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There are several usages of the formal language L. In each usage, we assume a certain range of discourse and a certain complete interpretation of the constants and the free variables. Thus, each usage corresponds to a certain system. We regard some of the usages of L as the standard usages. There may be several standard usages of L. Several assignments of a denotation to a constant may, for instance, be consistent with standard usage because of the ambiguity of the constant. We shall note also that L's constants may be vague in the standard usage: we are not necessarily required to know the denotations of the constants. Now, let
is a language. We shall make the assumption that every system in
. In the sequel, let 'V*' be a variable for standard primary valuations. We say that a name < c, V* > of < L,
is vague in r if there is no effective method available to decide for each member x of r whether or not x = V*(r, c). We say that an n-place predicate , and non-naturalistic otherwise. K.: Yes. Ph. : It follows from assumptions I and II of the preceding section that there are non-naturalistic deontic statements. But we may raise the problem whether or not there are non-naturalistic deontic propositions of < L, . The answer to this problem depends on the choice of . In any event, if we turn to standard English and raise the analogous problem, then a negative answer would certainly be unwarranted - it would be, we may argue, an expression of the so called
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Clearly, one source of vagueness in < L,tp> is an incomplete specification of tp. Another source is an incomplete specification of the relation RM occurring in the valuation of modal statements. We shall not, however , regard a complete specification of tp and RM as a desideratum. On the contrary, we shall think of tp and RM as if they were specified in such a way that the names, predicates and judgements of < L, tp > acquire the vagueness they have in the standard usages of L - so to speak. (There are also other and deeper sources of vagueness, but we may leave these without notice in this connection.) Some further definitions: A sentence S holds, a statement A is true (false), and an imperative sentence C is correct (incorrect) if S holds, A is true (false) and C is correct (incorrect) in some standard system < r*, v* > . S is analytic if some judgement < S,V* > is analytic. A judgement < S, v* > holds , a proposition
v,».
(1) (2)
For each Ll and each
The relation R is unique : /ffor each r and
=
1),
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then for each r' and r R(r',r) == (B)(V)(T(r, V,MB) = 1 :::> T(r', V,B) = 1).
If M is Ought and if R is ROught, then the first result provides a justification of the valuation for the deontic statements. The second result provides a kind of explanation of Rought. Perhaps we may paraphrase it as follows : Rought is the relation which holds between any two universes r' and r such that every proposition that ought to be true in r is true in r'. We shall now make three assumptions concerning the relation Reught: I II III
(r)(Er ')Raught(r " r) (Er) - Rought(r, r)
(Er')«Er)Rought(r',r) & (Er) - Rought(r',r»
Assumption I is equivalent with the assumption that (OughtA :::> RightA) is always valid. Assumption III is equivalent with the assumption that there are synthetic deontic propositions of the form < OughtA, V> . It is also clear that III is an expression of a kind of moral relativism. I shall now list a few valid sentences. The validity of these sentences does not depend on the assumptions just made . 1 2 3 4 5 6 7 8 9 10 11
Ought A, where A is valid (Ought(A:::> B) :::> (Ought A :::> Ought B» (Ought(A & B) == (Ought A & Ought B» (Right(A :::> B) == (Ought A :::> Right B» (Right(A V B) == (Right A V Right B» (Ought UxFx :::> Ought FD (Ought UxFx :::> Ux Ought Fx) (Ex Ought Fx :::> Ought ExFx) (Right UxFx :::> Ux Right Fx) (Ex Right Fx :::> Right ExFx) (Ought A == fA)
We note that the converses of 7 and 10 are not always valid. This fact depends primarily on the properties of quantification. Cf. the note at the end of section 3. We note also that «a = b) :::> (Ought
Fa
:::> Ought
Fg»
is not always valid . An assumption to the contrary would lead to a paradox of the same kind as the well-known Morning Star paradox.
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We may, perhaps, deny the truth of assumption III. A denial of III is equivalent with the assumption that 'Ought' is definable thus: Ought A
==df N(Q :)
A)
where N is the notion of analytic necessity : T(r, V,NB) = 1
==
(r')(T(r', V,B) = 1)
and Q is a propositional constant with a fixed valuation: T(r', V,Q) = 1
== (r)Roughlr',r)
(We may think of Q as a constant stating what morality prescribes.) If we accept this definition of 'Ought', we may obtain a deontic logic by extending a logic for analytic necessity with Q as a new primitive symbol and with the statement ( - NQ& - N - Q) as a new postulate. Such a logic was given in a paper on deontic and imperative logic (including a theory of unavoidability), which I wrote in 1950 and submitted in partial fulfillment for the lic.phil. degree at the University of Stockholm. Almost the same kind of deontic logic has been given in A.R. Anderson, The Formal Analysis of Normative Systems (New Haven 1956). A summary of some main ideas in Anderson's paper may be found in A.R. Anderson and O.K. Moore , "The Formal Analysis of Normative Concepts" (American Sociological Review 22 (1957), 9-17) and in A.N. Prior, Time and Modality (Oxford 1957), Appendix D. I am now inclined to reject this definition of Ought; my main reason is the fact that some deontic propositions < OughtA, V> seem to be synthetic. But I also think that the vagueness of such deontic propositions excludes the possibility of making a definite decision in this case. There is another assumption which perhaps might be adopted: (r)(r')(r")(Rought(r',r) :) (Rought(r",r')
== Rought(r ",r)))
If we assume assumption I, this assumption is equivalent with the assumption that Ought A :) Ought Ought A and Right A :) Ought Right A are always valid. It implies that Ought(Ought A :) A) is always valid. Thus, it may be regarded as an expression of what might be called moralism.
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7. A Dialogue The complete moral philosopher : Excuse me for interrupting you, Mr. Kanger, but would you admit a short interview before you proceed? Kanger: Yes. Ph. : According to a wellknown theory in moral philosophy, known as the emotive theory, deontic propositions are neither true nor false. Now, I understand, you have the opposite view. K.: Yes. Ph.: Of course you and the adherents of this theory may have in mind two different notions of truth. But your notion of truth seems to be in agreement with scientific semantics and I am sure that the adherents of the emotive theory would adopt it if they were met with the problem . K.: Yes. Ph.: Now, clearly, deontic propositions with the valuation they got in Section 3 must be either true or false. So, if the adherents of the emotive theory wish to sustain their standpoint they have to reject the valuation clause for Ought or the equivalent thesis that Oughtzs always entails < OughtA, V> when a entails
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imperative, and this fact is in full agreement with everything else in this paper . K.: Yes. Ph.: Now let me return to the notion of truth. Perhaps our agreement on this point was a little rash. Could we not restrict the application of the truth predicate to, say, non-deontic statements and call the formerly true deontic statements correct instead? And couldn't we do this and still be in agreement with scientific semantics? K. : Yes. Ph.: Do you mean that the choice of the range of applicability of the truth predicate is, to some degree at least, conventional? K.: Yes. Ph.: So the adherents of the emotive theory have a chance to be right after all by a terminological trick . K. : Yes. Ph.: I now turn to a new problem. There is a wellknown distinction between so called natural properties and non-natural properties. Some authorities believe that value is a non-natural property, while others disagree. In this paper, the distinction is difficult to maintain because of your tendency to do away with all kinds of spurious entities. The predicates do not refer to properties in the sense we may have in mind in this connection, but to classes of individuals or to classes of ordered sets of individuals. And I cannot see how one class can be less natural than another. This fact does not, of course, mean that the distinction between naturalism and nonnaturalism cannot be maintained at all. We may perfectly well distinguish between naturalistic and non-naturalistic statements and propositions. Thus , we say that A is naturalistic if A or - A is logically equivalent with a sequence of ordinary statements, and non-naturalistic otherwise . We say that
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naturalistic fallacy. But the naturalist position is easier to defend on the meta-level, so to speak. For each deontic statement A and each < r, V> there is a naturalistic proposition in the meta-language, which states a necessary and sufficient condition for the truth of A in < r, V> . K.: Yes. Ph.: There is another problem which should not be confused with the preceding one: Is there a statement OughtA that is a logical consequence of a contingent ordinary statement and that also has a contingent ordinary statement as a logical consequence? An affirmative answer to this question would mean that we can draw ethical conclusions from non-ethical premisses in a non-trivial sense. K .: Yes. Ph.: According to assumption I of the preceding section, there are contingent statements of the form OughtA; but, by saying this, we do not provide a strict answer to our questions. Perhaps we should leave it undecided and allege the vagueness of 'Ought' as an excuse. K.: Yes. Ph.: I have a final problem. The thesis equivalent with the valuation for 'Ought' and given in the preceding section seems to me to be indubitable. Hence we have to accept the valuation. Furthermore, (OughtA ::> RightA) is clearly always valid, and hence, we have to accept assumption L But this assumption implies, roughly speaking, that there is a universe r' which is a moral standard for our universe r*. K. : Yes. Ph. : But what is this universe, if I may inquire? Heaven? Or do I have to review my thoughts on this ultimate matter once more? K. : Yes. Ph.: Well, good-day. We must forgive our complete philosopher for his incomplete references. The emotive theory originates with A. Hagerstrom, Om moraliska forestallningars sanning (Stockholm 1911). A later, but independent, expression of the theory is given in C.K. Ogden and LA. Richards, The Meaning of Meaning (London 1923), p. 125. Still later versions of the theory may be found, for instance, in B. Russell, Religion and Science (New York 1935), Chapter 9; A.I. Ayer, Language, Truth, and Logic (London 1936), Chapter 6; L Hedenius, Om rdtt och moral (Stockholm 1941); C. Stevenson, Ethics and Language (New Haven 1944); A.J. Ayer, "On the Analysis of Moral Judgments" (Horizon 20 (1949» ; and R.M. Hare, The Language of Morals (Oxford 1952). The importance of the emotive theory lies particularly in its
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emphasis on the phenomenon of emotive meaning. Thus, the emotive theory cannot be properly characterized as the theory which denies truth and falsehood to ethical judgments. The phenomenon of emotive meaning cannot be analyzed unless we consider the concrete instances of expressions in the context of human communication, but that is not the concern of this paper. The prominence of the naturalistic fallacy originates with G.E. Moore, Principia Ethica (Cambridge 1903). The idea that we cannot draw ethical conclusions from factual premisses goes back to David Hume, A Treatise of Human Nature, Book 3 (1740), Part 1, Section 1. The non-naturalist ethics and moral philosophy has been developed in G.E. Moore, Principia Ethica (Cambridge 1903) and Ethics (London 1912), and in D. Ross, The Right and the Good (Oxford 1930) and Foundations of Ethics (Oxford 1939) (to mention some of the most important contributions only). Among recent naturalistic works we may note R.B. Perry, General Theory of Value (New York 1926) and B. Russell , Human Society in Ethics and Politics (London 1954). The naturalist - non-naturalist controversy and the controversy about the emotive theory have given rise to many studies . We may mention particularly G .E . Moore, "A Reply to My Critics" (in The Philosophy ofG.E. Moore (ed. by P.A. Schilpp), Evanston and Chicago 1942, pp. 535-677). Other studies may be found in W. Sellars and J. Hospers, Readings in Ethical Theory (New York 1952).
STIG KANGER AND HELLE KANGER
RIGHTS AND PARLIAMENTARISM
INTRODUCTION
It is almost a truism that the idea of having a right is vague and ambiguous, and that it can be approached from many angles. It is also clear that good explications of this idea are needed in many fields, but one can hardly say that the attempts to provide them have been very successful. The best attempts, so far, at an explication or analysis of the notion of a right are found in jurisprudence. Hohfeld's contribution should especially be mentioned in this connection. The object of the first part of this essay is to give an analysis of the concept of a right which, in certain respects, is a further development of Hohfeld's distinctions. 1 The analysis of the notion of a right appears to have some applications in political science. The object of the second part of the essay is to provide an example of such an application. First, the notion of a position structure in government is introduced. Roughly speaking, the position structure is the system of rights to appoint and to dismiss members of the government and to dissolve the parliament which parties like the head of state, the parliament, and the government have towards each other. There is a large number of logically possible position structures, but only a few of them are politically feasible. These are characterized by certain political axioms. Among the feasible position structures, we shall finally distinguish those that are characteristic for parliamentarism . PART I. THE CONCEPT OF A RIGHT
1. The Simple Types of Rights Consider the following two examples: X has a right to have back what he has lent to Y. X has a right to publish in Sweden a manuscript he has written.
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G. Holmstrbm-Hintikka, S. Lindstrom and R. Sliwinski (eds.), Collected Papers ofStig Kanger with Essays on his Ufe and Work, Vol. I. 120-145. © 200 1 All Rights Reserved. Printed by Kluwer Academic Publishers. the Netherlands. Originally published in Theoria 32 (1966),85-115.
RIGHTS AND PARLIAMENTARISM
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The concept of right which appears in these two examples can be made precise in at least two ways: (1) One generally regards a right as a relation which a party X has versus a party Y, and which concerns a given state of affairs between X and Y. The examples can be made more precise so that the relational character of the right appears more distinctly. Thereby, they get a somewhat schematic and artificial formulation: X has versus Y a right to the effect that X receive from Y what X has lent to Y. X has versus the Swedish state a right that a manuscript written by X be published in Sweden. (2) Rights can be of different types. X's right versus Y appears to be of the type 'claim,' and X's right versus the state can perhaps be said to be of the type 'power.' The examples can be made more precise so that the types of rights under consideration are indicated: X has versus Y a claim to the effect that X receive from Y what X has lent to Y. X has versus the Swedish state a power to the effect that a manuscript written by X be published in Sweden. We shall now distinguish eight types of rights, which we shall call the
simple types of rights: (a) (b) (c) (d) (a') (b') (c') (d')
claim freedom power immunity counterclaim counterfreedom counterpower counterimmunity
The types (a') to (d') have a close connection with the types (a) to (d), respectively . To say, e.g ., that X has versus Y a counterclaim which concerns a state of affairs S between X and Y is (by definition) the same as to say that X has versus Y a claim which concerns the opposite to S between X and Y. We have thus the following four synonym pairs (where S(X,Y) stands for the state of affairs S between X and Y):
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X has versus Y a counterclaim to the effect that S(X,Y) { X has versus Y a claim to the effect that not-S(X,Y) versus Y a counterfreedom to the effect that S(X,Y) { XX has has versus Y a freedom to the effect that not-S(X,Y) versus Y a counterpower to the effect that S(X,Y) { XX has has versus Y a power to the effect that not-S(X,Y) versus Y a counterimmunity to the effect that S(X,Y) { XX has has versus Y an immunity to the effect that not-S(X, Y) 2. An Explication of the Simple Types of Rights We shall now give an interpretation of the following four rights-propositions: (la) (lb) (lc) (ld)
X X X X
has has has has
versus versus versus versus
Y Y Y Y
a claim to the effect that S(X,Y) a freedom to the effect that S(X,Y) a power to the effect that S(X,Y) an immunity to the effect that S(X,Y)
The interpretation we intend to set forth yields an explication of the simple types (a) to (d) of rights, and thereby also of the types (a') to (d') . It will be formulated in a semiformalized language and it has the advantage that the differences between the simple types of rights are carried over into logical differences which can be expressed by the positions of the symbol, "not, " of negation and the variables X and Y. The interpretation of (la) through (ld) is as follows: (2a) (2b) (2c) (2d)
It shall be that Y sees to it that S(X,Y) Not: it shall be that X sees to it that not-S(X,Y) Not: it shall be that not: X sees to it that S(X,Y) It shall be that not: Y sees to it that not-S(X,Yf
Instead of the expression: Not: it shall be that not: ... we may use the synonymous expression: It may be that ... With this interpretation, the examples of § I become: It shall be that Y sees to it that X receives from Y what X has lent to Y.
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It may be that X sees to it that a manuscript written by X be published in Sweden. The concepts 'shall' and 'seeing to it' , which are used in the interpretation, are vague and admit different specifications in different contexts . The question of how they are to be made precise, however, we may leave open here . We shall assume only that they are interpreted in a reasonable way, and that they satisfy certain logical principles . Let F and G be arbitrary propositions or conditions. Let us writeshall-F in place of the longer expression : it shall be that F. Let X be an arbitrarily chosen party. Let the arrow ~ denote the relation of logical consequence." Among the logical principles which the concepts 'shall' and 'seeing to it' are assumed to satisfy, we then have the following five: I. If F ~ G, then shall-F ~ shall-G II. (Shall-F and shall-G) ~ shall-(F and G) III. Shall-F ~ not shall-mot-F) IV. If F ~ G and G ~ F, then X sees to it that F it that G V. X sees to it that F ~ F
~
X sees to
Finally, our notion of a state of affairs needs some comment. By a state of affairs we shall, in this essay, always mean a relation between parties . If X and Yare parties, S(X,Y) means that the party X stands in the relation S to the party Y. S may, for instance, be specified as the relation between any two parties PI and P2 such that PI receives from P2 what PI has lent to P2 . Then, of course, S(X,Y) means: X receives from Y what X has lent to Y. We shall note that X or Y need not always occur in S(X,Y) when S is specified . Thus - to allude to a specification with which we shall be concerned subsequently - S(X,Y) may become: X resigns as prime minister. Here Y does not occur .
3. Some Relations among the Simple Types of Rights As a direct consequence of the logical principles I - V and the explication of the simple types of rights, we get the synonymity of certain rights-propositions. Thus we get the following four pairs of synonyms: X has versus Y a claim to the effect that S(X,Y) { Not: Y has versus X a freedom to the effect that not-S(X,Y) X has versus Y a freedom to the effect that S(X, Y) { Not: Y has versus X a claim to the effect that not-S(X,Y)
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STIG KANGER AND HELLE KANGER
X has versus Y a power to the effect that S(X,Y) { Not: Y has versus X an immunity to the effect that not-S(X,Y) X has versus Y an immunity to the effect that S(X,Y) { Not: Y has versus X a power to the effect that not-S(X,Y) Another consequence of the principles I - V and the explication is that certain rights-propositions of the kinds (la)-(ld) are logical consequences of others of these kinds. In order to exhibit these relationships of logical strength in a simple diagram, we introduce some abbreviations. We shall write : CI(X, Y, S) Fr(X, Y, S) Po(X, Y, S) Im(X, Y, S) CI(Y, X, S) Fr(Y, X, S) Po(Y, X, S) Im(Y, X, S)
for for for for for for for for
X has versus Y a claim to the effect that S(X,Y) X has versus Y a freedom to the effect that S(X,Y) X has versus Y a power to the effect that S(X,Y) X has versus Y an immunity to the effect that S(X,Y) Y has versus X a claim to the effect that S(X,Y) Y has versus X a freedom to the effect that S(X,Y) Y has versus X a power to the effect that S(X,Y) Y has versus X an immunity to the effect that S(X,Y)
Note the difference between CI(Y,X,S) and CI(Y,X,S). The latter expression means: Y has versus X a claim to the effect that S(Y,X). We now have the following strength diagram:
Im(X, Y, S) ~ Fr(Y, X, oS)
Fr(X, Y, S) +- Im(y, X, oS)
4. Atomic Types of Rights
A right which a party X has versus a party Y and which concerns a given state of affairs S between X and Y is not completely characterized by
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merely saying, e.g ., that X has versus Y a power that S(X,Y) . In order to illustrate this, let us return to the second example of § 1: X has versus the Swedish state a right to the effect that a manuscript written by X is published in Sweden. The right we are considering here is the.so-called freedom of the press, and what we wish to say in the example is not only what was stated in § 1, viz.: X has versus the Swedish state a power to the effect that a manuscript written by X be published in Sweden (i.e., X may see to it that the manuscript is published in Sweden), but also that: X has versus the Swedish state an immunity to the effect that a manuscript written by X be published in Sweden (i.e., the state may not see to it that the manuscript is not published in Sweden), and, naturally, also that: X has versus the Swedish state a counterpower to the effect that a manuscript written by X be published in Sweden (i.e., X may see to it that the manuscript is not published in Sweden, and X has versus the Swedish state a counterimmunity to the effect that a manuscript written by X be published in Sweden (i.e., the state may not see to it that the manuscript is published in Sweden). The type of right in this example can thus be more closely stated not only as power, but as: Power, immunity, counterpower, counterimmunity, and we can restate the example by saying that X has versus the Swedish state a right of the type: power, immunity, counterpower, counterimmunity to the effect that a manuscript written by X be published in Sweden. If we go back to the first example of § 1: X has versus Y a right to the effect that X receive from Y what X has lent to Y, we may assert that it states two things: X has versus Y a claim to the effect that X receive from Y what X has lent to Y,
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STIG KANGER AND HELLE KANGER
and Not : X has versus Y a power to the effect that X receive from Y what X has lent to Y. The type of right here is: Claim, not power , and we may restate the example by saying that X has versus Y a right of the type : 'claim, not power' to the effect that X receive from Y what X has lent to Y. Both the type : power, immunity, counterpower, counterimmunity, and the type : claim, not power, are complete in a certain sense : any additional specification of them with the help of simple types of rights or negated simple types is either unnecessary or inconsistent. We shall now list all the types of rights which are complete in this way. We call them the atomic types of rights . Our method is as follows : We start with the list: CI(X, Y, S) Fr(X, Y, S) Po(X, Y, S) Im(X, Y, S) Counter-CI(X, Y, S) Counter-Fr(X, Y, S) Counter-Po(X, Y, S) Counter-Im(X, Y, S) and every list we can obtain from the above by negating one or more lines of it. There are 256 such lists, but some of them are inconsistent according to the strength diagram. A list which contains, e.g .• the lines: not Fr(X, Y. S) not Counter-Po(X, Y, S) is inconsistent, since the negation of the last line, i.e. Counter-Po(X,Y,S), is synonymous with: not IM(Y,X,S), which follows from : not Fr(X,Y,S), by the strength diagram. We omit the inconsistent lists and then go through every one of the remaining 26 consistent lists and reduce them by striking out each unnecessary line, i.e , each line of the list which, according to the strength diagram, is a logical consequence of another line of the list. In a list where the lines: CI(X, Y, S)
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not Counter-Po(X, Y, S) appear, we thus cross out the latter line since it is synonymous with Im(Y,X,S), which, according to the diagram, follows from Cl(X,Y,S). Each reduced list now indicates exactly one atomic type of right, and each atomic type of right is indicated by exactly one reduced list. The 26 atomic types of rights are the following: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
Power, not immunity, counterpower, not counterimmunity. Not power, immunity, not counterpower, counterimmunity. Claim, not counterfreedom. Not claim, power, immunity, counterfreedom, not counterpower, not counterimmunity. Power, immunity, counterpower, counterimmunity. Claim, power, counterfreedom. Claim, not power. Power, immunity, counterfreedom, not counterpower, counterimmunity. Power, immunity, counterpower, not counterimmunity. Power, not immunity, not counterpower, counterimmunity. Not freedom, counterclaim. Freedom, not power,' not immunity, not counterclaim, counterpower, counterimmunity. Freedom, counterclaim, counterpower. Counterclaim, not counterpower. Freedom, not power, immunity, counterpower, counterimmunity . Power, not immunity, counterpower, counterimmunity. Not power, immunity, counterpower, not counterimmunity. Not power, not immunity, not counterpower, not counterimmunity . Not claim, not counterfreedom, not counterimmunity. Not counterfreedom, counterimmunity. Not claim, not power, immunity, not counterpower, not counterimmunity. Power, not immunity, not counterpower, not counterimmunity. Not freedom, not immunity, not counterclaim. Not freedom, immunity. Not power, not immunity, not counterclaim, not counterpower, counterimmunity. Not power, not immunity, counterpower, not counterimmunity.
These 26 types of rights can be displayed together as in the following diagram:
128
STIG .KANGER AND HELLE KANGER not counterimmunity claim
not counter- { freedom power
{
2
7
21 18 25 14 2
20
3
19
8
6
4
20 22 10
5
9
15
17 26 12 13 15
24
23 11 24
2
7
1
16
8 5
counterpower not } freedom
}
21 18 25 14 2
counterclaim not immunity
We see that we have here three atomic types of rights which come under the simple type of claim, namely, nos. 3, 6, and 7. We see also that we have 23 atomic types which come under freedom, namely, all the atomic types except 11, 23, and 24. To say that X has versus Y a freedom that S(X,Y) is, therefore, to say something which, in a certain sense, is very indefinite.
5. Inversion, Conversion, and Co-ordination We say that a type of right, T I , is the inverse of a type T2 , if it is always the case that: X has versus Y a right of type T I that S(X,Y) if and only if Y has versus X a right of type T2 that S(X,Y). We say that a type of right T I' is the converse of a type T2 if it is always the case that: X has versus Y a right of type T I that S(X,Y) if and only if X has versus Y a right of type T2 that not-S(X,Y). We may now ascertain for ourselves that the atomic type 18 to 26 are inverses of 5 to 9 and 13 to 16, respectively. We can also establish that 11
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129
to 17 are the converses of 3,4, and 6 to 10, respectively. From the first 10 atomic types of rights, we can thus obtain all the others by conversion and inversion. Instead of saying, e.g., that: X has versus Y a right of type no. 24 to the effect that S(X,Y), we may as well say that: Y has versus X a right of type no. 14 (i.e., the inverse of 24) to the effect that S(X,Y), and instead of saying this, we could as well say: Y has versus X a right of type no. 7 (i.e., the converse of 14) to the effect that not-S(X,Y). We see in the diagram of the types of rights that the square for the inverse of an atomic type T is the mirror image of the square for T with respect to the line of symmetry: 2,3,4,1, 12, 11,2. Thus, 20 is the inverse of7 , 3 is the inverse of 3, etc. We see also that the square of the converse of T is the square we reach from T by rotating 180 0 about the central square 1. Thus, 14, is the converse of 7, 11 is the converse of 3, 1 is the converse of 1, etc. We say that an atomic type of right T is symmetric if it is its own inverse, and we say that T is neutral if it is its own converse. The symmetric atomic types are nos. 1, 2, 3, 4, 11, and 12, and the neutral atomic types are nos. 1,2,5 and 18. We say that a type of right T is the co-ordinate of a type T 1 paired with a type Tz if it is always the case that: X has versus Y a right of type T to the effect that S(X,Y) if and only if X has versus X a right of type T 1 to the effect that S(X,Y) and Y has versus Y a right of type T z to the effect that S(X,Y). Each atomic type of right T can be constructed as the coordinate of a symmetric atomic type T 1 paired with a symmetric atomic type T z- In the diagram, the square for the co-ordinate of T 1 paired with Tz then lies in the same row as T 1 and in the same column as T z. We see, for example, that 7 is the co-ordinate of 2 paired with 3 and that 20 is the co-ordinate of 3 paired with 2. Note that some types are incompatible in the sense of not having a co-ordinate. Types 3 and 1, for instance, have no co-ordinates.
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6. The Concepts of Rights-Type and Right
We can now make precise the concept of a type of right and the concept of a right, which we used above. In order to define the concept of a type of right, or rights-type, as we shall call it henceforth, we return to the 26 consistent reduced lists of § 5. We now think of a rights-type as a relation of three variables: a party X, a party Y, and a state of affairs S. We say that such a relation T is an atomic rights-type if there is a consistent list L which defines T, i.e . if there is a consistent list L such that T is the relation of all X, Y, and S which satisfy L. We have, for instance, the following reduced list: CI(X, Y, S) not: Po(X, Y, S) If T is the relation of all X, Y, and S which satisfy this list (i.e. the relation of all X, Y, S such that CI(X,Y ,S) and not: Po(X,Y ,S», then T is an atomic rights-type . More precisely, T is the atomic rights-type no. 7. It can be shown for each X, Y, and S that X, Y, S occur in exactly one atomic rights-type. In other words, given X, Y, and S, it is always the case that X has versus Y a right of some atomic type to the effect that S(X,Y), but it is never the case that X has versus Y rights of two different atomic types that S(X,Y). It can also be shown with the help of examples that no atomic rights-type T isempty, i.e., there are always parties X and Y and a state of affairs S such that X, Y, and S stand in the relation T - in other words, there are X, Y, and S such that X has versus Y a right of type T to the effect that S(X,Y). By a rights-type (which need not be atomic), we shall understand a relation T which is defined by a disjunction of one or more consistent lists. We can, in other words, say that a rights-type is a union of one or more atomic rights-types. Among the rights-types, there are the simple rightstypes: Claim, for example, is the union of the atomic rights-types nos. 3, 6, and 7. Cf. the diagram of rights-types. We have defined a rights-type as a relation of three variables: a party X, a party Y and a state of affairs S. On the other hand, we have thought of a right as a relation of two variables: a party X and a party Y. We say now that such a relation R is a right if there is a specified consistent list L which defines R, i.e , if there is a specified consistent list L such that R is the relation of all X and Y which satisfy L. By a specified consistent list, we mean a list which can be obtained by taking one of the 26 consistent lists and in some way specifying S in it. S can, e.g., be so specified that S(X,Y)
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becomes: X receives from Y what X has lent to Y. It follows from this definition that each rightis uniquely determined by an atomic rights-type T and a state of affairs 5, and the right determined by T and 5 is identical with the relation of all X and Y such that X, Y, and 5 stand in the relation T, i.e., the relation of all X and Y such that X has versus Y a right of atomic type T to the effect that 5(X,Y) . By the scope ofa right R, we shall understand the set of all ordered pairs (X,Y) of parties such that X has the right R versus Y. If each pair (X,Y) belongs to the scope of a right R (i.e. if each party X has R versus each party Y), then we say that R is a universal right. If no pair (X,Y) belongs to the scope of R, we say that R is an empty right. The right of atomic type 3 to the effect that a person be tortured is an example of an empty right. 7. Rules of Rights
By a rule of rights we may often understand a proposition which says that each pair of parties which satisfies a certain condition belongs to the scope of a certain right. A rule of right very often can take the form: For every party X and every party Y such that F(X,Y), it is the case that X has versus Y a right of atomic type T to the effect that 5(X,Y), where F(X,Y) is a condition characterizing X, Y. The proposition : For every X and Y such that X is a pedestrian and Y is a motorist who encounters X, it is the case that X has versus Y a right of atomic type: claim, power, counterfreedom, to the effect that Y does not run into X. is an example of a rule of rights of this form. We say that a party Z breaks a rule of rights if either of the following alternatives holds: (1) according to the rule, it shall be that Z sees to it that .. ., but actually it is not so that Z sees to it that ... , (2) according to the rule, it shall be that Z does not see to it that ..., but actually it is so that Z sees to it that ... . According to our example of a rule of rights the following holds for each pedestrian X and each motorist Y who encounters X: it shall be that Y sees to it that Y not run into X, and hence also by the strength diagram: it shall be that X does not see to it that Y run into X. Thus, if Y is a motorist who does not see to it that Y not run into an en-
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countered pedestrian, then Y breaks the rule. And if X is a pedestrian who sees to it that an approaching motorist runs into X, then X breaks the rule. We can seldom say that a rule of rights is unconditionally valid or true, but we can often say that it is valid according to, for example, a certain political or legal ideology, doctrine or practice; and we say this if the rule follows in a certain sense from an authoritative codification K of the political or legal ideology, doctrine or practice. We obtain examples of this conditional validity if we let K be Swedish law. We call a rule of rights which follows from the law a legal rule of rights . When we speak here of a rule of rights following from K, we mean that the rule is a consequence of an elucidation K' of some parts of K. An elucidation of a part of K can be obtained by making precise and completing that part of K in a way which admits applications of logic and which is reasonable and in line with the spirit and purpose of K. And when we say that the rule is a consequence of K', we mean that it follows from K' by ordinary logic extended in a suitable way with logical principles for concepts such as 'shall' and 'seeing to it.' An example of an elucidation will be given in § 11.
8. Background and References In analytical jurisprudence there are some interesting analyses, or attempts at analyses, of the concept of a right along the lines of the analysis we have set forth in this essay. John Austin (the founder of the analytical school of jurisprudence) emphasized in his Lectures on Jurisprudence (1861) that a right is a relation of a certain kind between two parties : All rights reside in persons, and are rights to acts or forbearances on the part of other persons.
By a right, Austin usually meant a claim or a claim which followed from the law: A party has a right, when another or others are bound or obliged by the law, to do or to forbear, towards or in regard of him.
