistency by inspecting all the disjuncts: 1. 2. 3. 4.
p and r and not p OR not: if p then p OR not: either q or not q r and not r OR not both s and not p sand not s OR u and neither to nor u. p and (q or notp) (hint: use the law of distribution) 5. sand neither q nor not q. (hint: a disjunction may consist of a single 'disjunct'.)
******************************************************************* 9.
Inconsistency and Validity
In doing logic we are often concerned with the question of deciding whether some given argument is valid or not. Now we have just learned how to tell whether a certain kind of disjunctive statement is inconsistent or not. It is important to be able find inconsistency in conjunctions as well as disjunctions. For we know that an argument is valid if and only if its counterclaim is inconsistent. The counterclaim of any argument is the conjunction of its premises and the denial of its conclusion. So knowing how to tell whether a counterclaim conjunction is inconsistent or not is tantamount to knowing how to tell whether the argument is valid or not. Thus suppose that S1, S2 ... /Sn is an argument with n1 premises. Assume we have a way of determining
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177
whether its counterclaim conjunction, Sland S2 ... and Sn, is inconsistent. Then we should ipso facto have a way of deciding whether 'Sl,S2 .. /Sn' is valid. For 'Sl,S2 .. ./Sn' is valid if and only if 'Sl&S2 ... &Sn' is inconsistent. Now in most cases, a direct examination of a conjunction does not tell us whether it is consistent or inconsistent. On the other hand, as we shall soon be seeing, any conjunction is equivalent to a normal form disjunction. And we already know how to inspect any normal form disjunction for inconsistency. Of course, if a conjunction is inconsistent, any statement that is equivalent to it is also inconsistent. So if we had a way of transforming any conjunction into an equivalent DNF disjunction, we could use the method of inspection for contradiction to decide on the consistency or inconsistency of conjunctions. For all we should have to do is inspect their DNF equivalents. We shall soon learn how to transform any conjunction into its disjunctive normal form equivalent. The ability to get a disjunctive equivalent for any conjunction will give us a tool for evaluating any argument to see whether it is valid. The general method is this: Suppose we have an argument A. A is valid if and only if the conjunction of its premises and the denial of its conclusion (its conjunctive counterclaim) is a contradiction. Let C(A) be the conjunctive counterclaim) of A. Then A is valid if and only if C(A) is inconsistent. Let D(A) be the DNF equivalent to C(A). Then A is valid if and only ifD(A) is a contradiction. Now by the law of distribution, each disjunct ofD(A) is a conjunction. In that case D(A) is a contradiction if and only if each and every disjunct contains a conjunction ofthe form 'x&x'. So we examine each disjunct ofD(A) to determine whether D(A) is a contradiction. If D(A) is a contradiction then so is C(A). But if C(A) is inconsistent A is valid. If D(A) is not a contradiction then neither is C(A). In that case A is invalid. One key step in this procedure has yet to be learned: how to transform any counterclaim conjunction into its equivalent DNF disjunction. We can do this for very simple cases by using the law of and/or distribution. But we need to be able to apply the law of distribution to conjunctions of any length and complexity. For that purpose we shall now learn a new way of way of representing the law of distribution that graphically exhibits the equivalence of any conjunction to a disjunction of conjunctions. We have so far been using an algebraic notation to represent compound statements like 'p and q' and 'p or q'. We have also learned, but not used, a symbolic notation which employs the symbol '&'for 'and', 'v' for 'or' and '=>' for 'if then'. A third notation which we shall now learn is
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graphic. By representing compound statements on a tree figure, it has the advantage of revealing certain relationships that are very useful for logical reckoning. In choosing a notation for logic we keep two desiderata in mind. The first is brevity: in this respect 'p=>q' is better than 'if p then q'. The second is the capacity of the notation to reveal logical relationships. For example using the algebraic notation the equivalence relationship between 'if not p then q' and 'q or p' is revealed as an equation of two covalent statements which are therefore equivalent by peq: (p)+q = (q)(p) By contrast the equivalence is not perspicuously shown in symbolic notation where it is represented as: p>q =qvp We are about to learn yet a third way to represent compound statements. This new way is graphic.
10. Graphic Representation of Compound Statements The graphic method uses spatial relations to represent the connectives 'and' and 'or' that join simple statements to form compound statements. Roughly, when two statements are vertically positioned, they are conjoined. When two statements are horizontally positioned they are disjoined. Thus when p and q are vertically arrayed the array represents the conjunction 'p and q' or 'q and p' depending on whether one reads it down or up. Spatial Notation p andq qandp
p q
Symbolic Notation p&q q&p
Algebraic Notation +p+q +q+p
When 'p ' and 'q' are horizontally arrayed this represents 'p or q' or 'q or p', depending on whether one reads the array from left to right or from right to left.
Statement Logic
p q
p orq q orp
pvq qvp
179 pq qp
Notice that the spatial notation for both conjunctions is a vertical array, while the spatial notation for both disjunctions is a horizontal array. In what follows we shall make liberal use of all three notations. By the principle of commutation left to right and right to left readings are equivalent. So too are updown and downup readings.
p
q
= q
p&q =q&p
p
=
p q
q p
pvq=qvp
p =
p&p
p
p p = pvp
p p
<=> (p v p)&(q v r) = (p&q) v (p and r) q r
Note that the vertical reading of
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An Invitation to Formal Reasoning p p q r
interprets it as the conjunction '(p v p)&(q v r)' while the horizontal reading interprets it as the disjunction '(p&q) v (p&r)'. Since p = p v p, we write the array 'p p' simply as 'p' and so represent 'p&(q v r)' by the array
p I \ q
r
By convention this array is usually represented as a 'tree.' +p+[qr]; p&(q v r)
p
I \ q r
 [+p+q] [+p+r]; (p&q) v (p&r)
Reading down and then left the tree represents the conjunction 'p&(q v r)'. But the tree can also be read sideways as the disjunction '(p&q) v (p&r)'. Thus the 'and/or' equivalence of 'p&(q or r)' and '(p&q) or (p&r)'is represented in the two equivalent ways to read the tree: vertically or horizontally. In effect the tree graphically depicts the equivalence of the conjunction to the disjunction. We may state the law of distribution as an equivalence between two trees: p
1\ q r
=
1\ p p q r
The equivalence shows that we may drop a common term downward to the bottom of every path that it dominates. Conversely if (as in the diagram at the right) a common term is at the bottom of several paths we may lift it upwards to a node that dominates the common paths. The move downward corresponds to iterating 'p' to give us 'p or p'. Moving pup corresponds to reducing 'p or p' to its equivalent, 'p'.
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A simple statement has no binary connectives. It is either positive (of the form 'p'), or negative (or the form' p'). Any conjunction or disjunction of simple statements can be 'treed.' For example, the conjunction 'not p and (q or not r)' is 'treed' as
p I \ q r
Any simple statement can be treed. But to be treed, a compound statement must be overtly conjunctive or disjunctive: only statements of the form 'x&y' and 'x v y' can be treed. For example, 'not both p and q' [=> (+p+q)] cannot be treed as it stands since it is not of the proper form. However it is always possible to apply peq to any compound statement that is not a conjunction or disjunction to get an equivalent conjunction or disjunction that is 'treeable'. For example 'not both p and q' is first represented as ' (+p+q)' and then transformed (by peq) into (p)(q) which we tree as
I \ p
q
Similarly 'ifp then q and r' is first represented as 'p+(+q+r)' and then as '(p)(+q+r)' [=> p v q&r]:
I \ p
q r
The tree for '+p+(p+q)' is: p
I \
p
q
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An Invitation to Formal Reasoning
Note again that 'p+q' istreatedasthedisjunction,' (p) q', [==>either not p or q], to which it is in fact equivalent. Only conjunctions and disjunctions can be represented in tree notation. If two statements are connected by a path that can be traversed by going either up or down, they are conjoined. If the path connecting them requires us to go both up and down, they are disjoined. Henceforth we shall treat all conditional statements of the form 'ifp then q' as ifthey are disjunctions ofthe form 'either not p or q'. In effect, we shall look on all conditionals as 'disjunctions'. Note that if we drop 'p' down, the tree for 'p and ifp then q' is seen to be equivalent to
p
I \ p
p
q
a disjunction of two conjunctions one of which is a contradiction. A path to or from the top of the tree is either 'open' or 'closed'. We call a path closed if it contains a contradictory pair of statements. A closed path represents an inadmissible alternative. Thus the only open path on the above tree consists ofthe disjunct 'p and q'. To repeat: Statements that consist of a single letter with or without a minus sign in front are called simple. Statements that contain binary connectives such as 'and', 'or' and 'if then' are called compound. Representing compound statements on a tree is subject to a strict limitation: only affirmative compounds can be represented on a tree and then, only those that have 'and' or 'or' as their binary connective. Denials of compounds must therefore be transformed by equivalence into an affirmative disjunction or conjunction before we can tree them. For example if we want to tree ' (+p+q)' we must first clear the minus sign of denial by driving it inward to get the equivalent affirmative statement, the disjunction 'pq' which is equivalent to ' ( p) ( q)' and which trees as:
I \
p
q
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11. Regimenting Statements for Treeing We have noted that statements like 'ifp then q' or 'not ifp then q' cannot be treed without first being transformed into disjunctive or conjunctive form. The process of getting a statement into a form suitable for 'treeing' is called regimentation. In regimenting any compound statement we first transcribe it and then follow two rules: 1. Clear all external minus signs by driving them inward. 2. Represent the result as a statement of the form +x+y or   x  y. For example, given 'not ifp then q' we first transcribe it as '{p+q)'. We next drive the external minus sign inward to give us '+pq'. We next transform this into a statement of form '+x+y', which in this case is the statement '+p+(q)'. This can be treed by putting p over 'q' in vertical juxtaposition. The statement 'ifp then not q' is first transcribed as 'p+{q)' which is then transformed into ' ( p) ( q). This trees by putting ' p' and ' q' in horizontal juxtaposition. When a formula is cleared of all external minus signs it is easy to see how it will be treed as a conjunction (vertically) or disjunction (horizontally). For ' p+q' can be treated as 'not p or q' being equivalent to '  ( p)  q' . So we tree 'p+q' as 'not p or q' putting 'p' on the left leg and 'q' on the right leg. Similarly, 'pq' can be read as 'not p or not q' and treed as 'p or q':
I \
p
q
More complicated statements are easily handled in piecemeal fashion. Given 'not both p and not q and not if not r then not s' we first transcribe it as '+[(+p+{q))]+[((r)+(s))]'. We then use peq to get an equivalent conjunction,+[ p+q] + [+{ r)+s], a formula that is clear of external minus signs and which we tree thus: r
s I \
p
q
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An Invitation to Formal Reasoning
Note that we put the simple statements on top. This makes for a smaller tree. But we could also tree the compound thus:
I \
p
q
r
r
s
s
The above example shows how a statement that cannot be treed as it stands can be transformed into a treeable statement. The following conditions determine that a statement is 'treeable: I. Any simple statement is treeable 2. Ifx andy are simple, then '+x+y' and' x y' are treeable. 3. Ifx andy are treeable then '+x+y' and' x y' are treeable. No statement that is not of the kind specified in 13 can be treed. On the other hand any compound statement has a treeable equivalent (called its 'Tform'). Consider, for example, the statement '(p& (ifq then r&s))'. This transcribes as  (+p+( q+(+r+s))) which, by peq, is equivalent to  ( p) (+q+( ( r) ( s))). In this form the statement is treed as
I \
p
q I \
r
s
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******************************************************************** Exercise: Give the Tform equivalents ofthe following statements.
I. 'ifp then not q' 2. 3. 4. 5.
not both p and (r or s) p and not either m or k not if not p then not q if p or not q then both r and not s
Tree each Tform.
******************************************************************** 12. Large Trees According to the Law of and/or Distribution, (AOD), a conjunction 'p&(q v r)' is equivalent to the disjunction '(p&q) or (p&r)'. Stated in tree form the equivalence holds between the two trees:
I \
p
I \ q
r
p q
p r
Consider now a conjunction with three conjuncts: p&(q v r)&(s v t) Applying and/or distribution to the first two conjuncts gives us ((p&q) or (p&r))&(s or t) Applying distribution yet again gives us [((p&q) or (p&r))&s] or [((p&q) or (p&r))&t]
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An Invitation to Formal Reasoning
which again by distribution gives us [(p&q&s) v (p&r&s)] v [(p&q&t) v (p&r&t)] which is equivalent to p&q&s v p&q& t
v
p&r&s
v
p&r & t.
The reader may agree that the process of and/or distribution for more than two conjuncts can get fairly tedious. Fortunately, there is a much quicker way to get the DNF equivalent of any treeable conjunction. Put the conjunction on a tree and then read the tree disjunctively. The following trees depict the equivalence of 'p&(q v r)&(s v t)' to '(p&q&s) v (p&q&t) v (p&r&s) v (p&r&t)' at a glance: p I \ r I \ I \ s t s t q
1\
=
I \ I \ I \ p p p p q q r r s t s t
The downward reading is conjunctive, while the left right reading gives the disjunctive equivalent. Note again that the common term p may be dropped to the bottom and similarly for q and r. And once again we see that this allows the horizontal reading to give us the disjunctive equivalent to the original conjunction. To remind ourselves that the law of distribution is the ground of the equivalence that is depicted on the tree, we might call these trees on the right 'distribution trees'. In effect, a distribution tree shows how a conjunction is equivalent to a 'normal form disjunction' that is the result of applying the law of distribution by dropping all terms down to the bottom of the tree and then reading from left to right disjunctively. Our ability to get a DNF equivalent to any conjunction makes it possible to decide on the validity of any argument in statement logic. Suppose we are given the argument:
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Al 1. both r and p 2. ifp then q I both rand q This argument is valid and one way of showing this is to show that the C(Al ), the counterclaim conjunction of its premises and the denial of the conclusion, is a contradiction. C(Al) consists of the following three conjuncts:
1. both r and p 2. ifp then q 3. not both rand q
+r+p p+q (+r+q)
To show that C(Al) is inconsistent we will represent it on a tree and then examine its disjunctive equivalent. This needs a bit of tinkering since the second and third statements are not in T form. So we replace them by their Tequivalents to give us: +r+p  (p)q (r)(q)
both rand p notp or q notr or notq
The distribution tree for the conjunction of these three statements is:
I \ I \ I \ r r r r p p p p p p q q r q r q
r p
I \ p I \
q
I \
r q r q
We get the right tree by dropping dominating terms to give us the DNF equivalent to C(Al): r&p&(p)&(r) r&p&q&(q)
v
r&p&(p)&(q)
v
r&p&q&(r)
v
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An Invitation to Formal Reasoning
Each path on the tree represents a disjunct. Note that each disjunct contains a contradictory pair of conjuncts; in other words, each path is closed. Since all paths are closed, the whole disjunction is a contradiction of type 3. Now A1 is valid if and only ifC(A1) is inconsistent. Since C(A1) is inconsistent, A 1 is valid. Generally then, a path that contains a contradictory pair of simple statements is closed and when each and every path of a tree is 'closed' the tree represents a contradiction. We could also show that the three statements represented on the tree are jointly inconsistent by applying Modus Ponens and peq to derive a contradiction: 4. +r+q 5. (rq) 6. +(rq) +((rq)
1, 2 MP 4,peq 3, 5, Conj
6 is a contradiction. Our ability to use distribution trees for showing the equivalence of conjunctions and disjunctions gives us a powerful tool for determining the validity of any argument in statement logic. To evaluate an argument we proceed as follows:
1. Form the CC of the argument. 2. If any conjunct in the CC is not treeable, transform the statement into its T form equivalent. 3. Tree the CC. 4. Read the CC tree disjunctively taking each path as a disjunct. 5. Check each path to see whether it contains a contradictory pair of simple statements. If any path is open the argument is invalid; if all paths are closed, then the CC is inconsistent and the argument is valid. Example: evaluate the following argument: A2
ifr then p and q not both s and if r then q not s
r+[+p+q] (+s+[ r+q])
1s
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1. C(A2): +[r+[+p+q]]+[s+[+r+(q)]] + ( s) 2. We give the Tfonn equivalent ofC(A2): +[(r)[+p+q]] + [(s)  [+r+(q)]] + s 3.Tree the Tfonn:
s I \ r
I I \
s
p q
I \
r s r q q
Reading each path we see that it contains a contradictory pair, which shows that C(a2) is a contradiction. This, in turn, shows that A2 is valid. Equivalently we could show that the tree represents a contradiction by dropping all tenns to the bottom:
I I \ s r s
s r r q
\ I \ s s p p q q s r q
Each conjunction presents a disjunction. The whole disjunction is a nonnal form type 3 contradiction. We could also show the validity of A2 by a direct proof:
1. r+[+p+q] 2. 3. 4. 5.