Austin also distinguished the inverse of this type of right and called it a relative duty: The term "right" and the term "relative duty" are correlating expressions . They signify the same notions, considered from different aspects , or taken in different series. The acts or forbearances which are expressly or tacitly enjoined, are the objects of the right as well as of the corresponding duty. But with reference to the person or persons commanded to do or forbear, a duty is imposed. With reference to the opposite party, a right is conferred .
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Along with right and relative duty, Austin also distinguished a type of right which he called: Political or Civil Liberty : ... a term which, not infrequently, is synonymous with right; but which often denotes simply exemption from obligation, conferred in a peculiar manner : namely by the indirect or circuitous process which is styled "permission".
These distinctions between different types of rights were further developed in H.T. Terry, SomeLeading Principles ofAnglo-American Law (1884) and J. Salmond, Jurisprudence (1902).4 The development was completed by W.N . Hohfeld. In the article, "Some Fundamental Legal Conceptions as Applied in Judicial Reasoning" (Yale Law Journal, 1913 and 1917),5 Hohfeld distinguished the following types: (at the right is given our rightstype which most closely corresponds): Claim, Right Duty Privilege No-right Power Liability Immunity Disability
Claim Inverse of Freedom Inverse of Power Inverse of Immunity Inverse of
claim freedom power immunity
Hohfeld gave rigor to the distinctions among these eight types by establishing the following logical relations between them: (1) Duty, No-right , Liability, and Disability are to be correlatives of, L e., identical with the inverse of, Claim, Privilege, Power, and Immunity, respectively. (2) Noright, Duty , Disability, and Liability are to be opposites of, i.e ., identical with the negations of the converses of, Claim, Privilege, Power, and Immunity, respectively. To say, e.g., that X has a Duty versus Y to the effect that S(X,Y) is thus according to (1) the same as to say that Y has a Claim versus X to the effect that S(X,Y), and according to (2) the same as to say that X does not have a Privilege versus Y to the effect that not-S(X,Y). These logical relations are Hohfeld's most important contribution to the analysis of the concept of a right. They imply that each of the concepts Claim, Duty, No-right, and Privilege can be defined in terms of any other of these concepts. They imply also that each of the concepts Power, Liability, Immunity, and Disability can be defined in terms of any other of them. Thus , in order to give explications of Hohfeld's concepts we need to explicate only one concept of each of the two groups - e.g . the concepts Claim and Immunity.
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Hohfeld did not provide any such explications, except in an indirect way by giving examples and discussions of court cases. So much, however, is clear from Hohfeld's text - that what he calls Claim and Immunity cannot without further ado be interpreted as what we have called claim and immunity. Hohfeld's concept Claim can more accurately be interpreted in way such that X is said to have a Claim versus Y to the effect that S(X,Y), if it follows from the law that it shall be that Y sees to it that S(X,Y). Hohfeld's concept Immunity can in certain cases be approximately interpreted by saying that X has an Immunity versus Y if it follows from the law that it shall be that Y does not see to it that nbt-S(X, Y). The cases in question are those in which S(X,Y) is a state of affairs which concerns X's rights versus Y or versus a third party. Other cases are not discussed by Hohfeld. Hohfeld's distinctions have had great influence. They have, for instance, been adopted in The American Restatement of Law, which is published by the American Law Institute. They have also been adopted in subsequent jurisprudence, especially by the so-called American analytical school. Early expositions and applications of Hohfeld's system may be found, for instance, in A.L. Corbin, "Legal Analysis and Terminology" (Yale Law Journal, 1919), J.R. Commons, Legal Foundations of Capitalism (1924), and A. Kocourek, Jural Relations (1927).6 Among later, more logical, approaches to Hohfeld's distinctions there is A.R . Anderson, "Logic, Norms and Roles" (Ratio, 1962). In Scandinavian jurisprudence, accounts of Hohfeld's system are given in T. Eckhoff, Rettsvesen og Rettsvitenskap i USA (1953), and in A. Ross, Om Ret og Retfaerdighed (1953). A more thorough exposition is M. Moritz, Ueber Hohfelds System der juridischen Grundbegriffe (1960) . The logical core of Hohfeld's distinctions has created some difficulties for analytical jurists. Hohfeld says, for example, that Privilege is the opposite of Duty or the negation of Duty. But he emphasizes that:
a
Some caution is necessary at this point: for, always when it is said that the given privilege is the mere negation of a duty, what is meant, of course , is a duty having a content or tenor precisely opposite to that of the privilege in question.
To say that X has a privilege versus Y that S(X,Y) can therefore be interpreted as: X does not have a Duty versus Y to the effect that not-S(X,Y), but not as: X does not have a Duty versus Y to the effect that S(X,Y). This point has not, as a rule, been perceived by jurists. One reason is that jurists seldom take the trouble to put the various notions of rights into a proper context. Sweeping formulations like: Privilege = the negation of Duty, are too elliptic to exhibit the Hohfeld distinctions. Another reason is that many
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jurists wanted to explain what Claim, Duty, Privilege, etc. really mean without being able to uphold the logic of the distinctions in their explanations. All this has caused the logical rigour of Hohfeld's distinctions to be partly lost sight of in jurisprudence. This concludes our discussion of the notion of a right. We now turn to some applications. PART II. THE GOVERNMENT POSITION STRUCTURE AND PARLIAMENTARISM
9. The Notion of Government Position Structure
In the study of a political system there is often good reason for distinguishing the following single parties: H P M G C
the head of state the prime minister member of the government other than P the government as a whole the congress or parliament
and the joint parties that consist of two or more of the parties H, P, M, C or the parties H, G, C in union, as, for instance, the union HG of Hand G or the union HPC of H, P and C. Union of the type PG or MG are, of course, redundant; they do not differ from G. Thus, we get 19 parties : M, P,PM,G,H,HM,HP,HPM,HG,C,CM,CP,CPM,CG,CH,CHM, CHP, CHPM, CHG. By the position structure in a government, we mean (in this part of the essay) the system of rights to appoint and to dismiss members of the government and to dissolve the congress which these 19 parties have versus each other. The position structure can be given in tabular form, one table for each of the following five states of affairs: 51 52
53 54
55
that C is dissolved that P resigns as prime minister that M resigns as member of the government that the candidate X for the prime ministerial appointment is appointed that the candidate Y for a ministerial appointment is appointed
Each table has the following form:
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M
P
P M
G
H
C H G
M P PM G H
CHG
in which atomic types of rights are indicated by number. If in the table for 52 we have number 4 in the row for C and the column for HG, we may read: The congress C has versus HG (i.e. the head of state in union with the government) a right of atomic type no. 4 to the effect that P resign as prime minister. Thus , each table may be interpreted as a set of 19x 19 rules of rights of this kind. 10. Some Methodological Simplifications
To construct one of the five tables which determine the position structure, we only need find the diagonal from the left downwards (i.e. , the types of rights which the parties have versus themselves - note that these types must be compatible). When we have the diagonal, we get the rest of the table by means of co-ordination . For instance, if we have the atomic type no. 2 in the diagonal at M and no. 4 at C, then we have the co-ordinate of 2 paired with 4 (i.e., no. 21) in the row M at the column C. We shall note also that the atomic types in the diagonal are uniquely determined by atomic types outside the diagonal. For instance, if we have no. 18 (i.e., the co-ordinate of2 paired with 1) in row H at column CG, we have 2 in the diagonal at H and 1 at CG. Thus, to construct a table we only need to determine 10 of its 361 places, but these 10 places must, of course, be chosen in a suitable way. The construction of the table can be further simplified if we add the following logical principle to the principles I - V in § 2:
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(a) (b) (c)
137
X sees to it that F ~ XY sees to it that F X sees to it that F ~ YX sees to it that F XY sees to it that F ~ XZY sees to it that F
We call this principle the principle ofjoint parties . By means of the principle of joint parties (added to I-V) we may prove facts like: If X has versus Y a right of atomic type no. 1 to the effect that S, then so has every party that includes X. Clearly , facts of this kind may further reduce the number of places we must determine to acquire the whole table. 11. An Example: The Position Structure in West German Government
As an example of a position structure we shall now describe the position structure of the present West German government. To do this it is sufficient to give the diagonals of the tables for the five states of affairs St-Ss. An asterisk is put at those places in the diagonal which we determined by means of the German constitution. These places then yield the rest of the diagonal by logic, i.e. , the principles I - VI. The diagonals are these:
M P PM G H HM HP HPM HG C CM CP CPM CG CH CHM CHP CHPM CHG
SI
S2
S3
S4
S5
2 2 2 2* 12* 12 12 12 12* 12* 12 12 12 12* 1* 1 1 1 1
2 4* 4 4 2 2* 4 4 4* 4* 4 1* 1 1 4 4* 1 1 1
4* 4* 4 4 2* 4 4 4 4* 4* 4 4 1* 1 4 4* 4* 1 1
2 2 2 2 2 2 2 2 2* 1* 1 1 1 1 1 1 1 1 1
2 1* 1 1 2 2 1 1 1 2 2 1 1 1 2 2* 1 1 1
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The numbers with an asterisk can be justified by references to the following articles of the German constitution : 58, 62, 63, 64:1, 67:1, 68:1 , and 69:2. To demonstrate the method, we shall carry through the justification in case of 52 (i.e. , the fact that P resigns as prime minister) . The articles we shall rely on in this case are these: Art . 58 Anordnungen und Verfugungen des Bundesprasident bediirfen zu ihrer Giiltigkeit der Gegenzeichnung durch den Bundeskanzler oder durch den zustandigen Bundesminister. Dies gilt nich fur die Ernennung und Entlassung des Bundeskanzlers, die Auflosung des Bundestages gemass Art . 63 und das Ersuchen gemass Art. 69:3 . Art. 67 1. Der Bundestag kann dem Bundeskanzler das Misstrauen nur dadurch aussprechen , dass er mit der Mehrheit seiner Mitglieder einen Nachfolger wahlt und den Bundesprasidenten ersucht, den Bundeskanzler zu entlassen . Der Bundesprasident muss dem Ersuchen entsprechen und den Gewahlten ernennen . The method of justification is that of an elucidation (in the sense of § 7) of these articles. The elucidation will involve three kinds of data: (i)
Facts that are explicitly or almost explicitly stated in the articles of the constitution. We shall be free to formulate these facts in a language suitable for applications of logic. (ii) Political principles which hold in every feasible position structure of the type here in question. These principles are in most cases too obvious to be worth explicit statement in a document like the constitution , but they are often implicitly assumed in interpretations of the constitution . (iii) Hypotheses to fill in lacumas of the constitution which cannot be filled in with consequences of data of kinds (i) and (ii). These hypotheses shall be in line with the spirit and purpose of the constitution and with the actual political life. The facts of kind (i) that are relevant for 52 are these: (1) C may see to it that 52' (2) HG may not see to it that not-5 2. These facts are almost explicitly stated in art . 67: 1 combined with art. 58. We need four principles of kind (ii): (3) (4)
P may see to it that 52. If X and Yare two parties in the position structure without any common member and if X may see to it that 52' then Y may not
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(6)
139
see to it that not-Sj. If every party of the position structure which may see to it that S2 is identical with or contains one of the parties X and Y, then XY may see to itthat not-Sj. CHG may see to it that not-Sj.
We need one hypothesis of kind (iii): (7)
HM may not see to it that S2'
Now, from the elucidation (1) - (7) we can obtain all the facts necessary to determine the numbers with an asterisk. The only means we need is logic involving the five logical principles of § 2 and the principle of joint parties. According to (1) and (3), C and P may see to it that S2; it follows logically that every party that includes C or P may see to it that S2' Thus we get: (8) HG may see to it that S2' (9) CP may see to it that S2' (10) CHM may see to it that S2' We note also that neither H nor M may see to it that S2 since according to (7), HM may not see to it that S2' Hence , C or P must occur in every party that may see to it that S2 ' Then according to (5), we get: (11) CP may see to it that not-Sj. Next, according to (3) and (4), we get: (12) CHM may not see to it that not-Sj. (13) C may not see to it that not-Sj. We also derive logically from (2) : (14) HM may not see to it that not-Sj. (15) P may not see to it that not-Sj . Since, according to (6), CHG may see to it that not-Sj, it is not the case that CHG shall see to it that S2' Hence, no party included in CHG shall see to it that S2' Thus we have : (16) (17) (18) (19)
It It It It
is is is is
not not not not
the the the the
case case case case
that that that that
CHM shall see to it that S2' C shall see to it that S2' HG shall see to it that S2 ' P shall see to it that S2'
Now (3), (15), and (19) jointly state :
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P has versus P a right of atomic type no. 4 to the effect that P resign as prime minister. Further, (7) and (14) state: HM has versus HM a right of atomic type no. 2 to the effect that P resign as prime minister. (8), (2), and (18) state: HG has versus HG a right of atomic type no. 4 to the effect that P resign as prime minister. (1), (13), and (17) state: C has versus C a right of atomic type no. 4 to the effect that P resign as prime minister . (9) and (11) state: CP has versus CP a right of atomic type no. 1 to the effect that P resign as prime minister. Finally, (10), (12), and (16) state: CHM has versus CHM a right of atomic type no. 4 to the effect that P resign as prime minister. Thus we have determined all the asterisked numbers in the diagonal for 52. We can easily derive the rest of the diagonal from these numbers by means of logic.
12. Some Main Types of Parliamentarism The main characteristic of a parliamentary political system is the fact that its government position structure is of a certain kind, which we shall call parliamentary position structures. In this section we shall make some preparatory comments on these structures and we shall distinguish some main types or levels of parliamentarism. First, a distinction is made between what may be called control parliamentarism and delegation parliamentarism. Control parliamentarism is characterized by the fact that the parliament can dismiss the government, but not necessarily by the fact that the parliament has an influence on the appointment of the government members. Control parliamentarism is compatible with a strong position for the head of the state with authority to appoint the government. The political system of
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the fifth French republic is a good example of this kind of parliamentarism. Delegation parliamentarism is a control parliamentarism in which the parliament also has a decisive influence on the formation of the government. The government will then be a kind of delegate of the parliament. A good example of delegation parliamentarism is found in the political system of West Germany . Here the Bundeskanzler is elected by the Bundestag. Next, we distinguish what we may call minister parliamentarism from government parliamentarism. Minister parliamentarism is characterized by the fact that the parliament can dismiss single members of the government. Denmark provides an example of this kind of parliamentarism . In a system with government parliamentarism, on the other hand, the parliament can dismiss the government as a whole only. This is often done by a vote of censure on the prime minister. Example: West Germany. Parliamentary political systems are characterized mainly by the fact that the parliament can dismiss the government. Besides these political systems there is a category of possible systems which are parliamentary in a wider sense - we may call them pseudo-parliamentary. As an example, we may take a system where the parliament appoints the members of the government for a period of, say, four years. During this period they cannot be dismissed, but every fourth year the parliament can dismiss a minister simply by refusing to reappoint him. Switzerland has a pseudo-parliamentarism of this kind. In most parliamentary systems the government or the head of state may dissolve the parliament. In political science this right is usually considered a necessary feature of parliamentarism. But the agreement is not complete. There are also adherents of an opposite opinion." In this essay, we shall hold the view that the question whether or not a position structure is parliamentary is independent of the question whether or not the parliament can be dissolved. We shall also disregard pseudo-parliamentarism and minister parliamentarism. Hence, in our study of parliamentarism, we may confine ourselves to a simplified kind of position structure where the parties P and M are not distinguished from G, and where the dissolution of parliament is left out of account. These position structures - we shall call them reducedpositionstructures - can be given by one table for each of the following two states of affairs: R that the government G resigns S that the candidate K for government is appointed government. 8
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Each table has the form: G
H H G G HGCGCCC
H G C HG HC GC HGC
in which, just as before, atomic types of rights are indicated by number. And, as before, we only need to establish the diagonals in order to determine the tables. 13. Feasible Reduced Position Structures
The number of logically possible reduced position structures is very large." But most of them are politically unfeasible. In order to define those that are feasible we shall set forth some principles or axioms of political feasibility. The feasible structures will then be defined as those which fulfill the principles . The principles are these: 10 (1)
(2)
The (a) (b) The (a) (b)
(3)
The (a) (b)
principle of joint sovereignty: HGC may see to it that R, and HGC may see to it that not-R. HGC may see to it that S, and HGC may see to it that not-So principle of non-obstruction: If X and Yare disjoint parties and if X may see to it that R, then it is not the case that Y may see to it that not-R. If X and Yare disjoint parties and if X may see to it that S, then it is not the case that Y may see to it that not-So principle of non-compulsion: If every party that may see to it that R is identical with or contains one of the parties X and Y, then XY may see to it that not-R. If every party that may see to it that S is identical with or contains one of the parties X and Y, then XY may see to it that not-So
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143
The principle of non-competition : If X and Yare disjoint parties and if X may see to it that S, then it is not the case that Y may see to it that S. The principle of resignation : G may see to it that R. The principle of preservation: If a party X may see to it that not-R, then X may see to it that not-
S.
(7)
The principle of elimination: If HC may see to it that R, then HC may see to it that S.
When we refer to parties in these principles, we always mean parties involved in the reduced position structures. Among the tables for Rand S that are consistent with the logical principles I - VI, there are 324 tables for each of Rand S that contain No . 1 in the diagonal at HGC. These tables are consistent with the principle of joint sovereignty. We may easily examine these 324 tables one by one and omit those which are inconsistent with the principles (2) -(5). As a result, we get 5 tables for Rand 11 tables for S. Thus , we have 55 combinations of a table for R with a table for S. But we do not get 55 position structures , since only 25 of the combinations are consistent with the principles (6) and (7). These 25 combinations are displayed in the table: H
A B C D E F G H
1, 2, 2, 12, 2, 2, 12, 12, 2, I J 2, K 12,
a G
2, 2, 1, 2, 12, 2, 12, 2, 12, 2, 12,
C
2, 1, 2, 2, 2, 12, 2, 12,
12, 2, 12,
HG
1, 2, 1, 1, 1, 12, 1, 12, 12, 1, 12,
HC
1, 1, 2, 1, 12, 1, 12, 1, 12, 1, 12,
GC 2, 1, 1, 12, 1, 1, 12, 12, 1, 1, 12,
HGC
2 1 2 1 2 1 1
b
e
4
4 4
2
2 1
2 1
4
4 4
1 1
1
Ae Ca Ea Ga 1a Ka
d
e
4 4 4 4
4 4 4 4 4 4
2
1 1
1
Bd
Ae Be
Db
De
Dd
De
Fb
Fe
Fd
Fe
Hb
He
Hd
He
Ib
Jc
Jd
Ie
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In the table, a, b, c, d, e are the 5 diagonals for R, and A,B,C, ... ,K are the 11 diagonals for S. The table shows that diagonal A can be combined only with diagonal c and diagonal e. We call the position structures determined by these combinations Ac and Ae, respectively. Thus, we have exactly 25 feasible reduced position structures. Of these, 11 are parliamentary, namely those indicated in the d- and e-columns. Ae is purely control parliamentary without delegation parliamentarism, while Bd is completely delegation parliamentary. To see the delegation parliamentarism of Bd, we note that diagonal B has no. 1 at C and no. 2 at HG. Hence C has versus HG a right of atomic type no. 5 (i.e., the co-ordinate of 1 with 2) to the effect that S. In other words, the parliament has versus the head of state joined by the government a power, counterpower, immunity, and counterimmunity to appoint a new government. We note also that diagonal D has no. 4 at C and at HG. Hence C has versus HG a right of atomic type 4 to the effect that R. This implies that C has versus HG a power and an immunity to dismiss G. 14. Some Concluding Remarks
So far we have considered two types of position structures : the unreduced and the reduced type. Of course, there are also other possible types. There are, for instance, types of extended position structures involving new single parties like the supreme court, the central committee of the communist party , the army, etc. In the study of a political system, the position structure is a main topic. But, of course, the type of the structure must be adequate in the sense that it permits the structure to mirror the political life of the system. The type of structure suggested in § 9 with 19 parties and 5 states of affairs is fairly adequate in the case of many western political systems. But it is clearly inadequate in the case of the Soviet system, for instance, where the central committee of the communist party has a decisive influence on the appointment and dismissal of the government. We shall note also that the validity of the principles of political feasibility depends on the adequacy of the position structure. For instance, the principle of joint sovereignty is clearly false in the case of political systems where the army can dismiss the government. Finally, we shall note that not all the 25 feasible reduced position structures we have defined above are politically attractive . The structure Ca, for example, is certainly not attractive from a democratic point of view. But the problem of indicating those that are attractive is a problem of political ideology and not a problem of political science.
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NOTES The first part is a translation (with minor changes) of the essay "Rattighetsbegrepper" by Stig Kanger, which is contained in Sju filosofiska studier tilliignade Anders Wedberg (Stockholm, 1963). 2 The interpretation is, in all essential respects, identical with that which was given in S. Kanger, New Foundations for Ethical Theory (Stockholm, 1957). 3 Thus F ~ G if G follows from F by ordinary logic extended in a suitable way with logical principles for the concepts shall and seeing to it. Note that the relation ~ is assumed to fulfill principles like: (i) if F and if F ~ G, then G; (ii) if F ~ G, then not-G ~ not-F: (iii) ifF~ G and G~ H, then F~ H. 4 A summary of these authors' distinctions is given in J. Hall, ed., Readings in Jurisprudence (1938). 5 Reprinted in W.N. Hohfeld, Fundamental Legal Conceptions as Applied in Judicial Reasoning, and Other Legal Essays (1923), ed. by W.H. Cook. 6 A summary of Corbin's and Kocourek's ideas may be found in J . Hall, ed., Readings in Jurisprudence (1938). 7 For instance R. Fusilier in Les Monarchies Parlementaires (1960); "Le caractere non fondamental du droit de dissolution dans Ie fonctionnement du regime parlementaire est demontre non seulement par l'usage generalernent de moins en moins frequent de la dissolution dans les pays consideres, mais encore par la pratique norvegienne, qui l'ignore, et de l'etude de laquelle iI ressort nettement que la dissolution ne constitue pas un facteur necessair du regime parlementaire" (p. 32). The standard view is maintained by D.V. Verney in The Analysis of Political Systems (1959): "The power of the Government to request a dissolution is a distinctive characteristic of parliamentarism. ... Certain States generally regarded as parliamentary severely restrict the right of the Executive to dissolve the Assembly. In Norway the Storting dissolves itself, the Head of State being allowed to dissolve on special sessions, but this is a departure from parliamentarism inspired by the convention theory of the French Revolution" (pp. 31-32). 8 The somewhat odd formulation of S should not cause confusion. We note that S is not synonymous with: Some government is appointed. It might very well happen that H may see to it that K not be appointed, but H may not see to it that no government be appointed at all. 9 If we take into consideration the logical principles I-V of § 2 and the principle of joint parties given in § 10 we get 619 different tables for each ofR and S. Hence, the number of logically possible reduced position structures is 619x619=383161. As a curiosity we may mention that the number would have been larger than 430 millions - (47+37+37)2 to be precise - if we had not taken into account the principle of joint parties. And if we also had kept the unreduced structures with 19 parties and 5 states of affairs, the number would have been astronomical: (419+319+31~5 . 10 Note that principles corresponding to the principles I, 2, 3 and 5 were applied in our elucidation of some parts of the West German constitution. In principles 2 and 4 we say that X and Yare disjoint parties if X and Y have no member in common.
LAW AND LOGIC*
1. INTRODUCTION
By a system of law we shall mean - in this paper - any system of rules which has the purpose of regulating human action under certain conditions. Examples : A nation's constitution, the traffic laws, club's statutes, recipes in a cook-book, etc. A system of law can be more or less well-written; indeed, we can even speak of unwritten laws. When we say, in this context, that a system of law is well-written, we shall take into consideration only the logical criteria: well-written = logically well-written . The combination of law and logic is highly problematic, and the results are few and far between. One of the reasons for this is that very few logicians are interested in law, and very few jurists are interested in logic. Moreover, the purpose of such a combination, as well as suitable approaches to the study of it, is a bit unclear . However, it appears suitable to start by distinguishing two categories of problems and theories, namely, (1) problems and theories which are relevant for the application of logic to already existing, not well-written, systems of law, and (2) problems and theories which are relevant for the creation of well-written systems of law. In general, the first category represents the jurist's approach to the combination of law and logic, while the second category represents the logician's. In the second category the logical aspects can be developed without the hindrance of logical flaws and complications which often accompany existing systems of law. Of course, both categories are vague and have much in common. In this paper, we shall concentrate on the second category of theories and problems . (The reader interested in the first category should consult the general introductions listed in the bibliography .) In the second category the main theme is that of the formalization of a system of law, i.e ., the system's logical codification, deductive development, and semantics. Another theme is that of the applicability of the system of law and thus, among other things, the definability of juridical or normative ideas which we want to express within the framework of a well-written system. 146
G. Holmstrom -Hintikka, S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. t, /46-/69. © 2001 All Rights Reserved. Printed by Kluwer Academic Publishers. the Netherlands. Orig inally published in Theoria 38 (1972). 105-129.
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This paper is addressed to the reader with a philosophical background . It presupposes an elementary knowledge of logic, as well as some familiarity with the philosophy of law. Its purpose is to give a brief sketch (not a systematic account) of some scattered logical or philosophical ideas which do not lie within the main stream of logic, but which nevertheless appear to be important in the attempt to create logically well-written systems of law. 2. CODIFICATION
As we have conceived it, a system of law is a set of rules which has the purpose of regulating human action under certain conditions. The linguistic framework within which the rules of such a system of law are formulated cannot be too narrow . Among other things, it should contain sentences by means of which one can (1) describe states of affairs or conditions, including numerical conditions, (2) state that a state of affairs is unavoidable, and (3) prescribe that something shall be, or ought to be, the case. It should also contain sentences by means of which one can (4) state that an agent, i.e ., a person or an ordered set of people, does something or sees to it that something is the case. In addition, there is often a need for sentences by means of which one can (5) state that an agent decides upon a certain state of affairs, and (6) state that a person knows about, or is convinced of, some particular state of affairs. Finally, we often also need (7) measures for states of affairs and conditions - for example, measures of probability. We shall assume that sentences of type (1) can be formulated within the framework of the language of many-sorted elementary logic extended by elementary algebra. The language of many-sorted elementary logic differs from the usual one-sorted type due to the fact that it has several kinds of individual variables . We shall assume that there are at least four kinds, namely, variables x, y, Z, .. . for things, r, s, t, ... for time, a, b, C, .• . for numbers, and p, q,... for people (or for agents regarded as units). It is not necessary for either the relation symbols or the operation symbols in this language to be homogeneous: a relation symbol can denote, for example, a relationship between time and people. The algebraic extension of this 4sorted elementary logic deals primarily with the numerical variables. We may, however, include in it also some notions dealing with time, say, an order relation and a relation ordering time intervals. We form the sentences of type (2) by means of the modal operator it is unavoidable that. Type (3) sentences are formed by means of the modal operators ought and shall. As far as the sentences of types (4) and (5) are concerned, we shall, for simplicity's sake, let Greek letters a, {3, ... stand
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for a sequence of person variables. Thus, for example, ex can be p, otpq, or qpq, etc. We now form sentences of type (4) by means of the operators ex sees to it that and ex sees to it at t that. Sentences of type (5) are formed with the operators ex decides that and ex decides at t that. Type (6) sentences are formed with the operators p is convinced that and p is convinced at t that. Finally, measures for conditions (7) are expressed in the usual way: m (---), where m denotes the measure, and (---) is a sentence. A language with these means of expression shall be called an L-Ianguage. Example: A necessary condition for the person p to have borrowed an object x from the person q at time r can be formulated in a language L. (Let r< t mean that r precedes t, let G(x,q,t) mean that x is given or transferred to q at t) . This is the condition: (3s) (r < s & p,q decide at r that it shall be so that (3t) (r < t & s 4:. t & P sees to it that G(x,q,t))). It is not at all certain that the means of expression so far introduced in the L-Ianguages are sufficient for the formulation of 'ordinary' systems of law. It is possible that, in certain contexts, still another kind of variable is needed; for example, variables for classes of individuals of some of the sorts already introduced . It is also possible that other types of modal operators need to be introduced. Moreover, perhaps imperative formulas should be added to the indicatives of the L-Ianguage - in the philosophy of law legal sentences are sometimes considered to be imperatives. Finally, it may happen that some of the operators introduced can be analysed in terms of more basic operators. A good example is that of the operator seeing to it. We can define ex sees to it that (---) as the conjunction: (---) is necessary for something which ex does and (---) is sufficient for something which ex does. Similarly, we define ex sees to it at t that (---) as the conjunction: (---) is necessary for something which ex does at t and (---) is sufficient for something which ex does at t. The bearings of these definitions will be more apparent in the sequel. 3. DEDUCTIVE DEVELOPMENT
Many of the L-Ianguage formulas are valid in the sense that their truth depends only on the meaning given to elementary logical and algebraical concepts and to the operators which were introduced in the preceding section. A number of examples of formulas which seem to be valid will be given in the sequel. Some rules of inference will also be presented. Thereby a system of law formulated in the L-Ianguage gets a deductive structure: by using elementary logic, given valid L-Ianguage formulas and the rules of inference, we can deduce laws in the system from laws which already exist there .
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In order to facilitate the formulation of the valid formulas and of the rules of inference, we use the letters F, G, and H for formulas in the Llanguage. In the rules of inference, the symbol I- F is also used to indicate that the formula F is given as valid or that it can be deduced from given valid formulas. Let us also use the following abbreviated notations: UnavF ShallF OughtF D6(ex, F) D6(ex,t,F) Do(ex,F) Do(ex,t,F) Dec(ex,F) Dec(ex,t,F) Conv(p,F) Conv(p,t,F)
for for for for for for for for for for for
it is unavoidable that F it shall be the case that F it ought to be the case that F F is necessary for something ex does F is necessary for something ex does at t F is sufficient for something ex does F is sufficient for something ex does at t ex decides that F ex decides at t that F p is convinced that F p is convinced at t that F
We also use the notation Do(ex,F) for D6(ex,F) & Do(ex,F) and Do(ex,t,F) for D6(ex,t,F) & Do(ex,t,F). The letter 0 will sometimes be used to represent any of the above eleven operators, with agent and time variables where applicable . OF may, thus , stand for ShallF in one context, but for Conv(p,t,F) in another . Let us now consider the following two rules of inference : I.
II.
If I-F, then I-OF. If I- (F == G), then
I- (OF
==
OG).
along with the following types of formulas : 1. 2. 3. 4.
5.
OF & O(F:J G) :J OG. OF & OG :J O(F & G). O(F & G) :J OF & OG. OF:J F. OF:J -O-F.
Note that 1, together with I, implies II, 2, and 3; and that 4 implies 5. These rules and formulas seem to be valid for the operators to the following extent: Unav Shall
I I
II II
1 1
2 2
3 3
4
5 5
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Ought D6 Do Dec Conv
STIG KANGER
I I
II II II II
1 1 1
2 2 2
3 3
1
2
3
4
5 5 5 5 5
The assumption that the formulas 1, 2, 3, and 5 are valid for Conv is perhaps a bit uncertain. Indeed, it is conceivable that an irrational person is convinced that F and that (F :J 0) without being convinced that G. We assume, however, that a person always possesses a certain amount of rationality so that at least 1, 2, 3, and 5 are valid. It is also assumed that formulas like, for example, OF == O--F OF == OaF
are valid for Conv. We may also assume that 1 and 2 are valid for Dec in case of individual decisions, even if 1 and 2 are not always valid in case of collective decisions. There are several other formulas which we assume to be valid. For example: UnavF:J - Do(a,F). UnavDo(a,F) :J - Ought- Dofo.F). OughtDo(a,F) :J -'- Unav- Do(a,F) . Unav(F == 0) :J OughtF == OughtG. Unav(F == 0) :J Do(a,F) == Do(a,O) . Do(a,F) & Do(a,O) :J Do(a,(F V 0». In some cases, the validity of a formula is connected with the unavoidability of the past. Let r< t mean, as before, that time r precedes time t. Then, as an example of this connection, we have: Do(a,r,F) & r
< t :J Unav- Do(~,t, - Do(a,r,F).
The following formulas also seem to be valid: a = ~ :J [Do(a,F) == Do(~,F)]. a = ~ & r = t :J [Do(a ,r,F) == Do(~,t,F)] . a = ~ :J [Dec(a,F) == Dec(~,F)]. a = ~ & r = t :J [Dec(a,r,F) == Dec(~,t,F)].
p = q :J [Conv(p,F) == Conv(q,F)]. p = q & r = t :J [Conv(p,r,F) == Conv(q,t,F)].
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This does not mean, however, that formulas like, for example, Dec(a,F) or Conv(p,F) are extensional. Eleven types of operators have now been added to the elementary logical concepts of the language L. Some of these operators are well-known from logical contexts, and their logic has been studied in detail. This applies especially to the deontic operator Ought. Classical references are von Wright 1951a, Anderson 1956, Kanger 1957d, and Hintikka 1957. Among later contributions, von Wright 1964 and Hansson 1969a should be mentioned. The combination of a deontic operator and the operator seeing to it was first introduced in Kanger 1957d. In Kanger & Kanger 1966 certain logical principles for the operator seeing to it were suggested. More advanced proposals can be found in Chellas 1969 and in Porn 1970, 1971. The main difference between our theory and those of Chellas and Porn is that the latter contain an inference rule of type I for the operator seeing to it, thus identifying with our 06 rather than with Do. As far as we know, the operator p decides that has never been explicitly introduced as a logical operator. It has been studied, however, within the framework of the theory of individual decision. An excellent survey of decision theory can be found in Luce & Raiffa 1957, chapter 13. The operators of the type a decides that, when a represents several people, have to do with group decisions. The classical reference here is Arrow 1951. For several more recent results, see Luce & Raiffa 1957, chapter 14 and Hansson 1969b, c. The operators of the type p is convinced that (or, p knows that or p believes that) and the difficulties resulting from the highly intensional character of these operators, have attracted the logicians' interests for a long time. See, among others, Carnap 1947, Church 1950, and Kanger 1957b. See also Hintikka 1962, where the operator p knows that is treated as a modal operator satisfying inference rule I. 4. SEMANTICS
The fixing of truth conditions for the formulas in a logical language is an important part of its semantics, and can be done in different ways. A brief account of how it has been done in some well-known semantic theories will be given in the following. This account will be limited to the language of classical two-valued elementary logic, setting intuitionistic logic and manyvalued logic aside. It will also, for simplicity's sake, be limited to onesorted logic without symbols for operations in the domain of individuals.
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As far as truth is concerned, the semantic theories have certain things in common: 1. The truth of a formula F is defined relative to an interpretation Int. Truth is a function T which assigns either the truth value t (true) or I (false) to a formula F with an interpretation Int. 2. Truth is defined recursively over the length of the formulas. The following always holds true : T( - F, Int) = tiff T(F, lnt) = f. T«F & G), Int) = tiff T(F, Int) = T(G, Int) = t. T«F V G), Int) = I iff T(F, Int) = T(G, Int) = f. T«F :) G), Int) = I iff T(F, Int) = t and T(G, Int)
= f.
3. Validity is defined thus: a formula F is said to be valid if T(F,Int)=t for all Int. The semantic theories of which we shall now give an account differ from one another when it comes to the construction of Int and to the truth conditions for the atomic formulas and for the quantification and modal formulas . They also differ somewhat concerning the selection of formulas for which truth is defined . For instance, some theories define truth for a larger variety of modal formulas than others, and some theories define truth only for closed formulas, i.e., formulas which may contain individual constants but no free individual variables. This latter difference is not essential and we shall pay little attention to it, letting free individual variables when possible play the role of individual constants. In our presentation of the semantic theories, which now follows, let us use x, y, xl>... ,xn as notations for individual variables, k, k}, .. .,kn for individual constants, and B", S" as notations for n-place relation symbols. Tarski's theory. In Tarski 1936 a theory of truth is presented which is essentially as follows. Let V be a non-empty domain of individuals (or 'universe') . Let Wu and Vu be functions which assign denotations and values to the relation symbol and individual variables, respectively, in such a way and Vu(x)E U. that Wu(Rn) £ An interpretation, in Tarski's sense, is a system or a 'possible realization' (V, Wu, Vu)' The truth conditions for atomic formulas, for example, (x=y) and R2(x,y), are the following:
o,
» »
T«x=y), (V,Wu, Vu = tiff Vu(x) = Vu(y) . T(R2(x,y), (V, Wu, Vu = tiff < Vu(x), Vu(Y»
E WU(R 2 ) .
The truth conditions for other atomic formulas, for example R 3(x },X2 ,X3) , are completely analogous.
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The truth condition for universal quantification is the following:
w
T(VxF, (U, Wu, Vu» = tiff T(F, (U, u, Vb» = t for all Vb such = Vu(y) for each individual variable yother than x.
that Vb(Y)
Tarski's theory does not present any truth conditions for modal formulas. Carnap's theory. Another construction of Int can be found in Carnap 1943, 1946, and 1947. According to Carnap, an interpretation or a 'state description' consists of a class E of closed formulas which satisfy the following two conditions: (1) each formula in E is of the form Rn(k., ... ,kn) or of the form -Rn(k., ... ,kn), and (2) an atomic formula Rn(k1, ... ,kn) is a member of E if and only if its negation is not a member of E. The truth conditions are the following: T(Rn(k1, ... ,kn),E) = tiff Rn(k., ... ,kn) E E. T«k=k),E) = t. T«k 1=kz),E) = f when k 1 and kz are different individual con-
stants.
T(VxFx,E)
= tiff T(Fk ,E) = t for each choice of the individual
constant k substituted for all free occurrences of x in Fx '
A consequence of the last two conditions is that each individual is denoted by exactly one individual constant. In Carnap 1946 and 1947, truth conditions are given for modal sentences of the type 'It is universally necessary that F (symbolically: OF). The condition is: T(OF,E)
= tiff T(F,E) = t for all E.
Kanger 's theory I. In Kanger 1957a, b, and d, a modification and extension of Tarski's theory was made with the purpose of obtaining semantics for modal formulas. Let W be a binary function which assigns meanings to relation symbols with respect to a domain of individuals U. Similarly, let V be a binary function which assigns values to the individual variables with respect to U. The functions W and V shall always fulfill the following requirements: W(~, U) ~ tr: and V(x, U) E U. An interpretation in Kanger's sense, or a 'valuation ', is a system (U, l¥, V). The truth conditions for atomic formulas, for example (x=y) and R2(x,y), and for quantification are the following: T«x=y) , (U, l¥, V» T(R2(x,y), (U, l¥, V»
= tiff V(x, U) = V(y, U). = tiff < V(x, U), V(y, U»
E W(R2 ,U).
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STIG KANGER
T(VxF, (U, lV, V» = tiff T(F, (U, lV, V'» = t for all V' such that V'(y, U) = V(y, U) for each U and each individual variable y other than x.
The truth condition for formulas with a modal operator M is formulated by means of an 'alternative' relation RM between domains of individuals (or 'possible worlds ') as follows: T(MF, (U, W, V» = t iff T(F, (U', lV, V» = t for all U' such that U'RMU.
By varying RM one obtains truth conditions for different modalities M. For example, if RM is the universal relation between domains of individuals, then M is a universal necessity 0; if RM is reflexive and transitive, then M is a necessity of Lewis' 54 type; etc. Kanger's theory II. This theory differs from the preceding one when it comes to truth conditions for quantification (see Kanger 1957c, d). The condition theory II is as follows: T(VxF, (U, W, V» = tiff T(F, (U, W, V'» = t for all V' such that (i) V'(y, U) = V(y, U) for each U and each individual variable y other than x, and (ii) V'(x, U') = V(x, U') for all U' other than U.
Of course, this theory combines with the previous one if we use two kinds of universal quantifiers in the language. It ought to be noted that the two quantifications are logically equivalent whenever F is a formula of elementary logic without modalities. Hintikka's theory. In Hintikka 1957, 1961, 1962, and 1963, a semantic theory is presented which is essentially a modification and an extension of Carnap's theory . We shall here formulate Hintikka's theory so that its relationship to Carnap's becomes explicit - or maybe over-explicit. (In fact, we are depriving Hintikka's theory of one of its virtues.) A state description in Hintikka's sense is, thus, a class E of closed formulas which satisfies the following four conditions: (1) Each formula in E is of the form Rn(klo ,kn)' (k 1=k2 ) , - Rn(k1, ... ,kn), or - (k) =k2 ) . (2) An atomic formula Rn(klo ,kn) or (k) =k2) is a member of E if and only if its negation is not a member of E. (3) Each atomic formula of the form (k=k) is a member of E. (4) If (k 1=k2 ) and the atomic formula F is a member of E, and if G is like F except for containing occurrences of ~ at one or more places where F contains occurrences of k), then G is also a member of E. An interpretation in Hintikka's sense is a system (H,E) where H is a nonempty class of state descriptions in Hintikka's sense and where EEH.
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The truth conditions for closed atomic formulas and quantification are these: T«k l = kZ) ' (H,E» = tiff (kl = kz) E E; T(Rn(k!> ... ,kn), (H,E» = tiff Rn(kl,... ,kn) E E; T(VxFx ' (H,E» = tiff T(F k, (H,E» = t for each choice of k substituted for all free occurrences of x in Fr
The truth condition for formulas with a modal operator M is formulated by means of an alternative relation RM between state descriptions in H . The condition is as follows : T(MF, (H,E» E'RME.
=
t iff T(F, (H,E'»
=
t for all E' EH such that
By varying RM and H one obtains truth conditions for different modalities M. Universal H and RM' for instance yield universal necessity O . Kripke's theory I. In Kripke 1959, Kripke defines an interpretation as a system (K,A) such that (i) (ii)
K is a class of possible realizations, and A E K; if (U, Wu' Vu) and (U', Wu', Vu') are members of K, then U= U'
u"
and Vu=V
The truth conditions for atomic formulas and quantification are analogous to those of Tarski's theory. Truth conditions are also given for a kind of universal necessity: T(OF, (K,A»
= tiff T(F, (K,A'» = t for
all A'EK.
These truth conditions make OF equivalent to a formula in a logic of a higher order. Assume, for example, that F contains occurrences of three relation symbols, let us say R I , Rz, and R3 . Let Fxrz be the formula obtained from F by replacing these relation symbols by relation variables X, Y, and Z respectively. Let 4> be a relationship among relations such that 4>(X,Y,Z) iff there is a possible realization (U, Wu, Vu)EK such that X=WU(R I ) , Y= Wu(Rz), and Z= WU(R3) . In that case, OF is always equivalent to (X)(Y)(Z)(4>(X, y,z) ~ FXlZ) . Note that in this theory OF is not an intensional formula. Kripke's theory II. In Kripke 1963, a semantic theory is presented which is essentially as follows . An interpretation in Kripke 's sense 2 is a system (I,j,(U, W u, V u in which (l) I is a class of possible worlds and j E I - each possible world j contains a non-empty domain of individuals d(j); (2) U is the union of all
»
156
STiG KANGER
domains of individuals d(j) with j E I; (3) (U, Wt» Vu) is a possible realization, except that Wu is a binary function defined for relation symbols paired with possible worlds as in Kanger's theory - not for relation symbols alone as in Tarski's theory. We presume that Wu(Rn,i) £ U" and that Vu(x)E U always hold true . The truth conditions for atomic formulas, for example (x=y) and R2(x,y) , are as follows: T«x=y) , (/,j,(U, Wu,Vu))) = tiff Vu(x) = Vu(y) . T(R2(x,y) , (/,j,(U, Wu,Vu))) = tiff < Vu(x), Vu(y» E WU(R2 J) .
E
The truth condition for quantification is: T(VxF, (/,j,(U, Wu, Vu))) = tiff T(F, (/,j,(U, Wu, Vb))) = t for all Vb such that Vb(x)Ed(j) and Vb(Y) = Vu(y) for each individual
variable yother than x.
Thus, quantification does not cover all of U: it covers only d(j), i.e., the individuals in the 'actual' world j. Finally, the truth condition for modal formulas is: T(MF, (/,j,(U, Wu,vu))) iEI such that iRMj .
= tiff T(F, (I, t.(U. Wu,Vu))) = t for all
By choosing interpretations with different relations RM, and different 1 we obtain truth conditions for different modalities. For example, with the universal RM , we get a kind of universal necessity. Scott 's theory. Scott has made several contributions to the development of the semantics for modal expressions - see, especially, Scott 1970. A somewhat simplified version of Scott's theory is given in Chellas 1969. If one disregards certain details, which are unimportant in this context, then Chellas' version is equivalent to the following theory. An interpretation in Scott-Chellas' sense is a system (/,j,(UI , WUI, VUI) in which (1) 1 is a class of possible worlds or indices, andjE I; (2) UI is a class of individual concepts, i.e., a class of functions from 1 into the domain U of possible individuals; (3) (U I , WUI,vUI) is a possible realization modified in the same way as in Kripke's theory II, i.e ., WUI is a function of two variables : relation symbols and indices. The truth conditions for atomic formulas are the same as in Kripke's theory. However, we note that the relations are now relations between individual concepts . The relation of identity, therefore, becomes a stronger form of identity - let us denote it by '~'.
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T«x ~ y), (l,j,(UI,WUI,VUI») = tiff VUI(X) = VUI(y); 2(x,y) , (l,j,(UI,WUI,vul») = tiff < VUI(X), VUI(y) > E T(R E Wu1(R2 ,j) .
The truth condition for the weaker form of identity = is as follows: T«x=y), (I,j, (U I, WUI, VUI»)
atj.
= tiff
VUI(X) and VUI(y) coincide
The truth conditions for quantification and modal formulas are the following: I, T(\;fxF, (l,j, (U I; Wyr/' VUI») = tiff (l,j, (U WUI, ~/))) = t for all ~I such that VUI(Y) = VUI(y) for each individual variable y
other than x;
T(MF, (l,j,(UI,WUI,vul») for all iEI such that iRMj.
= tiff
T(F, (l,i, (UI,WUI,vu1»)
=t
Scott points out that the truth condition for quantification can be limited so that quantification is over concepts for actual (instead of possible) individuals only. (Compare Kripke's theory II and Kanger's theory II.) Scott also points out that the truth conditions for modalities can be given other forms than those with an alternative relation. An interesting development of this observation has been given in Segerberg 1970 and 1971. A comparison. The following sentences are valid according to all of the semantic theories accounted for above. Moreover, they are valid regardless of how the alternative relation RM is chosen. M(F ~ G) ~ (MF ~ MG). M(F& G) ~ MF& MG. MF& MG ~ M(F& G).
Further, the following rules always hold true, where -.=F means that F is valid: -.=F -.=MF
== G) == MG)
-.=(F -.=(MF
According to Kanger, Hintikka, Kripke II, and Scott-Chel!as, it also holds true that certain sentences are valid if RM fulfills certain requirements. For example: MF ~ - M - F, if xRMy always for some x; MF ~ F, if RM is reflexive; F ~ - M - M - F, if RM is symmetric;
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STIGKANGER
MF ::> MMF, if RM is transitive.
Thus, if RM is an equivalence relation - for example, if RM is universal then all of these sentences are valid. In both Carnap's theory and Kripke's theory I, where no alternative relation was introduced (which is equivalent to having a universal alternative relation), all of these sentences are valid. The theories differ, however, when it comes to sentences having both modalities and quantification or identity. The following selection of such sentences (where 0 stands for universal necessity, and F is a sentence without modalities) illustrates the differences. '+' means that the sentence is valid, '-' that it is not. The symbols u and w stand for individual constants in Carnap's and Hintikka's theories and for individual variables in the rest. Camap Kanger Kanger
'v'FOFx ::> OFw 'v'xOF::> O'v'xF O'v'xF ::> 'v'xOF (u
=
w) ::> (OFu
== OFw)
+ + + +
Hintikka
I
II
+ + +
-
+
+ + +
-
-
-
Kripke Kripke scoUI II Chellas
+ + + +
-
+ + +
+
-
+
5. SEMANTICS FOR SOME OF THE OPERATORS IN LANGUAGE L
Let us see how the semantics for some of the operators introduced in the language L has been or can be constructed. Sentences containing the concepts Unav, Shall, and Ought receive truth conditions formulated in the standard way by means of the alternative relations R Unav' R Shall' and ROught' respectively. For example, the truth condition for Ought in Kanger's theory is: T(OughtF, (V, l¥, V)) = t if T(F, (V', W,V)) = t for all V' such that V' ROughtV, (where ROught is a relation between universes such that V' ROughtV iff everything that ought to be the case in V is the case in V').
The truth conditions for Unav and Shall are analogous. R Shali is a relation such that V' ROughtV iff everything that shall be the case in V is the case in V' . R Unav is a relation such that everything that is unavoidable in V is the case in V' . It is natural to conceive of R Unav as a reflexive and, perhaps,
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even as a transitive relation. In Kanger 1957d, it was proposed that ROught be a relation such that: (U)(3V')(V'RoughtU) ; (3U) - (V ROughtU); (3V')[(3U)(V' ROughtU) & (3U)-(V' ROughtU)] .
Similar criteria should hold true for RShali. Furthermore, the following connection between ROught and RShall seems to hold true as well: V" ROughtV ~ (3V')[V" RShaIlV' & V' ROught U].
In other words, the following sentence is valid: Ought Shall F :::> Ought F.
In Hintikka 1957, a similar semantics (but of Hintikka type, of course) for Ought is presented . Fitting 1969 gives a semantics for both Ought and Shall within the framework of Kripke's theory II and with a connection between Ought and Shall which differs slightly from the one here suggested. The semantics for Do(P,F) and Do(P,F) can be constructed in Kanger's theory as follows : T(Do(P,F), (V, W; V» = tiff T(F, (V ', W; V» = t for all V' such that R0 6(V(P, U), V', U),
where R0 6 is a 3-place relation such that R0 6(V(P, U) , V', U) means that everything the person V(p, U) does in V is the case in V'. Note that the assignment V of values to the variables applies to person variables as well as to individual variables of other sorts. The semantics for Do is the following: T(Do(P,F), (V, W; V» = tiff T( - F, (V ', W, V» such that Roo(V(P, U), V', U),
= t for all V'
where Roo(V(P, U), V', U) means that the opposite of everything V(p, U) does in V is the case in V' . We assume that R0 6(V(P, U), V, U) and (3V')R oo(V(p, U) , V', U) always hold true. The semantics for action at time t , Do(p,t,F), is entirely analogous to that of Do(P,F) . Instead of the person p , we have the person p at time t. The alternative relations are accordingly supplied with a time variable: R0 6 [yep, U), V(t, U), V', U] and Roo[V(P' U), V(t, U), V', U]. Semantics for the concept Do is found in Chellas 1969 and in Porn 1970.
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The latter is of Hintikka type and only individual actions are considered. A semantics for the concept Dec has - as far as we know - never been given. However, perhaps it could be made analogous to the semantics for Do, i.e ., by means of two alternative relations ROee and ROlle, except that ROee is not reflexive. 6. ACTIONS
We shall now look into the problem of expressing some fundamental juridical notions within the framework of the language L. We begin with the notion of action. In the philosophy of action there are four main distinctions which have not always been observed in spite of their obvious importance. These distinctions are: Act-Acting. An act is an action expressed by a noun phrase; for instance, moon-walking, murder, handshaking, etc. Acting is action expressed by a sentence; for instance, p walks on the moon, p kills someone, p and q shake hands, p and q shake hands at time t, etc. Acting- Pseudo-acting. Acting involves some sort of activity performed by an agent; for instance, p walks on the moon. Pseudo-acting does not have to involve activity; for instance, p=p, p does not walk on the moon, etc. We also have cases involving acting for some of the agents, but mere pseudo-acting for others; for example, p walks on the moon but q doesn't. l-person acting, 2-person acting, etc. l-person acting is acting in which exactly one person is active; for example, p walks on the moon. Pseudoacting of other agents may be involved. Thus, p walks on the moon but q does not is also an example of l-person acting. 2-person acting is acting in which two persons are active; for instance, p and q shake hands at time t. In general, n-person acting is acting in which n different persons are active. The degenerate case of O-person acting is possible. We have, for example, some p walks on the moon, and mere pseudo-actings such as p = p . Acting- Instances of acting. A type of acting is acting regardless of time; for instance, p and q shake hands. An instance of acting is acting at a certain time; for example, p and q shake hands at time t. Note, however, that there are all sorts of borderline cases between types of acting and instances of acting; for example, p takes his morning walk. So, the distinction we need is rather that between acting (in general) and instances of acting (i.e ., acting at a particular given time t). In the philosophy of action there are three main problems which, so far, have not been observed to the extent which they deserve. These problems
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are the characterization problem for acting, the elimination problem for acts and the identity problem for acts. The characterization problem for acting. What distinguishes acting from mere pseudo-acting? A good solution to this 'characterization' problem for acting is needed to make the notion of n-person acting meaningful. The problem is a good problem for Oxford-philosophical investigations , but, in this context, we prefer a solution given within a certain theoretical framework, namely the language L. And there is perhaps such a solution. A formula F without occurrences of person names expresses an n-person acting if and only if there is a choice of n person variables PI " ..,Pn such that (PI)" '(Pn)(F == DO(Pt> .. ·,PwF» & (3PI).. .(3Pn)(PI,. .. -P« are distinct & F)
is true. A formula F with occurrences of person names expresses an nperson acting if and only if the formula F' obtained from F by substitution of person variables for the names expresses an n-person acting. The case of instances of n-person actings is analogous, except that a time variable is involved. For example, "Kim opens this window at 10 o'clock" expresses an instance of a 1-person acting, since, for each P and t, P opens this window at t if and only if P sees to it at t that P opens the window at t, and there is also a (distinct) P and a t such that P opens this window at t. We assume that the variables PI,oo .,Pn (and t) are uniquely determined in case F express an n-person acting (or acting instance). These variables will be said to stand for the active persons (and time) in the acting (instance) expressed by F . The elimination problem for acts. The elimination problem for noun phrases denoting acts is very similar to the elimination problem of class abstraction and definite description . First we put the phrases denoting acts in a standard form. For example, 'moon-walking ' will be rephrased in standard form as 'the act done by every P such that P walks on the moon' . We also have pseudo-acts; for instance, the pseudo-act done by every P such that P does not walk on the moon. In general, a noun phrase denoting an act (or pseudo-act) will be rephrased in standard form as: the (pseudo-)act done by PI" " ,Pn such that F. Such an expression denotes an n-person act if and only if F expresses an acting and PI>' ,Pn stand for all the active persons in that acting. Similarly, a noun phrase denoting an instance of an act (or pseudo-act) will be rephrased in standard form as: the (pseudo-)act instance done by Pt>''',Pn at t such that F . Such an expression denotes an instance of an n-person act if and only if F expresses an acting at t and PI" " ,Pn stands for all the active persons in that acting at t. Now, the elimination problem 00
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is simply that of giving a contextual definition of noun phrases for acts and instances of acts and also, perhaps, of pseudo-acts and instances of pseudoacts. This problem is almost trivial with respect to one particular kind of context, namely, the context of agents so and so doing the act such and such. To be more precise: persons qlo .. . ,q n do the act done by Plo " "Pn such that F p1," . rpn if and only if, F q1,··· »qn: (Here, of course, the bound variables in Fp 1, ' " rpn are assumed to be written in such a way that the variables ql, ... ,qn when substituted for the free occurrences of Pl, ... ,Pn in Fp1""'pn are not bound in Fq1""'qn') The elimination problem is also easily solved with respect to contexts such as 'moon-walking is forbidden' . This is reduced to either: (p)Shall- (p walks on the moon), or : Shall(p) - (p walks on the moon). But elimination in other contexts is more difficult. For instance, it is not clear how to eliminate 'moon-walking' in the context 'moon-walking is nice' unless we introduce some queer new operators in the language. The identity problem. There is a third main problem in the philosophy of action - when are acts identical? In other words, the problem is to find a non-trivial necessary and sufficient criterion for acts to be distinct. This is the identity problem for acts. We shall note that the identity problem does not arise in the framework of the language L (as developed so far), except, perhaps, as a special case of the elimination problem. It will arise, however, if we introduce acts into our ontology and extend the L-Ianguage with variables (and quantifiers) for acts and predicates and relations of acts; for instance, with the predicates of being forbidden or being nice and the relation of identity. But, since we have not chosen to extend the language in this way, we may perhaps leave the identity problem open for the time being. 7. RIGHTS
The next fundamental legal conception is that of a right. The classical reference is Hohfeld 1913, in which the first attempt to make a logical analysis of rights is made. Two modern expositions and closely related explications of Hohfeld's theory may be found in Ross 1953 and Moritz 1960. An interesting logical analysis is given in Anderson 1962. A different logical development of the theory of rights is presented in Kanger & Kanger 1966 and in Porn 1970. It is based on an explication given in Kanger 1957d of Hohfeld's distinctions between four simple types of rights, viz., Claim, Power, Immunity , and Freedom. These types of rights were conceived as 3-place relations between two parties ex and {3 and a state
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of affairs, or condition, involving IX and {3. The explication can be formulated in the framework of the language L: (1) (2) (3)
(4)
IX has a claim against (3 with respect to F if and only if Shall Oo({3,F). IX has a power against (3 with respect to F if and only if May OO(IX,F). (Note that May = - Shall- .) a has an immunity against (3 with respect to F if and only if Shall- Oo({3, - F). a has a freedom (or liberty) against (3 with respect to F if and only if - Shall Oo(a, - F).
Example: Consider four cases: (i) (ii)
(iii) (iv)
p P p p
has has has has
a claim against q to get back what he lent q. a power against q to read q's latest book. an immunity against q to read q's latest book . a freedom against q to visit q in his home.
According to the explications suggested, these four cases are, respectively, equivalent to : (i') (ii') (iii')
(iv')
q shall see to it that p gets back what he lent q. P may see to it that p reads q's latest book. q may not see to it that p does not read q's latest book. it is not so that p shall see to it that p does not visit q in his home.
Four other simple types of rights, being the 'correlatives' (to use Hohfeld's term) of claim, power, immunity, and freedom, can be defined by a permutation of the two parties involved. Thus , IX has a duty, liability, disability, or exposure (or 'no-right') against {3 with respect to F if and only if {3 has a claim, power , immunity, or freedom, respectively, against IX with respect to F. When we talk about rights, we usually mean either a claim (as in the case of p having a right against q to get back what he lent q) or a power combined with an immunity (as in the case of p having a right against q to read q's latest book). The combination of power and immunity is rather frequent . When, for instance, the U.N. Declaration of Human Rights states that everyone has the right to leave his own country, what is presumably meant is that each p - or at least each ordinary mature person p - has a power and an immunity against the authorities of his own country to leave the country.
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There are several other possible combinations of the simple types of rights. A systematic survey is given in Kanger & Kanger 1966 and in Porn 1970. Note the distinction between a type of rights and a right. The latter is a two-place relationship between two parties which is defined once the type of rights and the state of affairs involved are specified. 8. INFLUENCE
The notion of influence seems to be very similar to that of a right. We may distinguish simple types of influence which a party may have in relation to another party with respect to a state of affairs or condition. These types are similar to the simple types of rights . The main difference is that we replace Shall by Unav, and May by Can = - Unav - . The influence type corresponding to duty, for instance, is that of being forced. Thus, ex is said to be forced in relation to (j to F if Unav Do(ex,F) . And ex may be said to have a power in relation to (j with respect to F if Can Do(ex,F). And ex has an irresistible power in relation to (j with respect to F if and only if Can Do(ex,F) & Unav - Do({j, - F) . This type of influence corresponds to the combination of power and immunity in the case of rights. There are, of course, several possible combinations of simple types of influence. A systematic survey can be carried out in the style of Kanger & Kanger 1966. There is another sort of distinction which has to do with the overlapping of the Do-operator. In the case of power, for example, we must distinguish 'power in relation' to a party from 'power over' a party. The party ex is said to have power over the party (j with respect to F if Can Do(ex, Do({j,F). This must, in tum, be distinguished from the excercising of power. ex is said to excercise power over (j with respect to F if Do(ex, Do({j,F) . Again, this has to be distinguished from Do(ex, Can Do({j,F), etc. Finally, time variables may be introduced in order to define types of instances of influence, or influences at a time t. The notions of influence and power play an important role in political science. (See, for example, Cartwright 1965.) The first attempt at a classification of these notions along lines which in some respects are similar to those we have sketched here was made in Oppenheim 1961. A more systematic attempt is that of Porn 1970 and 1971.
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9. RESPONSmILITY
The notion of responsibility is probably more complex than the notions discussed above. To begin with, we have to distinguish responsibility, in general, from responsibility to an agent (or authority). The latter notion seems to contain little of interest in addition to the first one. In most cases, responsibility to p for F boils down to responsibility for F plus a duty to report to p about F. The preliminary analysis of a is responsiblefor F, which we, suggest, is: a is blameworthy for F or a is praiseworthy for F. The next step is a definition in the L-Ianguage of a is blameworthy for F . The definition which we suggest is the one given in Kanger 1957d, when a consists of one person p: (1) (2) (3)
(4) (5) (6)
Shall-F; Do(p,F) ;
- Unav Do(p,F); Can Conv(p, Shall- F); Can Conv(p, Do(p,F»; Can Conv(p, - Unav Do(p,F».
The definition of praiseworthiness is similar, except that 'Shall- F' in (1) and (4) is replaced by 'ShallF'. 10. PREFERENCE
In a system of law we often need to express facts like ---- is better than .. .. . The logic of the relation better than (preference logic) has been extensively studied in recent years. (Some of the contributions of interest in this connection are listed in the bibliography.) There is a less orthodox approach to preference logic: instead of using a binary preference relation, we use a quarternary - F is for the purpose of P better than G is for the purpose of Q. Then, we ask, How good is F for the purpose of P? A measure can, perhaps, be constructed in the following way: g(F,P) =
pr(PI F)
pr(PI-F)
.
Here, pr(PIF) is a conditional probability measure of P given F. For example, How good is sunshine today for the purpose of my having a nice day on the beach? The measure is the probability of my having a nice day
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on the beach, given sunshine today, divided by the probability of my having a nice day on the beach, given no sunshine today . The preference relation is then defined by a comparison of the measures. NOTE •
This work was supported by a grant from the Wenner-Gren Foundation. BIBLIOGRAPHY
Journal MULL (Modern uses of logic in law), 1959-66. Continued 1966 as Jurimetrics Journal.
General bibliography Conte, A.G. 1961. Bibliograjia di logica giuridica 1936-1960, Rivista internazionale di jilosojia del diritto, vol. 38, 1961. (See also MULL 1962.)
General introduction to jurisprudence Ross, A. 1953. Om ret og retfardighed, Copenhagen, 1953. Translation: On lawandjustice, London, 1958.
General introductions to logic applied to law Kalinowski, J. 1965. Introduction a la logique juridique, Paris, 1965. Klug, U. 1958. Juristische Loglk, 2d ed. , Berlin, 1958. Schreiber, R. 1962. Logik des Rechts, Berlin, 1962. Tammelo, I. 1969. Outlines of modern legal logic, Wiesbaden, 1969. Weinberger, O. 1970. Rechtslogik, Vienna, 1970.
Semantics Carnap , R. 1946. "Modalities and quantification," The journal of symbolic logic, vol. 11 (1946), pp. 33-64. Carnap, R. 1947. Meaning and necessity, Chicago, 1947. Chellas, B. 1969. The logicalform of imperatives, Stanford, 1969. Hintikka, J. 1957. Quantifiers in deontic logic. Societas Scientiarum Fennica, commentationes humanarum Iitterarum 23: 4, Helsinki, 1957. Hintikka, J. 1961. "Modality and quantification," Theoria, vol. 27 (1961), pp. 119-128. Hintikka, J. 1962. Knowledge and belief' An introduction to the logic of the two notions, Ithaca, N.Y. , 1962. Hintikka, J. 1963. "The modes of modality," Acta philosophica Fennica , fasc. 16 (1963), pp.65-81. Kanger, S. 1957a. Provability in logic, Stockholm, 1957. Correction: "On the characterization of modalities," Theoria, vol. 23 (1957), pp. 152-155 . Kanger, S. 1957b. "The Morning Star paradox," Theoria , vol. 23 (1957), pp. 1-11.