(+s+[ r+q]) (+s+[ [+p+q]+q]] s + [+p+q+( q)] [+p+q+(q)] +(s)
premise premise
1, 2,MP 3, peq 4, peq
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An Invitation to Formal Reasoning 6.  [p+q+( q)] 7. s
Tautology 6,5,MP
As this example of a proof shows, in moving toward the conclusion of an argument we use certain rules called 'rules of inference'. Applying these rules to the premises, we may proceed step by step to the conclusion. The sequence of statements consisting of the premises, the intervening steps and the conclusion is called a proof. For convenience we shall speak of every statement in the proof as a step in the proof. Thus the premises are the first steps of the proof and the conclusion is the last step on the proof. We have learned several ways of evaluating a given argument. 1. check for validity by treeing the counterclaim and checking for inconsistency. 2. check for validity by seeing whether we can derive a contradiction from it by appealing to accepted rules of inference. 3. Check for validity by seeing whether we an derive the conclusion from the premises by appealing to accepted rules of inference. Methods 1 and 2 are indirect: we show validity by showing that counterclaim is inconsistent. Methods 2 and 3 are 'derivational': we move to a conclusion by a series of justified steps. In the case of method 2, the conclusion we seek is a contradiction. In the case of method 3 the conclusion we seek is the conclusion of the argument itself. In both cases we move from step to step by appealing to rules of inference. But what ifthe argument is invalid? In that case we might go on looking and not know when to stop. Since it cannot be applied to tell us that an argument is invalid the derivational method does not automatically render a decision about an argument's validity. By contrast Method (1) tells us about invalidity as well as validity. If the argument is invalid, Method 1 delivers a decision to that effect since the tree will show an open path. Because Method 1 delivers a decision on invalidity as well as on validity, logicians call this method a 'mechanical decision procedure'. It is in fact easy enough to program a computer to apply method 1 to any argument of statement logic and to determine whether it is valid or not. The steps it would follow would be the steps we outlined: 1. Form the counterclaim ofthe argument 2. Get it into Tform
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3. Check the disjunction to see whether each disjunct contains a
contradictory pair of conjuncts. 4. If each disjunct is a contradiction, judge the argument valid; otherwise judge it invalid. Because the procedure is decisive and simple, it is deservedly popular. On the other hand logicians have soft spots in their hearts (some would say in their heads) for direct proofs and method 3 has its devotees. The real advantage of the direct method is for handling incomplete arguments where we are called upon to derive a conclusion from premises. Decision procedures are not used in such inferential problems since they are not designed for drawing inferences (to complete an incomplete argument) but for evaluating arguments already complete with conclusion. We have so far learned how to use distribution trees in decision procedures. In the next section we shall see how trees can be used to deal with inferential problems.
******************************************************************** Exercises: Evaluate each of the following argument for validity by using Method 1: 1.
if p then not q if q then r or s I ifp then s
2.
p and q or r and not s ifrthen s /p
3.
p if p and q then not r /ifqther
4.
not r p and q orr /q
192 5.
An Invitation to Formal Reasoning ifp or q then r ifr then q and s I p and not s
******************************************************************** 13. Drawing Conclusions Reasoning is a twofold affair. We evaluate arguments that are given to us to see whether they are valid. But we also make arguments by inferring or drawing conclusions from premises. In evaluating an argument we seek a yes/no type of answer: is the argument valid? and we may apply a decision procedure to gives us ends in the yes/no decision we seek. In drawing a conclusion we have no complete argument to evaluate. Instead it is up to us to produce the argument by supplying a conclusion to some given premises. Of course the conclusion we seek is one that follows from the premises. The process of reasoning here is direct. The question to which we seek the answer is: Given these premises, what nontrivial conclusion can we validly infer? In making an argument we usually want conclusions that are not trivial. For example, given two premises 'p' and 'q' we may infer 'q' from them, thereby giving us the complete argument: 'p, q I q'. But that argument is trivial. Similarly, given a single premise 'p', we can draw from it any number of conclusions of form 'p or q'. But that too is trivial. In the discussion that follows we shall assume that the trivial and obvious conclusions are not what we are seeking. For example suppose we are given the premises 1. ifp then q 2. not q and we want to know what follows. One nontrivial conclusion that may be drawn from the premises is 'not p'. But the following may also be drawn: i) not q ii) either r or not q iii) not q and if p then q iv) either p or not p
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193
We call such conclusions 'trivial entailments'. An entailment is nontrivial if and only if a. it is not a premise (as (i) is). b. it is not a tautology (as (iv) is). c. it is not a disjunctive addition to a premise (as (ii) is). d. it is not a conjunction of premises (as (iii) is ). e. it is not an addition to any nontrivial conclusion. Any conclusion that does not offend against any of the first four triviality conditions is nontrivial. But if c is a nontrivial conclusion, then the fifth condition requires us to count as trivial any conclusion offonn 'cor x'. Thus suppose that 'not p' is a nontrivial conclusion because it does not trespass rules a to d. Then the fifth condition for nontriviality rules out adding new entailments such as 'not p or s', 'not p or t' and so forth which are disjunctive additions to 'not p'.
14.
Partial Disjunctions
In what follows we shall give a general procedure for getting nontrivial entailments from any set of premises. One useful notion that we shall need is that of 'partial disjunction'. Given a disjunction such as 'p and q and r or s and t and u or m and rand d' we may fonn partial disjunctions by taking conjuncts from each and every one of its disjuncts. Thus 'q or tor d' is one partial disjunction and 'r or s or r' is another. A disjunction entails all of its partial disjunctions. Our problem, in its general form, is: Given a set of premises, what (nontrivial) conclusions may we infer from them? For example, suppose we are given the premises 1. (r and t) or (s and not q) 2. p 3. if s then (not t and not r) What follows? It may seem that we are here limited to using the rules of inference of statement logic to move step by step to conclusions that are, hopefully, non trivial. But we can do better than this. The following procedure
194
An Invitation to Formal Reasoning
shows how to represent a given set of premises in a form that allows us to see graphically what nontrivial conclusions may be drawn from them. 1. Tree the conjunction ofthe premises. 2. Mark the closed paths and ignore them. 3. Taking each open path as a disjunct, write the disjunction that is equivalent to the conjunction of premises. Each disjunct is a conjunction. 4. Form partial disjunctions by taking conjuncts from each disjunct. 5. From the partial disjunctions given by the fourth step, select those that are nontrivial by not breaching any of the triviality conditions. The selected disjunctions are nontrivial entailments.
In the above example we have the premises 1.  [+r+t]  [+s+( q)] 2. p 3. s+[+( t)+( r)] [=>  ( s) [+( t)+( r)] whose tree is p
I r t
s q
I \ s
\
I \
t s r
t r
The tree shows that the conjunction of the premises is equivalent to 'p&r&t&s v p&r&t&t&r v p&s&q&s v p&s&q&t&r', each disjunct representing a path on the tree. However, two paths are closed, which means that we may ignore them (steps 1 and 2). This leaves p&r&t&s v p&s&q&t&r This disjunction is equivalent to the conjunction of the treed premises and so is entailed by them.
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195
We may now (step 4) form partial disjunctions by taking a conjunct from each ofthese disjunctions. Among these are pvp pv q tv t
s v s Each partial disjunction in the above list is entailed. But they are all trivial entailments. However, the following partial disjunctions are nontrivial: t or not r (if r then t) (ift then r) r or not t not s or not q (if q then not s) That these partial disjunctions are entailed can easily be verified. For suppose we denied any one of them and added the denial to the tree. In each case the tree will 'close' entirely and show contradiction. Since the consequence of denying any partial disjunction is inconsistency, all of the partial disjunctions are entailed by the original disjunction that is equivalent to the conjunction of the premises. Thus the answer to the question 'what follows from these premises' can be given by treeing the premises and looking at the possibilities for forming partial disjunctions that cover all of the open paths. The nontrivial entailments can be picked off the tree directly, care being taken to choose a partial disjunction that exhausts all open paths. Consider for example the following tree representing the two premises 'if s then (rand not p)' and 'tor not r or (sand p)':
I \ s \ r I I \ r s p t p I I \ t r s p
196
An Invitation to Formal Reasoning
The first, second and fourth paths are open. Two of the open paths are dominated by ' s'; the remaining path has a 't' on it. Thus ' s v t' is a partial disjunction that covers all open paths. Another exhaustive disjunction is 't v r'. Both of these are nontrivial. Another is 'r v tv r', but that is tautologically trivial. Yet another is 'p v r v t'. But that only adds to the previous entailment, 't v  r', and so does not give us more information than we already have. Thus we may complete the argument in two non trivial ways: 1. if s then r and not p 2. tor not r or (sand p) 3. /not s or p 3*. /not r or t Since (s)+p = s+p and (r)t = r+p, our two alternative conclusions could be stated as 'if s then p' and 'if r then t'. Of course real premises come in English sentences and not in neat symbols. We might want, for example, to know what the following premises entail: 1. If you marry a rich girl you'll be wealthy but unenterprising 2. If you marry a poor girl you'll be enterprising but not wealthy 3. You will marry, but only once. Let r transcribe 'you'll marry a rich girl', p transcribe 'you'll marry a poor girl', e transcribe 'you'll be enterprising', and w transcribe 'you'll be wealthy'. In algebraic notation, then:
1. r+[+w+(e)] 2. p+[+e+(w)] 3. [+r+(p)][+p+(r)] The third premise states that you'll either marry a rich girl and wont marry a poor girl or else you'll marry a poor girl and won't marry a rich girl. The tree for these conjoint premises is
Statement Logic
I I r I \ p I I p r
197
\
\ w e
I \ e p w \ e w \ I \ I \ r p r p r I \ p r p r p p r r p
This tree has two open paths representing r&e&w&p or w&e&p&r from which the following nontrivial entailment may be extracted: w v e; e v w
The first of these is equivalent to 'if w then not e', the second to 'if not e then w'. Thus we may conclude that you'll be wealthy if and only ifyou won't be enterprising. Other nontrivial entailments include : e v r; w v p; w v p or their conditional equivalents 'if e then not r;' 'if not w then p' ; 'if w then not p'. In the process of constructing a tree for a set of statements we will often close a path without further ado because that path has a contradictory pair of statements on it. For example, in constructing the tree for the premises 1. ifp then q
2. p orr 3. not r
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An Invitation to Formal Reasoning
r
I p
I p
\ r
\ q
we close the third path without going farther because we know that all paths below it will close. This tree contains a single disjunct 'r&p&q' and three nontrivial entailments: 'p', 'q', 'p&q'. Entailments that include 'r' are trivial since that is given as a premise. 'p or q' is trivial since it is an 'addition' to a nontrivial entailment.
****************************************************************!** Exercises: I. Tree the premises given and draw two nontrivial conclusions from them by inspecting the tree and exhausting the open paths. 1. p&s v p&q (q>r)vt p > (r v q)
I?? 2. not (p or q) ifr or s then q
I?? II. For each example in (I) provide a conventional annotated proof for one of the two conclusions you got.
********************************************************************
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199
15. Using the Tree Method for Annotated Proofs The tree method for deriving conclusion can be used in annotated proofs. The following rules are needed. 1. When first given a set of premises we use peq as needed to transform them into a conjunction of disjunctions of simple statements. We call this process 'treeing the premises'. 2. The tree read downwards represents the conjunction of the (transformed) premises. Read across the bottom the same tree represents an equivalent disjunction since the tree pictures the operation of the law of distribution according to which any conjunction of disjunctions is equivalent to a disjunction of conjunctions. This licenses us to restate the conjunction of premises as a disjunction. In stating the equivalent disjunction, we appeal to the law of distribution, putting 'tree distribution' (TD) as our justification. In applying TD to get a disjunction, we throw out all disjuncts that are contradictory taking all and only those that are at the bottom of open paths. Call this disjunction D 1. 3. By the law of disjunctive entailment, any disjunction D 1 entails a partial disjunction 02 consisting of disjuncts containing statements taken from each disjunct of D 1. Thus from D 1 we may move to 02, annotating this step as justified by 'disjunctive entailment' (DE). 4. The last step consists of eliminating those disjunctive entailments that are trivialpremises, 'additions' to premises, tautologies as well as any addition to a nontrivial entailment. See above section. The following is an example of an annotated proof that uses the tree method of derivation: l.p 2. q > r 3. p > s 4. (s&t) > q 5. q v r 6. p v s
p q+r p+s [+s+t]+q (q)r (p)s
premise premise premise premise 2,peq 3,peq
200
An Invitation to Formal Reasoning 7. (s&t) v q 8. (s v t) v q
[+s+t] q [(s)(t)]q
4,peq 7,peq
Steps 58 are needed to render all of the premises 'treeable'. p
I
\
q
I
9. Tree of 1,5,6,8
r
I
\
p
I I s t
\
p
s
s
I I
\
q
s t
\
q
10.  [+p+( q)+s+( t)]  [+p+r+s+( t)]  [+p+r+s+q] 11. (t)  (t)  r 12. tr 13. s 14. tq
TD 10, DE 11, DI 10, DE lO,DE
The last three lines are the nontrivial entailments.