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Kanger, S. 1957c. "A note on quantification and modalities," Theoria, vol. 23 (1957), pp. 133-134. Kanger, S. 1957d. New foundations for ethical theory, Stockholm, 1957. Reprinted in Hilpinen (1971), pp. 36-58. Kripke, S. 1959. "A completeness theorem in modal logic," Thejournal of symbolic logic, vol. 24 (1959), pp. 1-14. Kripke, S. 1963a. "Semantical analysis of modal logic I: Normal modal propositional calculi," Zeitschrijt fur mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 67-96. Kripke, S. 1963b. "Semantical considerations on modal logic," Acta philosophica Fennica, fasc. 16, Helsinki, 1963, pp. 83-94. Porn, I. 1970. The logic ofpower, Oxford, 1970. Scott, D. 1970. "Advice on modal logic" in K. Lambert (ed.), Philosophical problems in logic: Some recentdevelopments, Dordrecht, Holland, 1970, pp. 143-173 . Segerberg, K. 1971a. "Some logics of commitment and obligation" in Hilpinen (1971), pp.148-158. Segerberg, K. 1971b. An essay in classical modallogic, Uppsala, 1971. Tarski, A. 1936. "Der Wahrheitsbegriff in den formalisierten Sprachen," Studia philosophica, vol. 1 (1936), pp. 261-405.
Deontic logic Anderson, A.R. 1956. The formal analysis of normative systems. Technical report no. 2, contract no. SARlnonr-609 (16), Office of Naval Research, Group Psychology Branch, New Haven, Conn., 1956. Reprinted in N. Rescher (ed.), The logic of decision and action, Pittsburgh, 1967, pp. 147-213. Fitting, M. 1969. "Logics with several modal operators," Theoria, vol. 35 (for 1969, publ. 1970), pp. 259-266. Hansson, B. 1969a. "An Analysis of Some Deontic Logics," Nous, vol. 4 (1970), pp. 373398. Reprinted in Hilpinen (1971), pp. 121-147. Hilpinen, R. 1971 (ed.). Deontic logic: Introductory and systematic readings, Dordrecht, Holland, 1971. Hintikka, J. 1957. Quantifiers in deontic logic. Societas Scientiarum Fennica, commentationes humanarum Iitterarum 23: 4, Helsinki, 1957. Kanger, S. 1957d. New foundations for ethical theory, Stockholm, 1957. Reprinted in Hilpinen (1971), pp. 36-58. von Wright, G.H. 1951a. "Deontic logic," Mind, n. s. vol. 60 (1951), pp. 1-15. Reprinted in G.H. von Wright, Logical studies, London, 1957, pp. 58-74. von Wright, G.H. 1951b. An essay in modal logic, Amsterdam, 1951. von Wright, G.H. 1964. "A new system of deontic logic," Danish yearbookof philosophy, vol. 1 (1964), pp. 173-182. Correction in Danish yearbook ofphilosophy, vol. 2 (1965). Reprinted in Hilpinen (1971), pp. 105-120.
Commented bibliography in deontic logic and related topics von Wright, G.H. 1968. "The logic of practical discourse" in R. Klibansky (ed.), Contemporaryphilosophy: Logicandfoundations of mathematics, Florence, 1968, pp. 141-167.
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Action Chellas, B. 1969. The logical form of imperatives, Stanford, 1969. Davidson, D. 1967. "The logical form of action sentences" in N. Rescher (ed.), The logic of decision and action, Pittsburgh, 1967, pp. 81-95. Kenny, A. 1963. Action, emotion, and will, London, 1963. Porn, I. 1970. The logic ofpower, Oxford, 1970. Porn, I. 1971. Elements of social analysis, Uppsala, 1971. von Wright, G.H. 1963. Norm and action, London, 1963. von Wright, G.H. 1967. "Logic of action" in N. Rescher (ed.), The logic of decision and action, Pittsburgh, 1967, pp. 121-136.
Rights, power, and influence Anderson, A.R. 1962. "Logic, norms and roles," Ratio, vol. 4 (1962), pp. 36-49. Cartwright, D. 1965. "Influence, leadership, and control" in J.G. March (ed.), Handbook of organizations, Chicago, 1965, pp. 1-47. Hohfeld, W.N. 1913. "Some fundamental legal conceptions as applied in judicial reasonin," Yale lawjournal, vol. 23, 1913, pp. 16-59. Reprinted in W.N. Hohfeld, Fundamental legal conceptions as appliedin judicial reasoning, and other legal essays, New Haven, Conn., 1923. Kanger, S. 1957d. New foundations for ethical theory, Stockholm, 1957. Reprinted in Hilpinen (1971), pp. 36-58. Kanger, S. and Kanger, H. 1966. "Rights and parliarnentarism ," Theoria, vol. 32 (1966), pp.85-115. Moritz, M. 1960. Uber Hohfelds System derjuridischen Grundbegrijfe, Lund, 1960. Oppenheim, F. 1961. Dimensions offreedom, New York, 1961. Porn, I. 1970. The logic ofpower, Oxford, 1970. Porn, I. 1971. Elements of social analysis, Uppsala, 1971.
Decision and preference Arrow, K.J. 1963. Social choice and individual values, 2d ed., New York, 1963. Danielsson, S. 1968. Preference and obligation, Uppsala, 1968. Hallden, S. 1957. On the logic of "better", Lund, 1957. Hansson, B. 1968a. "Fundamental axioms for preference relations," Symhese, vol. 18 (1968), pp. 423-442. Hansson, B. 1968b. "Choice structures and preference relations," Synthese, vol. 18 (1968), pp. 443 -458. Hansson, B. 1969b. "Group preferences," Econometrica, vol. 37 (1969), pp. 50-54. Hansson, B. 1969c. "Voting and group decision functions," Synthese , vol. 20 (1969), pp. 443 -458. Kanger, S. 1968. "Preferenslogik" in Hj. Wennerberg (ed.), Nio filosofiska studier tilliignade Konrad Marc-Wogau, Uppsala, 1968. Luce, R.D. and Raiffa, H. 1957. Games and decisions, 1957. Luce, D. and Suppes, P. 1965. "Preference, utility and subjective probability" in R.D. Luce, R.R. Bush and E. Galanter (eds.), Handbook of mathematical psychology vol. 3, 1965.
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Porn, I. 1971. Elements of social analysis, Uppsala, 1971. Rescher, N. 1967. "Semantic foundations for the logic of preference" in N. Rescher (ed.), The logic of decision and action, Pittsburgh, 1967. Segerberg, K. 1970. "Kripke-type semantics for preference logic" in T. Pauli (ed.), Logic and value. Essays dedicated to Thorild Dahlquist on hisfiftieth birthday, Uppsala, 1970.
SOME ASPECTS ON THE CONCEPT OF INFLUENCE
1. INTRODUCTION
In the essay "Rattighetsbegreppet" (The Concept of Rights) (1963), I conceived of rights as relations between pairs of parties with respect to states of affairs . In a similar vein, we can conceive of influence as a relation between two parties with respect to a state of affairs. And just as there are several different rights-types, there are also different influence-types. Rights relations were analysed (or could have been analysed) in terms of the notions of May and Sees to it that. The rights type Power, for example, was interpreted so that the statement X has a right versus Y of the type power with respect to the state of affairs S(X,Y) became synonymous with the statement It may be the case that X sees to it that S(X,Y). We can now preliminarily conceive of analysing influence in an analogous manner, in terms of the concepts of Possible and Sees to it that. The influence type Ability, for instance, would then be interpreted such that the statement X has versus Y an influence of the type ability with respect to S(X,Y) becomes synonymous with the statement It is possible that X sees to it that S(X,Y). When I say that something is possible, I mean that it is practically possible. Thus it is not a question of only being conceivable in principle.
170 G. Holmstrom-Hintikka. S. Lindstrom and R. Sliwinski [eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. J. /70-/78. © 2001 All Rights Reserved. Printed by Kluwer Academic Publishers. the Netherlands. Originally published in Swedish as 'Nagra synpunkter p! begreppet inflytande' in Filosojiska smulor tillagnade Konrad Marc -Wogau, 75 11r, Filosofiska smulor utgivna av Filosofiska flireningen och Filosofiska institutionen vid Uppsala universitet 27, Uppsala 1977, 12-23. Translated by Sharon Rider.
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171
Since the logic of Possible resembles that of May , one might think that the theory of influence-types could be developed by analogy with the theory of rights-types as it was presented in "Rattighetsbegreppet". A sugge stion along these lines was made in my essay, "Law and Logic " (Theoria, 1972). But I rather soon realized that such a theory would not be satisfactory, and that the preliminary analysis of the concept of influence sketched above had to be revised. The aim of this essay is to propose an improved and more adequate theory for influence-types. 2. A COUNTER-EXAMPLE
Let us first take a closer look at the concept of Sees to it that. When X sees to it that the state of affairs S is the case, it is reasonable to assume that X sees to it that S by (or with the help ot) some measure which , within the scope of the natural and social order that leads to it that S is the case . Example: When X sees to it that the light is switched off, X sees to it by turning off the switch. When the state sees to it that the speed of traffic is limited , this occurs with the help of a traffic ordinance. Notice that a measure need not consist of an active doing. If the light is already off, X can see to it that it remains switched off, for example, by preventing someone from turning on the switch . But if nobody comes by wishing to turn it on, then X's measure need not consist in anything more than a certain readiness to intervene. We now interpret the sentence X sees to it that S to be equivalent with the sentence There is a measure A such that X sees to it that S by means of A. Let us now use the usual symbolism "(3A)" for "There is a measure A such that", and let "St(X,S)", "St(X,S,A)" be abbreviations for "X sees to it that S" and "X sees to it that S by means of A", respectively. The preliminary interpretation of the sentence X has versus Y an influence of the type ability with respect to S(X ,Y) can then be explicated as Poss (3A) St(X,S(X, Y),A) .
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An example now shows that this preliminary interpretation is too weak to be adequate: almost anyone gets the ability to do almost anything . In Sweden, like most other people, I have the ability to stay abroad for a period of time. A simple measure that leads to this is to join some suitable tour . But comrade X in the closed country Z, where foreign travel is effectively prevented, cannot reasonably be said to have the ability to stay abroad for a period of time . Despite this, however, he must be said to have such an ability, if one maintains the preliminary interpretation of the concept of ability. For even if no measures leading to staying abroad for X are available in Z, they are entirely possible all the same : the regime in Z might, for example , with a simple change in the regulations, allow Cook's travel agency or some other charter company to establish themselves in Z. The example gives an indication that the interpretation of the sentence X has versus Y an influence of the type ability with respect to S(X,Y) ought to be strengthened to (:3A) Poss St(X,S(X,Y),A) . This interpretation seems adequate; in any case, it no longer allows for the claim that comrade X has the ability to stay abroad. In order for that to be the case, it is now required that there actually are such measures A that it is practically possible that X, with the help of A, sees to it that X stays abroad, and of course there existed no such measures in Z. 3. ATOMIC TYPES OF INFLUENCE
In "Rattighetsbegreppet", the so-called atomic types of rights were defined by non-contradictory maximal conjunctions of formulas which have either the form May St(X,S,(X,Y» or May St(Y,S(X, Y» or which were obtained from these formulas by the insertion of a negation symbol in front of "May", "St' or "S(X,Y)" . These conjunctions were 26 in number. We can now in a similar manner define atomic influence-types through non-contradictory maximal conjunctions of expressions like
SOME ASPECTS ON THE CONCEPT OF INFLUENCE
173
(:=JA) Poss St(X,S(X, Y),A) or (:=JA) Poss St(Y,S(X,Y),A) or which are obtained from these expressions by inserting a negation symbol in front of "(:=JA)" , "Poss", "St" or "S(X, Y)". Let us call those expressions simple influence-sentences. At first sight, it might appear that we would then get significantly more than 26 atomic types. But due to two almost selfevident but important principles, the number is reduced drastically. The principles are: I. II.
(:=JA) not Poss St(X,S,A) Poss not St(X,S,A).
The first principle establishes the trivial fact that there are always measures by means of which it is impossible to see to it that S. It is, for example, impossible to see to it that the light is turned off by counting to three. The second principle states that it is always possible that X refrains from seeing to it that S by means of A, and this is of course the case, since X can always remain passive. This principle is not as entirely self-evident as is the first one. For instance, one could ask, whether there are no cases in which X is forced by some other party or by external circumstances to see to it that S by means of A. But in such cases, in my mind, claiming that X is the one who sees to it that S is a mistake - X is rather to be regarded as an instrument in the hands of the other party or as a cog in the wheel of circumstances. In other words, the presence of constraint does not alter the validity of Principle II. We now see that when atomic influence-types are defined with the help of non-contradictory conjunctions of simple influence-sentences, there is no need to take into account those simple influence-sentences that have a negation sign in front of neither "Poss" nor "St" . For these expressions, by virtue of Principles I and II, are either always true, and therefore redundant in conjunctions, or always false, and therewith excluded in non-contradictory conjunctions. The only simple influence-sentences we need to care about are those which lack the negation sign before "Poss" or "St". Let us introduce a simplified notation for these sentences and write CanSt (X,S) in place of the longer expression (:=JA) Poss St(X,S,A).
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STIGKANGER
CanSt(X,S) reads as: X can see to it that S will be the case . Note that CanSt(X ,S) is not synonymous with Poss St(X;S). We can now give an overview ofthe influence-types . We begin with four simple types of influence which may be called Ability, Security , CounterAbility and Counter-Security. The sentences X has versus Y an influence of the type ability with respect to S(X,Y) X has versus Y an influence of the type security with respect to S(X,Y) X has versus Y an influence of the type counter-ability with respect to S(X, Y) X has versus Y an influence of the type counter-security with respect to (S(X, Y) are interpreted, respectively, as CanSt(X,S(X, Y» Not: CanSt(Y, not-S(X,Y» CanSt(X,not-S(X,Y» Not: CanSt(Y ,S(X,Y» . The atomic influence-types can now be construed as combinations of simple influence-types or negations thereof. The combination Ability, Security, Counter-Ability and Counter-Security, for example, constitutes an atomic influence-type, and the other influence-types can be obtained from this by the denial of one or several of the simple types included. For instance, the sentence X has versus Y an influence of the atomic type Ability, Security, Not Counter-Ability, Not Counter-Security with respect to S(X,Y) will be synonymous with the conjunction CanSt(X,S(X,Y» and Not: CanSt(Y ,not-S(X,Y» and Not: CanSt(X,not-S(X, Y» and CanSt(Y ,S(X,Y». The number T of atomic influence-types that can be construed in this manner is 16. None of the atomic types is empty : examples of everyone of the 16 types of influence can be given. They are , furthermore, exhaustive :
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175
given two parties X and Y, and a state of affairs S, there is exactly one atomic influence-type T, such that X has versus Y an influence of type T with respect to S. My influence versus my colleague Y with respect to a certain window at our office being open is of the type Ability, Not Security, Counter-Ability, Not Counter-Security . My influence versus Y with respect to the withdrawal of 100 crowns from my checking-account is of the type Ability, Security, Counter-Ability, Counter-Security , etc. 4. INFLUENCE OF HIGHER-ORDER
The state of affairs to which an influence applies need not always be simple and settled, as for example, that I stay abroad. In many interesting cases, the state of affairs is itself a matter of influence, for instance, my ability to stay abroad. What we have then is something which perhaps could be called influence of the second order . As an example of second-order influence, let us take the influence which the closed country Z has versus comrade X with respect to X's ability to stay abroad for a period of time. Some little reflection tells us that Z has versus X an influence of the atomic type Ability, Not Security, Counter-Ability and Counter-Security with respect to the state of affairs .that X versus Z has an influence of the type Ability with respect to X's staying beyond Z's borders for a certain period of time. Of course , a second-order influence need not apply only to the ability of a party. It can also apply to his security, counter-ability or counter-security. These influences of the second-order are of special interest when one wishes to give a deeper perspective on the influence one party , X, has versus a counterparty, Y, with respect to a state of affairs S(X, Y), or if one wishes to specify more thoroughly in which type of influence-state X stays in relation to Y with respect to S(X, Y). Comrade X, for example, had an influence versus the state of Z of the type Not Ability, Not Security, Counter-Ability and (presumably) Not Counter-Security with respect to the state of affairs that X stays beyond the borders of Z. This tells a good deal but not all about the state in which X finds himself with respect to travel abroad. Further information about the situation is obtained , for instance, by the addendum that X has the same weak type of influence versus Z also with respect to X's ability to stay outside Z's borders. An answer to the question about which type of influence state X finds himself in versus Y with respect to S(X,Y) could be thought to consist in that one specifies which atomic types of influence X has versus Y with respect to
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STIGKANGER
each and every one of the following five states of affairs: (1)
(2) (3) (4) (5)
S(X, Y), CanSt(X,S(X, Y», Not: CanSt(Y, not-S(X,Y», CanSt(X, not-S(X,Y», Not: CanSt(Y,S(X,Y».
The last four express X's ability, security, counter-ability, and countersecurity, respectively, versus Y with respect to S(X,Y). This is not, however, sufficient. We should also specify the extent to which the counterparty Y actually exercises his influence versus X. As regards comrade X's situation, for example, it is not insignificant that country Z, being closed, actually exercises its counter-ability with respect to X's ability to travel abroad, i.e. that Z actually sees to it that X cannot see to it that X stays outside of Z. The distinction between influence (or potential influence) and exercised influence is simple. Only influence of the type ability or of type counterability can be exercised. To say that Y exercises his ability versus X with respect to S(X, Y) means that Y sees to it that S(X, Y). That Y exercises his counter-ability versus X with respect to S(X, Y) means that Y sees to it that not-S(X, Y). Y's ability versus X corresponds to X's lack of counter-security versus Y, and Y's counter-ability versus X corresponds to X's lack of security versus Y. When Y exercises his ability (counter-ability) versus X with respect to S(X, Y), we say that X, in his lack of counter-security (security), is subjected versus Y with respect to S(X,Y) . Thus a more comprehensive answer to the question of which type of influence-state X finds himself in versus Y with respect to S(X, Y) ought to do more than specify which atomic type of influence X has versus Y with respect to the five states of affairs mentioned above . In the specification of these atomic types, it should also be indicated when X is subjected in his lack of counter-security or security (should such a lack be the case) . If we now limit ourselves exclusively to the possibility of specifying influence-states in this way, the number of conceivable states would be quite large. But, in reality, the number of influence states is significantly smaller, since most of the conceivable specifications are inconsistent with a couple of self-evident principles. According to the first principle, if X sees to it that the state of affairs S is the case, then it is the case:
SOME ASPECTS ON THE CONCEPt OF INl<'LUEt-.lCE
III
177
If St(X,S) then S.
This principle entails, among other things, that one cannot simultaneously exercise both ability and counter-ability. The next principle entails that only ability and counter-ability can be exercised: IV
If St(X,S) then CanSt(X,S).
Next three principles concern second-order influence: V VI VII
If CanSt(X, CanSt(X,S» then CanSt(X,S) If CanSt(X,S) then CanSt(X,CanSt(Y,S» If CanSt(X,S) then CanSt(X, not-CanSt(Y,not-S».
Principle VI says that if X can see to it that S, then X can also see to it that Y can see to it that S; X can, for example, place himself at Y's disposal. This principle does not apply generally, without exception . It does not apply, for instance, if X cannot have contact with Y. But if we are willing to restrict our theory for second-order influence to cases in which the parties can communicate with each other, then, I think, the principle can be accepted . The last principle, VII, says that if X can see to it that S, then X can also see to it that Y cannot see to it that not-S; X can do this, for example, by actually seeing to it that S (and sticking to it) and, in this way, presenting Y with a
fait accompli.
The above list of principles which reduce the number of influence-states which are in practice possible makes no claim as completeness. There are certainly further such principles to be found. Aside from them, we also have principles of a more logical character, for example, those which show that expressions which are provably equivalent can be exchanged for one another in the kind of influence-expressions which we have made use of in the characterization of influence-types or influence-states. But I shall not go into this here. 5. SOME REFERENCES
The outline of a typology for the concept of influence that I have sketched above has many affinities with a theory put forward by Ingmar Porn in his book, Action Theory and Social Science (1977). I warmly recommend that book to anyone interested in the concept of influence and related concepts, such as power and control, which play such an important role in social philosophy and political science. (An earlier interesting study of influence, power and control can be found in Felix Oppenheim's book, Dimensions of
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Freedom (1961) . Oppenheim's ideas, however, are only distantly related to Porn's .) Naturally, influence can concern rights as well as other states of affairs. On this , see Chapter 6 of Lars Lindahl's dissertation, Position and Change (1976), in which, among other things, Lindahl gives an account of Hohfeld's theory of power.
ON REALIZATION OF HUMAN RIGHTS
BACKGROUND
In my paper "Rattighetsbegreppet" (1963) I adopted the view that a right often is (or can be conceived as) a relation which one party has to another in respect of a state of affairs. Rights can be of different types; for example, claim, power, or immunity. When a person X, for instance, has a claim to recover what he has lent to Y, this can be understood as X's having a rights-relation of type claim to Y in respect of the state of affairs that X receives from Y what he has lent to Y. The development of a typology for the various rights-relations has been a concern of legal philosophy ever since Bentham.' My own work in this area took as its starting point the idea that types of rights-relations could be defined in terms of the notions Shall, May and Seeing to it, where Shall and May are interdefinable in accordance with the scheme: May = not-Shall-not Thus, Claim, for instance, is interpreted so that party X is said to have a claim in relation to Y with respect to the state of affairs S(X, Y) if it shall be the case that Y sees to it that S(X,Y). This could be expressed by means of the formula Shall Do(Y,S(X,Y)) where the Do-operator stands for the Seeing to it. The rights-type Power (not to be confused with strength or the influencetype of Ability) is interpreted so that X is said to have a power in relation to Y with respect to S(X,Y) if it may be the case that X sees to it that S(X,Y) . In formulas: May Do(X,S(X,Y)) X is said to have an Immunity in relation to Y with respect to S(X,Y) if it shall be the case that Y does not see to it that non-S(X,Y): 179
G. Holmstrom-Hintikka, S. Lindstrom and R. Sliwinski [eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. I. 179-1 85. © 2001 All Rights Reserved. Printed by Kluwer Academic Publishe rs. the Netherlands. Originally published in Ghita Holmstrom and Andrew J.I. Jones (eds.), Action. Logic and Social Theory. Festschrift Dedicated to Ingmar Porn on the Occasion ofHis Fiftieth Birthday, Acta Philosophica Fennica 38, Helsinki. 1985.71-78.
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Shall - Do(Y, - S(X, Y» where - stands for negation. Types of rights-relations are very often combined types . Article 13.2 of the United Nations Declaration of Human Rights , for instance, states that : Everyone has the right to leave any country, including his own, and to return to his country. This certainly implies that everyone has, in relation to the authorities of his country, both a power and an immunity to leave the country. Such combined types of rights-relations which X can have in relation to Y with respect to S(X, Y) could be specified by means of conjunctions of formulas like Shall Do(X,S(X,Y» Shall Do(Y,S(X,Y» or formulas obtained from these two by inserting a negation symbol before S(X,Y) or the Do-operator or the Shall-operator or before two or all three of these expressions. An "atomic" type of rights-relations is a combined type specified by a maximal consistent conjunction of (eight) such formulas . If we assume that Seeing to it is a success verb (i .e., that Do(X,S) always implies S) we need little more than a fragment of standard deontic logic, and some instantiating examples, in order to show that there are exactly 26 such consistent non-vacuous conjunctions and , consequently, 26 atomic types of rights-relations . It follows from these constructions that, given any two parties X and Y and a state of affairs S(X, Y), there is exactly one atomic type T of rightsrelations such that X has to Y a rights-relation of type T with respect to S(X,Y) .2 A rule of rights most often can be put in the form: (R)
For each party X and Y satisfying the condition C(X,Y), X has to Y a rights-relation of atomic type T in respect of the state of affairs S(X,Y).
A simple, but illustrative, example is article 13.2 of the United Nations Declaration quoted above. This article expresses several rules of rights amongst which is the following: (A)
For each individual X and state Y such that X is a normal noncriminal adult citizen of Y residing in Y, X has to Y a rights-
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181
relation of the type power and immunity with respect to L(X,Y) and also with respect to non-L(X,Y) where L(X, Y) is the state of affairs that X leaves the territory of Y. The atomic type here is sufficiently specified by the following conjunction of formulas: -Shall Shall - Shall Shall
-Do(X,L(X,Y» - Do(Y, - L(X,Y» - Do(X, - L(X,Y» - Do(Y,L(X,Y»
(The remaining formulas of the maximal conjunction need not be added since they follow logically from these four .) Some rules of rights are valid (in the sense of being true). To be valid, three ingredients of a rule must be in harmony: (i) (ii) (iii)
the the the the
condition C which specifies the category of parties to which rule applies, type T of rights-relations and state of affairs S which the rule is all about.
The rule (A) derived from article 13 of the U.N. Declaration stated above is, I am convinced, a good example of a valid rule. The problem, if a rule is valid or not, should (of course)not be confused with the problem of why a valid rule is valid, or the problem of how we know that a valid rule is valid or that a non-valid one is non-valid (if we can know that at all). Neither should it be confused with the problem of what it is for a rule to be valid. COMPLYING WITH A RULE OF RIGHTS
Let us again consider the schematically formulated rule (R)
For each X and Y such that C(X,Y), X has to Y a rights-relation of atomic type T in respect of S(X,Y)
Now, given two parties X and Y such that C(X,Y), rule R might imply, depending on the type T involved, that (say) Y shall see to something or refrain from seeing to something. If Y does what he shall, Y complies with the rule in relation to X. Or, to be slightly more formal: If C(X,Y), Y is said to comply with R in relation to X if:
182
(1) (2) (3) (4)
STIG KANGER
Do(Y,S(X,Y», if Shall Do(Y,S(X,Y» follows from C(X, Y) and R -Do(Y,S(X,Y», if Shall -Do(Y,S(X,Y» follows from C(X, Y) and R Do(Y,-S(X,Y», if Shall Do(Y,-S(X,Y» follows from C(X, Y) and R -Do(Y,-S(X,Y», if Shall -Do(Y,-S(X,Y» follows from C(X, Y) and R
We should perhaps note that a party Y such that C(X,Y) could fail to comply with R in relation to X without breaking R in the sense of seeing to it that he fails to comply. If, for instance, the rule implies that Shall Do(Y,S(X,Y» and Y fails to comply because - Do(Y,S(X,Y», this fact does not necessarily mean that Y breaks the rule in the sense of Do(Y, - Do(Y,S(X ,Y»). REALIZATION OF RULES OF RIGHTS
The idea of realization in connection with rights plays an important role in many contexts. In the V. N. Declaration, for example, article 28 asserts that Everyone is entitled to a social and international order in which the rights and freedoms set forth in this Declaration can be fully realized. The main purpose of this paper is to suggest an explication of this idea of realization of rights. First of all, I think we must distinguish realization from compliance. The realization of rule R (or the right expressed by it) for X in relation to Y implies Y's compliance with R in relation to X - but not always vice versa. An example might show this. Consider rule A derived from article 13 of the V.N. Declaration concerning the right ofleaving one's own country. Let Y be the state of India and X a poor peasant in central India such that X and Y satisfy the C-condition of the rule. The rule then implies that Shall - Do(Y, -L(X,Y» Shall - Do(Y,L(X,Y» There is no doubt that the state Y of India complies with this rule in relation to the peasant X, i.e. , - Do(Y, - L(X,Y»
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183
- Do(Y,L(X,Y». But in spite of this fact, the rule can hardly be said to be realized for X, simply because X lacks the means for leaving India. The main point in the explication of realization that I would like to suggest is simply this: If rule R implies that X has both a power and immunity but not a claim in relation to Y with respect to S(X,Y), then to be realized for X in relation to Y it is not sufficient that R is complied with by Y. There must also be some measure by means of which it is possible for X to see to it that S(X,Y). Let us write Cando(X,S(X, V»~ for: There are measures M such that it is possible in practice that X sees to it that S(X,Y) by means of M. 3 Then, if C(X,Y) the rule R is said to be realized for X in relation to Y if: (1) (2) (3)
Y complies with R in relation to X Cando(X,S(X,Y», if C(X,Y) and R imply that -Shall -Do(X,S(X,Y», Shall -Do(Y,-S(X,Y» and -Shall Do(X,S(X,Y», Cando(X, -S(X,Y», if C(X,Y) and R imply that -Shall -Do(X,-S(X,Y» , Shall -Do(Y,S(X,Y» and - Shall Do(X, - S(X,Y».
In case of rule A, for instance, if Y is a state and X is a normal noncriminal adult citizen of Y residing in Y, the rule is realized for X in relation to Y if and only if (1) (2) (3)
-Do(Y,-L(X,Y» and -Do(Y,L(X,Y» Cando(X,L(X,Y» Cando(X, - L(X,Y».
Finally, we will say that the rule R is realized for X, if R is realized for X in relation to each Y such that C(X,Y). Our explication of realization, if we accept it, provides a framework for spelling out many questions concerning rights. Suppose we ask: For whom in Europe is article 13.2 of the U.N. Declaration realized? The main part of this question, spelled out, would be: For which states Y in Europe and categories of normal non-criminal citizens X of Y residing in Y does it hold that (i) Y does not see to it that X stays in Y or that X leaves Y and (ii) X can see to it that he leaves Y and also (iii) that he stays in Y? A critic of the framework might argue that we do not need this kind of spelling out of our questions in order to answer them. This might perhaps
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be true in case of article 13.2. But I invite the critic to put to the test some of the other articles of the Declaration before I yield ." In conclusion of this paper I would like to make a general remark on the U.N. Declaration of Human Rights. This Declaration could, and should, be regarded as an authoritative list of important valid rules of rights . These rules state that people have - not only ought to have - certain rights in relation to each other or to the State authorities . But it is more than that. It is also a direction, expressed in its article 28. This direction could, I believe, be formulated as a normative statement: It ought to be the case that the rules listed in the Declaration be realized for everyone. This normative statement is by no means trivialized by the fact that the rules are valid, since it is easy to find examples of obviously valid rules that clearly ought not to be realized for anyone. NOTES I An excellent survey and analysis of some of the classical works in this area is given in Lindahl (1977). This work also contains a substantial contribution to the development of a typology of rights-relations. Another important recent contribution is given in Porn (1970). 2 The state of affairs S(X,Y) could be of any kind including such in which the parties X and Yare redundant, as for example: 2+2=5. In ordinary cases , however, X and Y figure in a non-redundant way, like in: X recovers from Y what X has lent to Y. Note that in many such cases, S is a rights-relation between X and Y. 3 There is much more to say about notions like Cando in theories of influence and action. The pioneering work in this field has been done by Ingmar Porn. See especially Porn (1970), (1971) and (1977). In this particular context, see also Kanger (1977). 4 An attempt at a spelling out of all the articles in the U. N. Declaration and the conditions for their realization has been made by Helle Kanger (1981), (1984).
REFERENCES Kanger , H. (1981), Human Rights and their Realization. Department of Philosophy, University of Uppsala , Uppsala . Kanger, H. (1984), Human Rightsin the U.N. Declaration . Acta Universitatis Upsaliensis, Uppsala . Kanger, S. (1963), "Rattighetsbegreppet," Sjujilosojiska studiertilliignade AndersWedberg den 30 mars 1963. Philosophical Studies published by the Department of Philosophy, University of Stockholm, No 9, Stockholm. Kanger, S. (1977), "Nagra synpunkter pa begreppet inflytande ," Filosojiska smulor tilliignade Konrad Marc-Wogau. Philosophical Studies published by the Department of Philosophy, Uppsala University , Uppsala.