******************************************************************** Exercises: Use the four step method above to derive at least two nontrivial conclusions:
1. p ::> q t => r (t&s)&p
s=>w I?? 2. (pvq) (q&r) v (r&s)
I??
********************************************************************
Statement Logic
16.
201
Statement Logic as a Special Branch of Syllogistic Logic
The section that follows is optional, being of historical interest and of special concern to the student who may be interested in some of the reasons that have led logicians to choose different approaches to their subject. On one approach, the logic of statements is looked upon as a special branch of the logic of terms. On another approach the logic of statements is fundamental and the logic of terms is treated as grounded in the logic of statement. Our approach in this book has not taken a stand on this question. We have however shown that term and statement logic have the same basic syntax so that an elementary sentence of form 'some A is B' and an propositional sentence of the form 'p&q' can both be transcribed as sentences of form '+x+y'. More generally, we have shown that all statements are built out of dyads of this form. We have already shown that such dyads occur as subsentences in sentences that contain compound or relational terms. In what follows we shall show that the dyads representing compound sentences can also be construed as subjectpredicate sentences with a subject of form 'some/every x' and a predicate of form 'is/isn't y'. By showing this we shall have carried out a program of reducing propositional logic to term logic. This terminist program has been announced by many logicians who hold that term logic is the basic logic and that propositional logic is simply a special branch ofterm logic. The question ofwhether term logic or statement logic is primary is our topic and we introduce it by citing two remarks of the great eighteenth century logician Gottfried Leibniz. Leibniz was a 'terminist'. 1.
Thomas Hobbes rightly stated that everything done by the mind is a computation by which is understood either the addition of a sum or the subtraction of a difference. So just as there are two primary signs in algebra,'+' and'' in the same way there are as it were two copulas.
2.
If, as I hope, I can conceive all propositions as terms, and hypotheticals as categoricals, and if I can treat all propositions universally, this promises a wonderful ease in my symbolism and analysis of concepts, and will be a discovery of the greatest importance.
The first remark of Leibniz has been developed by us in the plus/minus notation for the logical 'functors' that join terms and statements. Thus we
202
An Invitation to Formal Reasoning
represented 'being P characterizes (some) S' as 'P+S' and 'being P characterizes every S' as 'PS' with the plus and minus signs that connect the terms serving as 'logical copulas' in place of the predicative formulas 'being (a so and so) characterizes a/every (such and such)'. The same signs are used in forming compound propositions: 'q+p' for 'q and p' and 'qp' for 'q ifp'. We later introduced the more natural mode of representing English sentences, e.g., transcribing 'every Sis Pas 'S+P' and 'ifp then q' as 'p+q'. Leibniz' s second remark alludes to the fact that a compound statement like 'p and q' logically behaves like an elementary statement 'some Pis Q' and that a hypothetical compound 'if p then q' logically behaves like the categorical universal form 'every p is q'. This led Leibniz to look for a way to 'read' propositional statements categorically, construing for example 'ifp then q' as a statement of form 'every xis a y; he hoped thereby to treat statement logic as a special branch of the logic of terms. We call this approach 'terminism'. The terminist logician holds that term logic is primary logic. It should be said that Leibniz's belief that term logic was more basic than statement logic was traditional. Historically, term logic came first with Aristotle's epochal study of the syllogism; the study of statement logic was developed by the Stoics quite a bit later. Before the twentieth century, the teaching oflogic followed the historical order: term logic including syllogistic was taught first and statement logic second. In recent years, however, the historical order was judged to be misleading or irrelevant and today the vast majority of logicians think of statement logic as primary logic. Thus the contemporary doctrine reverses Leibniz's priorities by giving 'every A is B' a hypothetical reading, construing it to say something like 'for any thing x, if x is an S then x is a P'. If this is the right way to construe 'every A is B', then one cannot learn the logical behavior of this and other universal sentences before one learns how the sentential connective expression 'if then' behaves. And indeed, today, the students oflogic almost always begins with statement logic where they study the logical behavior of the sentential connectives 'if then', 'and', 'or' etc. After mastering the logic ofthe sentential connectives students go on to study how to construe 'every A is B', 'some A is B' and the other forms of'elementary' sentences that enter into syllogistic arguments that were once the starting point of logical study. The present book rejects the modem thesis ofthe priority of statement logic. We have in fact shown that the two logics may be taken on equal terms. But if priorities are to be assigned, we would favor Leibniz's idea of treating statements as terms. We will now show how this may be done and examine the implications of doing it.
Statement Logic
203
In the plusminus notation the elementary and compound forms are isomorphic. In this section we shall construe 'p+q' as a statement of form 'every x is a y'. The idea is to treat statements as terms and compound statements as if they were ordinary categorical statements. To implement this idea we shall exploit the doctrine presented in Chapter 1 that a statement is just like a term in signifying a characteristic and denoting what has the characteristic signified. We shall show that a statement may be construed as a term that signifies a world characteristic and denotes the world that has that characteristic. And we shall then show how this makes it possible to construe compound forms such as 'p and q' (or 'ifp then q') as statements of the form 'some x is y' or ('every xis y'). In using the algebraic mode of transcription we see that an elementary statement like 'some A is B' and a compound statement like 'p and q' have the same formal structure: +X+Y. Similarly, 'every A is B' and 'ifp then q' have the structure 'X+Y'. But even if we did not represent the statements the same way the similarities would be brought home to us in other ways. Logicians have long ago noted that an argument of form 'if p then q, if q then r/ if p then r' exhibits the same kind of transitivity as a syllogism like 'every A is B, every B is C/ every A is C'. It was facts like this that led Leibniz and others to speculate that 'ifp then q' might somehow be construed 'categorically', as making a claim of form 'every X is Y'. The problem is that we do not seem to have a natural way oftreating 'ifp then q' as a categorical statement. Or to put the difficulty another way: we have no way of construing whole statements as if they were terms within a statement. Similarly we should like to give a categorical reading to '+p+q', reading it as saying something of the form 'some X is a Y' but, again, we do not know how to construe·the terms of such a statement. A further difficulty presents itself: Leibniz believed that by treating statements as terms and reading propositional statements categorically we could subsume the laws of statement logic under the laws of term logic. But treating statements as terms would be pointless unless it could also be shown that the categorical reading of compound statements preserved such familiar laws of statement logic as modus ponens, iteration, distribution and so on. It is not at all clear that this can be done; we have, for example, noted that some of the formal constraints of term logic such as the covalence and the equality conditions for equivalence, do not hold for statement logic. Any attempt to show that statement logic is like a branch of term logic must account for some crucial differences as well as for the formal similarities of term and statement logic.
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We are about to take the step that Leibniz recommended but before we show how to interpret propositional statements categorically by treating statements as terms we will remind ourselves of the ways that terms and statements are alike. Consider a woman who makes the following two statements:
1. p (=there are elks) 2. q (=there are no elves) Assume that she has in mind the real world. As it happens, (1) and (2) are both true of the real world. Let us call a world a 'pworld' if the statement 'p' is true of it. If 'p' is nottrue of the world, we call it a 'notp world'. For example if p =there are elks in Canada and q =there are elves in Canada, then W (the real world) is a world characterized by the existence of Canadian·elks and the nonexistence of Canadian elves. In other words, the world is a p world and a notq world. The question that Leibniz has set us is how to rephrase propositional statements such as 'p and q' as saying something like 'some pis q' and 'ifp then q' as saying something like 'every p is q'. And the answer we argue for is: to understand 'p and q' to say 'some pis q' by reading this as 'some pworld is a qworld' and to understand 'ifp then q' as 'every pis q' by reading 'every p is q' as 'every pworld is a qworld'. In reconstruing propositional statements as categorical statements we bear in mind the following facts: 1. Propositional statements and ordinary statements have the same basic structure. Singular and general statements have the same basic structure. These facts are perspicuous in algebraic transcription where the basic structure is represented as a dyad of form '#+@'. 2 Ordinary statements are about things in a domain of things (a universe of discourse); when the terms of an ordinary statement denotes, it denotes one or more things in that world. 3. Propositional statements are about one thing only, the world they all claim to characterize; when a propositional statement denotes, it denotes that world. These points suggest that if we are to give categorical readings to
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205
propositional statements, then we must understand them all as making claims about one and the same thing: the world. To see the significance of this, let us imagine that some domain, D, has a as its sole member. Suppose that a is a pthing, authing , an rthing but not an sthing. Now suppose that an outside observer, Molly, is characterizing the objects in D. Unbeknownst to Molly, whenever she describes any object in D, it is always the same object. Molly makes the following true assertions I. 1. some u is r 2. something is p 3. nothing iss
true true true
Having satisfied herself that these are true Molly now wonders whether any of the following are true: II. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
every p is r every nonq IS u everything is p every q is r every sIS p no r is p and s every p and sis r every u is p everything is u some r 1s p 14. some r is not s 15. every sis r
Had Molly known that D contains no more than one thing, she would have seen that the truth of every statement in II is entailed by some statement in I. For example, (6) 'everything is p' follows form (2) 'something is p', there being only one thing and that thing being p. Moreover since everything is p, it follows that (11) every u is p and (8) every sis p since nothing in Dis authing or an sthing without being a pthing. Also, since (1) some u is r it follows that both that something is u and that something is r. But from 'something is u', 'everything is u' follows, which, in turn, entails (5) 'every
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nonq is u'. Similarly from 'some thing is r' 'everything is r' follows. But if 'everything is r, then (4) every pis r, (7) every q is rand (8) whatever is p and sis r. Finally, from 'nothing iss', Molly would infer (11) 'nor is p and s'. In a universe of discourse containing only one thing 'everything is X' and 'some thing is X' are equivalent. Now this fact about singular domains has an important logical application. For while ordinary statements are claims about the things of a domain that contains many things, propositional statements may be understood to make categorical claims about the domain itself. We are now in a position to say how we may construe propositional statements terministically as categorical. The idea is to construe all propositional statements as being about the one world. We now read 'p and q' as saying something like 'some pis q' and 'ifp then q' as saying something like 'every p is q'. Let 'p' be any sentence and let '' be the state of affairs in which p (i.e., the state signified by 'p') and let the term 'p' denote any domain or world characterized by
(i.e., any pworld). By convention, lower case letters are statement letters. But in our new way of construing these letters they are a special type of term that apply to the members of a domain of worlds, 'p' applying topworlds, 'q' to qworlds and so forth. But in this case, the domain of worlds, which is the domain of the claim of any propositional statement, contains only one world as its member (by hypothesis the actual world). In effect, all propositional statements are now construed as singular statements about the one actual world. Since in any context of discourse only one domain is under consideration, all propositional statements are semantically singular. It is as if all propositional (lower case) letters were starred as uniquely denoting terms. We are saying that all propositional statements are about one and only one thing: the world itself. We may read 'p' categorically as the statement: 'the world is a pworld'; moreover 'p' is true if and only ifthe world is a pworld. We now read our formulas '+p+q', ' p+q', etc., as 'some pworld is a qworld. 'every pworld is a qworld', etc. Since the categorical version of a propositional statement is a statement of the form 'some/every xis a y', the bare assertion 'p' may be understood as the categorical assertion 'some world is a pworld'. However, since all statements are about the one world, 'some pworld is a qworld' entails 'every pworld is a qworld'. We rewrite propositional statements in categorical form in accordance with the following rules:
Statement Logic 1. 2. 3. 4.
p&q p ::> q
p p
some pis a q every pis a q some thing is a p (some pis a p) no thing is a p (no p is a p)
207 +p+q p+q +p+p (+p+p)
Because of their singularity, propositional assertions are subject to special laws that distinguish them from ordinary categorical statements. The latter are about many things. But propositional assertions are all about one and the same thing: the world itself. In particular, the following two propositional laws are especially important: 1. some world is pworld = every world is a pworld 2. some pis q =every pis q. According to the first law any statement 'p' claims that 'some world is a pworld'. But since only one world is under consideration, this claim is tantamount to the claim that every world is a pworld. The truth of the second law is also obvious; there being only one world, if that world is both a pworld and a qworld, then every world is a qworld and in particular every pworld is a qworld. By construing every propositional statement as a categorical statement about worlds, we carry out Leibniz's terminist program. Leibniz correctly believed that statement logic could be understood as a special branch of the logic of terms. This terminist doctrine is encouraged when we note (a) that compound statements have the same formal structure as categorical statements whose elements are terms and (b) that many of the laws of statement logic (e.g., the law that denying a conjunction is equivalent to affirming the disjunction of denials) are the same as the laws of term logic (e.g., that denying that some A is B is equivalent to affirming that every A is not B since both laws have the structure: (+x+y) = +(xy)). We noted, however, that terminism seemed wrong when we consider that the categorical versions of propositional statements led us to a branch of term logic that differs in fundamental ways from the rest ofterm logic. For example, we know that in the ordinary logic of terms covalence and equality are necessary conditions for logical equivalence. But covalence and equality are not necessary conditions for the equivalence of propositional statements. Other discrepancies are equally serious. For example, we noted that 'some A is B /every A is B' is not valid in ordinary term logic, but its analogue, 'p and
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q /p::::~q', is valid in statement logic. We note also that the pair of statements 'some A is A; some nonA is nonA' can both be true in ordinary term logic but their propositional analogue in statement logic is the pair 'p and p' and 'not p and not p', which can't be jointly true. But we now see that all differences between term and statement logic can be explained once we recognize that propositional statements are unique in having a singleton domain as their universe of discourse so that all propositional terms denote one and the same world. A logic of terms for a singleton domain has just the properties that distinguish statement logic from ordinary term logic. In a context where all statements apply to only one thing, 'every A is B' does follow from 'some A is B' and 'some A is A' is incompatible with 'some nonA is nonA'. Interpreting the propositional statements of statement logic by treating the statement letters in the formulas '+p+q' and 'p+q' as propositional term letters does not violate any of the laws of a term logic whose statements have a universe of discourse that has exactly one member. Thus, while the move from 'some A is B' to 'every A is B' is prohibited, this isn't the case where the terms are UDTs. Thus from 'some pis q' to 'every p is q' is permitted. For if it is true that something (viz., The World) is both p and q then it is true that everything is both p and q. And if every thing is both p and q then everything is either not p or q. This explains why '+p+q' entails 'p+q'. For, where there is only one world to consider, 'some pword is a qworld' does entail 'every pworld is a q world'. The singularity of propositional statements also accounts for their divergence from the law of equivalence governing general statements. According to that law two statements are logically equivalent if and only if they are covalent and equal. But in propositional logic, when two statements are both covalent and equal, that suffices for their equivalence. However, neither of these conditions is necessary. One of the divergent cases is the law equating 'p&p' to 'p v p ', which in categorical form is the equivalence of 'some pis p' to 'every nonp isn't nonp' . For ordinary terms this equivalence between divalent statements is invalid: 'some P is P' is not equivalent to 'every nonP isn't nonP'. The right hand side is equivalent to 'no nonP is nonP'. (Consider 'some ape is an ape', which is true and 'no nonape is a nonape' or 'every nonape is an ape', which are false.) But in propositional logic the divalent equivalence holds. Here again the validity of the propositional equivalence is due to the circumstance that only one thing is being talked about. In a singleton domain of worlds 'some pworld is a pworld' = 'no pworld is a nonpworld', since
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209
where something (the actual world ) is a pworld it follows that nothing fails to be a pworld. Thus the crucial difference between ordinary term logic and propositional 'term' logic is that the latter makes claims about a singular domain and in a singular domain 'something is X' is equivalent to 'everything is X'. This crucial difference gives statement logic its special character. And, once this is understood, we are free to understand statement logic as a special branch of the logic of terms that we have all along been studying. We have shown that the logic of propositional statements can be construed as a special branch of the logic of terms. But having shown this, it still remains true, as a practical matter, that the techniques we have learned for dealing with arguments in statement logic are especially suited to it. In any case, even though '+p+q' has the same form as a statement '+P+Q', which we read 'some P is Q', it would be pointless to change our way of expressing ourselves, by reading, say, 'roses are red and violets are blue' as 'some world in which roses are red is a world in which violets are blue'. We therefore continue to read '+p+q' as 'p and q' and 'p+q' as 'if p then q'. For all intents and purposes, we continue to treat statement logic as an independent logic even as we bear in mind the terminist interpretation that Leibniz adumbrated.