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Lindahl, L. (1977) , Position and Change: A Study in Law and Logic. Reidel Publishing Company, Dordrecht. Porn, I. (1970) ; The Logic of Power. Basil BlackwelI , Oxford . Porn, I. (1971), Elements of Social Analysis. Philosophical Studies published by the Department of Philosophy, Uppsala University , Uppsala. Porn, I. (1977) , Action Theory and Social Science. Reidel Publishing Company, Dordrecht.
UNAVOIDABILITY INTRODUCTION
In this article I would like to suggest a non-modal explication of the notion: Condition P is unavoidable for agent A. I The set of conditions, as I see it, is closed under Boolean operations and quantification over domains of individuals. Examples of conditions are: Agent A turns on the electric switch Some men are immortal (x)(x
+y
~
x)
We will use the letters M, P, Q, R as variables for conditions. The set of agents comprises normally most human beings. But it could also contain other phenomena that cause deterministic systems to produce outputs of some particular kind. We may form judgements about conditions and agents. An important type of judgement, in this connection, is formed by means of the Do-predicate: Do(A,P,Q), to be read as: By means of the fact that P, the agent A brings about the result Q, or as: With P at hand, A sees to it that Q, or: By means of P the agent A causes it to be the case that Q. Other similar readings are conceivable. Example: By means of the fact that Per turns on the switch, Per sees to it that the lights are on. The adequacy of a particular sort of reading depends often on what kind of conditions P and Q and agent A are involved. We shall assume that every condition is ajudgement, and that (P=Q) and Do(A,P,Q) always are judgements . More judgements may be formed by means of quantification over agents and over conditions, and by means of classical connectives of propositional logic. When these connectives are applied to conditions , they coincide with the corresponding Boolean operations . Note that I do not assume that every judgement is a condition. Neither do I rule out the possibility that judgements of the form Do(A,P,Q) are conditions . 186 G. Holmstrom-Hintikka, S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. I, 186-191. © 2001 All Rights Reserved. Printed by Kluwer Academic Publishers. the Netherlands. Originally published in Logic and Abstraction: Essays Dedicat ed to Per Lindstrom on His Fiftieth Birthday, Acta Philosoph ica Gothoburgensia I, Goteborg , 1986,227-236.
UNAVOIDABILITY
187
DEFINITIONS
The central concept in this study is an equivalence relation P as follows: Def
(P -
A
A
Q defined
Q) = «R)(Do(A,P,R) B Do(A,Q,R» & (M)(Do(A,M,P) B Do(A,M ,Q»)
Read P - A Q as: As far as A's activities are concerned P is on a par with Q. Or more briefly: P and Q are on a par for A. In terms of this relation we may define a stricter equivalence relation which presumably is a congruence relation with respect to the Boolean operations : Def
(P ::::: A Q) = (R)«P
~
R) -
A
(Q
~
R»
Read P ::::: A Q as: As far as A's activities are concerned P and Q are equal. Or more briefly : P and Q are equal for A. For most agents, many conditions are, so to speak, out of reach, or avoided, in the sense of not being involved in any of the agent's activities as means or as results. Thus, for example, for me all false conditions as well as true conditions such as: Kiwis breed on Kapiti island, are clear-cut instances of avoided conditions. Obviously, the more passive an agent is, the more conditions would be avoided. This notion of being avoided is defined in the straightforward way: Def
Avd(A,P) = «R) ... Do(A,P,R) & (M) ... Do(A,M ,P»
Now the definition of unavoidability for A, that I have in mind, is the following: Def
Unav(A,P) = (Q)«P & Q) -
A
Q)
Trivially true conditions and conditions that always are at hand for the agent are often unavoidable. Thus, for me, 2+2=4 and the fact that I am mortal are clear-cut instances of unavoidable conditions. Among the corollaries of these definitions we may note: [T is the universal conditionf Unav(A, T) Unav(A,(P & Q» Unav(A,(P B Q»
B ~
(Unav(A,P) & Unav(A,Q» (P ::::: A Q)
They also imply some alternative, but equivalent, ways of defining unavoidability:
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STIGKANGER
Unav(A,P) B (Q)«P ~ Q) Unav(A,P) B (P == AT)
A
Q)
SOME ASSUMPTIONS
The Do-predicate may be understood in several ways. In this article I will assume that it always satisfies the following principles : (1)
(2) (3)
Do(A,P,Q) ~ (P & Q) -,Do(A,P, T) (3Q)Do(A, T ,Q)
The last assumption might need some explanation. Do(A, T ,Q) could be understood as: A brings about Q in a passive way without doing anything in particular as a means for Q. For instance, if the light is already lit, it often happens that I see to it that the lit light is lit without doing anything. Do(A, T ,Q) presupposes however that the agent A is somehow involved in a deterministic system in which Q is brought about and that Q is not avoided. Thus, the fact that Kiwis breed on Kapiti island is something I do not bring about with T at hand. The distinction between Do(A, T ,Q) and -,Do(A, T ,Q) is sometimes subtle. Example: Suppose Mr . A meets on the street an acquaintance B of whom he is not very fond, and suppose that A does not raise his hat (call this condition -,R) . Now consider two alternative cases: Either Mr. A observed B or he happened not to observe B when they met. In the first case we may suspect that A actually saw to it that he did not raise his hat, but, of course , without doing anything as a means for this. In other words: Do(A, T , -, R). In the second case -,Do(A, T , -, R) is more plausible. Mr. A's not raising his hat was nothing he did or saw to. From the principles (1)-(3) we may deduce that the following chain of implications holds: [.1 is the empty condition.] (P - AT) ~ -'(P - A .1) ~ P
~(-'P-A.l) ~
-,(-,P - AT)
We may also deduce: Avd(A,P)
B
(P -
A
.1)
Do the converses of the above implications hold for some agents? In other words : Are there maximally passive agents and are there maximally
UNAVOIDABILITY
189
active agents? I will assume that the answer to these questions is negative: (4) (5)
(3P)(""'(P - AT) & ....,(P - A L) (3P)«P - A L) & P)
From the definitions it follows that Unav(A,P)
(P - AT)
~
Does the converse of this implication hold? I will assume that it does. Hence we obtain a new equivalent to our original definition of unavoidability: (6)
Unav(A,P)
~
(P - AT)
A noteworthy consequence of this, and some preceding principles is: (P ::::. A Q)
~
Unav(A,(P
~
Q» .
It is now easy to see that ::::. A is a congruence relation with respect to the Boolean operations, as I have presumed it should be. We have, in particular, (P ::::. A Q)
~
(P ::::. A Q)
~
«P & R) ::::. A (Q & R» (....,p ::::. A ""'Q)
In conclusion of this section I would like to mention one principle which I will assume to be valid and which will playa role in the sequel: The principle says that (P & Q) is avoided for A if Pis: (7)
(P -
A
.l)
~
«P & Q) -
A
.l)
AN EXTENSION OF THE THEORY
Let us now extend the set of conditions with every judgement of the kind Do(A,P,Q) and (P=Q), and let us assume that this extended set is closed under elementary logical operations. Thus «P +A .l) - AT) would be a new example of a condition . We let the variables M, P, Q, R range over the extended set of conditions and we shall let this decision have a retroactive force as far as the previous definitions and assumed principles are concerned. Of course, ordinary logical precautions should be observed. For instance, Do-expressions substituted for P or Q in the definition of - A should not contain free variables M and R which are bound in the definition. Let I- ~ mean that the formula ~ denoting a condition is deducible from our definitions and principles . We shall now add to our deductive system the following rule of deduc-
190
STIG KANGER
tion: (8)
If
~
<1>, then
~ (<1>
= T)
A useful derived rule is: If
~(<1> ~
'1') then
~(Unav(A,<1» ~
Unav(A,'1'».
By means of this rule we infer a version of the idea that a condition is unavoidable (for A) if it is inevitable in the sense of being unavoidably not avoided: «P +A .L) - AT)
~
(P - AT)
Another corollary expresses the idea that a condition P is unavoidable only if the opposite -,p is unavoidably avoided: (P - AT) ~
«-,p - A .1) - AT)
The notion «-,P - A .L) -AT) is worth more attention. It could be read as: Condition P is irrevocable for agent A: Def
Irrev(A,P) = «-,P - A .1) - AT) .
The logical behaviour of Irrev is, I believe, similar to that of Unav. Thus it follows from the assumptions already made that Irrev(A , T) Irrev(A,(P & Q»
~
(Irrev(A,P) & Irrev(A,Q»
Other analogues to Unav are, however, less obvious and depend on how we view unavoidability when applied, not to avoided conditions, but to the condition of being an avoided condition . But I believe we could safely assume that the following principles hold: (9) (10)
(Irrev(A,P) & IrrevrA.O) Irrev(A ,P) ~ P
~
Irrev(A,(P & Q»
In spite of these similarities, irrevocability should not be confused with unavoidability . A true condition such as: Kiwis breed on Kapiti island, which (as far as I am concerned) is avoided, is of course not unavoidable. But it is irrevocable . NOTES 1 I am grateful to Dr. W. Rabinowicz, Uppsala University, and Prof. I. Porn of the University of Helsinki, for constructive comments .
UNAVOIDABILITY
191
The proofs of these and subsequent corollaries are all very simple and involve nothing but elementary logic. As a typical example I will demonstrate:
2
(Unav(A,P) & Unav(A,Q»
~
Unav(A,(p & Q»
by showing that Unav(A,(P & Q» follows from the premisses Unav(A,P) and Unav(A,Q), i.e . from (i)
(ii)
(R)«P & R) -A R) (R)«Q & R) - A R)
From (i) by instantiation of (Q & R) for R we get: (iii)
«P & Q &R) -
A
(Q & R»
From (ii) and (iii) by the obvious transitivity of - A:
Since R is not free in the premisses we may conclude that (R)«P & Q & R)
-A
R)
which by definition is: Unav(A,(P & Q».
UNAVOIDABILITY APPENDIX
PROOFS 1
1
Unav(A, T)
2a
Unav(A,(P & Q» -) Unav(A,P) -) (R)(P & Q & R - R) Unav(P & Q) -) (P & Q & [P &] R - P & R) 0: trans P & R - R 0: (R)[P & R -R] Unav(P)
2b Assume Assume from from
Unav(A,P) & Unav(A,Q) -) Unav(A ,(P & Q» Unav(P): (R)(P & R - R) Unav(Q): (R)(Q & R - R) Unav(P) (P & Q & R - Q & R) Unav(Q) and trans (P & Q & R - R) °° (R)(P & Q & R - R) i.e. Unav(P & Q)
triv.
Corollary Unav(A,P) & Unav(A,(P -) Q» -) Unav(A, P & (P -) Q» -) Unav(P & (P -) Q» (by 2b) Unav(P) & Unav(P -) Q) = Unav(P & Q) -) Unav(Q) (by 2a) 3 Unav(A,(P -) Q» -) (P ::::: A Q) Assume Unav(P B Q) then Unav(P B R) ~ (Q -) R) by corollary above 192 G. Holmstrom-Hintikka, S. Lindstrom and R. Sliwinski (eds.}, Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. t. /92-/96. © 2001 All Rights Reserved, Printed by Kluwer Academic Publishers . the Netherlands .
193
UNAVOIDABILITY
since
[(P ~ Q) ~ (P ~ R) ~ (Q ~ R)] (S)[[(P ~ R) ~ (Q ~ R)] & S - S] ·. [[(P ~ R) ~ (Q ~ R)] & (P ~ R)] - (P i.e., [(P ~ R) & (Q ~ R) - P ~ R] .: [[(P ~ R) ~ (Q ~ R)] & (Q ~ R) - (Q i.e. , [(P - R) & (Q - R) - Q ~ R] (P ~ R) - (Q ~ R) trans R is not assumed to be free ·. (R)[(P ~ R) - (Q ~ R)] .. P=::Q
i.e.,
~
R)
~
R)]
Unav(P ~ Q) ~ Unav(P ~ ~ (R)(P ~ R - Q ~ R) ~ (P~.l - Q~ .l)~ (P - Q)
cn
4a Unav(A,P) ~ (R)[P V R - A R] Assume Unav(A,P) : (R)[P & R - R] & P V R - P V R) ·. ([P & P V] P & R - P V R) (P & P) = .1 trans PV R - R ·. (R)[P V R - R] 4b (R)(P V R - A R) ~ Unav(A,P) Assume (R)(P V R - R) (P V (P & R) - P & R) i.e., ([T &] P V R - P & R) trans P& R - R ·. (R)(P & R - R) i.e., Unav(P) 5 Assume i.e. , ·. i.e., •. i.e.,
Unav(A,P) ~ (Q)[P & Q =:: A Q] Unav(P) (Q)[(P ~ Q) - Q] (by 4) [P ~ (Q ~ R) - (Q ~ R)] [(P & Q ~ R) - (Q ~ R)] (Q)(R)[(P & Q ~ R) - (Q ~ R)] (Q)(P & Q == Q)
194
Assume ·. i.e., .• ·. ·. ·.
STIG KANGER
(Q)[P & Q =:: Q] (Q)(R)[(P & Q ~ R) - (Q ~ R)] (Q)(R)[P ~ (Q ~ R) - (Q ~ R)] (Q)[P ~ (Q ~ .L) - (Q ~ .L)] (Q)[(P ~ Q) - Q] (Q)[P ~ Q ., Q] Unav(P) (by 4)
Lemma T fA .L Assume T - .L i.e., (R)[Do(A , T ,R)
Do(A, .L ,R)] & (false) (M)[Do(A,M, T) ~ Do(A,M, .L)] (false) (false) .. ~ (R)-'Do(A, T ,R) i.e., -'(3R)Do(A, T ,R) which contradicts principle (3) .: T f .L ~
6 P -A T ~(PfA.L) ad observandum Assume P - T & P - .L then trans T - .L contradicts lemma
7 P fA .L ~ P Assume f> then -,Do(A,M,P) by principle (1) -,Do(A,P ,R) (R)[Do(A ,P ,R) ~ Do(A,.L ,R)] & (false) (false) (M)[Do(A,M,P) ~ Do(A,M, .L)] (false) (false) i.e., P - A .L
8 and 9 Duals to 6 and 7 10 Avd(A ,P) +-+ (P - A .L) follows from definition and principle (1)
UNAVOIDABILITY
Lemma a P ::::: A Q ~ (P & R ::::: A Q & R) Assume P ::::: Q (S)[P -+ S - Q -+ S] (S)(R)[P ~ (R ~ 5) - Q -+ (R ~ S)] (S)(R)[(P & R -+ 5) - (Q & R ~ S)] P&R:::::Q&R Lemma {3 P:::::Q-+P-Q Assume P ::::: Q i.e., (R)[(P ~ R) - (Q ~ R)] .. (P-+.l -Q-+.l) i.e ., (P - Q) 11
P ::::: A Q -+ Unav(A,(P 0 Q» Assume P ::::: Q We have that a P ::::: Q -+ (R)(P & R ::::: Q & R) {3P:::::Q~P-Q
by Ta lemma .. P&Q:::::Q&Q .1
.. P&Q - T by 7{3lemma i.e., (P -+ Q) - T .. also P&P:::::Q&P .. (Q ~ P) - T If we have (M - T) -+ Unav(M) i.e., principle (6) it follows that (P -+ Q) & (Q -+ P) - T by (2) i.e ., (P 0 Q) - T .. Unav(P 0 Q)
195
196
STIGKANGER NOTE
On some handwritten sheets of paper Kanger worked out the proofs for the theorems in this paper. Simple as they are Kanger never published them but they have been included in this collection as an appendix , for the sake of completeness. (The Swedish text was translated by the editors.)
APPLIED LOGIC: PREFERENCE AND CHOICE
PREFERENCE LOGIC
A CRITICAL PREFACE
Preference logic is difficult. Attempts at developing a theory of preference logic that goes beyond the limits of the trivial often end up in the absurd. Let me, to begin with, give a couple of examples of this. Preference logic is mainly a theory for the relations between at least as good as (z), better than (» and as good as ( z) between conditions. When I speak of the trivial part of preference logic, I am referring to that part which is determined by the following axioms: (AI) (A2) (A3) (A4)
P
~
P
ifp ~ q and q ~ rthenp ~ r p > q if and only if p ~ q and q i:. P P z q if and only ifp ~ q and q ~ p.
The first attempt at bringing preference logic beyond these limits was made by Soren Hallden in The Logic of Better (1957) . Hallden proposed that the following principles could be added to the trivial part: (SRI) p > q if and only if (p & -q) > (q & -p) (SR2) P z q if and only if(p & -q) z (q & -p) . These principles, however, are not entirely reasonable . SR2, for example, has an unnatural consequence: every condition that has a neutral supplement is as good as a contradiction. (By a neutral supplement to a condition p is meant a condition q such that p
z
(p & q) andp
z
(p & -q).)
Hallden also proposed a principle of comparability: (SR3)
p
> q or p
z
q or q
> p.
There are objections to this principle. These objections are not based, how199 G. Holmstrom-Hintikka. S. Lindstrom and R. Sliwinski (eds.], Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. I, 199-208. © 2001 All Rights Reserved, Printed by Kluwer Academic Publishers. the Netherlands . Originally published in Swedish in Hjalmar Wennerberg (ed.),Nio Filosofisca studier tilldgnade Konrad Man:-Wogau, Filosofiska studier utgivna av Filosofiska fOreningen och Filosofiska institutionen vid Uppsala universitet 6, Uppsala 1968, 1-13 . Translated by Sharon Rider.
STIGKANGER
200
ever, on some superficial absurdity with the principle, but are bound up with much deeper questions concerning the interpretation of the preference relations. I will return to this. The literature in preference logic after Hallden is not vast. There is a good bibliography in N. Rescher, "Semantic Foundations for the Logic of Preference", The Logic ofDecision and Action (ed. N Rescher, 1967). The most important works are G.H. von Wright, The Logic of Preference (1963; according to Rescher, it is the principal treatise on the subject), Rescher 's own essay, and Bengt Hansson, Topics in the Theory ofPreference Relations (1967). G.H. von Wright proposes that the trivial part of preference logic be augmented, in part with Hallden's principles, and in part with four principles of his own: (vWl) (p V q) > (r V s) if and only if (p & -r & -s) > (-p & -q & r) and (p & - r & - s) > (- P & - q & s) and (q & -r & -s) > (-p & -q & r) and (q& -r& -s) > (-p& -q&s) . (vW2) P > q if and only if (p & r) > (q & r) and (p & - r) > (q & - r)' (vW3) The same as vWl but with z in place of >. (vW4) The same as vW2 but with z in place of"> ." From these principles one may reason more or less as usual in logic. von Wright, however, makes the original restriction that the terms in a preference statement can freely be replaced by logically equivalent terms only if the preference statement is not thereby provided with any new variables in its conditions. von Wright's principles are not satisfactory. Let c stand for a contradiction. From vWl, we then get: if (c V q) > (c V s) then (c & - c & - s) > (- c & - q & c) which, through simple (and, according to von Wright, permitted) substitution, becomes: if q > s then c > c. For trivial reasons, we have that c
q
"j>
s.
"j>
c. Thus
PREFERENCE LOGIC
201
In other words, nothing is better or worse than anything else. Principle vW2 has the same absurd consequence. ' First we get: if p > q then (p & c) > (q & c) which becomes if p > q then c > c. Therefore
p
j>
q.
Principle vW3 has this unnatural consequence: Assume that p and q are distinct conditions in the sense that (p & q) = c. Then p '" q if and only if p '" c and q '" c. Each conditions that is as good as some distinct condition is thus as good as a contraction. Finally, principle vW4 has the same unnatural consequence. 4 Bengt Hansson proposes three non-trivial principles : (BH 1) P ~ q or q ~ P (BH2) if P ~ q and p ~ r then p ~ (q V r) (BH3) if P ~ rand q ~ r then (p V q) ~ r. I find it difficult to see how this system of principles can be justified. There are objections to the comparability principle BH1, and against principles BH2 and BH3, one may provide counter-examples of the following kind: It is at least as good to get ten crowns as to get a ten-crown book , and it is at least as good to get ten crowns as to get a ten-crown box of chocolate. But it is not clear that it is at least as good to get ten crowns as to get a ten-crown book or a ten-crown box of chocolates. The disjunction can entail that one gets both. In what follows, I shall nonetheless give a reasonable interpretation of the preference relation that satisfies BH2 and BH3, but not BH1. I shall also give a reasonable interpretation that satisfies BHl , but not BH2 or BH3. I must leave unsolved the problem of finding a reasonable interpretation that satisfies all three Hansson principles. THE VALUE PLANE
We can represent a condition by a volume, that is to say, a class of points in a wo-dimensional space in which the nth dimension fixes the domain of the nth individual variable. A point in the space is identified with an infinite sequence of individuals XI' x 2 , x 3 , .... A condition is represented, in other
202
STIGKANGER
words, by a class of such sequences . The condition, XI is a refrigerator, for example, is represented by the class of all sequences XI ' X 2, x 3, . .. such that XI is a refrigerator. We can well use formulae from the language of elementary logic to denote conditions that are represented in this way. The elementary logical operations - , &, V, and Ex, on conditions are represented by complement, intersection, union and cylindrification along the xn-dimension of volumes in the wo-dimensional space. Let us henceforth by conditions mean conditions that are represented in this way; let us, for simplicity's sake, identify the conditions with their representations." We can now imagine that we take conditions and place them in other spaces, for example, in a two-dimensional domain or value plane with utility (u) and disutility (d) as the two dimensions . What we have in mind, more precisely, is that each sequence of individuals XI' x 2 , x 3 . .. receives exactly one utility position and exactly one disutility position, and that the sequence with the utility position u and disutility position d is placed at point u,d on the value plane. Several sequences can be placed at the same point. By virtue of the sequences of individuals being so placed on the value plane, the conditions are of course also in place. It might look like it does in the figure where the sequences in the condition p have been circled: d
U
We do not assume that each utility and disutility position has some sequence assigned to it. We shall, however, make the following hypothesis: if a utility position U is given to some sequence, and a disutility position d is given to some sequence, then there exists a sequence that has both U and d assigned to it. ELEMENTARY VALUE-LOGIC
Let D be the cylindrification of conditions along the utility-dimension and let U be the cylindrification along the disutility-dimension. Up and Dp are then conditions corresponding to the shaded cylinders in the figures below.
PREFERENCE LOGIC
d
203
d
u
u
Dp determines unambiguously p ' s disutility position or, as I prefer to say, the
disutility of p. The utility of p, on the other hand, is left indeterminate. In the same way, Up unambiguously determines the utility of p, while the disutility of p is left entirely indeterminate . The elementary logical qualities of the operations U and D can be specified axiomatically. Let c, as earlier, be contradiction.
= c then (p & Uq) = c = c then (p &Dq) = c
(A5) (A6)
if (Up & q) if(Dp & q)
(A7)
UUp = Up DDp = Dp UDp = DUp = -c whenp
(A8) (A9)
* c.
To these axioms, we add (formulated in a suitably "algebraic" manner) axioms for the elementary logical operations -, & and V, and (if we so wish) for Ex; and for identity between individuals. We can then deduce inferences in the usual logical manner, and we obtain a calculus which I would like to call elementary value-logic . A SPECIFICATION OF THE VALUE PLANE
Let us imagine that we have a grouping of advantages and a grouping of disadvantages. Let us consider all classes of advantages and all classes of disadvantages . We imagine further that there exists a one-to-one correlation between the classes of advantages and the positions in the value plane's utility-dimension, and a one-to-one correlation between the classes of disadvantages and the positions in the disutility-dimension. In this way, we obtain an assignment of exactly one advantage-class and exactly one disadvantage-class to each sequence of individuals. We extend this arrangement to the conditions so that each condition p is assigned a class <j>(p) of advantages
204
STIGKANGER
and a class l\1(P) of disadvantages. We can call these classes the advantages with p and disadvantages with p, respectively . Thus every condition p receives a utility position and a disutility position and, therewith, a point on the value plane. We call this point p's value-point. Let u(P) be the set of all classes of advantages assigned to the sequences of individuals in p , and let d(P) be the set of all classes of disadvantages accorded to the sequences of individuals inp . We now make the following hypotheses: (i) (ii) (iii)
if u(P) = u(q) then <j>(P) = q,(q) and if d(P) = d(q) then l\1(P) = l\1(q) , [<j>(P) n <j>(q») ~ q,(p V q) c [q,(P) u <j>(q») and [l\1(P) n l\1(q») c l\1(P V q) c [l\1(P) u l\1(q)) q,(P V -p) = l\1(P V -p) = A.
Notice that an hypothesis for <j>(P & q) and l\1(P & q) that is analogous with (ii) is difficult to justify. We see this, if we identify <j>(P) and l\1(P) with the intersection or with the union of advantage-classes and disadvantage-classes, respectively, that have been assigned to the sequences of individuals in p identifications that cannot be excluded . ELEMENTARY PREFERENCE LOGIC
Let us now give a definition of the preference relation, at least as good as (z):
p
~
q if and only if <j>(q) c q,(P) and l\1(P) c l\1(q) .
This definition is cautious , and it leaves a great deal of room for incomparable conditions: p ~ q is valid only if p has at least all the advantages and at most all the disadvantages that q has. We see immediately that the trivial Axioms Al and A2 are satisfied. We see also that the following axioms, by virtue of hypothesis (i), also hold: (A 10) p ~ q if and only if Up (All) (Up & Dq) z p.
~
Uq and Dp
~
Dq
Further, Hansson's BH2 and BH3 hold, by virtue of hypothesis (ii): (AI2) (Al3)
if p ~ q and p if P ~ rand q
~
r then p
~
r then (P V q)
~
(q V r) ~
Hypothesis (iii) gives the following axioms :
r.
PREFERENCE LOGIC
(AI4) (AI5)
205
Uq ~ U(p V -p) D(p V -p) ~ Dq.
The principles of comparability BHI and SH3, as we said, are not verified. VALUE METRICS
Let me now (in a fit of value-theoretical enthusiasm) introduce a metrics onto the value plane. More precisely, I attach a numerical measurementA(p,q) to the distance betweenp's and q's value-points. The function A is a function that is defined for pairs of conditions, and which takes non-negative real numbers as values. It satisfies the following axioms: (AI6) (AI?) (AI8)
if P '" q then A(p,q) = 0, if P "" q then A(p,q) > O. if A(p ,q) = 0 then A(p ,r) = A(r,q). A(p,q) ~ A(p,r) + A(r,q).
The value-points for conditions of the type Up all have the same disutility position, and they form a disutility-axis on the value plane . The value-points for conditions of the type Dp all have the same utility position, and they form a disutility-axis on the value plane. The point of intersection of the axis is the value-point for a tautology. It is natural to think of this point as a zero-point on the plane which does not lie between other points on the axes. We then get: (AI9) (A20)
A( -c,Up) A( -c,Uq) A( - c,Dp) A( -c,Dq)
= = = =
A( -c,Uq) A( -c,Up) A( - c,Dq) A( -c,Dp)
+ A(Uq, Up) or + A(Up,Uq) . + A(Dq,Dp) or + A(Dp,Dq) .
I shall (before the enthusiasm subsides) assume that the utility- and disutilityaxes are perpendicular to each other : (A21)
[A(P,q)]2 = [A(Up,Uq)]2
+
[A(Dp,DqW
We assume that the advantage-classes are arranged in order of importance, and that two different classes are never of equal importance. We make the same assumption with respect to the disadvantage -classes . We further assume that the utility and disutility positions are placed and correlated with the advantage- and disadvantage-classes in such a way that the distance from point-zero is a measure of importance . (There may be difficulties with that, but let us not be discouraged .) The following axioms should now apply: (A22)
if Up
~
Uq thenA( -c,Up)
~
A( -c,Uq)
206
(A23)
STIG KANGER
if Dp
~
Dq then A( - c ,Dp) ~ A( -
C,Dq)
Finally, we imagine that advantages and disadvantages can counterbalance each other. Let the value of p be, roughly speaking, p's advantages minus p's disadvantages. We now define a measurement for the value of p : (A24)
V(P) = A(-c,Up) -A(-c,Dp).
It is then natural to define a new preference relation thus: (A25)
P z ' q if and only if V(P)
~
V(q).
This preference relation may be suitable if one wants to combine preference logic with a theory for value-measurement. But the preference logic one gets with z ' will be rather meagre . The trivial axioms will, of course, be satisfied, as will the comparability principle . But the principles BH2 and BH3 are not verified, nor are a number of other preference-logical principles that have been proposed in the literature and which in essential ways are based on occurrences of - , &, or V in the terms of the preference statements. MIXTURES OF CONDITIONS
A primitive thermometer can be constructed thus: We want to know the temperature of the water in a bowl, and we assume that we can determine whether two bowls of water are equally warm by feeling the water. We now combine ice water with boiling water in a second bowl, so that the water in both bowls will be equally warm. If the mixture consists of 1/2 ice water and 1/2 boiling water , the temperature will be 50°C. If the mixture consists of3/5 ice water and 2/5 boiling water, the temperature will be 40°C, etc. In von Neumann & Morgenstern, Theory of Games and Economic Behavior (1944; 2nd ed. 1947), this idea was utilized for the measurement of utility. If one wishes to find out where utility A lies in the interval between utility B and utility C, one combined Band C so that the combination was as good as A. If the combination consisted of 3/5 Band 2/5 C, A would lie 2/5 of the way from B to C. Naturally, the combination cannot be made in a literal sense, but must be done probabilistically, like a ticket in a lottery in which 3/5 of the tickets give a prize of B, and 2/5 of the tickets give a prize of C. The combinations can, of course, themselves be combined , for example, 3/10 of the combination just mentioned and 7/10 of D. This combination corresponds to a ticket in a lottery in which 3/10 of the tickets give a ticket in the first lottery as a prize, and 7/10 of the tickets give D. We now introduce combinations of conditions, that is to say, combinations
207
PREFERENCE LOGIC
of conditions or of combinations of conditions. The combinations are of probabilistic kind. We will use the letters P and Q for condition combinations, and the letters a and b for real numbers 0 ~ a, b ~ 1. The combinations are constructed with the help of two operations, . and +. We represent a combination thus: [a . P + (1 - a) . Q]. We can articulate it as: a parts P with 1 - a parts Q. Notice that we always combine so that the sum of the parts will be 1. The following axiom applies:" (A26) (A27)
[a . P + (1 - a) . Q] = [(1 - a) . Q + a . P] [a . [b . P + (1 - b) . Q] + (1 - a) . R] = [ab' P
(A28) (A29)
[a . P
+
(1 - abr : [a-ab . Q I-ab
+ (1 -
If [a . P and if a
a) . P] = P a) . R] = [a .
+ (1 -
=1=
0, then P = Q.
+
I -a . R]] I-ab
where ab=l=I
Q + (1 - a) . R]
A condition q is thus a special case of combination: q = [a' q + (I-a) . q]. The combination [a . q + (I-a) . -q] is a probability condition: q with the probability a. We shall now recursively define a measure W of the value of condition combinations. (A30)
W(q) = V(q) and W([a . P + (1 - a) . Q])
=
a . W(P)
+
(1 - a) . W(Q).
(We must not confuse multiplication and addition of numbers with operations in the combination.) We then extend the preference relation so that it also applies to combinations . We call the extended relation 2". (A3I)
P 2" Q if and only if W(P) ~ W(Q) .
This preference relation is extremely convenient if we want to combine preference logic with a theory for value-measurement. But the preference logic we get with 2 " , will be, of course , as meagre as that with 2 I •
A PESSIMISTIC CLOSING WORD
One may wish for a reasonable and clear interpretation of the preference relation that offers an interesting preference logic and which (at least for essential classes of conditions) satisfies the principle of comparability and brings preference logic closer to mainstream preference theory . (It is largely concerned with decisions and the measurement of value). Without such an interpretation, or the promise of such an interpretation, preference logic in its
208
STIGKANGER
present form is reduced to a fairly isolated and fruitless branch of philosophy. I believe that this wish is in vain. In any case, it would seem hopeless to fulfill within the framework of preference logic coming out of Hallden's pioneering work. Preference logic appears to be in need of a new turn, with new frameworks and new ideas. But what these should look like is not easy to say.7 Preference logic is actually quite difficult. NOTES von Wright makes the restriction here that we let " r be a state which is different from p and
q and which is not itself a truth-function of other states." Now every r is a truth-function of "other states"; we have, for example, that r is a disjunction of sand t where S = (r & u) and t = (r & - u). Thus the meaning of the restriction is rather unclear. 2 See above. 3 I ignore the unclear restriction that von Wright placed on the principle. 4 See above. 5 A more adequate representation, at least in certain respects, can be obtained by representing the conditions as unions of separate classes of sequences of individuals, where the classes are separate because the individuals in the sequences in different classes are taken from different universes and are indexed with their respective universes. This representation makes it possible to introduce modal operations on conditions . It also gives us a neat representation theorem for the elementary logic. 6 The axioms have in essence been given by M. Hausner in "Multi-dimensional utilities", in Thrall, Coombs & Devis (eds.), Decision Processes (1954) . 7 One idea that is perhaps worth trying is to build preference logic around a more complicated preference relation : Rip.s.q.t) , meaning: p is for the purpose s at least as good as q is, for purpose t. The special cases R(p,s,q,s) and R(p,s,p,t) are obviously relevant in a valuetheoretical context. Comparability could be limited to these special cases : R(p,s,q,s) or R(q,s,p,s) and: R(p,s,p,t) or R(p,t,p,s).
A NOTE ON PREFERENCE-LOGIC
Let E be the domain of states of affairs and assume that (I)
E is closed under Boolean operations.
Let >- be a strict preference relation in E and let non-strict preference A ~ B and preferential indifference A "" B be defined as B 1- A and (A ~ B) & (B ~ A) respectively. In preference logic it is often taken for granted that preference is a weak ordering of E, i.e., (II)
>- is asymmetric and
is transitive in E.
~
Let - and =1= be the Boolean operations of negation and exclusive disjunction in E. The following two principles seem to be evident: (III) (IVa) (IVb)
There are states of affairs A and B in E such that A, B, B are pairwise preferentially different. If A >- B, then A ~ (A =1= B) If A >- B, then (A =1= B) ~ B.
A and
The aim of this note is to show by means of a simple argument that the assumptions (I) - (IV) are inconsistent. Make the hypothesis p>-q>-q>-p Now by IVa and IVb starting from the hypothesis p >- q we have p z tp w q)
and
(p
es
qj z q
Moreover, since >- is assumed to be a weak ordering and since p >- q p >- (p
=1=
q)
or
(p
=1=
q) >- q
If p >- '(p =l= q) we get (p =1= (p=l=q» ~ (p=l=q) by IVb (with (p=l=q) as B) which is q ~ (p=l=q) (since (p =1= (p=l=q» = q by tautology) . If (p=l=q) >- q we get by IVa (p =1= q) zp . We conclude that either 209 G, Holmstrom-Hintikka; S. Lindstrom and R. Sliwinski ieds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. I. 109-210. © 2001 All Rights Reserved. Printed by Kluwer Academic Publishers. the Netherlands . Originally published in ThD60: Philosophical Essays Dedicated to Thorild Dalquist on His Sixtieth Birthday. Filosofiska studier utgivna av Filosofiska foren ingen och Pilosofiska institutionen vid Uppsala universitet 32. Uppsala 1980,37-38.
210
STIGKANGER
p ::::: (p
=1=
q) or
q::::: (p
=1=
q)
In a similar way, starting from the hypothesis
q :::::
(p
=1=
q) or
p:::::
(p
=1=
q >- P we conclude that either
q)
But these conclusions (independently of how p and q are chosen) are jointly incompatible with the hypothesis. So, in case preference is a weak ordering, then either the hypothesis p >- q >- q >- P is always false or the principles IVa and IVb are not valid. By the same kind of argument this can be shown to be true of each of the 24 hypothetical strict preferential orderings of p, p, q and q. The disjunction of these 24 orderings is of course equivalent with p, p, q and q being pairwise preferentially different. Our final conclusions will then be: If preference is a weak ordering in E, then either the hypothesis that p, p, q and q are preferentially different is always false, or the principles IVa and IVb are not valid . In other words : Given the assumption I (which was implicitly used in our argument) then the assumptions II, III and IV are inconsistent.
CHOICE AND MODALITY
Let A = {x,y, ... } be a non-empty set of alternatives. Let 0 be an operator with subsets of A as arguments and values which satisfies the following axioms: I. II. III. IV. V.
If X £; Y, then O(X) £; O(Y). nXEPO(X) £; O(nXEPX) where F is any non-empty family of subsets of A. O(A) = A. O(X) £; O(O(X». If X =1= 0, then (X n O(A - X» =1= 0.
Clearly, the axioms I - III determine a modal logic which is normal in the Segerberg sense and I - IV determine a logic of the type called K4 by Segerberg. Given some well-known results in the theory of Boolean algebras with operators, it is not difficult to show that the axioms I-V hold for each X,Y £; A if and only if there is a strict partial ordering > in A without infinite ascending chains such that O(X)
= {x I (y) (y >
x
~
Y E X)} for each X
£;
A,
and y
> x~x
ff O(A - {xy}) for each x, yEA.
Let us define an operator C in the following way: C(X)
=X
n O(A - X).
Clearly C(X) = {x
Ix
E X & (y)(y E X
~
Y 1> x)} for each X,
and x > y
~
x ff C({xy}) for each x, y. 211
G. Holmstrom-Hintikka, S. Lindstrom and R. Sliw inski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. I. 21 1-213. © 2001 All Rights Reserved. Printed by Kluwer Academic Publishers. the Netherlands. Originally published in Dand 0 : Mini-e ssays in Honor of Krister Segeberg of His Fortieth Birthday April 26, 1976, Publications of the Group in Logic and Methodolog y of Real Finland, vol 4, 19 76. 25-32.
212
STIG KANGER
Now, if we take > to be a relation of preference in A, (read x > y as: x is better than y), then C is a choice function in the sense of Sen and others. And we get as theorems of our theory all the principles governing such a choice function that have recently been proposed or studied by Sen except those which presuppose that > is a strict weak ordering . For instance: C(X) ~ X. X =1= 0 ~ C(X) =1= 0. (C(X) n Y) ~ C(X n Y) (Sen's property a). nXEFC(X) £; C(UXEFX) (Sen's property -y). C(C(X) U C(Y» = C(X U Y) (Principle of path independence). In case we think that > is a weak ordering we should add an extra axiom: VI.
If (C(X) n C(Y»
=1=
0, then C(X n Y)
~
(C(X) n C(Y».
As theorems we then get the remaining principles : Sen's property (3, the axioms of revealed preference etc. If we wish we could also add: VII.
O(O(X»
£;
O(X).
This axiom implies that the chains of the preference order > will be converse well-orderings of a formidable sort : wherever a cut is made in such a chain, there is always a next element below but an infinite descending chain above. This axiom may seem absurd at a first glance. But we might need it if we wish to extend our theory of choice and preference to a theory of measurement of value in which the value scale will not be unique (up to, say, linear transformations) unless the alternatives of A form a continuum with respect to value. Our way of treating choice functions does not of course solve any of the well-known problems of choice - for example choice amalgamation. But it shows that there is a close and direct connection between choice theory and modal logic which might be worth some further exploration. REFERENCES
J6nsson, B. & A. Tarski, "Boolean Algebras with Operators," American Journal of Mathematics 73, 1951. Plott, C.R., "Path Independence, Rationality and Social Choice," Econometrica 41, 1973. Segerberg, K.K., An Essay in Classical Modal Logic. Uppsala, 1971.
CHOICE AND MODALITY
213
Sen, A.K. , "Choice Functions and Revealed Preference," The Review of Economic Studies 38, 1971. Sen, A.K ., Social Choice Theory: A Re-examination . Paper presented at the third World Econometric Congress, August 1975.
CHOICE BASED ON PREFERENCE
1. INTRODUCTION
In this paper, I shall develop an axiomatic theory for binary choice functions C defined for subsets of a grand domain U of alternatives. When V,X £;; U, I intend to interpret C(V,X) as the set of those alternatives of (V n X) which, compared with alternatives of V, are regarded as not being worse than any alternative of (V n X). In other words: x E C(V,X) iff x E (V nX) & (y)(yPyx -) Y E (V-X)) where P, is a preference relation included in VxV. (Read yPyx as: y is better than x in V.) If this interpretation of C holds for each x and X we say that C is based on P, in V. Neither C nor P, will, however, be primitive notions of our theory. Instead we shall start with a binary function D defined for subsets of U and interpreted so that D(V,X) is the set of those alternatives of V which, compared with alternatives of V, are regarded as not being worse than any alternative of (V-X). In formulas : x E D(V,X) iff x E V & (y)(yPyx -) y E (VnX)) Then we define C in terms of D:
Definition 1.0
C(V,X) = X n D(V,(V-X))
In the first part of the theory (sections 3 - 8) we deal with cases of C(V,X) in which V is kept fixed . The axioms and theorems of this part will be formulated in terms of the functions D or C and elementary notions of set theory and logic . The axioms are arranged in groups. The axioms of the first group amount to the fact that the function D, and thereby also C, is based on a binary relation Pv - Step by step the axioms of the next five groups impose more structure on Py. In group 2, P, is assumed to be irreflexive and in group 3, P v is assumed to be a strict partial ordering in V which includes no infinite ascending Py-chains. Group 4 adds to this in that P, should not include any infinite descending chains either . In group 5, Py is assumed to be not only 214 G. Holmstrom-Hin tikka, S. Lindstrom and R. Sliwinsk i (eds.], Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. I. 214-230. © 2001 All Rights Reserved. Printed by Kluwer Academic Publishers, the Netherlands.
CHOICE BASED ON PREFERENCE
215
a partial ordering but a semiorder, and in group 6, finally, this assumption is strengthened to that of P, being a strict weak ordering . Other orderings of interest in connection with preferences, semilattice orderings, for example, will not be considered. The axioms of groups 1-4 are basic as far as this part of the theory is concerned. Groups 5-6 are optional-.anyone of them could be omitted. If one does not believe in a weak preference order in V, for instance, group 6 should be omitted. To mark the distinction between the basic and the optional axioms we shall label the latter "propositions" rather than "axioms" . The axioms and propositions of this part of the theory should be compared with related conditions on choice functions suggested in the literature; for instance, the two conditions known as Sen's property ex and 'Y which, given certain non-controversial preliminaries, imply that the choice function is based on an acyclic binary relation of preference without infinite ascending chains. (Cf. Sen, 1971.) The main difference between our approach to choices and preferences and the more standard approaches lies in the fact that our choice functions are functions of two variables, V and X - not of X only - and that they are based on preference relations of the type: on comparing alternatives of V, x is better than y. This way of relativizing the preferences to a background V involves certain philosophical difficulties which, however, do not seriously affect the development of our theory. We shall simply take for granted that for each background V we can fix a strict partial preference order (with no infinite chains) of the alternatives in V. As in the case of most backgrounds, this ordering is nothing but a partial ordering which contains some scattered chains. But in some cases the result may be a semiorder or a weak ordering. This is true in particular of backgrounds in which the alternatives differ mainly in only one dimension, for example, in economy or in efficiency. Axioms and conditions such as Sen's properties or those we will suggest in the first part of this paper are often regarded to be principles of rationality of choices. In my opinion this view is justified in so far as the principles imply that the choice is based on a partial order of preference (which we assume contains no infinite chains). But imposing more structures on the order has little to do with rationality. It has to do rather with the character of the background. For instance, if V is fixed as a set of alternatives selected at random from different fields, it would probably be more conceit than rational to base one's choice on a weak preference order. There are, however, (as we will try to explain in the second part of the
216
STIG KANGER
paper) rationality principles of another kind in which the background V is no longer kept fixed. Some of these are stability or consistency conditions which have to do with the way choices or preference orders are affected by a shift in background. A well-known - but dubious - example is the principle of independence of irrelevant alternatives which in this context could be given the following simple formulation: C(V,X) = C(X,V) It is dubious because it implies the extreme view that the choices are completely stable - in fact the choices will always be based on a preference relation P, which is the restriction to V of a "dictatorial" preference order Pu among all the alternatives in the grand domain U. On the opposite extreme there is the view that the choices are completely unstable. According to this view C(V,X) is the set of those alternatives of (V n X) which compared with alternatives of V, are regarded as not being worse than any alternative of (V n X) with respect to criteria of preference that happen to be current in V. But these criteria are related to the criteria current in other backgrounds in a loose way only, so that no stability rules determine how they are affected by a shift in background . The course we pursue in this paper goes midpoint between the two extremes. The basic idea, to be explained in section 9, is very simple. Let T be the V-part of the transitive closure of the union of certain basic preference orders. Then we assume that for each background V, the preference order P v on which C is based in V should (roughly speaking) be obtainable from T by resolving cyclical patterns in this closure. . This stability principle is, of course, not very strong . But it could be reinforced by certain consistency conditions for the resolutions of the cycles. We could require, for instance, that the resolutions should be carried out with some respect to the importance of the preference orders corresponding to the backgrounds. The more important an order is, the more immune it is to the clippings necessary to resolve the cycles. Now, a very natural relation of importance is definable in terms of the C-function and the consistency condition we have in mind reduces to the requirement that this relation be a quasi-ordering. This, and some other stability or consistency conditions, will be proposed in sections 10- II. In the concluding section we shall point out the close connection between choice functions - the D-function, in particular - and some kinds of operators studied in certain extension of Boolean algebras and in modal
CHOICE BASED ON PREFERENCE
217
logic. Some of the results given in sections 3 - 5 could be reviewed as simple modifications of results obtained in these fields . 2 . NOTATIONS
We will - without further ado - use standard notation for basic notions of elementary logic and set theory (including Boolean algebra). Lower case letters x.y .z, v, w will be used as variables ranging over the grand domain U, capital letters X,Y,Z, V,W will be variables ranging over the set
irreflexive in V iff .(xPx) for each x E V asymmetric in V iff xPy ~ xRy for each x,y E V transitive in V iff xPy & ypz ~ xpz for each x,y,z E V strict partial ordering in V iff P is asymmetric and transitive in
(v)
strict semiorder in V iff P asymmetric in V and for each x,y ,z, w
V
(vi) (vii) (viii)
E V, xRy & ypz & zPw ~ xPw and xPy & yRz & zPw ~ xPw strict weak ordering of V iff P is asymmetric in V and for each x,y,w E V, xRy & yPw ~ xPw strict linear ordering of V iff P is a strict partial ordering in V and xRy ~ xPy for each x,y E V, x¢y chain iff P is a strict linear ordering of some V
Clearly, (ii) implies (i) and (i) & (iii) implies (ii) and (vii) implies (vi) which implies (v) which implies (iv). Note that in a weak ordering a non-strict R-step can always be compensated by a strict P-step in order to secure a strict total step . In a semi-order, however, this is not the case. A non-strict step must sometimes be compensated by two strict steps in order to secure a strict total . This is because a strict P-step upwards in a semiorder has to be definitely noticeable, in order to count, and consequently a non-strict R-step upwards could be an unnoticeable, but nevertheless real , downward step . Note also that if all the chains of the strict ordering P are continuous or dense (i.e. there is a z such that ypz and zPw whenever yPw) then the distinction between a semiorder and a weak ordering collapses .
218
STIGKANGER
A strict partial ordering P in V is said to be neat in V iff each non-empty subset X of V contains a maximal element x such that (y)(y E X
~
.(yPx»
P is said to be grounded in V iff each non-empty subset X of V contains a "minimal" element x such that (y)(y E X
~
'(xPy»
In other words : a strict partial ordering is neat iff it contains no infinite ascending chains and it is grounded iff it contains no infinite descending chain. By the converse of P we mean the relation P such that xPy iff yPx for each x,y. By the transitive closure of P we mean the least relation p* such that for each x,y (i) (ii)
xP*y if xPy xP*y if xp*z & zP*y for some z
C is said to be a choice function based on Pv in V iff for each x and X x E C(V,X) iff x E (VnX) & (y)(yPyx ~ Y E (V-X» We also say that 0 is based on P, in V iff for each x and X x E D(V,X) iff x E V & (y)(yPyx ~ y E (VnX» 3. CHOICES BASED ON BINARY RELAnONS
The first group of axioms in our theory of choice functions consists of four principles:
Axiom 3.0 If X S; Y, then D(V,X) S; D(V,Y) Axiom 3.1 nxEpD(V,X) S; D(V,nxEpX) where F is any non-empty family of subsets of U Axiom 3.2 D(V,X) S; V Axiom 3.3 D(V,V) = V The axioms 3.0 and 3.2 are of course meant to hold for each X,Y S; U. Among the immediate corollaries of these axioms and definition 1.0 I would like to mention: (i) (ii)
(D(V,X) n D(V,Y» C(V,X) S; (V n X)
= D(V,(Xn Y»
CHOICE BASED ON PREFERENCE
(iii) (iv) (v)
219
(C(V,X) n Y) £; crv.rxrm (Sen's property a) (Sen's property 1') nXEFC(v,X) £; C(V,UxEFX) C(V,(XUY» £; C(V,(C(V,X) U C(V,Y»)
Theorem 3.4. The axioms 3.0-3.3 hold for each X,Y £; U if and only if D is based on some binary relation P, in V. (Jonsson & Tarski, 1951.) Proof. Assume that the axioms hold for each X,Y £; U. Assume also that V is non-empty. For the sake of notational simplicity let us in this proof restrict the range of the two variables v and w to V. Let P' be a binary relation defined in such a way that wP'v
B
(Y)(v E D(V,Y)
~
w E Y)
always holds. Let c be an arbitrarily chosen member of V. Let F be the family of subsets Y of U such that c E D(V,Y). By the definition of P' we then get: (Y)(Y E F
~
w E Y)
~
Since, because of axiom 3.3, nyEFy nyEFy
£;
w E {wlwP'c} £;
V, we may conclude that
{w l wl':c}
Then by axiom 3.0 D(V,nyEFY)
£;
D(V,{wlwP'c})
£;
D(V,{wlwP'c})
and hence by axiom 3.1 nyEFD(V,Y)
Since by the definition of F c E nyEFD(V,Y) we get c E D(V,{wlwP'c}) From this and axiom 3.0 we conclude that
{wlwl-'c}
£;
X
~
c E D(V,X)
In other words: (w)(wP'c
~
w E X)
~
c E D(V,X)
Since the converse implication c E D(V,X)
~
(w)(wP'c
~
w E X)
220
STIGKANGER
holds by the definition of P' we get: c E D(V,X)
(w)(wP'c
B
~
w E X)
Now, remembering the fact that c was an arbitrarily chosen member of V and that v,w E V, we get: (v)(v E D(V,X)
B
V
E V & (w)(wP'v
~
w E (Vn X»
Let P" be the restriction of P' to V. Then, in view of axiom 3.2 we may conclude that (x)(x E D(V,X)
B
X
E V & (y)(yP"x
~
Y E (V n X»
This conclusion implies the condition of 0 being based on some Pv in V under the assumption that V was non-empty. But since this condition is trivial when V is empty (because of axiom 3.2) we have now proved that the axioms 3.0-3.3 imply that 0 is based on some binary relation P, in V. The converse implication offers no problem. The only case that is not completely trivial is that of showing that the condition implies axiom 3.1 and we will restrict the proof to this case. Assume that there is a relation Pv such that for each x and X x E D(V,X)
B
x E V & (y)(yPyx
~
Y E (VnX»
We want to show that for each x x E nxEFD(V,X)
~
x E D(V,nxEFX)
In view of our assumption, the antecedent formula of this implication says that (X)(X E F
~
x E V & (y)(yPyx
~
y E (VnX)))
But this is logically equivalent to the following pair of statements: (3X)(X E F) ~ x E V (y)(yPyx ~ y E V & (X)(X E F
~
Y EX»
Since F is supposed to be non-empty, we then directly get, using our assumption, the succedent formula x E D(V,nxEFX) This concludes the proof of theorem 3.4 . The relation P, of theorem 3.4 is unique and definable in terms of 0 :
Theorem 3.5. If 0 is based on P, in V, then for each x,y E V
CHOICE BASED ON PREFERENCE
yPyx iff (X)(x E D(V,X)
~
221
Y E (VnX))
Proof. Assume the premise of the theorem andassume that x,y E V and that (X)(x E D(V,X)
~
Y E (VnX))
Then But since x E D(V,{ylyPyx}) is a direct consequence of the premise when x E V we get yPyx and conclude that the only-if-part of the conclusion of the theorem follows from the premises. Since the if-part of the conclusion is a direct consequence of the premise when x E V, theorem 3.5 is proved. The next theorem embodies our principal result on choice functions in this section. The theorem is a direct consequence of theorem 3.4.
Theorem 3.6. The following two conditions are equivalent: (i)
C is a choice function based on some relation Py in V. There is a function D in terms of which C is definable in the manner of definition 1.0 which satisfies the axioms 3.0-3.3 for each X,Y £; U. We note finally that if U is finite then axiom 3.1 can be replaced, without loss of strength, by the simpler principle: (ii)
(D(V,X) n D(U,Y))
£;
D(V,(Xny))
4. CHOICES BASED ON IRREFLEXIVE RELATIONS
Axiom 4.0 x E D(V,(V-{x})) for each x E V. Theorem 4.1 . If D is based on Py in V then Py is irreflexive in V iff axiom 4.0 holds. Proof. Trivial. Theorem 4.2. If C is a choice function based on P, in V and if P, is
irreflexive in V, then for each x,y E V xpyy iff .(y E qv,{x,y}))
222
STIGKANGER
Proof. Trivial, given the irreflexivity assumption • (yPS)· When C is a choice function based on an irreflexive relation in V, it follows from theorem 4.2 that this relation is unique. Since irreflexivity may be taken for granted in view of axiom 4.0 we may now speak of the relation on which a given choice function C is based. We shall introduce a special notation for this relation as well as for the corresponding D-function.
Definition 4.3.
P~ is the relation defined in terms of C, and DC is the function defined in terms of P~ in the following way: for each x, y and X:
xP~y iff .(y E C(V,{x,y})) x E DC(V,X) iff x E V & (y)(yP~x
~
Y E (VnX»
Note that the assumption of irreflexivity is essential in this connection. Without irreflexivity a choice function C might very well be based on several different relations. The choice function based on the identity relation in {a.b}, for instance, is also based on the universal relation in [a,b}. These facts constitute the reason why we have singled out irrefle.x.ivity as a separate case in the development of our theory.
Theorem 4.3. The following two conditions are equivalent: (i)
C is a choice function based on some irreflexive relation Pv in V. There is a function D (the function DC, to be precise) in terms of which C is definable in the manner of definition 1.0 which satisfies the axioms 3.0-3.3 and 4.0 for each X,Y £ U. (ii)
Proof. The theorem follows directly from theorems 3.4, 3.6 and 4.1. The next theorem is due to Sen.
Theorem 4.4. The corollaries (ii), (iii), and (iv) stated in section 3 combined with the principle
4J whenever (V n X)
¢
imply that C is a choice function based on
P~
C(V,X)
¢
4J in V.
Proof. Cf. Sen (1971). Sen's result does not hold if corollary (iv) is weakened to: (C(V,X) n C(V,Y» £ C(V,(XUY» because of counter-examples such as the following: Let V = U be the set of ordinal numbers. Every subset of V is in a natural way well-ordered by the relation <. When (V n X) is finite let C(V,X) be (V n X). When (V n X) is infinite, let C(V,X) be the set of
CHOICE BASED ON PREFERENCE
223
ordinal numbers contained in the initial infinite sequence (of type w) in the natural wellordering of (V n X). We readily verify that this choice function satisfies the corollaries (ii) - (iv) as well as the additional principle stated above. But it is not based on P~ because x E (VnV) & (y)(.(y E C(V,{x,y}) obviously holds for each x but x
e
~
C(V,V) when x
Y E (V-V»
~ w.
5 . CHOICES BASED ON NEAT PARTIAL ORDERINGS
Axiom 5.0 D(V,X) S; D(V,D(V,X» Axiom 5.1 If (V n X ~ ¢ , then (X n D(V,(V-X»)
~
¢
Theorem 5.2 . If D is based on P, in V, then P, is transitive in V iff axiom 5.0 holds for each X; and P, is a neat strict partial ordering in V iff both axioms 5.0 and 5.1 hold for each X. Proof. Assume the premise of the theorem and spell out the axioms in terms of R; Then, the transitivity of P, trivially implies axiom 5.0, and axiom 5.0 implies transitivity when X is instantiated to {ylyPyx}. Axiom 5.1, when spelled out, coincides with the condition of neatness explained in section 2. Moreover, it implies the irreflexivity of Py when X is instantiated to singletons {x} S; V. From these facts we get the conclusions of the theorem.
Theorem 5.3. The following two conditions are equivalent: (i)
C is a choice function based on some neat strict partial ordering in V. (ii) There is a function D (the function DC, to be precise) in terms of which C is definable in the manner of definition 1.0 which satisfies the axioms 3.0-3 .3 and 5.0-5 .1.
Proof. The theorem follows from theorems 3.4, 3.6, and 5.2. Now that the choice functions C are based on neat strict partial orderings in V, their behaviour is easily visualized. The elements of V are ordered by Pv in the pattern of diagrams of a well-known kind. Then, given the intersection of V and a set X, C selects all top elements of ascending chains of the diagram inside this intersection not excluding top elements that are "joins" of several chains or are contained in degenerate chains consisting of only one element. Since the partial ordering is neat we can always be sure that every chain contains a top element. For instance, in the following diagram Pv
224
STIG KANGER
0
I
I
a
v
0
0
b
c
I I
I /~ r ~d/ O 0
I
0
e
0
0
I 0
x
I
I
C(V,X) = {a,b,c,d,e}.
We note that if we keep axiom 4.0, the axioms 5.0 and 5.1 could be replaced by the following "principle of path independence" (Plott, 1973). C(V,(C(V,X) U C(V,Y») £ C(V,(XUY»
Theorem 5.4. If D is based on an irreflexive relation Pv in V and if C is defined in the manner of definition 1.0, then P, is a neat strict partial ordering in V iff the principle of path independence holds for each X,Y. Proof. Cf. Plott (1973). Note also that the converse principle of path independence was a corollary of the axioms in section 3. 6. CHOICES BASED ON GROUNDED PARTIAL ORDERINGS
= (Y n D(V,(V-Y») for which it holds that there is no Z£(VnX) such that Y = (ZnD(V,(V-Z))) but Y ;iI! Z.
Axiom 6.0. There is a Y £ (VnX) such that Y
Theorem 6.1. If D is based on a strict partial ordering Pv in V, then Pv is grounded iff axiom 6.0 holds for each X. Proof. Assume the premise of the theorem. Then, if Py is grounded in V every descending chain in the ordering Pv or in the part of Pv included in X has a minimal element. Axiom 6.0 is verified when Y is the set of these minimal elements. If P, is not grounded , P, contains an infinite descending chain. The axiom is falsified when X is the set of elements in this chain. Theorem 6.2 . The following two conditions are equivalent: (i)
C is a choice function based on some grounded and neat strict partial
CHOICE BASED ON PREFERENCE
225
ordering in V. (ii) There is a function D (DC, to be precise) in terms of which C is definable in the manner of definition 1.0 which satisfies the axioms 3.03.3,5.0,5.1, and 6.0 . Proof. The theorem follows from theorems 5.2 and 6.1.
The axiom 6.0, when added to the preceding axioms, implies that the chains in the ordering P; are all finite. This assumption of finiteness is of course realistic in view of the fact that P; is supposed to be an order of preference . In any event, we never need to use infinite ascending or descending chains of the preference order in the practice of ordinary life. In theory, however, we may be interested in infinite preference chains. This is true especially when we want to extend the theory of preference and choice to a theory of measurement of value. It is well-known that some kinds of measures cannot be shown to be unique unless we assume that the property to be measured is continuous: Between every two points of the property there shall be an intermediate point. The Fahrenheit scale of temperature, for instance, cannot be shown to be unique (up to linear transformations) unless we take for granted that between two temperatures there is always an intermediate point of temperature. Thus in a theory of value that is analogous to the theory of temperature scales we may need to assume the following principle: (x)(y)(xPyy
~
(3z)(xPyz & zPyy))
This principle could also be expressed in terms of the D-function : D(V,D(V,X))
~
D(V,X)
This principle is, of course, incompatible with the assumption of P, being neat and grounded. But it can be added to the axioms 3.0-3.3,5.0, and 5.1 without involving any inconsistencies. It then implies that every Py-chain is a converse well-ordering of a formidable sort: Wherever a cut is made in such a chain, there is always a next element below but an infinite descending chain above. Moreover, whatever position in such a Py-order, one can always reach a top position in a finite number of steps. But once there one can never get back - no matter how many steps are climbed down, one never returns to any position passed on the way up. In the development of our theory of choice we have now come to a crossroad, and our decision to go in the direction of finiteness is somewhat arbitrary . But this decision is not particularly crucial. The axiom 6.0 of finiteness plays only a minor role in the theory and it could be omitted.
226
STIGKANGER 7. CHOICES BASED ON SEMIORDERS
From now on we shall sometimes formulate our axioms or propositions in terms of C instead of D. In doing so, we always, in accordance with definition 1.0, regard C(V,X) as an abbreviation of the longer expression (X n D(V,(V-X»). The main reason for this notational change is the fact that it makes the content of the propositions more apparent.
Prop. 7.0.
If X,Y £; V and X ~ rfJ, and if (X U C(V,Y» = C(V,(X U C(V,Y»), then X = C(V,(X U «Y-C(V,Y») - C(V,(Y-C(V,Y»»).
Prop.7.1.
If X,Y £; V and (X-C(V,X» ~ rfJ, and if «X-C(V,X» U C(V,Y» = C(V,«X-C(V,X» U C(V,Y»), then C(V,X) = C(V,(C(V,X) U (Y-C(V,Y»».
Theorem 7.2. If 0 is based on a neat strict partial ordering P, in V, then P, is a semiorder in V iff 7.0 and 7.1 hold for each X,Y. Proof. Assume the premise of the theorem and consequently also that C is a choice function based on the neat strict partial ordering Py in V. Let X be {x} and Y be {y,z,w} in prop . 7.0 and let X be {x,y} and Y be {z,w} in prop. 7.1. These two instantiations imply, in view of the premise, the two conditions which (together with asymmetry) define a semiorder : .(yPyx) & yPyz & zPyw xPyy & .(zPyy) & zPyw
~
~
xPyw xPyw
Conversely, these two conditions imply prop. 7.0 and 7.1 respectively. To see this, in case of 7.0, assume that X,Y £; V and X ~ rfJ and that (X U C(V,Y» = C(V,(X U C(V,Y») Then, clearly in view of the premise • (yPyx) when x E X and y E C(V,Y), and yPyz & zPyw when y E C(V,Y) , z E C(V,(Y-C(V,Y))) and w E «Y-C(V,Y» C(V,(Y -C(V, Y»» . Hence, by the first of the conditions defining semiorders for each x E X and w E «Y-C(V,Y» - C(V,(Y-C(V,Y»». Since it follows from our assumptions that
CHOICE BASED ON PREFERENCE
227
x = C(V,X) we may conclude, using the premise again, that x = C(V,(X u «Y-C(V,Y» - C(V,(Y-C(V,Y»»» The proof of prop . 7.1 from the second condition is similar to that of 7.0.
Theorem 7.3 . The following two conditions are equivalent: (i) C is a choice function based on some neat strict semiorder in V. (ii) There is a function D (DC to be precise) in terms of which C is defined in the manner of definition 1.0 which satisfies the axioms 3.0-3 .3, 5.0, 5.1 , and prop . 7.0 and 7.1. Proof. The theorem follows from theorems 5.3 and 7.2 . Prop . 7.0 and 7.1 are, of course, not the only conditions that in this context are equivalent with Py being a semiorder. Among the alternatives to 7.0 and 7.1 I would like to mention axioms 3 and 5 in Fishburn (1975) (or rather 3 and 5 adapted to binary choice functions). 8. CHOICES BASED ON WEAK ORDERINGS
Prop. 8.0 .