17.
Venn Diagrams for the Singleton Universe of Propositional Logic
Consider the Venn diagram for 'someS is P' where 'S' and 'P' are ordinary terms: Figure 23
s
p
The cross mark inside the SP segment indicates the presence of an SP thing. But what about the other segments? Since no information is given we leave
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An Invitation to Formal Reasoning
this empty~ nothing may or may not be in any one of them. Now consider the Venn diagram for 'some pworld is a qworld' where it is understood that no more than one world is under consideration.
Figure 24
Note that we have shaded everything except the pq area. Note now that an unshaded area represents the presence of a pqthing (in this case a pqworld). The difference between a regular Venn diagram and a Venn diagram for a singleton universe is this: on a regular diagram we have three possibilities for any given area:
1. it has nothing in it (marked by a cross) 2. it is definitely empty (shaded) 3. it may be empty or not (unmarked and unshaded)
In a Venn diagram for a singleton universe we have only two possibilities: 1. it has something in it (unshaded) 2. it is empty (shaded) There is no third possibility. Consider the Venn diagram for 'p pis aq':
Figure 25
:;:~
q' in its categorical form 'every
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Looking at Figure 24 we note that Figure 25 is already depicted: the p(not q) area is shaded. This graphically shows that 'some pis q' entails 'every pis q' in the singleton universe of discourse of propositional logic.
***************************************************************** Exercises: Construct Venn diagrams for each of the following, showing that the conclusion follows from the premises. l.p=>q
s q=>s I p&s 2. p&q I P => q 3. not p I ifp then q 4. not (p and not q) not q lnotp 5. ifp then q ifq then r I ifp then r
*******************************************************************
8 Modem Predicate Logic
1. Syntax The logical system that we have studied in this book is known as Term Logic or Term/Functor Logic (TFL). TFL parses sentences as 'dyads' made up of two terms and a functor that connects them. Not only sentences, but relational terms, compound terms and compound sentences are given dyadic parsings. Term Logic goes back to Aristotle. In this chapter we shall acquaint ourselves with another logical grammar and another system oflogic known as Predicate Logic or Modern Predicate Logic (MPL). For discussion of the differences between TFL and MPL, the reader may wish to look again at Chapter 2 section 11, where we distinguished two approaches to logical syntax: 'the Term Way' and 'the Predicate Way'. We have so far followed the Term Way. In this chapter we follow the Predicate Way. The grammar of a logical language is called logical syntax. Looking at a sentence, the logician distinguishes between its material elements, which carry its meaningful content, and its formative elements, which determine its form. For example, looking at 'some apes are omnivores' the logician who follows the term way distinguishes between the two material elements 'apes' and 'omnivores', on the one hand, and the functor expression 'some are' that joins these terms to give us a sentence of the form 'some X are Y', on the other. The material elements ofTFL are nouns or noun phrases. By contrast, the material elements of Predicate Logic are verbs or verb phrases that serve as predicates. Let us recapitulate the parsing style of the Term Way. Consider the fact that 'some bird sings' entails 'some singer (singing thing) is a bird'. The terminist represents this inference as '+B+S I +S+B'. In so representing it one first paraphrases 'some birds sing' as 'some bird is a singer'. In so 'regimenting' the sentence, we treat the verb 'sing' in the premise as if it were the predicate verb phrase 'is a singer'. We then use 'singer' as a term that could take either subject or predicate position. In effect, TFL splits the verb
213
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An Invitation to Formal Reasoning
'sing' into a copula, 'is (a)', and a term, the noun or noun phrase 'singer' or 'one who sings'; this term then reappears in the subject position of the conclusion. We similarly regiment 'all birds sing' as 'all birds are singers', again analyzing it as a two term sentence, ' B+S'. In TFL each sentence has two terms and a functor that connects them. The same two term parsing is given to a singular sentence like 'Caruso sings'. Here the (singular) term 'Caruso' and the general term 'singer' are connected by the commutative functor 'some is a' and the sentence transcribes as '+C*+S'. In TFL a singular sentence such as 'Caruso sings' has 'wild' quantity; it is particular, but because its subject term, 'Caruso', denotes uniquely, 'some Caruso sings' entails its corresponding universal, 'every Caruso sings'.
2.
MPL: The Predicate Way
An alternative to the two term way of construing sentences follows the linguist's familiar parsing of sentences into Noun Phrase and Verb Phrase. Consider 'Caruso sings'. The linguist parses this as 'Caruso/sings' with 'Caruso as subjectname and 'sings' as predicateverb. This style of parsing is adopted by MPL for all singular sentences. Note that in dividing the sentence into the two material parts consisting of 'Caruso' and 'sings', MPL does not further analyze the verb 'sings' as 'is a singer'. Thus in 'Caruso sings' there is no 'connective' expression that ties the verb to the noun. Indeed, that is one very important difference between TFL and MPL; in TFL we always have a formative element, a 'connective' that ties the two material elements. But in MPL the verb 'sings' follows the noun 'Caruso' without benefit of a connective joining one to the other. In the relational sentence 'Paris loves Helen', MPL recognizes the verb 'loves' as a two place predicate. Both 'Paris' and 'Helen' are subjectnames. By contrast, TFL parses 'Paris loves Helen' as consisting of two 'subsentences': 'Paris loves' [+P* 1+L 12] and 'Helen is loved' [+H*2+L 12], both implicit in 'P 1+L 12+H* 2'. Note that 'P* 1+(L 12+H* 2)' transforms into '+P* 1+(+H2+L 12+)' [read 'Paris is what Helen is loved by'], a dyad whose second term is itself a dyad: the subsentence '+H* 2+L 12 '.
Modern Predicate Logic
3.
215
General Sentences in MPL
Predicate Logic treats a general sentence like 'some birds sing' as a two verb sentence. The contrast between TFL and MPL can be better appreciated when we look at the different way they approach a simple inference like some birds sing I some singers are bird
In representing this little inference the logician must identify two 'material' (nonformal, 'extralogical') expressions that appear in converse order in the two sentences. As we said, TFL chooses nouns; MPL chooses verbs. In line with its terminist parsing policy, TFL regiments 'some birds sing' as 'some birds are singers'; the two expressions that convert are the nouns 'singers' and 'bird. MPL takes verbs (predicates) as the interchangeable expressions. A verb may be simple like 'ran' or 'sings' or it may be formed by taking a noun and putting a copula in front of it.' For example, 'is a singer' is a verb formed in this manner. Thus any predicate of the form 'is an X' is a verb. Using verbs as the basic material elements, MPL regiments 'some bird sings' as a sentence containing the two verbs or predicates, 'is a bird' and 'sings'. So analyzed, the inference looks like this: something is such that it is a bird and it sings I so something is such that it sings and it is a bird. Note that the sentence, as parsed in MPL, is now understood to contain pronouns as well as predicates; the word 'something' behaves as the antecedent to the pronoun 'it', which appears as the subject in the two component (singular) sentences 'it is a bird' and 'it sings'. Because the MPL construal of 'some bird sings' is a bit cumbersome, practitioners ofMPL have introduced certain symbols to make it tidier. (MPL is often popularly called 'symbolic logic'.) Let '(Ex)' abbreviate the phrase 'something, x, is such that', let 'Bx' stand for 'xis a bird' and let 'Sx' stand for 'x sings'. Then MPL represents the inference thus: (Ex)(Bx&Sx)
I (Ex)(Sx&Bx) In words:
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An Invitation to Formal Reasoning something x is such that x is a bird and x sings I something x is such that x sings and x is a bird
The expression '(Ex)' is sometimes read as 'there exists an x such that'. Grammatically, this expression serves as the antecedent to the pronoun, 'x', that comes after it in 'x is a bird' and 'x sings'. Logicians call '(Ex)' 'the existential quantifier'. It serves to introduce the pronoun variable 'x', which is said to be 'bound' by the quantifier antecedent. Thus, in MPL pronouns are 'bound variables' and antecedents are 'quantifiers;' in the logical language of MPL, the relation of binding is the relation that an antecedent expression (e.g., the quantifier 'something') has to its pronouns (the variables it binds). That '(Ex)' is related to the variables it binds in the way that an antecedent is related to its pronouns is evident if we reads '(Ex)(Bx&Sx)' thus: there exists a thing [antecedent] such that it (that thing) [pronoun] is a bird and it [pronoun] sings Note (again) that the letter 'B' in 'Bx' does not stand for the noun 'bird'; it is not a term letter but a predicate letter representing the verb 'is a bird'. By contrast, in the terminist formula '+B+S' the letter 'B' is a term letter representing the noun 'bird'.
4.
The Logical Language of MPL
Compare '+S+P' as a way of representing 'some Spaniard is a painter' to '(Ex)(Sx&Px)'. The TFL formula is a straightforward transcription ofthe English sentence. The MPL formula is more complicated. And indeed, the language of MPL is not easy to master. To help us in learning it we shall make use ofthe more natural and familiar (to us) formulas ofTFL. Thus in 'translating' a sentence like 'some boy loves a girl' into MPL we shall first transcribe it as 'B 1+L 12+G2 ' and then use the transcribed formula as a bridge to get to '(Ex)(Bx&(Ey)(Gy&Lxy))', treating the TFL formulas as a bridge to the corresponding formulas of MPL. Before learning how to reckon arguments in MPL, we must learn the logical language ofMPL. The symbolic formulas ofMPL are not algebraic: they contain no plus or minus signs. Thus 'p and q' is not represented as '+p+q' and 'ifp then q is not represented as 'p+q'. Instead, the symbolic ways of representing compound sentences is exclusively used:
Modern Predicate Logic
English form pandq ifp then q p orq p if and only if q notp
Algebraic Form
Symbolic Form
+p+q p+q +[ p+q]+[ q+p]
p&q p=>q pvq pq
p
p
(p)(q)
217
The following symbolic expressions are used in forming elementary sentences: English Form X is P SomeS is P Every Sis P
Algebraic Form +X*+P +S+P S+P
Symbolic Form Px (Ex)(Sx&Px) (x)(Sx => Px)
We use the TFL algebraic formulas as bridges to MPL formulas. The rules for translating any sentence of TFL into a sentence of MPL are called 'TR' (for 'translation rules'). The first of these rules, TRI, tells how to translate particular TFL sentences into MPL. The TR rule for particular sentences is TRI: TRJ:
English TFL MPL some A is B ==> +A 1 +B1 ==> (Ex)(Ax&Bx)
For example, applying TRI to 'some American is a banker' we use the TFL bridge, '+A1+B 1', to get to '(Ex)(Ax&Bx)' as the MPL formula. Note the use of numerical pairing indices in the TFL formula. These are usually omitted in transcribing nonrelational sentences since it is obvious in such simple sentences that the two terms of the sentence are the only terms being paired. Nevertheless, in using a TFL sentence as a bridge for translating into MPL, we shall give each term pair (arbitrarily chosen) numerals; these indices are then replaced by bound variables in the corresponding MPL formulas. The expression that follows a quantifier is called the 'matrix'. For example the matrix of the formula '(Ex)(Ax&Bx)' is '(Ax&Bx)'. In translating '+A 1+B 1' as '(Ex)(Ax&Bx)' we treat the numeral as the bound variable pronoun 'x' and we treat '+A 1+B 1' as a conjunction of form '+p+q'. This gives us the matrix formula 'Ax&Bx' or 'it is A and it is B', which needs
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An Invitation to Formal Reasoning
a quantifier antecedent to the pronoun, 'x'. MPL prefixes the conjunctive formula (the matrix) with the quantifier antecedent '(Ex)' or 'something, x, is such that'. There is a corresponding TR for universal sentences in MPL. Consider the universal form 'every A is B', which the terminist transcribes as ' A+B'. MPL gives ita conditional reading, treating it as 'Ax::JBx'. But now the quantifier antecedent that 'binds' the pronouns is universal. The MPL way of expressing 'every A is a B' is every thing is such that if it is an A then it is a B Using the bound variable 'x' in place of 'it' we have the formula: every thing x is such that: if x is an A then x is a B As we will see quite shortly, a translation rule applied to the 'matrix' gives us (Ax::JBx). The symbolic abbreviation for 'every thing xis such that' is '(x)'. Thus the MPL formula corresponding to ' A+B' is (x)(Ax ::J Bx) The expression 'every thing xis such that', which is symbolically abbreviated as '(x)' is called 'the universal quantifier'. The universal quantifier in '(x)(Ax ::J Bx)' binds the variables of the component sentences 'Ax' and 'Bx' in the matrix, 'Ax ::J Bx'. The TR for translating 'every A is B' into MPL is:
TR2:
every A is B
==>
A 1+B1 ==> (x)(Ax
~Ex)
For example, to translate 'every American is a banker' we use the bridge 'A 1+B 1'. Applying TR2, yields '(x)(Ax>Bx)' as the MPL formula. There are three TR rules for translating compound forms. Wherever 'p' and 'q' are sentences, the formula '+p+q' translates as 'p&q'. The rule that takes us from '+p+q' to 'p&q' is 'TR/and', the And rule:
TR!and:
p and q ==> +p+q ==> p&q
A second rule, called, 'TR/if, or the If rule, is used to translate conditional forms. Thus, given 'ifr then s', we first transcribe it as 'r+s' and then, by TR/if, translate it as 'r ::J s' :
Modern Predicate Logic TR!if
219
ifp then q ==> p+q ==> p:::xJ
A third rule, called 'TRior', tells us how to render 'p or q' into MPL: TRior:
p or q ==>
pq
==>
pvq
In applying TRI to a compound form like 'some roses are pink and some are white' we first transcribe it in the usual way +[+R+P]+[+R+W] which, byTR/and, becomes '[+R+P]&[+R+W]'. WenextapplyTRl to each conjunct. However, in applying TRI we must treat each conjunct separately giving each conjunct its own quantifier; we may not bind all the pronoun variables by a single antecedent quantifier. The correct MPL translation is (Ex)(Rx&Px)&(Ex)(Rx&Wx) Although 'x' is used throughout, it has different antecedent quantifiers in the two conjuncts. Another (easier to read) MPL version would use 'y' for the second conjunct: (Ex)(Rx&Px)&(Ey)(Ry&Wy) These two versions are equivalent. But the following formula which binds 'x' by only a single quantifier is not a correct translation: (Ex)((Rx&Px)&(Rx&Wx)) TR/and and TRior apply to compound terms as well as to compound sentences. In translating a sentence like 'some A is either B or C', we first apply TRI to give us '(Ex)(Ax&(B or C)x)' and then apply TRior to the disjunctive expression. The result is '(Ex)(Ax&(B v C)x)'. A later rule {TR4) allows us to represent '(B v C)x' as 'Bx v Cx'.