If (C(V,X) n C(V,Y» ~ cP, then ~ (C(V,X) n C(V,Y»
crv.rxnv:
Theorem 8.1. If D is based on a neat strict partial ordering Pv in V, then Pv is a strict weak ordering in V iff prop. 8.0 holds for each X, Y.
Proof. Assume the premise of the theorem and assume first that .(yPy x) and yPyz. Now either xpyy or .(xPyy). In the first case we immediately get xPyZ since Pv was a strict partial ordering. In the second case let (V n X) = {x,z} and (Vr't Y) = {xyz}. Then clearly x,y E C(V,Y) but z fE C(V,Y). Hence, by prop . 8.0, z fE C(v,(XnY» . But then x E C(V(XnY» and consequently xPyZ. Assume then that Pv is a weak ordering in V and that (C(V,X) n C(V,Y» ~ cP - let us say that x E (C(V,X) n C(V,Y» . Assume also that y E C(v,(XnY». Hence, in view of the premise, '(xPyy). Assume now that y somehow should not be a member of C(V,Y). Then there must be a z E C(V,Y) such that zpvY and • (zPyx). Since P, was a weak ordering, xpyy and we get a contradiction. We conclude that y E C(V,Y) and also, for similar reasons, that y E C(V,X) . Theorem 8.2 . The following two conditions are equivalent: (i) (ii)
C is a choice function based on some neat strict weak ordering in V. There is a function D (DC to be precise) in terms of which C is defined
228
STIG KANGER
in the manner of definition 1.0 which satisfies the axioms 3.0-3 .3, 5.0, 5.1, and prop. 8.0. Proof. The theorem follows from theorems 5.3 and 8.1.
We shall note, finally, that prop. 8.0 could be replaced by the principle which expresses Sen's property (3: If X ~ Y, then either C(V,X) C(V,X) ~ «VnY) - C(V,Y»
~
C(V,Y) or
Theorem 8.3 . If D is based on a neat strict partial ordering Pv in V, then Pv is a strict weak ordering in V iff Sen's property (3 holds for each X,Y . Proof. Cf. Sen (1969). 9. THE STABILITY AXIOM
Let us use the formula SwordiPfn) - read P~n is a strict weak ordering of Vn - as an abbreviation of the following sentence: For each X and Y, if (C(Vn,X) n C(Vn,Y» then C(vn,(XnY» ~ (C(Vn,X) n C(Vn,Y»
~
cP,
which is Prop. 8.0. n = 1,2, .. . Let S* be the transitive closure of the relation Sand S the converse of S. Then the stability axiom we have in mind could be formulated as follows: Axiom 9.0. There are V1 ,... ,Vm ~ U such that Sword(P~ I )&.. .&Sword(P~m) ~ and such that «T- T) n (WxW» S;; P; S;; T, where T = (P~ I u ...u P~m)*
In other words: Each preference order P; is included in the transitive closure T of the union of certain basic preference orders P~ I ,... ,P~m and includes the W-part of the non-controversial part (T- T) of this closure. Note that (T- T) is what remains of T after removal of all cycles in T . The basic preference orders should correspond to simple aspects with respect to which the comparison of the alternatives are made. We assume that these aspects are finite in number and that the basic preference orders are strict weak orderings . The latter assumption is perhaps doubtful - perhaps it should be weakened in the direction of semiorders. For instance, if price is one of the basic aspects we often get only a semiorder of prices, since many prices could not be calculated without a margin of error. But it could be argued on the other hand that such errors need not be taken account of explicitly in the theory. They are external to it and concern ~
CHOICE BASED ON PREFERENCE
229
only our difficulties in estimating the real preferences (which are weak orderings) when we wish to apply the theory. We leave this problem unsettled. The strength of the axiom of stability depends on the number m of basic preferences. If there is only one, say pe , the preferences are completely stable and unaffected by a shift in background. p~ will always be the restriction to W of pel. But if there are several conflicting basic preferences, then the axiom says very little about the structure of P~. lO. THE CONSISTENCY AXIOM
The stability axiom 9.0 could be reviewed as a principle requiring that a preference order Pw should always be obtainable from the transitive closure T of the union of certain basic preference orders by resolving cyclical patterns in T . When there are several conflicting basic orders , the extension of Pw may depend to a large extent on how the cycles are resolved. These resolutions could be carried out in any way as far as the axiom 9.0 is concerned. There are, however, some restrictions on the resolutions which seem rational. First we define a relation ~ between preference orders.
Definition 10.0 pe
~ p~
iff
P~nw ~ P~ ~ P~uw
means that the preference orders P~nw and peuw conform to P~ rather than to P~ in case P~ and P~ should be in conflict. When it comes to resolutions of cycles in which both pe and p~ are involved, the resolution should be made at the expense of P; rather than ~ . We could interpret P~ ~ P~ as P~ being a stronger or more important preference order than P~ when they are in conflict in a cyclical preference pattern. The relation ~ is obviously reflexive. We shall now require as a condition of rationality that it is also transitive .
P~ ~ P~
Axiom 10.1. If pel
~
pe2 and pe2 ~ pe3 , then P~ l ~ pe3
Unpublished manuscript. The paper is written sometime during the 1970's and is unfinished. Section 10 is incomplete and sections 11 and 12 are missing. There is also no reference list. The list provided below is made by the editors.
230
STIG KANGER REFERENCES
Fishburn, P.C ., [the editors were unsuccessful in identifying this item], 1975. Jonsson, B. & A. Tarski, "Boolean Algebras with Operators, " American Journal of Mathematics 73, 1951. Plott, C.R., "Path Independence, Rationality and Social Choice," Econometrica 41, 1973. Sen, A.K., [the editors were unsuccessful in identifying this item], 1969. Sen, A.K., "Choice Functions and Revealed Preference," The Reviewof Economic Studies 38, 1971.
DECISION BY DEMOCRATIC PROCEDURE
Is it possible to arrive at a parliamentary decision method , for example , in the form of a voting procedure, that is democratically satisfactory? This is one of the more interesting problems in theoretical political science and political philosophy, and it is worth more consideration than it has received . Roughly speaking, the work that has been done to deal with the problem can be divided in the area into two groups: works with a positive orientation, which aim at finding a satisfactory voting system, and works with a negative orientation, which aim at showing that a satisfactory parliamentary decision process is impossible . The most important piece of work with a negative orientation is Kenneth J. Arrow' s Social Choice and Individual Values (New York , 1951). A good source for survey and information within this area of research is Luce & Raiffa's Games and Decisions (New York, 1957), chap. 14. This essay is a contribution to the negative approach. First, I give an example that illustrates certain unsatisfactory features of a common voting procedure. I then present Arrow's basic negative result and a consideration that reduces its negative effect. Finally, I prove a (as far as I know) new and (it would seem) more negative variant of Arrow's result. (I want to thank Mr . Goran Enger for contributing to the result.) AN EXAMPLE OF VOTING
We imagine a parliament with 101 members and 4 parties: Left with 48 representatives, Right with 47, Extremists with 4 and Independents with 2 representatives. The parliament is to choose and ratify one of four proposed alternatives : A, B, C and D. The parties rank the alternatives thus:
231 G. Holmstriim -Hintikka, S. Lindstrom and R. Sliwinsk i (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. I. 231-236. © 2001 All Rights Reserved, Printed by Kluwer Academ ic Publishers , the Netherlands . OriginaIly published in Swedish in Ann -Man Henschen-Dahlqvist et a1. (eds .), Sann ing Dikt Tro. Till Ingemar Hedenius, Stockholm: Albert Bonniers fiirlag AB , 1968. Translated by Sharon Rider.
232
STIG KANGER very good
good
poor
very poor
2
A
B
C
D
Left
48
B
A
D
C
Right
47
C
A
B
D
4
D
C
B
A
Independents
Extremists
We suppose that the parliament votes on the alternatives according to the simple majority rule, and that the alternatives meet in pairs in some order . If, for example, B first meets C, the winner meets D, and the winning alternative meets A, we have a voting order that we represent with BC, -D , -A. There are twelve different voting orders of this kind. Let us now make the (somewhat unrealistic) assumption that parliament representatives always vote according to the party line. In that case, only one of the twelve voting orders (namely, BC, -D, -A) gives the final victory to alternative A, which all parties except the extremists considered good. (C first beats B with 51 votes to 50, D then beats C with 52 votes to 49, and finally A wins over D with 97 votes to 4.) Three voting orders give victory to the controversial B, and five to the more than controversial C. Three voting orders give the final victory to alternative D, which all parties except the extremists considered poor. The distribution looks somewhat different if we do not assume that the party line is always followed, but imagine instead that tactical voting does occur. Tactical voting on the part of a party aims at giving the final victory to an alternative that is as good as possible according to the party's own ranking of the alternatives. With tactical voting, the voting order Be, -D, -A no longer gives the final victory to A. First e beats B. But when e meets D, the extremists vote tactically for C, against their own preferences for D. In the final round, e then wins over A. With this tactic, the extremists give the final victory to the "good" alternative C instead of the "very poor" alternative A. If we assume that every party always votes tactically in this manner, and takes into account in their tactics that all parties vote tactically, the distribution will be this: alternative B wins in three voting orders, e wins in six, and D in three. (The assumption that the parties always vote tactically is naturally a bit unrealistic . Common sense tells us that in voting orders in which alternative D triumphs, Left and Right agree upon a compromise, such as alternative A.)
DECISION BY DEMOCRATIC PROCEDURE
233
Our example has illustrated a couple of democratically unsatisfactory aspects of voting as a parliamentary decision method: the outcome of the voting is often dependent upon the voting order (who decides that?) and the outcome need not be an expression of the so-called will of the people (any more than alternative D was). ARROW 'S RESULT
Let us imagine that a parliament is going to rank at least three alternatives. The ranking takes place through one decision method or another, on the basis of the members' own ranking of the alternatives. Arrow advanced four axioms that can be understood as necessary conditions that must be fulfilled for the parliamentary decision method to be considered satisfactory . The axioms are these: (i)
The parliament's ranking must not be negativistic: if any alternative is moved up in a parliament member's ranking, but no other changes are made in the members ' rankings, that alternative is not moved down in the parliament's ranking. (ii) The parliament's ranking must not be too variable: the reciprocal order between two alternatives in the parliament's ranking can be Changed only if a member changes it in his own ranking. (Thus it is not sufficient that the members of parliament move entirely different alternatives up or down in their rankings) (iii) The parliament's ranking must not be inflexible: the reciprocal order between two alternat ives can be changed if the members of parliament change them in their own rankings . (iv) The parliament's ranking must not be too biased: it must not constantly follow the ranking of any single member . Arrow proves that no parliamentary decision method is satisfactory such that it fulfills all four axioms. In the theory of democracy, it is thought that the will of the people will in some way manifest itself in the decisions of parliament. There is no doubt that Arrow's result constitutes a difficulty for this theory (or, if one so wishes, a negative contribution to the theory). The negative effect is mitigated, however, by the fact that Axiom (ii) may seem unnecessarily strong. An example can illustrate this. Suppose that 45 percent of the members of parliament rank five alternatives thus: Al > A 2 > A3 > A4 > As (we write Al > A2 in place of: Al is put ahead of A2 ; we will also write
234
STIG KANGER
Al = A z in place of: AI is ranked on the same level as A z). Let us also suppose that 45 percent of the members of parliament have the ranking: Al > A z > A4 > As > A3 • It is not unreasonable to suppose then that the parliament has the ranking: AI > A z > A3 = A4 = As. Let us now suppose that 90 percent of the members changed their rankings to: A4 > AI > A z > A3 > As. The parliament ought not then retain its old ranking. Rather, the ranking should be changed so that it coincided with the 90 percent majority's unanimous ranking. But this change. which affects the reciprocal order between A4 and As, is brought about without the need for any member of parliament to change the order between A4 and As in his own ranking. It suffices that other alternatives are moved up or down. We can observe that voting according to the majority rule does not satisfy Axiom (ii). The voting order Be, -D, -A in our example (without tactical voting) gives the victory to A. The parliament accordingly puts A ahead of e. But if the extremists should change their ranking from D > e > B > A to e > D > B > A, the victory would go to alternative e. Parliament would therewith put e ahead of A, and the change of parliament's order between A and e depends exclusively on the extremists demoting alternative D. If one considers Arrow's Axiom (ii) too strong, there immediately arises the following problem: can it be shown that no satisfactory parliamentary decision process exists even if Arrow 's Axiom (ii) is weakened? I will show that this can easily be done, if the remaining axioms are made somewhat stronger. The reinforcement is entirely within the scope of what must reasonably be demanded of a democratically satisfactory decision method. A VARIANT OF ARROW 'S RESULT
As earlier, we imagine that the parliament is going to rank at least three alternatives . The ranking is to occur with the aid of some decision method on the basis of the parliament members' own rankings . We can call the set of these rankings the parliament's opinion. We assume (as earlier) that the decision method gives exactly one ranking for every opinion. One group, G, of (one or several) members of parliament is said to be decisive for the alternative pair Ai' Aj if the parliament's ranking always has Ai > Aj when every member of G has Ai > Aj and every other member of parliament has Aj > Ai' A parliamentary decision method is said to be consistent regarding Ai relative to Aj if the order between Ai and Aj in the parliament's ranking is
DECISION BY DEMOCRATIC PROCEDURE
235
changed only if some member of parliament changes it in his own ranking . The decision process is said to be consistent regard ing Ai> quite simply, if it is consistent regarding Ai relative to every other alternative in question. We now advance the following conditions for the parliamentary decision process : I.
II.
If all the members of parliament, or all but one, have Ai > Aj , then the parliament also has Ai > Aj in its ranking. The decision process shall be consistent regarding at least one alternative .
Condition I is stronger than Arrow's corresponding condition. It means that no individual member of parliament may be decisive for any pair of alternatives. Arrow was contented, in Axiom (iv), to require that no individual member of parliament may be decisive for every pair of alternatives. Condition I also demands that the number of members of parliament shall not be too small with respect to the number of alternatives. Suppose, for instance, that we have only three members of parliament, and that they rank three alternatives as follows: Member 1: Al > A2 > A3 Member 2 : A z > A3 > Al Member 3: A3 > Al > A2 Parliament must then have Al > A2 , since every member except one thinks so. For the same reason, the parliament has A2 > A3 , and therewith, also Al > A3 • But all members but one are of the opinion A3 > AI' The unreasonableness disappears if the number of parliament members is increased. A parliament of the usual size can handle all arrangements of alternatives that occur in practice without the risk of conflict with condition I. Condition II is weaker than Arrow 's Axiom (ii), which required that the decision process be consistent regarding every alternative. We can now show that no parliamentary decision method can satisfy both Condition I and Condition II . The proof is very simple and builds upon ideas that are well-known from , among other things, the proof of Arrow's result rendered in Luce & Raiffa's book . Let us look at a parliamentary decision method, B, and assume that it satisfies I and II. Let G be a group of members of parliament such that: (1) G does not include every member of parliament; (2) G is decisive for some pair of alternatives, let us call it Ai' Aj ; and (3) G is minimal in the sense that no smaller subgroup of G is decisive for any pair of alternatives. According to
STIG KANGER
236
Condition I, G must consist of at least two members of parliament. Let q be one of them . G-q is G minus q, and P-G is parliament minus G. Let Ak be an alternative other than Ai and Aj • such that the decision method B is consistent regarding at least one of the alternatives Ai' Aj , and Ak • (There must be such an alternative by virtue of Condition II). We now have two cases: (1) decision method B is consistent regarding Ai' and (2) B is consistent regarding Aj or Ak • We will now show that in both cases, the assumption that B satisfies Conditions I and II leads to an absurdity. Case 1: Let 0 be the class of all parliamentary opinions that, with respect to the alternatives Ai' Aj , and Ak , appear thus : q: G-g: P-G:
Ak > Ai> Aj Ai> Aj > Ak Aj > Ak > Ai
We see that with the 0 opinions, parliament must have Ai > Aj (G decided it) and Aj > A k (since all but q had that). Thus parliament has Ai > Aj > Ak and, therewith, Ai > Ak • Since decision method B in this case was consistent with regard to Ai' parliament must also have Ai > A k with every opinion that resembles the opinions in 0 with respect to the ordering of Ai and A k , that is to say, with every opinion where q and P-G have Ak > Ai and G-q has Ai > Ale' But this means that G-q would be decisive for the pair of alternatives Ai' A k , which is absurd considering that G was a minimal decisive group. Case 2: Let 0' be the class of opinions such that q: G-q: P-G:
Ai> Aj > Ak A k > Ai > Aj Aj > Ak > Ai
We see that with these opinions, parliament must have Ai > Aj and Ak > Ai and, therewith, Ak > Aj • Since B in this case was consistent regarding Aj or A k , parliament must have Ak > Aj with all opinions that resemble the opinions in 0' as regards the order between Aj and A k • But this means that G-q would be decisive for the pair Aj , Ak , which is absurd. We draw the conclusion that a parliamentary decision method cannot satisfy both Condition I and Condition II.
PHILOSOPHY OF SCIENCE
MEASUREMENT: AN ESSAY IN PHILOSOPHY OF SCIENCE l
MEASUREMENT AND LOGIC
1. The concept of measure In this essay, we shall say that m is a measure in M if m and M fulfill the following conditions: I. M is a finite, or countably infinite, non-empty class; II. m is a one-place operation, defined for finite subclasses of M, and with real numbers as possible values; and III. for each finite subclass, Y, of M, it holds true that m(Y)
=
E m({x}).
xEY
We say that m is a measure with the unit N in M, if m is a measure in M and IV. N is a finite subclass of M; and V. m(N) = 1. Example. Let M be the class {a,b,c} - i.e., the class which has a, b, cas its only members - and let m be a measure in M such that m({a})=-l, m({b}) =0, m({c})=2. We can then see that , according to III, m({a ,b})=-l, m({a,c})=l, m({b,c})=2, m(M)=l, and m({})=O. ({} denotes the empty class.) We see that m is a measure with the unit {a,c} or with the unit M .
2. Aspects and numerical structures In this essay, we shall say that S is a structure of type (nl ,n2" " ,nk ) if S is an ordered set (U,Rl>R 2 , •• • ,Rk) in which U is a non-empty domain or universe, and Rj is an n.rplace relation between elements of U, i.e., Rj is a class of ordered n.rtuples of elements of U (j = 1,2, ... ,k; and nj= 1,2, ...). S is said to be finite if the domain U of S is finite . Let M be a non-empty finite or countably infinite class . It will be said that a structure (U,R 1" •• ,RJ is an aspect of M if U is a domain of finite subclasses of M . Note that U need not contain all finite subclasses of M . 239
G. Holmstrom-Hintikka, S. Lindstrom and R. Sliwinsk i (eds.}, Collected Papers of Stig Kanger with Essays on his Life and Work. Vol. I. 239-273. © 2001 All Rights Reserved. Printed by Kluwer Academic Publishers. the Netherlands. Originally published in Theoriea 38 (1972),1-44.
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Example. Let M be the class of all concrete things, and let V be the domain of all finite subclasses of M . Let == and ~ be 2-place relations between elements of V defined in such a way that for each X, YE V, it holds that
X == Y if and only if X is exactly as heavy as Y; X ~ Y if and only if X is lighter than Y. (To say, for example, that X is lighter than Y is synonymous with saying that the elements of X taken together are lighter than the elements of Y taken together .) The structure (V, == , ~) is an example of an aspect of M . We shall later assume that qualities such as length, weight, volume, density, subjective loudness, etc., can be identified with aspects. We assume, for example, that weight will be identifiable with precisely the aspect (V, == , -e) ofM.
A structure (V,R1,...,Rk ) in which V is the domain of all real numbers or the domain of all non-negative real numbers will be called a numerical
structure. Example. The structure (R, =, <) in which R contains all non-negative numbers, and = and < are the relations 'identity' and ' less than', respectively.
3. Measures for aspects Let S=(V,R1, ... ,Rk ) and S' =(U',R; ,.. .,RD be two arbitrarily chosen structures of the same type (nl,' " ,nk) . We say that S' is a substructure of S or, to be exact, that S' is a confinement to V' of S, if V' is a subdomain of V and if for each j = 1,2, .. . ,k, the relation RJ is identical with the relation Rj confined to V', i.e., RJ is the class of all ordered njtuples of elements of U' in Rj • The mapping h is called a homomorphism from S onto S' if (I) (2)
(3)
h is a l-place operation defined for at least all elements of V; V' is identical with the set of all h(X) with XE V; and for eachj= 1,2,...,k and each njtuple (X!> ... ,Xn .) of elements of V, it holds true that R/X1,... ,Xn) if and only if RJ(h(Xl) ,... ,h(Xn) .
S' is called a homomorphic image of S if there is a homomorphism from S onto S' . Note that our definition of homomorphism is slightly stronger than the one ordinarily used in algebra, which at (3) requires only that RJ should be the class of all ordered nrtuples (h(X1) ,... ,h(Xn) such that R/X!> ...,Xn)-
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m is called a measure (with the unit N) in M for A with respect to S if (1) (2) (3)
A is an aspect (U,R 1, ••• ,R0 of M (with NE U), and S is a numerical structure of the same type as A; m is a measure (with the unit N) in M; and
m is a homomorphism from A onto a substructure of S.
4. Scales and measurement We shall call certain numerical structures scales. S =(R, =, < ,...) is said to be an ordinal scale if S is a numerical structure in which = is the relation 'identity' and < is the relation 'less than' in the domain R. We say that S=(R,=,<,B, ... ) is a bisectional scale, if S is an ordinal scale, and B is a 3-place relation such that for each x,y,zER it holds true that B(x,y,z) if and only if y+y=x+z. A structure S = (R, = , < ,I, ...) is said to be an interval scale if S is an ordinal scale, and I is a 4-place relation such that for each x,y,z, wER, it holds true that/(x,y,z, w) if and only if [x-y I =:;; [z-w] . A structure S=(R , =, < ,D,...) is called a difference scale if S is an ordinal scale, and D is a 4-place relation such that for each x,y,z,wER, it holds true that Dtx.y.z,w) if and only if x+y=:;;z+w. A scale (R+, = , < ,...), where R+ contains only the non-negative numbers, is said to be non-negative. S = (R+ , =, < ,D,Q, ...) is called a quotient scale if S is a non-negative difference scale, and Q is a 4-place relation such that for each x,y,z, wER+, it holds true that Q(x,y,z,w) if and only if x ·y=:;;z ·w. The question of how we shall more exactly delimit the set of scales from other numerical structures is left open. But we can connect this question with a second problem, namely the definition of the abstract concepts measurement and measurability. We can accordingly say that a measurement is a system (M,A,S,m) in which m is a measure for A with respect to S, and S is a scale. Furthermore, it will be said that an aspect A of M is measurable if there is a measure, m, in M for A with respect to a particular scale. The scales we have defined above are important in connection with measurement. But it should be pointed out that they are not the only ones; there are several other types of numerical structures which are well qualified as scales. One such structure is the semi-ordinal scale (R, =', < '), where =' and <' are defined thus: For each x,yER,
x =' Y if and only if neither x <' y nor y <' x; x <' y if and only if x+ 1 < y-l.
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(Note that our terminology, when it comes to scales, differs from a more orthodox terminology; see § 24, below.)
5. A classification of aspects Let A=(U,Rl> .. .,Rk) be an aspect of M. We shall first classify aspects in regard to the nature of U, making the following definitions:
A is said to be a purely structural aspect of M if U is the domain of all unit classes {x} with xEM. A is a structural aspect of M with zero-element if U is the domain of all classes {x} with xEM in addition to the empty class A is an additive aspect of M if U is a domain of finite subclasses of M such that whenever X and Yare elements of U, their intersection xn Y, union XU Y, and difference X- Yare also elements of the domain U.
n.
We then classify aspects of M in regard to their measurability and make the following definitions :
A is said to be a comparative aspect if there is a measure in M for A with respect to an ordinal scale. A is a (non-negative) pseudo-intensity if there is a measure in M for A with respect to a (non-negative) bisectional scale. A is a (non-negative) quasi-intensity if there is a measure in M for A with respect to a (non-negative) interval scale. A is a (non-negative) intensity if there is a measure in M for A with respect to a (non-negative) difference scale. A is a capacity if there is a measure in M for A with respect to a quotient scale. Observe that a quotient scale always is non-negative. A is a quantity if A is an additive aspect of M, and if there is a measure in M for A with respect to a non-negative ordinal scale . We say, for the sake of simplicity, that a scale corresponds to a quantity if it is a non-negative ordinal scale, to a capacity if it is a quotient scale, to an intensity if it is a difference scale, and so on. We complete the classification of aspects in the following way: Let A be a comparative aspect, pseudo-intensity, quasi-intensity, intensity, capacity, or quantity . A is said to be dense if there is a measure, m, in M for A with respect to a corresponding scale such that for each X, YE U, there is a 'bisectional' element ZE U such that m(Z)+m(Z)=m(X) +m(Y). A is a compact quantity if A is a dense quantity , and if there is a
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measure, m, in M for A with respect to a corresponding scale (i.e ., a nonnegative ordinal scale) such that for each X, YE U with m(X) < m(y), there is a ZE U such that m(Y) =m(Z) and X is a proper subclass of Z. A is said to be linear if for all measures m and m' for A with respect to a corresponding scale, it holds true that m' is a positive linear transformation of min U, i.e., there is a real number p>O and a real number q such that for each XE U it holds that m'(X)=m(X)·p+q. A is uniform if for all measures m and m' for A with respect to a corresponding scale it holds true that m' is a similarity transformation of m in U, Le., there is a real number p > 0 and such that for each XE U, it holds that m'(X)=m(X)·p. Note that if A is linear and the empty class {} is an element of U, then A is also uniform, since it always holds true that m({})=m'({})=O. For example, a linear quantity is uniform.
Example. We can easily construct examples of additive aspects of M which are capacities but not linear. Let M be the class {a,b}, and let U be the domain of all subclasses of M, and let the aspect A=(U,R1,R2,R3,R4 ) of M be a capacity such that there is a measure, m, in M for A with respect to a quotient scale such that m({a})= 1 and m({b})=3 . Then A is not linear. (If we had assumed instead that m({b}) =2, then A would have been uniform.) 6. Some theorems
Theorem 1. Let A=(U,R1,.. .,Rk ) be an aspect ofM. If A is a dense pseudointensity, then A is also a linear pseudo-intensity. Proof. Let m and m' be two measures in M for A with respect to a bisectional scale. We shall show that there are numbers p and q, where p>O, such that for each XE U it holds true that m'(X) = m(X) . p+q. Let us define
f(Y,Z)
m'(Z)-m'(Y) m(Z)-m(Y)
YZ - m'(y) · m(Z)-m(Y) . m'(Z) g( , ) m(Z)-m(Y) We shall first show the following:
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(1)
STIG KANGER
For each X,Y,ZE U such that m(Y) < m(Z) and m(Y) ~ m(X) ~ m(Z), it holds true that m'(X)=m(X)·ftY,Z)+g(Y,Z) .
Let us make the hypothesis that (1) does not hold true, i.e., let us assume that X,Y,Z are some elements of U such that m(Y) < m(Z); m(Y) ~ m(X) ~ m(Z); and m'(X) = m(X) . ftY,z) + g(Y,Z)
+ e, where e=l=O .
Since A is a pseudo-intensity, it follows from the hypothesis that m'(y)
< m'(Z).
Let us now perform a reiterative bisection of the interval from m'(y) to m '(Z); in each step, we bisect all of the subintervals obtained in the previous steps. We perform the bisection in n steps, and choose n so that if r' and r are two bisectional points which lie next to each other after then n-th step, and if r< r', then r'-r< Ie]. Since A is dense, it holds, for each bisectional point r, that there is an element X'E U such that m'(X')=r. We choose such an element for each of the 2n-l bisectional points. The chosen elements, as well as Yand Z, are called the bisectional elements. Let us write 10 for Y, and Y2n for Z. Let Yo,Y"Y2 , . .. , Y2n _" Y2n be the 2n + 1 bisectional elements, and let m'(}j)<m'(}j+t), where i=O,I, .. .,2n- 1. We have that m'(l)+ +m'(l)=m'(l)-I)+m'(l}+I)' wherej=I ,2, ... ,2n- 1. Since A is a pseudointensity, we also have that m(}j) <m(}j+t), and that m(l})+m(l})= m(l}-t)+m(l}+t). We see now that for k=O ,1,.. .,2n , m'(Yk ) = m'(Yo) m(Yk )
.s.2 . n
= m(Yo) +k2n
»; o».
(m'(Y2n)-m'(Yo
. (m(Y2n)-m(Y
From that it follows by simple algebra that m'(Yk ) = m(Yk ) • j(Y,Z) + g(Y,Z) .
We now have two cases: (I) There is a bisectional element Yk such that m(X)=m(Yk ) and, consequently, m'(X)=m'(Y0 . Our hypothesis that (1) does not hold must, in this case, be false. (II) For some i=O,1, ... ,2n-l, it holds that m(}j)<m(X)<m(}j+I)' and, consequently, m'(}j) <m'(X) <m'(}j+t) . Since m'(}j+t)-m'(}j) < lei, we get m'(X)
<
m'(}j)
+
e = m(}j) . ftY,Z) < m(X) . ftY,z)
+ g(Y,Z) + e + g(Y,Z) + e;
MEASUREMENT: AN ESSAY IN PHILOSOPHY OF SCIENCE
m'(X)
> m'(Y;+l) + e
>
245
+ g(Y,Z) + e g(Y,Z) + e;
= m(Y;+l) . f(Y,Z)
m(X) .f(Y,z)
+
if e>O and e
For each X, Y, Z E U such that m(Y) m(X) 'f{X,Z) + g(Y,Z).
<
m(Z), it holds that m'(X)=
If there are Y,ZE U such that m(Y) < m(Z), then our theorem follows from (2). If there are no such Y, Z, then our theorem is trivial. Theorem 1 is thereby proved. The following two theorems are proved in the same way as Theorem 1. Theorem 2. Let A be an aspect of M. If A is a dense quasi-intensity, then A is linear. Theorem 3. Let A be an aspect of M. linear.
If A is a dense intensity, then A is
Theorem 4. Let A be an aspect of M. uniform.
If A is a dense capacity, then A is
Proof. Let m and m' be two measures in M for A with respect to a quotient scale. If there are no X,ZE U such that m '(X) < m '(Z), then the theorem is trivial. If there are such X, Z, then we can choose two elements Yo'y2n E U such that m '(Yo) < m '(Y2n), and such that the arithmetic mean of m '(Yo) and m '(Y2n) does not coincide with the geometric mean; let us say that the difference between the two means has the value d. We let the interval from m'(Yo) to m'(Y2n) be bisected into n steps (see the proof for Theorem 1), and we choose two bisectional elements Y; and Y;+ 1 such that (1)
m'(Y;) ' m'(Y;) ::;; m'(Yo) . m'(Y2n) ::;; m'(Y;+l) . m'(Y;+l)'
Since A is a capacity, we also have (2)
m(Y;) ' m(Y;) ::;; m(Yo) . m(Y2n) ::;; m(Y;+l) . m(Y;+l)'
By Theorem 3, we have
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(3)
STIG KANGER
m'(X) = m(X) . p
+ q, for each XE V.
Now, assume that q4=O. It can then easily be shown, with the help of (1), (2) and (3), that the difference between the arithmetic mean and the geometric mean has an absolute value which must be smaller than Id I, provided that we let n, the number of steps in the bisectional process, be sufficiently large. Thus q=O, and Theorem 4 is proved .
Theorem 5. Let A be an aspect of M. If A is a compact quantity, then A is uniform . Proof. The proof is entirely analogous with the proof for Theorem 1. It reads "non-negative ordinal scale" instead of "bisectional scale", and "comparative aspect" instead of "pseudo-intensity". The only essential difference in the proofs is that it does not follow immediately that the bisectional elements exist or that m(lj)+m(lj)=m(lj_l)+m(lj+l) if m '(lj) + +m'(lj)=m'(lj_l)+m'(lj+l)' See the proof for Theorem 1. But we obtain this by means of the following: Lemma. Let m and m' be two measures in M for the compact quantity A with respect to a non-negative ordinal scale. Let X, Y, Z be arbitrarily chosen elements in V such that (1) (2)
m'(X) m'(Z)
< m'(y), and
+
m'(Z) = m'(X)
+ m'(y) .