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An Invitation to Formal Reasoning
****************************************************************** Exercises: Translate into MPL: Example: if every A is B then some C is D 1.  [ A+B)+[+C+D] Algebraic Transcription 2. [ A1+BJ1 => [+C2+D2] TR/if 3. (x)(Ax=>Bx) => (Ey)(Cy&Dy) TR2, TRl i) if any A is B then every C is D ii) no C is D but some C is K iii) only A is B iv) every J is K unless some J is M (hint: If 'ifu't' were a word, 'unless' would be its synonym.) v) some rose is pink and it is white ********************************************************************
5. Singular Sentences in MPL
In TFL singular and general sentences are syntactically alike. In MPL singular sentences are treated as being radically different in form from general sentences. The TFL manner of transcribing sentences whose subjects are proper names or other singular subjects does not differ from the manner in which general sentences are transcribed. Thus 'Caruso is a tenor' transcribes as '+C* 1+T 1' and 'Caruso admired Puccini' as '+C* 1+A 12+P* 2 '. By contrast, where the MPL form of the general sentence 'some Italian is a tenor' is '(Ex)(lx&Tx)', the MPL form for 'Caruso is a tenor' is 'Tc', in which the lower case letter 'c' represents the subject 'Caruso' and the upper case letter represents the predicate 'is a tenor'. Singular sentences that have two singular subjects are given similar treatment. The MPL form for the relational singular sentence 'Caruso admired Puccini' is 'Acp'. This formula has two lower case letters in subject position, respectively representing the singular nouns 'Caruso' and 'Puccini'. The predicate letter 'A' represents the transitive verb 'admires', or the relational
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predicate expression 'is an admirer of. Predicates like these take two subjects; we call them 'two place predicates'. Translating sentences containing singular terms into the notation of MPL is straightforward and we do not need to use a TFL formula as a bridge. The translation rule is
TR3 S*isP S* is R toP* xis P xis R toy
==> ==> ==> ==>
Ps Rsp Px Rxy
Applying TR3 to 'Tom loves Ella' [T* 1+L 12+E*21 we get 'Lte'. (Note that we simply replace the numerals of the TFL formula, 'L 12 ' by the names of the lover and the beloved. Sentences with three subjects have a three place predicate. For example, the MPL formula for 'Italy ceded Fiume to Yugoslavia' is 'Cify', in which 'C' is a three place predicate relating Italy as the country that ceded, Fiume as the city that was ceded and Yugoslavia as the country to whom Fiume was ceded by Italy. A fmal translation rule takes compound terms into MPL formulas. The MPL formula for 'some gentleman and scholar is a farmer' is '(Ex)((Gx&Sx)&Fx)'. Its TFL formula is '+<+G+S> 1+F 1'. In getting from the TFL formula to the MPL translation, we first apply TR1 and then apply TR/and to '<+G+S>' to get '(G&S)'. This gives us '(Ex)(x&Fx)'. A new rule, TR4, allows us to distribute a variable inward, treating a form like 'x' as '(Gx&Sx)'.
TR4:
<+A+B>x ==> x ==> Ax&Bx <AB>x ==> x ==> AxvBx We get the MPL translation of 'some gentleman and scholar is a farmer' by the following steps: 1. 2. 3. 4.
+<+G+S>I+FI (Ex)(<+G+S>x+Fx) (Ex)(x&Fx) (Ex)((Gx&Sx)&Fx)
TFL 1, TR1 2, TR/and 3, TR4
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6. How the Logical Syntax of MPL is 'Ontologically Explicit' The logical grammar ofMPL is the standard logical grammar taught today. One reason for this is that formulas written in the language of MPL are explicit about what makes them true; each MPL formula tells us exactly what sort of things have to be present in the world or absent from it for the formula to be true. Consider again the TFL and MPL renditions of a sentence like 'some Spaniards are painters'. According to TFL this is to be rendered as '+S+P'. According to MPL this is to be rendered as '(Ex)(Sx&Px)'. The TFL formula is simpler and closer to the original English. However, MPL has a special virtue that has recommended it to most contemporary logicians. In the words of one the foremost exponents of MPL in this century: The grammar that we logicians tendentiously call standard is. a grammar designed with no other thought than to facilitate the tracing of truth conditions. And a very good thought this is. 0'/.V.O. Quine, Philosophy ofLogic, Prentice Hall, 1970, pp.35 36) That '(Ex)(Sx&Px)' has very explicit truth conditions is evident when we consider that for 'some Spaniard is a painter' to be true the following conditions must hold: a Spaniard must exist, a painter must exist, the Spaniard in question must be the painter in question. All these conditions for the truth of 'some Spaniard is a painter' are expressly present in the MPL formula, which says 'there exists a thing such that it is a Spaniard and it is {also) a painter'. The condition that a Spaniard exists is there. The condition that a painter exists is there. And the condition that the Spaniard in question is the painter in question is explicit in the use of a single pronominal subject for the two predicates 'is a Spaniard' and 'is a painter', both of these predicates being said to hold of that subject. More generally, an MPL formula is always explicit about what things in the world must exist or fail to exist in order for the statement to be true. This is even more clearly seen in more complicated sentences like 'some boy loves no girl', whose MPL form is '(Ex)(Bx& (Ey)(Gy&Lxy))' which says 'there exists an individual, x, such that x is a boy and there exists no individual, y, such that y is a girl and x loves y'. Here the existence of a boy and the nonexistence of any girl that is loved by that boy are the truth conditions of the sentence, and both ofthese truth conditions are explicitly expressed in the MPL formula. Quine is right in suggesting that one of the main reasons for the deserved popularity ofMPL lies in the 'ontological' explicitness of its
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logical grammar: any MPL statement as much as says what in the world there must be or fail to be in order for the statement to be true. Nevertheless, the virtue of explicitness is more than offset by the artificial complexity of the MPL formulas and by their 'distance' from the sentences of an ordinary language. The complexity is evident in the way it treats even the simplest sentences. Consider 'every horse is an animal', which TFL transcribes as 'H+A' but which MPL translates as '(x)(Hx =>Ax)'. The TFL formula is akin to a stenographic transcription (letters for terms, ' ' for 'every', '+'for 'is'). But the MPL formula paraphrases 'every A is B' as something like 'any individual is such that if it is a horse then it is an animal'. Such a paraphrase is aptly termed a 'translation' because it introduces novel syntactical elements not found in the original English sentence 'every horse is an animal'. In the first place, it treats 'every horse is an animal' as a pronominalization in which the quantifier expression 'any individual' serves as antecedent to the pronoun 'it'. The original sentence, of course, has no pronouns. In the second place, MPL introduces the sentential connective form 'if ... then' which connects two component 'subsentences', 'it is a horse' and 'it is an animal'. Again, the original sentence looks to be simple and not compound. The distance between the vernacular English sentence and the MPL formula is even greater for a relational sentence like 'some sailor is giving every child a toy'. TFL transcribes this in a stenographic way as '+S 1+G 123 C2+T3 '. By contrast, MPL creatively paraphrases it, or 'translates' it, as 'there exists an individual, x, such that x is sailor and for any individual, y, ify is a child then there exists an individual, z, such that z is a toy and xis giving y to z'. The symbolic formula for this is (Ex)(Sx&((y)(Cy => (Ez)(Tz&Gxyz))) Another example of an MPL rendering of a relational sentence is every owner of a cow owns a barn which MPL understands to say 'for every x if something, y, is such that y is a cow and x owns y then something, z, is such that z is a barn and x owns z'. Symbolically: (x)((Ey)(Cy&Oxy) => ((Ez)(Bz&Oxz)))
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Formulas like these are examples of 'multiple quantification'; which is to say, they are sentences with more than one quantifier expression, each of which binds a different pronominal variable. It is often not easy to see how to give MPL renderings of English relational sentences that involve multiple quantification. How, for example, is one to translate 'some boy envies every owner of a dog'? Or 'some one who owns a barn does not own any animal'? The language of MPL is the standard language of modern logic. But it is an artificial language, so it is not easy to learn. However, anyone who has mastered the language of TFL can easily learn how to translate any TFL sentence into MPL. Translation from English into MPL may be made simple by using TFL transcriptions as bridging formulas. With the aid of the TFL bridging formulas, and using the TR rules we have just learned, the translation of multiply quantified relational sentences can be done in a fairly straightforward mechanical way. The TR rules that get us from 'some/every A is B' to the corresponding MPL formulas apply to dyads ofthe form '+X+Y'. But any sentence of TFL can be written as a sentence that is either itself a dyad or analyzable into dyads.
7. Dyadic Normal Forms
The formulas of TFL that are especially useful for translating into MPL are what we call 'Dyadic Normal Forms' (DNF). The DNF of a sentence is a normal form in which compound and relational terms are dyads. We here review the method of transforming any standard TFL sentence into DNF first introduced in section 11 of Chapter 4. A dyad is an expression of the form '±X±Y'. For example, a compound term such as 'gentleman and scholar', which transcribes as an expression ofthe form '<+X+Y>', is a dyad. Any sentence is a dyad in which the first part '±X' is the subject and the second part '±Y' is the predicate. A sentence is in DNF when its structure is fully dyadic, so that every expression in it other than its terms is a dyad ofthe form '±X±Y'. As often as not, a given sentence is not initially given to us in dyadic form. Thus in 'some boy is teaching some girl'[==> '+B 1+(T 12+G2) 1' ], the expression 'T12+G2 ' has the subject 'a girl' on the right. To rewrite it as a proper dyad we commute the phrase 'T 12+G2 ', putting the subject term, 'G2 ', to the left. In effect, we rewrite the sentence as '+B 1+(+G2+T 12) 1'. In this formula each subject phrase has its predicate to the right and the whole formula is said to be in dyadic normal form. The expression '+G2+T 12 ' is a 'subsentence' that may be
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understood to say 'some girl is taught (by)'. Generally, when we put a relational sentence into DNF, we place each subject phrase to the left of its own predicate. Let us take the sailor sentence, 'some sailor is giving every child a toy', as another example. Its transcription is '+S 1+(G 123 C2+T3)'. This has three subject expressions: 'some sailor', 'every child' and 'some toy'. The first of these subjects, '+S 1', is already to the left, but the second and third subjects need to be relocated to the left of their own predicates. The resulting DNF formula is '+S 1+(C 2+(+T3+G 123))'. Here the expression that follows the subject '+S 1' is a predicate that says of some sailor that he gives every child a toy. 'C2 ' has its predicate; this says of every child that it gets a toy. And '+T3' too has its own predicate, which says of a toy that it is given. Each subject is now in its own dyad and the whole sentence is in 'dyadic normal form'. The subjectpredicate pairings may be made more explicit by giving each term pair its own numerical co indexing:
The rules of translation that take us from TFL forms to MPL forms apply to dyads. In applying these rules to relational statements, we first reformulate the TFL transcriptions in DNF and then apply each rule in a stepwise fashion to the dyadic forms of the TFL formula. Consider how we should use the DNF of the sailor sentence to get to the MPL formula: l.+St+( C2+(+T3+G 123)) 2.(Ex)(Sx& (C 2+(+T3+Gx23 ))
DNF TRl
Note that we began with the outermost dyad that includes all the other dyads. In it we replaced the numerical index that pairs the terms 'sailor' and 'giver' by the pronoun 'x', which now serv~s as common subject to the predicates 'is a sailor' and 'gives' in the formula 'he is a sailor and he gives'. Moving inward, we proceed by applying rule TR2 to the next dyad, ' C2+( +T 3+Gx23 ) ', this time replacing '2' by the bound variable pronoun 'y': 3. (Ex)(Sx&((y)(Cy => (+T3+Gxy3))
TR2
Finally, we again apply TRl to the remaining (innermost) dyad, '+T3+Gxy3'.
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An Invitation to Formal Reasoning 4. (Ex)((y)(Cy => (Ez)(Tz&Gxyz)))
TRI
The MPL translation is now complete. Let us see how the rules can be applied to give us the MPL translation of 'every owner of cow is an owner of a bam'. This first transcribes as '(0 12+C 2)+(0 13+B3)', but we need it in DNF so we put each subject to the left of its predicate:
1. (+C2+0 12)+(+B3+0 13) 2. (x)((+C 2+0x2) => (+B3+0x3)) TR2 3. (x)((Ey)((Cy&Oxy) => ((Ez)(Bz&Oxz))
DNF TRI (twice)
Sometimes one relational phrase is embedded in another. An example is 'some adolescent is a cousin of a murderer of a painter' which transcribes as '+A 1+(C1 2+(M23+P3))'. DNF TRI (thrice)
1. +AI+(+(+P3+M23)+C12)) 2. (Ex)(Ax&(Ey)((Ez)Pz&Myz&Cxy)
Some sentences have one singular subject and one general subject. An example is 'Tom is memorizing every poem', whose TFL transcription is '+T*1+(M12 P2)'. DNF TR3
1. +T*1+(P2+M12)
2.
(P 2+M~)
(Note that in applying TR3 we eliminate 'T* 1' and elsewhere replace the codenoting numerical index '1' by the name 't' .) We next apply TR2 to the formula that remains:
3. (y)(Py => Mty)
TR2
The MPL formula, 3, may be read thus: anything, y, is such that if y is a poem then Tom is memorizing y or, alternatively as for anything, y, if y is a poem then Tom is memorizing y
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and, more colloquially, as if anything is a poem, Tom is memorizing it Consider 'some boy envies everyone who owns a dog', whose TFL transcription is '+B 1+(E 12 (0 23+03))'. 1. +Bt+((+D3+023)+Et2)) 2. (Ex)(Bx&(y)((Ez)(Dz&Oyz)
=>
Exy)))
DNF l,TRl, TR2, TRl
(After a while, one gets to collapsing several steps into one.)
8. Translating Pronominalizations The general form of a pronominalization is ' ... some s ...the s'. In the natural languages, words such as 'it', 'him(self)', 'she', stand in for a pronominal subject 'the S (in question)'. Thus we do not normally say 'some barber shaves the barber in question' but 'some barber shaves himself. Most pronominalizations are paraphrases of sentences that contain no pronouns. A pronominalization such as 'some A is a B; it is also C' is merely a loquacious way of saying 'some A and B is a C'. Similarly, 'if any A is a B then it is a C' is reducible to 'every A and B (thing, person) is a C', which again is not a pronominalization. (See the rules for pronominal expansions in section 8 of Chapter 6 above.) The style ofMPL is to pronominalize. A sentence like 'some ape is hungry' is not naturally construed as a pronominalization. But the logical syntax of MPL construes any general sentence, i.e., any sentence containing 'some' or 'every', as a pronominalization. Thus 'some ape is hungry' becomes 'something is such that it is an ape and it is hungry'. And while 'every boy loves some girl' is not a pronominalization in natural language, its MPL rendering has two pronouns: 'everything is such that if it is a boy then there exists someone such that it is a girl and the boy (he) loves the girl (her)'. By contrast, TFL transcribes a natural language sentence of the form 'some A is B' or 'every A is R to some B' without treating it as pronominalizations. For example, ' B1+L 12+G2' is a pronounfree formula. The bridging rules that take us from TFL to MPL are thus designed to take us from a nonpronominal formula like 'some A and B is C' [ > '+<+A+B>+C'] to the
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corresponding quantified pronominalization formula '(Ex)(Ax&Bx&Cx)' of MPL. Some sentences are natural pronominalizations and TFL transcribes them with proterms. Oddly enough, when a TFL formula is itself a pronominalization, it is not really suitable as a bridge to MPL. For example, the transcription of'some A is a Band it is a C' is '+[+A' 1+BJJ+[+A\ +C 1]'. But this formula is not as good a bridge to the MPL formula as '+<+A+B>+C' which is pronoun free. For that reason, in using the formulas of TFL as bridges to MPL, it is best to confine oneself to pronounfree formulas wherever possible. We noted earlier that a pronominalizaion like 'some A is B; it is C' can be understood as a paraphrase of a pronounfree formula 'some A and B is a C'. But certain English pronominalizations are not merely stylistic paraphrases ofnonpronominal sentences. Examples are 'some barber shaves himself' and 'some student eloped with his aunt'. These sentences contain 'reflexive pronouns' and they cannot be naturally rephrased in a way that eliminates the pronouns. In transcribing such pronominalizations, TFL uses superscripted proterms. (See Chapter 4, section 8.) Thus, '+B' 1+S 12 +B' 2' transcribes 'some barber shaves himself' and '+S' 1+E 12+(A23 + S ' 3)' transcribes 'some student eloped with his aunt'. Note that in these formulas the common superscript signifies the common reference to the same barber again, the same student again. But the numerical indices of the proterms differ and we cannot automatically apply TRl and TR2 to the formulas by replacing the numbers by bound variables. The MPL version of'some barber shaves himself' is 'someone is such that it is a barber and it shaves it(self)'. Symbolically this is '(Ex)(Bx&Sxx)'. Suppose we tried to get this MPL formula by applying TRl to '+B' 1+S 12+B' 2', the transcription of 'some barber shaves himself. 1. +B' I+{+B' 2+S12) 2. (Ex)(B'x & (+B' 2 +Sx2)) 3. (Ex)(B'x&((Ey)(B'y & Sxy)
DNF TRl TRl
Arriving at 3 we can go no further. The reason is clear: the TFL formula has two numerical indices, which renders it unsuitable as a bridge to the MPL formula which has only one. If we are to apply our bridging rules to formulas that contain proterms, we shall need to modify the formulas in a way that renders them suitable as bridges to the MPL translation. As it stands, a formula like '+B' 1+{S 12+B' 2)' cannot serve as a bridge formula for purpose of
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translation to MPL. So we must modify it in a systematic way to serve that purpose.
9. Preparing the TFL Bridge We modify a TFL pronominalization such as +B' 1+(+B' 2+S 12) (the DNF form of 'some barber shaves himself) by applying to it our rule for internal pronoun elimination (IPE) [see section 17, chapter 4, and section 8, chapter 6]. Recall that this rule allows us to remove subsequent proterms and, in so doing, eliminate superscripts. Thus, our DNF formula for 'some barber shaves himself, viz., '+B' 1+(+B' 2+S 12)' becomes, by IPE, '+B 1+S 11 '. Applying TR1 to this formula we get '(Ex)Bx&Sxx)'. To translate 'some student eloped with his aunt' we begin with its transcription '+S' 1+(E 12+(A23 +S' 3))', and put that formula into DNF. 1. +S' 1+(+(+S\+A23 )+E 12) 2. +SI+(+A21+E12) 3. (Ex)(Sx&(Ey)(Ayx&Exy))
DNF IPE TR1 (twice)
which we read as 'there is an x such that x is a student and there is a y such that y is an aunt ofx and x eloped withy'
******************************************************************** Exercises: Translate the following sentences into MPL. 1. every A is Band C 2. everyone who kidnaps a son of a millionaire is a criminal 3. no mother hates her baby 4. some A or B is D but not E 5. every father of a bride mistrusts her groom
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10. Identity in MPL Consider how MPL treats 'some author is a Missourian' as opposed to a singular sentence like 'Twain is a Missourian'. In MPL 'Twain is a Missourian' is translated as 'Mt'. By contrast, 'some author is Missourian' is a general sentence that translates as '(Ex)(Ax&Mx)'. TFL marks the proper name, but gives both the same treatment: Twain is a Missourian > +T*+M some author is a Missourian => +A+M
An identity sentence in English is like any other singular sentence except for having singular terms in both subject and predicate positions (see section 23 of Chapter 3). And again, sentences affirming identity are not given special treatment in TFL. Consider 'Twain is Clemens', which TFL transcribes as '+T*+C*', in which 'T*' is the subject term and 'C*' is the predicate term. Apart from the fact that the proper names are starred to indicate that they are uniquely denoting terms, there is nothing special about this transcription. The form of 'Twain is Clemens' [=> '+T*+C*'] is the same as the form of 'some humorist is a Missourian' [=> '+H+M']. By contrast, MPL treats all proper names as subject terms and so it represents 'Twain' and 'Clemens' by lower case letters. But if both 't' and 'c' are subjects, they must both be attached to a two place predicate. What is the predicate? The answer given by MPL is that 'Twain is Clemens' should really read 'Twain is identical with Clemens'. More generally, in a statement of identity one would read 'is' as 'is identical with'. In translating 'Twain is Clemens' we first read it as 'Twain is identical with Clemens'. Letting 'I' represent the two place predicate 'is identical with', our translation is 'Itc'. It is also customary to use '=' for the identity predicate, in which case the translation is 't=c'. Thus, given an English sentence like 'Twain is Clemens', MPL treats it as a dyadic sentence ofthe form 'X*= Y*'. By contrast, TFL does not introduce a relation of identity. For TFL an identity sentence is simply one that has uniquely denoting terms in both subject and predicate positions. Thus 'Twain is Clemens' is 'monadic'. But since it is singular it has wild quantity. To put it bluntly; TFL does not recognize identity as any kind of relation. Even a sentence like '2 3 = 8' is read monadically as a particular sentence '(some) 23 is 8'. MPL, however, sees a sign of relation in 'Twain is Clemens'.
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Suppose A affirms that Twain is Clemens and B denies it. MPL represents their respective statements thus: A: Itc B: ltc The denial of identity comes up when I say 'Twain is funnier than everyone else'. Anyone who asserts this is saying in effect that Twain is funnier than everyone who is not (identical with) Twain. MPL translates this as (x)( Ixt => Ftx) i.e., 'for anyone x, ifx is not identical with Twain, then Twain is funnier than ' X.
To translate a sentence affirming identity or difference into MPL we need first to rephrase it in a way that explicitly brings in a two place predicate 'is identical with', 'is the same as' or 'is not identical with' or 'is other than'. Only then can we proceed to apply the rules of translation. Thus, given the sentence 'Twain is funnier than anyone else', we should first render it as 'Twain is funnier than everyone not identical with Twain'. 1. +T* 1+F 12 ((I23)+T*3)
2. +T* 1+( (+T*3+( I23))+F12)
DNF
Eliminating 'T*' and replacing all subscripts pairing with 'T*' by 't' we get: 3. (12t)+F2t 4. (x)( Ixt => Fxt)
TR2
This example also illustrates the general point that the bridge between TFL and MPL is more useful for sentences containing general terms than it is for sentences containing proper names or other uniquely denoting terms. The syntactical distance between the formulas of TFL and MPL is greatest when the sentences involved have proper names or other uniquely denoting subjects expressions (e.g., 'the King of France', Smith's eldest daughter'). In many cases it simply does not pay to use TFL as a bridge for translating singular statements (including identities) into MPL. Thus, given a simple singular sentence like 'Twain is an author', we should directly move to 'At'. (Why bother to get to it via '+T*+A'?)
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Sometimes the identity predicate comes between two pronouns or between a pronoun and a proper name. An example is there is only one divine being We understand this to say that some being is divine and no being other than that being is divine. TFL renders this as +[+B'+D]+[ (+( B')+D)] There is no natural way to eliminate the pronouns. So we don't bother to use the formula as a bridge. MPL renders the sentence as (Ex)(Dx&(y)( lyx
:J

Dy))
i.e., 'some being is divine and every being not identical with it isn't divine'.
******************************************************************** Exercises: 1. Translate 'nothing differs from itself into the language ofMPL. 2. Translate 'only Crick understands' into the language of MPL. (Hint: construe the sentence as 'Crick understands and no one else understands'.) 3. Translate into MPL: none except Teddy is loyal to Sally (Hint: no one not identical to Teddy)
******************************************************************** 11. Logical Reckoning in MPL Having learned how to translate into MPL language we will now learn some MPL techniques for evaluating arguments and drawing conclusions from premises. Consider the valid syllogistic argument 'every ape is hairy; some denizens of Madagascar are apes; so some denizens of Madagascar are hairy'. To show that it is valid in TFL we could transcribe it, deny its conclusion and derive a contradiction:
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1. A+H 2. +D+A 3. (+D+H) 4. D+(H) 5. +AH 6. +HH
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premise premise negation of conclusion 3,PEQ 4,2, DDO 5,1, DDO
This evaluates the argument as valid. For we have shown that denying the conclusion leads to contradiction. (Another method could be used: show that 1, 2, and 3 give you a P/Z conjunction (see section 5 of Chapter 5).) A similar technique may be used in MPL. First we should represent the premises and the denied conclusion in MPL thus: 1. (x)(Ax :::> Hx) 2. (Ex)(Dx&Ax) 3.  (Ex)(Dx&Hx)
premise premise negation of conclusion
But now we stop: we lack MPL rules analogous to the TFL rules such as PEQ and DDO that would enable us to proceed to derive a contradiction. In the next two sections we present the MPL rules that are used in evaluating arguments like the above.
12. Transformation Rules Rules of one important type are called Rules of Transformation (also called Rules of Substitution). A rule of transformation permits us to transform a formula into an equivalent formula thereby permitting us to replace the former by the latter. One such rule allows us to change a formula that has an initial sign of negation into one that has no such sign. For example, given the TFL formula '(+T+A)' [read: nothing is an A], we clear the negation sign by driving it inward algebraically by 'obversion' to give us 'T+(A)' [everything is a nonA]. The analogous move for MPL is  (Ex)(Ax) = (x) (Ax) Note that this transformation involves three changes: 1. we change the outermost sign
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An Invitation to Formal Reasoning 2. we change the quantifier 3. we change the sign of the Matrix
In MPL, the law that corresponds to TFL obversion is called 'The Law of Quantifier Interchange' (QI). The following transformations are justified by the law:
Law of Quantifier Interchange (QI): (x)( .. x...) = (Ex) ( .. x .. .) (Ex)( .. x... ) = (x) ( .. x.. .) (x)( .. x.. .) = (Ex) ( .. x ... ) (Ex)( .. x.. .) = (x) ( .. x.. .) Applying QI we may make the following simple inferences: not everything is created I something is uncreated (x)(Cx) I (Ex)Cx nothing is created I everything is not created  (Ex)Cx I (x) Cx everything is created I not: something is not created (x)Cx I (Ex)Cx something is created I not: everything is not created. (Ex)Cx I  (x) Cx A number of transformation laws of Statement Logic that deal with negation and the connectives ' & ', 'v', and '::J' figure prominently in reckoning with the formulas ofMPL. The following rules of transformation are known as DeMorgan's Laws:
DeMorgan 'sLaws (DML): Symbolic form (p&q) = ( p) v ( q) (p v q) = (p)&(q) p&q = ((p) v (q)) p v q = ((p)&(q))
Algebraic Form (+p+q) = pq (pq) = +(p)+(q) +p+q = (pq) pq = (+(p)+(q))
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Any conditional formula may be replaced by a disjunction. We call this the Conditional/Disjunction equivalence. Conditional/Disjunction (C/D): p ~ q = p v q p+q
=
 ( p) q
Another important rule oftransformation is the Law of Double Negation: Double Negation (DN): p = p
The following transformation laws are called Iteration Iteration (IT): p= p&p p= pvp
13. Rules of Inference Rules oflnference license the steps in a derivation leading to a conclusion from given premises. The following rules of inference are much used in justifying steps in a derivation. Modus Ponens (MP):
p p
~q
lq Example: if there is smoke, there is fire there is smoke so, there is fire Modus To/lens (MT): p ~ q
q I p
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Example: if there is smoke, there is fire there's no fire hence there's no smoke
Simplification (SIMPL): p&q lp Example: roses are red and violets are blue hence roses are red
Conjunction (Conj): p q lp&q Example: roses are red violets are blue hence roses are red and violets are blue
Addition (Add): p lpvq Example: Tony is home hence Tony is home or roses are red Some times an inference may be justified by simply appealing to a transformation rule. The transformation laws are then treated as rules of inference. For example, suppose we wished to show that 'not every A is B' entails 'some A is not B'. The MPL translation of 'not every A is B' is the premise of the following example. 1.  (x)(Ax => Bx)
prenuse
Modern Predicate Logic 2. 3. 4. 5.
(Ex) (Ax ::l Bx) (Ex)(Ax v Bx) (Ex)(( (Ax))&( Bx)) (Ex)(Ax&(Bx))
237
1, QI 2, C/D 3,DML 4,DN
The last line is MPL for 'some A is not B'.
14. Literal Formulas Literal formulas are simple sentences consisting ofa predicate and one or more singular subjects. The following sentences are examples ofliteral formulas: Caruso sings Plato admired Socrates x loves y xisnnullng Sally gave Tommy a cookie
Sc Aps Lxy
Rx Gstc
If a sentence is literal, so is its negation. For example, since 'Gstc' is a literal sentences, so is ' Gstc'. (Literal sentences that are positive are sometimes called 'atomic' sentences. Negative literals are not atomic.) We shall need four more rules before we get down to reckoning with the formulas ofMPL. The rules in question are called Rules ofInstantiation. A rule of instantiation permits us to replace a pronoun by an arbitrary name. For example, given the sentence 'someone is such that he is a Spani.ard and he is proud' we may decide to call that 'someone' 'Abe' and then say: 'Abe is a Spaniard and Abe is proud'. The formal rule justifying this in MPL is called 'existential instantiation' and looks like this:
Existential Instantiation (El): (Ex)( .. x.. .)
/ ... a... EI tells us that if we are given any step in a derivation that is existentially quantified we may remove its quantifier and substitute a name for the variables that it binds. Thus we may be given the premise 'some student is Pakistani', whose MPL form is '(Ex)(Sx&Px)'. If this sentence is true there must exist something (someone)call it (he or she) 'a' that is S and P. If so, 'Sa&Pa'
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is true. Again, this move eliminates the quantifier '(Ex)' and replaces the bound variable by an arbitrary name, giving us a formula that is free of quantifiers and variables. EI may be performed only once. For we have no right to assume that more than one student is Pakistani. Thus we cannot arbitrarily choose a second name, say 'Bob' and also infer 'Bob is a student and Bob is Pakistani'. Moreover, once 'a' has been introduced as a name via EI, it cannot be used in an another existential instantiation. Thus suppose we are now told that 'some student is a billionaire'. This translates as '(Ex)(Sx&Bx)'. Applying EI, we can derive 'Sb&Bb' by deciding to call the student in question 'b'. But we cannot instantiate '(Ex)(Sx&Bx)' by again using 'a' as the name for the billionaire student since that name has already been used for the student who is Pakistani. We have no grounds for assuming that they are the same person. A second rule, called 'universal instantiation', enables us to eliminate the universal quantifier by instantiation.
Universal Instantiation (UI): (x)(. .. x ...)
/ ... a... UI says that, given a universally quantified step in a derivation, we may remove its quantifier and replace the variables it binds by any name we choose. Thus, given the formula '(x)(Sx : : > Px)' as the translation of 'every Spaniard is proud', we may instantiate to 'Sa::::> Pa', since ifthe premise is true for any x, it is true for any arbitrary person, 'a'. For example, if it is true that every Spaniard is proud, then it is true that if Abe is a Spaniard, then Abe is proud. Universal instantiation is unrestricted; we can instantiate as many times as we please. For if it is true that every Spaniard is proud then it is true that if Bob is a Spaniard, then Bob is proud, and also true that if Zelda is a Spaniard then Zelda is proud, and so on for any name one cares to put in for the universally bound variable. The reverse of EI also holds. Given the premise 'Abe is a Spaniard and Abe is proud' we may deduce the conclusion 'someone is such that he is a Spaniard and he is proud', thereby replacing the name 'Abe' by pronouns. Because we move from a singular to a general sentence, the rule permitting this is called 'existential generalization':
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Existential Generalization (EG):
... a... I (Ex)( .. x .. .)
Applying EG to the premise' Adam was created' we may conclude 'something was created'. Thus: Ca I (Ex)(Cx) Finally, if we have a formula ' ... a .. .' in which 'a' appears as an arbitrary name so that ' ... b ... ' would also be true and ' ... c .. .' would also be true, etc., then we may 'generalize universally' to '(x)( ... x ... )': Universal Generalization (UG):
... a... I (x)(. .. x .. .) For example, suppose we chose 'Jupiter' as an arbitrary name to fill the gap in' ... was created'. Assume we could just as well have chosen any other name to fill that gap and that the resulting sentence would be taken as true. In that case we could have series of sentences of the form ' ... was created', all of which are assumed to be true, and we have the right to generalize to 'everything was created'. Thus, by applying UG, we have the inference 'Cj I (x)(Cx)'. As an example ofhow instantiation and generalization may figure in logical reckoning in MPL, consider how we may derive a conclusion from 'every A is B' and 'something is an A': 1. 2. 3. 4. 5. 6.
(x)(Ax => Bx) (Ex)(Ax) Aa Aa => Ba Ba (Ex)(Bx)
prenuse premise
2, EI 1, UI 3,4,MP 5,EG
We have derived the conclusion 'Something is aB'. Using 'T' for 'thing', the same conclusion may be quickly derived in TFL:
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1. A+B 2. +T+A 3.+T+B
prenuse prenuse 1+2, DDO
We offer now a summary of important rules for reckoning in MPL: 1. Rules dealing with negation: DN: p= p DML: (p&q) = p v q (p v q) = (p)&(q) C/D: (p&q)=p=>q=pvq 2. Rules of iteration: CI: p =p&p DI: p = p vp 3. Rules dealing with quantifiers:
QI: (Ex) = (x)(x) =(Ex)(Ex) = (x)(x) =(Ex)4. Rules of instantiation:
EI: (Ex)( ... x ... ) / ... a .. . UI: (x)( ... x ... ) / ... a .. . 5. Rules of generalization:
EG: ... a ... I (Ex)( ... x ... ) UG: ... a ... I (x) ( ... x ... )
15. Reckoning in MPL Consider the valid argument 'every Ape is furry, some denizens of Madagascar are apes, so, some denizens of Madagascar are furry'. To prove it is valid in TFL we might use a reductio proof, denying the conclusion and showing that conjoining the denial with the premises is inconsistent by P/Z. But a direct proof could also be given:
Modern Predicate Logic 1. A+F 2. +D+A 3. +D+F
241
premise premise 1+2, DDO
We now show how to do a direct proof in MPL: 1. (x)(Ax => Fx) 2. (Ex)(Dx&Ax) 3. Da&Aa 4. Aa=> Fa 5.Aa 6.Fa 7.Da 8. Da&Fa 9. (Ex)(Dx&Fx)
premise premise 2, EI 1, UI 3, SIMPL
4,5,MP 3, SIMPL 6, 7, Conj 8,EG
Step 9 is the conclusion 'some denizen ofMadagascar is furry'.
16. Canonical Normal Forms (CNF)
It is often useful to transform the sentences we are reckoning with into a 'standard' or 'normal' form. Any sentence ofMPL can, by transformation, be reduced to an equivalent sentence that has the following two characteristics: 1. Except for quantifiers, predicate letters and bound variables, the formula contains no signs other than ' ',' & ' and 'v'. 2. ''appears only in literal sentences. Any sentence that has these two characteristics is said to be in 'Canonical Normal Form' (CNF). If a sentence is not in CNF it is always possible to transform it into one that is. We want to learn how to transform sentences into CNF because CNF sentences are especially suited for logical reckoning in MPL. As an example of a CNF transformation we use QI, C/D, DML, and DN to tum ' (x)(Ax => Bx)' into the CNF formula '(Ex)(Ax&( Bx)' by the following series of steps. 1.  (x)(Ax => Bx) 2. (Ex) (Ax => Bx)
Given 1, Ql
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An Invitation to Formal Reasoning 3. (Ex)(Ax v Bx) 4. (Ex)(Ax&Bx) 5. (Ex)(Ax& Bx)
2,CD 3,DML 4,DN
Line 5 is a CNF formula. One technique for getting a CNF formula for an English sentence is to transcribe into a TFL 'bridge' formula and then to clear the minus signs from the bridge formula before translating it into MPL. Consider, for example, 'no boy hates every girl', whose TFL formula is '(+B 1+H 12 G2)'. If we applied TRI and TR2 to the DNF of this formula the result would be '(Ex)(Bx&(y)(Gy => Hxy))'. To get this into CNF we should have to apply QI and DML. But suppose we had first cleared the external minus sign from the bridging formula. We should then have' B 1(H12 G2)', from which we would again clear an external minus sign, giving us ' B 1+(( H 12)+G2)'. The DNF ofthis formula is 'B 1+(+G2+(H 12))', which we use as a bridge and which translates into '(x)(Bx => (Ey)(Gy& Hxy))'. Now this formula contains only literal components, but it just falls short of being in CNF since it contains the conditional sign'=>'. However, we may apply C/D to give us the CNF: (x)( Bx v (Ey)(Gy& Hxy)) The moral of this example is that the more we can simplify in TFL before we translate into MPL, the better off we are when it comes to doing logic in MPL.
17. Indirect Proofs in MPL Indirect proofs tell us whether a given argument is valid or not. But for that we need to be given a complete argument. In using an indirect method to evaluate an argument for validity, we (1) deny its conclusion, conjoining the denial to the premises, (2) get all conjuncts into CNF, (3) instantiate each conjunct, using EI or UI or both and (4) tree the instantiations and test the tree for inconsistency. An inconsistent result shows that the original argument is valid. Thus, the general method consists of denying the conclusion and then instantiating the counterclaim. This gives us a conjunction that has no quantifiers. We are then able to examine this conjunction in the usual way: by treeing it to see whether we have got a contradiction. Consider again the argument 'every A is F, some Dis A I some Dis F'. Denying the conclusion gives us the following counterclaim of three conjuncts:
Modern Predicate Logic 1. (x)(Ax ::1 Fx) 2. (Ex)(Dx&Ax) 3. (Ex)(Dx&Fx)
243
premise prenuse negation of conclusion
We now proceed by applying the rules of reckoning: 4. (x)(Dx&Fx) 5. (x)((Dx) v (Fx)) 6. (x)( Ax v Fx)
3, QI 4,DML 1, C/D
Lines 2, 5 and 6 are all in CNF. We now instantiate them: 7. Da&Aa 8. Da v Fa 9.AavFa
2, EI 5, UI 6, UI
The conjunction of 7, 8 and 9 can be treed: Da Aa
I Da
\ Fa
I Aa
\ Fa
The tree has all of its paths 'closed', revealing that 7, 8 and 9 are jointly contradictory. Since 7, 8 and 9 follow from the counterclaim, this shows that the counterclaim of the original argument is inconsistent. The argument itself is therefore valid. We could save ourselves several steps if we clear the minus from our TFL formulas before using them as the bridge to MPL. 1. A+F 2. +D+A 3. (+D+F) (== 0+(F))
=> => =>
(x)(Ax ::1 Fx) (Ex)(Dx&Ax) (x)(Dx ::1  Fx)
Applying C/D to 1 and 3 gives us three CNF formulas ready for instantiation and treeing:
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An Invitation to Formal Reasoning 1.* (x)(Ax v Fx) 2. (Ex)(Dx&Ax) 3.* (x)(Dx v Fx)
A word of advice is in order. It is better (and is often necessary) to use EI before using UI. This is due to the restriction that applies to EI but not to UI. Instantiating in the order 2, 1*, 3* gives us the same closed tree as before. The general technique for indirectly proving the validity of an argument in MPL is this: 1. Form the counterclaim of the argument in TFL. Clear all external minus signs from the formulas and translate into MPL. 2. Transform all sentences to CNF. 3. Instantiatebeginning with EI and then going on to UI. 4. Tree the resulting sentences. 5. Now check to see whether the tree closes. If it does close, the argument is valid. An invalid argument will not give a closed tree.
18. Relational MPL Arguments We now apply the technique to some more complicated arguments. Consider how we could show that the following argument is valid. every colt is a horse I every owner of a colt is an owner of a horse Denying the conclusion we have
1. C2+H2 2.  ( (0 12+C2)+(012+H2))
prenuse negation of conclusion
1 and 2 jointly form the counterclaim of the argument. Before proceeding to translate into MPL clear the external minus sign of 2: 2, peq
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245
A further simplification of 3 is available:
3,PEQ We now put 4 into DNF: 4,DNF We now translate 5 and 1. 6. (Ex)(+C 2+0x2)&( H 2+0x2) 7. (Ex)((Ey)(Cy&Oxy)&(y)(Hy => (Oxy))) 8. (y)(Cy => Hy)
5, TR1 6, TR1, TR2 1, TR2
Using C/D we get 8 and 7 into CNF: 9. (y)(Cy v Hy) 10. (Ex)((Ey)(Cy&Oxy)&(y)(Hyv Oxy))
8, C/D 7,C/D
Instantiating 10 gives us
11. Cb&Oab&(Hb v Oab)
10, EI,Ul
Instantiating 9 gives us 9, UI
12. Cb vHb Treeing 11 and 12 we get: Cb Oab
I Hb
I Cb
\ Oab
\ Hb
The tree for the instantiations closes; we have a contradiction. To show that 'every girl is loved by some boy' follows from 'some boy loves every girl' we counterclaim and get:
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An Invitation to Formal Reasoning 1. +B 1+(L 12 G2) 2.  ( G 2+(L 12+B 1)
pretruse negation of conclusion
We then simplify 2 by driving in the minus signs:
2,PEQ
3. +G2+(( L 12  B 1)) 4. +B 1+(G2+Ln)) 5. +G2+( B 1+( L 12))
1,DNF 3,DNF
4 and 5 are our bridging formulas. We now translate into MPL: 6. 7. 8. 9.
4, TR1, TR2
(Ex)(Bx&(y)(Gy;:, Lxy) (Ey)(Gy&(x)(Bx;:,  Lxy) (Ex)(Bx&(y)(Gy v Lxy)) (Ey)(Gy)&(x)( Bx v  Lxy))
5, TR1, TR2 6, C/D 7, C/D
8 and 9 are now in CNF so we instantiate them: 8, EI, UI 9, EI, UI
10. Ba&Gb&(Gb v Lab) 11. Gb&( Ba v Lab) We now tree 10 and 11: Ba Gb
I
\
Gb
Lab
I Ba
\ Lab
All paths close so we have derived a contradiction from the counterclaim. Here is a more complicated argument: anyone who ~nvies anyone who Qwns a §.lave is immoral every (captured) Akkadian was a slave I no Akkadian is owned by anyone that any moral person envies We will use a direct TFL proof to derive the conclusion and then follow this by an indirect MPL proof of validity.
Modern Predicate Logic 1.  (E 12+(0 23 +S 3))+( M 1) 2. A3+S3 I (+A3+(0 23+(+M 1+E12))) 3.  M~(E 12+(023+S3)) 4.  M 1+( E 12) (0 23 +S3) 5. (023 +S 3)+(E 12)M 1 6. (((E 12)M 1)))+(023 )S3 7. S 3+(0 23 )(EI2+MI) 8.  A3+( (0 23 ) (E 12+M1) 9. (+A3+0 23 +(E 12+M 1)
247
prenuse premise conclusion 1, peq 3, peq 4, LLC (twice), LLA 5, peq 6, LLC (twice), LLA 2+7, DDO 8, peq
Line 9 is the conclusion we sought to derive. In the indirect MPL proof we shall use steps 1 and 2 and the denial of the conclusion as bridges to the MPL formulas. 1* (+(+S 3+0 23 )+E 12))+(M 1) 1, DNF 1** (x)((Ey)((Ez)(Mx&Exy)&Oyz))) 1*, TR2, TR1 2, TR2 2* (z)(Az > Sz) 3* +A3+(0 23 +(+M 1+Ed negation of conclusion 3*, DNF 3** +A3+(+(+M 1+Ed+023 )) 3*** (Ez)(Az&(Ex)((Ey)(Mx&Exy)&(Oyz))) 3**, TR1 (thrice) We now tree instantiations of3***, 1**and 2*.
4*
Ac Ma Eab Obc
I  Sc
I Ac
I \ \  Obc  Eab  Ma
3***, EI
1**, UI
\ Sc
2*, UI
The tree closes, which shows that denying the conclusion entails a contradiction. To show that 'some Greek shaves himself follows from 'every barber shaves himself and 'some barber is a Greek' we deny the conclusion and
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An Invitation to Formal Reasoning
conjoin it with the premises:
1. no Greek shaves himself
negation of conclusion premise prenuse
2. every barber shaves himself 3. some barber is a Greek
To show that these three statements are jointly inconsistent we transcribe them, simplify, translate them, get them into CNF and then instantiate. Having simplified the transcriptions we arrive at:
1. G'1+(S12)G'2 2.  B' I+SI2+B' 2 3. +BI+GI
negation of conclusion premise premise
As a first step we apply IPE to 1 and 2: 1, IPE 2, IPE
4. GI+(SII) 5.  BI+SII
After transformation, instantiation and treeing, we have: Ga Ba
I
\
Ba
Saa
I Ga
\ saa
It is instructive to compare how one could do this example in TFL:
1. 2. 3. 4.
 B\+S 12  B' 2 +BI+GI  BI+SII +GI+S11
prenuse prenuse 1, IPE 2+3, DDO
[Note that an English language sentence like 'some boy envies every owner of a dog' has no pronouns in it. MPL 'translates' it with three pronouns [(Ex)(Bx&(y)((Ez)Dz&Oxy => Exy))], but TFL transcribes it as a proterm free formula [ B+(E(O+D)]. More often than not, even when a sentence
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249
explicitly contains pronouns, it can be rephrased and transcribed in a pronounfree way. For example, 'if any A is B, then it is C' can be rephrased as 'any A that is B is C' and transcribed without proterms as '<+A+B>+C'. Indeed, most logical reckoning can be done without the pronouns (bound variables) that are an indispensable feature of MPL. This gives TFL a decided advantage over MPL in naturalness and ease of reckoning. Having said this, it should be pointed out that reflexive pronouns are special; in transcribing a sentence like 'some barber shaves himself TFL needs proterms (prior to application of IPE). Though we prefer TFL to MPL, it must be conceded that in the area oflogical reckoning that involves reflexive pronouns, TFL has no special advantage over MPL.]
******************************************************************** Exercises: I. Show that 'Aristotle is not wiser than himself follows from 'no one is wiser than Aristotle'. (hint: use UI). 2. Using MPL, derive 'every senator admires a fool' from 'every senator admires himself and 'if any(one) is a senator, then he is a fool'.
******************************************************************* 19. Identity Arguments in MPL
As we have seen above (section 10 of this Chapter), statements such as 'Mark Twain is Sam Clemens' and 'the square root of twentyfive is five' are often called identities. An identity statement in TFL is simply a statement both of whose terms are uniquely denoting terms. Again, an identity is a statement of the form 'some X* is Y*'. Arguments that contain an identity premise or conclusion require special treatment in MPL. The reason for this is that in MPL a sentence like 'Twain is Clemens' is construed as a dyadic statement whose 'is' is understood to mean 'is identical with' a two place relational term that ties 'Twain' and 'Clemens' as two subjects in the way 'loves' ties 'Paris' and 'Helen' in 'Paris loves Helen'. Having introduced identity as a relation, MPL must also introduce special laws governing it. The following 'Laws of Identity' are often used when dealing with identity arguments.
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An Invitation to Formal Reasoning
Laws ofldentity: LI.l ifx=y, theny=x (I'he law ofsymmetry) LI.2 ifx=y and y=z, then x=z (I'he law oftransitivity) Ll.3 x=x (I'he law ofrejlexivity) Ll. 4 ifPx and x=y, then Py (Leibniz's Law)
LI.l asserts that identity is a symmetrical relation. According to LI.l, if Twain is Clemens, then Clemens is Twain. LI.2 asserts that identity is a transitive relation, so that if Clemens is Twain and Twain is the author ofLife on the Mississippi, then Clemens is the author of Life on the Mississippi. The third law asserts that identity is reflexive. According to LI.3, we can always assert that 'Twain is Twain', 'Socrates is Socrates' and so on. LI.4 allows us to replace 'x' by 'y' once we know that x=y. Thus, given that Twain is a humorist, and given that Twain is Clemens, we can conclude that Clemens· is a humorist. This law is often called Leibniz's Law. Thus, in MPL one or more special laws governing the identity relation must be added as 'axioms of identity'. In TFL identity statements are not relational and the analogous laws can be proved by ordinary means. For example, according to Leibniz's Law, 'Y* is P' follows from 'X* is P' and 'X* is Y*'. Since identities are singular statements, they have wild quantity. By assigning universal quantity to 'X* is P' and particular quantity to 'X* is Y' we can prove the fourth law syllogistically. Thus, an argument like the Twain example comes out as a valid syllogism: +T*+C* T*+H /+C*+H
Twain is Clemens Twain is a humorist I Clemens is a humorist
We now offer a pair of identity arguments to illustrate the way that MPL deals with them by applying the laws of identity. (i) Show that 'Mark is an author' follows from the premises 'Clemens is Twain', 'Twain is an author' and 'Clemens is Mark'. 1. c=t 2. At 3. c=m
4. m=c
premise premtse premise 3, symmetry
Modern Predicate Logic
5. m=t 6. t=m 7. Am
251
4, 1, transitivity 5, symmetry 2, 6, L.L. (Leibniz's Law)
(ii) Show that 'Twain is Clemens' follows from 'Twain is funnier than anyone else' and 'Twain isn't funnier than Clemens'. One way is to deny the conclusion and show that this gives us a contradiction:
1. (x)((x=t) 2. Ftc
:::>
Ftx)
3. (t=c) 4. (x)(x=t) v Ftx)
premise prenuse negation of conclusion 1, C/D,DN
4, U.I.
5. c=t v Ftc 6. t=c v Ftc
5, symmetry
The tree for 2,3, and 6 is: Ftc (t=c) I \ t=c Ftc The tree closes. Denying the conclusion leads to contradiction so the argument is valid.
******************************************************************** Exercises: 1. Show that 'the Evening Star is Hesperus' follows from 'Hesperus is Phosphorus', 'Phosphorus is the Morning Star' and 'the Evening Star is the Morning Star'. (in MPL and then TFL) 2. Given the premises 'only Clemens knows' and 'Twain knows', prove that Twain is Clemens (in MPL and then TFL).
*******************************************************************
Rules, Laws and Principles
TFL Logical Law of Commutation (LLC): +(+X+Y)
= +(+Y+X)
Law ofObversion (Obv): (+/X+/ Y) = +(/+X/+Y) Principle of Equivalence (PEQ): Two statements are logically equivalent if and only if they are covalent and equal. Logical Law of Association (LLA): +X+<+Y+Z> = +<+X+Y>+Z Law of Simplification (LS): Any wellformed dyad may be detached from an expression whose terms are connected by a binary commutative and functor). associative functor (viz., the'+ ... +' Internal
Pronoun Elimination (IPE): If P' m... P' n is an internal pronominalization, remove P 'n and replace any remaining occurrence of n by m·
The REGAL Principle (REGAL): A syllogism is valid if and only if its mood is regular and it 'adds up'. Principle of Validity (PV): An argument, A, is valid if and only if its counterclaim, C(A), is inconsistent. Principle of Transitivity: If being B characterizes every A and being C characterizes every B, then being C characterizes every A. P/Z Criteria: A syllogistic conjunction is inconsistent if and only if it is itself canonically inconsistent or else is equivalent to a canonically inconsistent conjunction.
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Laws of Identity: Law of Symmetry: A* is B* I B* is A* Law of Transitivity: A* is B* and B* is C* I A* is C* Law ofReflexivity: A* is A* Dictum de Omni (DDO): (Aristotelian version): Whatever characterizes every X characterizes any X. (Generalized version): What is true of every X is true of whatever is an X. (Host/Donor version): E(M), E*( M) I E(E*) Reassignment Rule (R A): The numerical index on a universally distributed occurrence of a term may be replaced by any other numeral. ... Ti··· I ... Ti··· (R A2): The superscript of any distributed proterm may be changed to form another pronominalization . ... Ti ... I ... Ti ... Pronominalization (Pl): some X is Y I so it (the X in question) is Y +X'+Y I +X'+Y (Pla): +X+Y I +X'+Y' Rule ofWild Quantity (WQ), (P2): +X'+Y I X'+Y Pronominal Expansions: some A is B = something is an A and it is a B +A+B = +[+T'+A]+[+T'+B] some A is B = an A exists (is a thing) and it is a B +A+B = +[+A'+T]+[+A'+B] every A is B = if any thing is an A, it is a B A+B = [+T'+A]+[+T'+B] every A is B = if an A exists, it is a B A+B = [+A'+T]+[+A'+B]
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255
STATEMENT LOGIC principle of equivalence (peq): Two compound statements are equivalent if (but not 'only if) they are covalent and equal. Modus Ponens (MP): Given any premise ofthe form E(ifp), where p has negative occurrence, and another premise M(p ), where p has positive occurrence, the conclusion M(E) follows. Conjunction (Conj): From any two premises the conjunction ofthose premises follows. Simplification (Simpl): From any conjunction either conjunct follows. Conjunctive Iteration (CI): Any statement is equivalent to its conjunction with itself. Disjunctive Iteration (DI): Any statement is equivalent to its disjunction with itself. Disjunctive Addition (DA): From any premise the disjunction of that premise and any other statement follows. Law of 'And/Or' Distribution (AOD): (p or p) and (q orr)= (p and q) or (p and r) +[ p p]+[ q r] =  [+p+q] [+p+r] Law of 'Or/And' Distribution (OAD): (p and p) or (q and r) = (p or q) and (p orr) [+p+p][+q+r] = +[pq]+[pr]
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MPL Translation Rules (TFL => MPL): TRI: some A is B > +A1+B 1 => (Ex)(Ax&Bx) every A is B >  A1+B 1 => (x)(Ax ;:, Bx) TR2: TR/and: p and q => +p+q > p&q TR/if: ifp then q > p+q > p;:, q TRior: porq > pq > pvq TR3: S* is P > Ps S* is R toP* > Rsp xis P > Px x is R to y > Rxy TR4: <+A+B>x > x => Ax&Bx <AB> >x >Ax vBx Law of Quantifier Interchange (QI): (x)( ... x ... ) = (Ex)( ... x ... ) (Ex)(... x ... ) = (x)( ... x ... ) (x)( ... x ... ) = (Ex)(... x ... ) (Ex)( ... x ... ) = (x)( ... x ... ) DeMorgan's Laws (DML): (p&q) = p v q (p v q) = p&q p&q = (p v q) p v q = (p&q) Conditional/Disjunction (C/D): p;:, q Double Negation (DN): p =   p Modus Ponens (MP): p ;:, q, p I q Modus Tolens (MT): p;:, q, q I p Simplification (Simpl): p&q I p Conjunction (Conj): p, q I p&q
(+p+q) = pq (pq) = +(p)+(q) +p+q = (pq) pq = (+(p)+(q))
= p v q
p+q = (p)q
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257
Addition (Add): p I p v q Existential Instantiation (EI): (Ex)( ... x ... ) I ...a ... Universal Instantiation (UI): (x)( ... x ... ) I ... a ... Existential Generalization (EG): ... a ... I (Ex)( ... x ... ) Universal Generalization (UG): given that 'a' is an arbitrary name so that ' ... b .. .', ' ... c .. .', etc. would also be true: ... a ... I (x)( ... x ... ) Laws of !~entity: Law of Symmetry (LI.l ): if x=y then y=x Law ofTransitivity (L1.2): ifx=y and y=z, the x=z Law of Reflexivity (LI.3): x=x Leibniz's Law (L1.4): ifPx and x=y, then Py
ANote on Further Reading
Having mastered An Invitation to Formal Reasoning, the reader may wish to pursue one or more of the topics discussed or alluded to here. Term Functor Logic (TFL) was given its present formulation by Fred Sommers in a series of articles published in philosophical journals. You might want to look at 'The Calculus of Terms', reprinted in The New Syllogistic, George Englebretsen, editor, New York, Peter Lang Publ., 1987. Sommers provided an extensive introduction to and philosophical foundation for TFL in his The Logic of Natura/Language, Oxford, ClarendonPress, 1982. David Kelley's textbook, The Art ofReasoning, 2nd expanded edition, New York, W.W. Norton & Co., 1994, includes a chapter presenting the main elements ofTFL. Englebretsen' s Something to Reckon With, Ottawa, University of Ottawa Press, 1996, offers a summary account of term logic as well as an extensive survey of the historical antecedents of Sommers' logic. The history of all logic, not just TFL, begins of course with Aristotle. The bold reader may wish to consult one of the many editions and translations of his Prior Analytics, the book in which Aristotle invented syllogistic logic. A very comprehensive history of logic is William and Martha Kneale's The Development ofLogic, Oxford, Clarendon Press, 1962. The reader interested in exploring further Leibniz's contributions to logic might examine some ofhis essays in Leibniz: Logical Papers, G.H.R. Parkinson, editor and translator, Oxford, Clarendon Press, 1966. While Aristotle was the first logician, and Leibniz was perhaps the greatest traditional logician, Lewis Carroll (pen name of the mathematician Charles L. Dodgson) was almost the last but certainly the most entertainingtraditional logician. His books, Symbolic Logic and The Game of Logic, first published a hundred years ago, have served as a source of delightful but challenging logic problems for generations of teachers and writers, including ourselves. Both books have been printed in a single volume by Dover Publ., Inc., New York, 1958. Modem Predicate Logic (MPL) has had a very brief (just over a century), but very rich and eventful, history. There are a very large number
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of textbooks available now presenting the standard version of that logic. I. Copi's Introduction to Logic, 9th edition, New York, Macmillian, 1994, and S. Barker's Elements ofLogic, 5th edition, New York, McGrawHill, 1989, are two of the more popular of these. W.V.O. Quine's Philosophy ofLogic, Englewood Cliffs, N.J., Prentice Hall, 1970, is an excellent, brief account of the philosophy behind MPL. The system of diagrams used here was invented by the mathematician John Venn in the nineteenth century. An interesting survey of systems of logic diagrams is found in Martin Gardner's Logic Machines and Diagrams, Chicago, University of Chicago Press, 1982. A system of diagrams quite different from Venn's, but built with Sommers' version ofTFL in mind, has been developed by Englebretsen. A full presentation of it is found in his Line Diagrams for Logic: Drawing Conclusion, Lewiston, N.Y., Mellen, 1998.