Then it also holds that (3)
m(Z)
+ m(Z) = m(X) + m(Y).
The lemma is verified thus: From (1) and (2) plus the fact that A is a compact quantity, we have that there are f',Z'E V such that m'(Z')=m'(Z) (and consequently m(Z')=m(Z») , and m'(f')=m'(y) (and consequently m(f')=m(y»), and XCZ'C f', which, with (2), yields m'(f'-Z')=m'(Z'-X) and consequently m(f'-Z') = m(Z'-X) , which yields m(f')-m(Z')=m(Z')- m(X) and thus also m(Y)-m(Z)=m(Z)-m(X), which is (3). In the proof of Theorem 5, note, finally, that the uniformity of A follows from the linearity, since the empty set {} E V. The following three, easily proved theorems give, in a certain sense, the inversions of Theorems 1- 5.
Theorem 6. Let the aspect A of M be a pseudo-intensity (V, == ,
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a measure in M and linear transformation of m in U. Then m' is a measure in M for A with respect to S.
Theorem 7. Let the aspect A=(U, ==,
I
The next theorem, which is also almost trivial, shows that linearity and uniformity are accidental properties of finite aspects which can disappear if the aspect is enlarged with new elements.
Theorem 9. Eachfinite pseudo-intensity (quasi-intensity, intensity, capacity, or quantity) is a substructure of some non-linear pseudo-intensity (quasiintensity, intensity, capacity, quantity, respectively). MEASUREMENT AND THE EMPIRICAL WORLD
7. Explication of comparative qualities By a comparative quality we shall understand, in a rough sense, a quality of the type which is exemplified by length, weight, density, temperature, loudness, subjective loudness, intelligence, utility, etc. From a comparative quality one can derive comparative relations such as, for example, as long as, shorter than, as heavy as, lighter than, etc. We can often identify a comparative quality Q with some aspect A = (U, R j,R 2 , . .. ) of some domain M , and thereby make a kind of explication of Q. To say, for example, that the comparative quality weight is identical with the aspect A = (U, == ,
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STIG KANGER
some methods and require that they shall be test-methods R1,R z,'" in given subdomains of U. By a test-method in U" ~ U for the relation R j , we shall roughly understand a method by means of which we can, as a rule, determine if a given ordered n-tuple <X!> ... ,Xn > of elements in U* is an element of R; The exceptions to the rule are due to technical circumstances like, for example, lack of precision in our observations. The operational specification should naturally be consistent with the axiomatic. For example, we must not, on the one hand, require axiomatically that R, be transitive and, on the other hand, establish as test-method for R, a method by which transitivity is confuted. A specification can be more or less detailed. An axiomatical specification of the weight aspect (U, == , -e) of M where we only require that (U, == , -e) be an additive aspect of M such that for each X, YE U, it holds that if X c:r:g Y, then not Y c:r:g X, and X == Y if and only if not X c:r:g Yand not Y
c:r:g
X,
is not a very detailed specification. But if we require instead that the aspect (U, == , -e) of M be a quantity, then our axiomatical specification would be much more detailed. In this essay we shall maintain the view that an explication of a comparative quality, Q, of the type: Q is identical with the aspect A of M , where A has been specified in a suitable way, is a fruitful first step in our endeavour to obtain a measurement of Q. In other words, when we try to obtain a measurement of Q, it is often fruitful to begin with a hypothesis of the following form: There is a domain M and an aspect A of M such that Q is identical with the aspect A of M and 4>(M,A) , where 4>(M,A) stands for those specifications of the aspect A of M which we can make without conflicting with our intuition about Q and which, at the same time, we want to make while keeping in mind our aim to obtain a satisfactory measurement of Q.
8. Scale-determination Assume that we made an explication of the comparative quality Q and identified it with the aspect A of M such that 4>(M,A) . The next step in our endeavour to obtain a measurement of Q can be called scale-determination of Q. A scale S is defined, and with 4>(M,A) presupposed, we show - or
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make it plausible - that there is a measure in M for A with respect to S. If we succeed with such a scale-determination, then we have shown, or made it plausible, that there is a measurement of Q, i.e., a measurement (M,A,S, m), where Q=the aspect A of M . The possibility of making a scale-determination of Q depends on how detailed (M,A) is. If in (M,A) in our explication of weight, for example, weight is assumed to be a quantity, then the scale-determination of weight is trivial: S is the non-negative ordinal scale (R+ ,=, <). If (M,A) is not that detailed, then we may proceed in the following way: We choose a suitable, representative subdomain U* of U and regard the confinement A * of A to U*. With the help of the specifications stated in (M,A), we show - or make it plausible - that there is a measure in M for A * with respect to the stated scale S. Through an inductive inference, we then conclude that there also is a measure in M for A with respect to S. Sometimes, without changing the explication of Q, we can make different scale-determinations of Q with different scales. The choice between two possible scale-determinations may depend partly on just how plausible we require it to be that there is a measure in M for A with respect to the scale, and partly on which scale we want to have, while keeping in mind our aim to obtain a satisfactory measurement of Q. Finally, when we say that we define a scale, we mean, in most cases, that the relations involved in the scale are defined by means of elementary real number algebra. In other words, the relations are defined in terms of addition and multiplication, integers, variables for real numbers, and elementary logical operations. However, we shall not presuppose that scales are always defined in this way. 9. Equivalent types of measurement
If there is a measure, m, in M such that (M,A,S,m) is a measurement, then we say that the system (M,A, S) is a measurement type. Two measurement types (M,A ,S) and (M,A ',S') are said to be equivalent if, for every measure m in M, it holds that (M,A,S,m) is a measurement if and only if (M,A ',S',m) is a measurement. The following, easily-proven theorem gives an example of equivalent types of measurement. Theorem 10. Let A. =(U, ==, ~ ,~) and A 2=(U, ==, ~ ,ff) and A 3=(U, ==, ~ , D) be aspects of M, and let A. be a dense pseudo-intensity, A 2 a dense quasi-intensity, and A 3 a dense intensity. Let S. =(R, =, < ,B) be a bisectional scale, ~ = (R, = , < ,f) an interval scale, and S3 = (R, =, < ,D) a
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STIG KANGER
difference scale. Then (M,Al>SI)' (M,Az,Sz), and (M,A 3,S3) are of equivalent measurement types .
The explication of a comparative quality and the scale-determination were aimed at fixing a type of measurement for the quality . Clearly, it is then not essential to choose one particular type of measurement instead of an equivalent type . As far as temperature is concerned, for example, instead of fixing a type of measurement where temperature is explicated as a dense, purely structural intensity, we could just as well fix a type of measurement where temperature is explicated as a dense, purely structural pseudointensity. The temperature measures are the same in both cases. 10. Metrical sentences
In colloquial language, we are often concerned with metrical sentences like, for example, (1)
(2)
the weight of X = 2 kg; the temperature of X = 20 degrees centigrade;
and sometimes with sentences like (3)
the subjective loudness of Y
= 10 sones;
and, if we have a bit of imagination, maybe also with sentences like (4)
the utility of Z = 100 utils.
Suppose now that we have fixed a measurement type (M,A,S) for weight, where S is a non-negative ordinal scale (R+ ,=, <), and where the aspect A of M is a compact quantity. We suppose also that we fixed a measurement type (M,A ',S') for temperature, where S' is a difference scale (R, =, < ,D) , and where the aspect A' of M is a purely structural , dense intensity. We can now give analyses of (1) and (2). We let kg be the prototype for kilogram and assume that {kg} is an element of the domain of the aspect of weight. We let i be one part ice water and b one part boiling water, and we assume that {i} and {b} are elements of the domain of the aspect of temperature. The analysis of (1) is then: (1') There is a measurement (M,A,S,m) of weight, where S is a nonnegative ordinal scale (R+, =, <), and where the aspect A of M is a compact quantity, and where X and {kg} are elements of A's domain, and, finally, where m({kg}) = 1 and m(X)=2.
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The analysis of (2) is: (2') There is a measurement (M,A ',S',m) of temperature, where S' is a difference scale (R, =, < ,D), and where the ' aspect A' of M is purely structural, dense intensity, and where X, {i}, and {b} are elements of the domain of A', and finally, where m( {i}) = 0, m({b}) = 100, and m(X)=20.
11. Meaningfulness A statement like, for example, "X is twice as warm as 1'" is vague : it can be true if the temperature is measured according to Celsius, but false if measured according to Fahrenheit. We say then that the relation 'twice as warm as' is meaningless as far as temperature is concerned. The statement "X is twice as heavy as 1"', on the other hand, is not vague in this way. We say that the relation 'twice as heavy as' is meaningful as far as weight is concerned. In any case, this holds as long as the measurements for weight and temperature are of the types suggested in the preceding paragraph. The concept of meaningfulness can perhaps be explicated in the following way: The relation R is said to be meaningful as far as the measurement type (M,A, S) is concerned if R is involved in an aspect A ' of M in a measurement type (M,A ',S') which is equivalent to (M,A,S). We easily see that the relation 'twice as warm as' cannot readily be meaningful as far as temperature is concerned. In any case, it can not be meaningful if we presuppose that the relation 'twice as warm as' is mapped into the numerical relation 'twice as large as', in the scale S'.
12. Defined measurement Let T be a theory which involves measurement. T can be , for example, a physical theory. In T, certain measurements can depend on others in the sense that they are characterized in terms of these other measurements. An especially important type of dependence is definitional dependence, and we shall distinguish between two main types of defined measurement: derivative measurement and derived measurement . We say that (M,A,S,m) is a derivative measurement of the comparative quality Q if (M,A,S,m) is a measurement of Q, and if the aspect A of M, in the explication of Q, is defined in terms of measurement of other comparative qualities. These measurements can in turn be defined.
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Example. As an example of a derivative measurement, we can take a measurement of density. Let (M,A,S,m) be a measurement of weight and (M,A',S,m') a measurement of volume. We can assume that M = the class of all concrete things and that A = (U, == , -s) is characterized as in the example in § 2 and A' = (U, == ', -s ') is characterized in an analogous way. We assume further that the aspects A and A' of M are compact quantities and that S is a non-negative ordinal scale. Let us adopt the abbreviation s({x}) for the quotient m({x})/m'({x}) and the symbol "-" for "if and only if" . We can now give the following explication of the comparative quality density : Density = the aspect A" =(U", ==", ~", D,Q) of M , where U" is the domain of all {x} with xEM, and where ==", -s", D , and Q are relations defined thus: For each x.y.z, wEM, it holds that {x} ==" {y} - s({x}) = s({y}); {x} ~" {y} - s({x}) < s({y}); D({x},{y},{z},{w}) - s({x}) + s({y}) ~ s({z}) + s({w}); Q({x},{y} ,{Z},{w}) - s({x}) . s({y}) ~ s({z}) . s({w}).
We see that the aspect A" of M is a purely structural capacity. In our explication of density, we can assume that A is dense and thereby also uniform. A measurement (M,A ",S",m), where S" is a quotient scale, can now be said to be a derivative measurement of density . It is easy to see that a measurement of density in which density is identified with an additive aspect is entirely absurd. We also see that a measurement in which density is identified with a purely structural intensity (U", ==", ~", If) is less satisfactory since the uniformity is then lost. We say that (M,A,S,m) is a derived measurement of the comparative quality Q if (M,A,S,m) is a measurement of Q, and if the measure m is defined in terms of measurement of other comparative qualities.
Example. In the measurement (M,A ",S",m) of density, let m be defined thus: m({x})=s({x}) for each xEM. Then the measurement (M,A ",S",m) is also a derived measurement of density . We shall observe that a derived measurement need not be derivative. We can very well imagine a measurement (M,A,S",m) of density where A of M is not defined in terms of weight and volume, but where m is defined as above. That m is a measure in M for A with respect to the quotient scale S" can then be an empirical result. Defined measurement plays an important role in physics. There, we have a complicated pattern of measurements, which can be ordered so that only the measurements of length, mass, and time are not defined. (Note that this
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does not mean that measurements of length, mass, and time are entirely independent of each other.) Also in psychology, defined measurement plays a certain role. A large part of psycho-physics can be said to aim at obtaining derived measurements of subjective comparative qualities (for example, subjective loudness), where the measure is defined in terms of physical measurements.
13. Satisfactory measurement and measuring-methods We have previously, on a couple of occasions, spoken about "satisfactory" measurement of Q; we can ask ourselves just when a measurement of Q is to be considered satisfactory. The answer to this question often seems to be a matter of opinion, and we shall here only try to give an incomplete and slightly vague answer. (1) A measurement (M,A,S,m) of Q is satisfactory only if the identification of Q with the aspect A of M is not incompatible with our intuition about Q, and only if we can show that it is plausible that there is a measure in M for A with respect to S. (2) A measurement (M,A,S,m) of Q is more satisfactory if it fulfils certain formal conditions than if it does not fulfil them. The measurement (M,A,S,m) of Q can thus be said to be more satisfactory if A is uniform than if A is only linear, and it is more satisfactory if A is linear than if A is not linear . (3) The question of whether or not a measurement is satisfactory also depends on the existence of measuring methods . By a measuring method for a measurement (M,A,S,m) of Q, we shall understand a method with whose help we can state, with sufficient precision, the value of m(x) for a large number of elements x in the domain U of A . An example of a measuring method for a measurement of weight is the usual method with lever balance and weights . We easily see that our possibilities of having a measuring method for a measurement (M,A, S, m) of Q may depend upon how detailed the operational specification of A of M is. If we assume, for example , that in the explication of weight, we did not fix a single test method for the relations == and ~, then we can hardly assert that the method with lever balance and weights is a measuring method for a measurement of weight.
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14. Explication
We let M be the class of all sounds which are audible to the human ear without discomfort. When we speak of sound, we are concerned with types of sound, not with the individual occurrences of sound. Two individual occurrences of sound belong to the same (type of) sound if they are acoustically equal; otherwise, they belong to different sounds . When we here say that two occurrences of sound are acoustically equal, we mean that they differ only in that they have different places in space or time; consequently, for example, they will not differ as far as frequency or amplitude is concerned. We can now identify the comparative quality subjective loudness with the aspect A=(V, ==,
==
Y - X is as loud as Y; Y - Y is louder than X; D(X, Y, Z, W) - the mean loudness of X and Y is less than or equal to the mean loudness of Z and W. X
X
When we here say, for example, that {y} is louder than {x}, this means that the sound y is louder than the sound x - or, rather, that an occurrence of x is subjectively louder than an occurrence of y . 0 can be understood as 'silence'. The wording "the mean loudness of X and Y" in the definition of D shall not be understood in such a way that we already now presuppose a measurement of subjective loudness. A is, consequently, a structural aspect of M with zero element. We specify A by requiring that the following axioms shall hold for each X,Y,Z, WE V and each x,yEM: (1) (2) (3) (4) (5) (6) (7)
X == Y - x ~ Yand Y ~ X. X -e Y - not D(Y, Y,XX). D(X, Y,Z, W) - D(Y, X,Z, W). D(X, Y,Z, W) - D(X, Y, W;Z). D (X, Y,Z, W) or D(Z, W;x, Y). D (X,X, Y, Y) - D(X,Z,Z, Y) .
not D({x}, {y},
Moreover, we require :
0 , 0)
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If the aspect A of M is an intensity, then it is a dense intensity.
Our axiomatic specification is not very detailed . We have not, for example, axiomatically required that the relation
n.
15. Scale-determination We carry out the following time-consuming experiment: We first let the person P decide for each sound x,yEM* which of the three alternatives {x}
»:
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differs only slightly from a non-negative intensity. We can then consider it plausible that also the confinement of A to U*, i.e., the aspect A *= (U*, == *, ~ * ,D *) of M*, is a non-negative intensity. By induction, we then conclude that the aspect A of M is a non-negative intensity. Our scale-determination of subjective loudness is thereby complete. The inference from the fact that AP differs only slightly from a nonnegative intensity to the conclusion that A is a non-negative intensity contains several steps which call for further motivation. The only step, however, which we shall discuss further is the starting-point, namely, we shall discuss what can be meant by the fact that AP differs only slightly from a non-negative intensity. 16. Difference scales with margin of error
We say that a numerical structure S=(R+,=',<',D') is a non-negative difference scale with the error f if (i) R+ is the domain of all non-negative real numbers; (ii) f is a 5-place operation defined in R+ and with real numbers as its values; (iii) =' and <.' are 2-place relations, and D' is a 4place relation among elements in R+ ; and (iv) for each u,x,y,z,wER+, it holds that (1)
(2) (3)
x =' y - not x <' y, and not y <' x; x < ' y - not D'(y,y,x,x); D'(x.y.z, w) - x-ftx.x.y.z, w)+y-f(y,x,y,z, w) ~ z+f(z,x,y,z, w)+w+f(w,x,y,z,w).
Keeping in mind the axiomatic specification we made of A, we assume also that (4) (5)
fiu.x.y.z, w) = f(u,y,x,z, w) f(u,x,x,y,y) = f(u,x,z,z,y) .
= fiu .x,y, w.z) = ftu,«,w,x,y) ;
Finally , we put a certain limit on the error which is motivated by the fact that the scale is non-negative: (6) (7)
-u < f(u,x,y,z ,w) < u, when u=l=O; f(O,x,y,z, w) = O.
We say that the number-structure S=(R+, =', <' ,D ') is a non-negative difference scale with the margin of error g if there is an operation f such that (i) S is a non-negative difference scale with the error f; (ii) g is a 1place continuous operation defined in U such that 0 ~ g(u) ~ u for each uER+; and (iii) for each u.x.y.z.wei R" ; it holds that If(u,x,y,z,w) I ~g(u).
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We can, a bit vaguely, talk about non-negative difference scales with small margins of error. A margin of error g such that g(u) =u/50 for each u E R+ can, in this context, be said to be an example of a small margin of error. We can now say that the aspect AP of M* (see the preceding section) differs only slightly from a non-negative intensity if there is a measure in M* for A P with respect to a non-negative difference scale with a small margin of error g. For example, if g(u)=u/20 for each uE U, then we can assert that A P differs from a non-negative intensity by at most 5 per cent, and that A P can be changed to a non-negative intensity by making a 5 per cent correction , at most, of P's judgements. We can easily imagine A P such that it differs from a non-negative intensity by more than 5 per cent. Let us assume, for example, that M contains, among other things, the sounds a, b, and c. Let us assume further that P asserts that not D({},{b},{a},{a}) and not D({a},{c},{b} ,{b}) and not D({a},{a},{},{c}). In other words, P wants, so to speak, to place {a} between the silence {} and {b}, but closer to {} than to {b}; and he wants to place {b} between {a} and {c}, but closer to {a} than to {c} ; and, furthermore, he wants to place {a} between {} and {c}, but closer to {c} than to {} . We can easily verify that zt" differs from a non-negative intensity by more than 5 per cent.
17. Measurement of subjective loudness Let us assume that we have succeeded in carrying out a scale-determination of subjective loudness, i.e., the aspect A=(U,= ,~ ,D) of M, so that the scale S is the non-negative difference scale (R, =, < ,D), and so that it is plausible that there is a measure in M for A in respect to S. We have then made it plausible that there is a measurement (M,A,S,m) of subjective loudness. We can very well assert that such a measurement is satisfactory. Our intuition about subjective loudness does not raise any objections against our explication of subjective loudness as the aspect A of M. The measurement also has satisfactory formal properties. We have made it plausible that A is a structural intensity with zero element, and we have assumed axiomatically that A is then also dense, and we then have as a consequence that A is uniform. In order that the measurement be entirely satisfactory, we shall also have a measuring method for it. Perhaps there is a derived measurement (M,A,S,m') of subjective loudness, where the measure m' is defined as a
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function of physical loudness and physical pitch. The function, if there is one, can be found with the help of well-known methods from experimental psychology. If we can show that there is such a derived measurement, then we also have a relatively handy measuring method.
18. Retrospect In our attempt to obtain a satisfactory measurement of subjective loudness, we run into - or risk running into - several difficulties: We began by identifying subjective loudness with the aspect A of M. One of the first difficulties is to decide how detailed the axiomatic specification of A should be. A more detailed axiomatic specification than the one we suggested above would make the scale-determination less time-consuming, since several of the judgements the person P now has to make would instead be determined by the axioms. But when the axiomatic specification is too detailed, we run the risk that the explicandum, subjective loudness, and explicatum, the aspect A of M , no longer correspond to each other. The decision as to how detailed the axiomatic specification should be is, to a great extent, a matter of opinion. A reasonable norm for the decision, however , is to avoid an axiomatic specification which is detailed in such a way that A P cannot differ considerably from a non-negative intensity. We easily see that an axiomatic specification which sharpens the one we suggested with the requirement that ~ be transitive conflicts with this norm. Another difficulty lies in the operational specification. Should we require the person P to have special qualifications? Instead of one person P, should we use a group of several people and let them make the decisions by, for example, majority rule? (We could then supplement the method of margins of error by a statistical technique: A correction of a decision is considered as a minor correction if and only if the decision was made by a bare majority .) A third difficulty concerns those requirements which should be placed on the inductive inference in scale-determination. How should M* be chosen? How may the margin of error look? We can expect larger percentile errors in P's judgements of very weak sound than of normal sound. Therefore, we cannot expect a small margin of error if it is linear. But, the margin of error can naturally have another shape. Maybe it looks like the so called discriminal dispersion for loudness. Finally, suppose that P's decisions did not turn out in a way which makes it plausible that A is an intensity - what do we do then? One possibility is to strengthen the axiomatic specification in the direction towards an
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intensity; we could, for example, simply require axiomatically that A is a non-negative intensity. But then again, we run the risk that the explicatum, A, will not cover to the explicandum, subjective loudness. Another possibility is to try to give subjective loudness another but equivalent type of measurement, for example, one in which subjective loudness is identified with a dense pseudo-intensity. The differences are not so essential, but perhaps the operational specification would become simpler and the risk for error in P's judgements would become smaller. A third possibility is to give up the hope for a type of measurement in which subjective loudness is uniform. We can then try some less satisfactory types of measurement with scales which are less orthodox than those we have described above. We couldtry, for example, some extension of the semi-ordinal scale. BACKGROUND AND REFERENCES
19. Introduction
Let us vaguely distinguish between (1) (2) (3)
theories of measurement in general, theories of measurement of given quality Q, and theories about measuring methods for Q.
This essay has been an attempt at a contribut ion to the theories of the first kind. In view of this attempt, a theory of measurement of Q (i.e ., a theory of the second kind) may be conceived of as consisting essentially of an explication and a scale determination of Q. Theories about measuring methods for Q have often a more specialized character, for example, theories about methods for measurement of distance in land surveying. Theories about measuring methods are perhaps the practical goal for all theories of measurement. But, the less sturdy the comparative quality Q is, the more difficult it is to reach the goal without help from theories of kind (2). And theories of kind (2) often profit from the frameworks and results obtained in measurement theory of kind (1). They may profit, for example, from general results concerning the representation problem or the uniqueness problem or the problem of meaningfulness in measurement theory . The representation problem is the following: A detailed axiomatic specification (M,A) of the aspect A of M can take place in two ways: (i) It can be formulated in terms of, among other things, measures. For example,
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it can contain the condition that there shall be a measure in M for A with respect to (let us say) a difference scale. (ii) It can be formulated entirely in terms of those concepts and relations which are involved in the aspect. In case (i), the scale-determination is most often trivial , but not in case (ii) . There, it is often a problem whether it follows from lJ>(M,A) that there is a measure in M for A with respect to a certain scale. This is the problem of representation in measurement theory. The problem of uniqueness is this : Can all measures in M for A with respect to the scale S be transformed into each other through some given, simple type of transformation , for example, linear transformations? We have given certain solutions to problems of uniqueness in Theorems 1- 5 above. The problem of meaningfulness is the problem of what is meant by meaningfulness, and what is meaningful as far as a certain type of measurement is concerned? Some suggestions as to the meaningfulness problem were given above in § 11. The general theory of measurement is an important part of the philosophy of science. Nowadays, it is rather difficult to get an over-all picture of the literature in this field. There are, however, some excellent introductions: Suppes & Zinnes 1963, Adams 1966, Ellis 1966, and Pfanzagl 1968. A comprehensive survey of measurement in psychology is given in Torgerson 1958. In the following sections, we shall give a short survey of some of the more important ideas within the general theory of measurement. In doing so, we shall first distinguish between two types of theories : the classical theory of measurement and the liberalized theory of measurement. 20. The classical theory of measurement
The origin of the classical theory is difficult to fix. But perhaps one can begin with Helmholtz 1887, and some early mathematical contributions: Veronese 1889, Bettazzi 1890, and Burali-Forti 1893. They were followed by, among others, Holder 1901, and Huntington 1902. The philosophical contribution at the same time came with Russell 1903. Among later philosophical contributions, Campbell 1920 and 1928, Carnap 1926, and Nagel 1930 and 1931 should be mentioned in the first place. Among modern contributions, Suppes 1951, Hempel 1952, and Wedberg 1963 should be noted. One of the distinctions made in the classical theory of measurement by Campbell is the one between fundamental and derived measurement.2 With a certain simplification, one can say that according to the classical
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theory , a comparative quality is fundamentally measurable only if it can be identified with a compact quantity. A comparative quality like, for example, subjective loudness would, consequently, not be fundamentally measurable according to the classical theory. A more true presentation of the theory is the following: First , we define the concept of combination structure or Csstructure.? By a C-structure, we shall understand a structure (M, ==, ~,o), where M is a non-empty domain, == and ~ are 2-place relations between elements in M, and 0 is a 2-place operation defined for elements in M such that xoyEM for each x,yEM. By a C-scale, we shall understand a C-structure (P, = , < , +), where P is a class of positive real numbers such that x-yEP when x,yEP and x> y, and where = and < are the relations 'identity' and 'less than', respectively, between numbers in P, and where + is the operation addition in the domain P. According to the classical theory of measurement, a comparative quality is now fundamentally measurable if and only if it can be identified with a Cstructure which has a C-scale as homomorphic image. To say that a Cstructure (M, ==, ~ , 0) has a C-scale (P, =, < , + ) as homomorphic image means that there is a homomorphism from (M, ==, -e , 0) onto (P, =, < ,+), i.e., that there is an operation m defined for M and with P as the class of all m(x) with xEM such that for each x,yEM, it holds that x == y - m(x) = m(y); x ~ y - m(x) < m(y); m(xoy) = m(x)+m(y).
A homomorphism from the C-structure (M, == , ~ , 0) onto some C-scale can be called a measure for (M, == , ~ , 0). We say that a C-structure is fundamentally measurable if there is a measure for it. With this terminology, we also say that according to the classical theory of measurement, a comparative quality is fundamentally measurable if and only if it can be identified with a fundamentally measurable C-structure. One can now prove the following representation theorem: A C-structure (M, == , ~ , 0) is fundamentally measurable if and only if the following conditions hold for each x,y,zEM:4 (1) (2)
(3) (4) (5)
== Y - not x
~ y & not y ~ x . Y ~ not y ~ x. Not x ~ Y & not y ~ Z ~ not x ~ z.
x
x
~
x==y~xoz==yoz.
xo(yoz)
== (xoy)oz .
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(6)
(7)
(8)
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=
xoy yox. x ~ Y - there is a zEM such that xoz=y. There is a number n= 1,2,3, ... such that y-snr, where
recursively thus: Ix = x; (n+l)x = nx
0
nx is defined
x.
One can also prove the following uniqueness theorem: If (M,
=,~ ,0) is a C-structure, and if m and m' are two measures for (M,
= , ~ , 0), then m' is a similarity transformation of m in M, i.e., there is a
number p>O such that m'(x)=m(x)'p for each xEM. 5 An example of a comparative quality which is fundamentally measurable in the classical sense is weight. We then let M be a domain of concrete things, =, and -e the relations 'as heavy as' and 'lighter than', respectively, and 0 the operation 'take together' . x 0 Y is thus x and y taken together. Another example is length. = and ~ are then the relations 'as long as' and 'shorter than', respectively, and 0 is the operation 'lay end to end'. x 0 y is thus, in this case, x and y laid end to end. A difficulty in these examples is how to interpret x ox. But, we can perhaps imagine that we have an unlimited amount of copies of each x which stand in the relation to x. We can then interpret xox as xox ' , where x' is some copy of x. There are a couple of versions of the classical theory of measurement. One version is formulated above, and the other can be formulated thus: A comparative quality is fundamentally measurable if and only if it can be identified with a C-structure which has a C-scale as isomorphic image (i.e., a C-structure for which there is a one-to-one measurej.P This version of the classical theory is fully plausible. Instead of identifying weight with the C-structure (M, =, ~,o), where M is the domain of and ~ are the relations 'as heavy as' and 'lighter than', concrete things, respectively, and 0 is the operation 'take together', we can identify weight with a C-structure (M', =', ~' , 0 '), where M' is the domain of all equivalence classes [x] with xEM ([x] stands here for the class of all yEM such that y =x) , and where =', -s', and 0 ' are defined so that, for each x,yEM, it holds that [x] = '[y]-x=y, [x]~ '[y]-x~y, and [x] 0 '[y]=[xoy]. These two versions of the classical theory of measurement have in common the fact that a measure for a C-structure always has a positive value. However, there are also versions in which we have non-positive as well as positive values . Thus, by an extended C-scale, we shall mean a Cstructure (R, = , < ,+), where R is a class of real numbers such that x-y E R when x,yER, and where =, < , and + are the relation ' identity' , the
=
=
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relation 'less than' , and the operation 'addition', respectively, in the domain R. We say then that the C-structure (M, = , -s , 0) is fundamentally measurable in the wider sense if there is an extended measure for (M, = ,
= is an equivalence relation in M; and
If we change the last condition in (iii) to: if x-ey then there is a zEM such that xoz=y, and if we add (v) x-e.r c y, then (i) -(v) become equivalent to (l) - (8) stated above. One can also prove the following uniqueness theorem: If m and m' are two extended measures for a C-structure (M, =, -s , 0), then m I is a similarity transformation of m.
21. The liberalized theory of measurement: Stevens Most of the subjective phenomena which psychology is interested in measuring are not fundamentally measurable in the classical sense, neither are phenomena like utility and value which are of some interest in economics . And the possibility of a derived measurement of these phenomena within the framework of the classical theory seems doubtful. This fact led up to a liberalized theory of measurement. The foremost originator of this theory was the psychologist Stevens." Among the contributions to the liberalized theory, we want, in the first place, to mention Stevens 1946 and 1951; Coombs 1950, 1952, and 1964; Suppes & Winet
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1955; Scott & Suppes 1958; Debreu 1960; Scott 1964; Luce & Tukey 1964; and Luce 1966 . With a certain simplification, it can be maintained that according to the liberalized theory of measurement, a comparative quality Q is measurable if it can be identified with a purely structural quasi-intensity. Q is measurable in a stronger sense if Q can be identified with a purely structural capacity. Q is measurable in a weaker sense if Q can be identified with a purely structural comparative aspect. Linearity and uniformity, respectively, are obtained in the first two cases by assuming (a bit vaguely) that Q is dense or continuous. In psychology, one speaks of, for example, a loudnesscontinuum or a pitch-continuum. A more faithful presentation (as far as measurability in the first sense is concerned) is the following : By an interval-structure let us understand a structure (M, == , ~ ,Il), where M is a non-empty domain, and where == and ~ are 2-place relations between elements in M, and where Jl is a 4-place relation among elements in M. By an interval scale, we shall understand (in this section) an interval structure (R, =, < ,Il), where R is a class of real numbers such that (x+y)/ZER when x,yER, and where = and < are the relations 'identity' and 'less than' , respectively, in R, and where Jl is a relation such that for each x.v.z. wER , Jl(x,y,z,w) .. Ix-yl =:;; [z-w] . A homomorphism from an interval structure (M, ==, ~ ,Il) onto an interval scale can be called a measure for (M, == , ~ ,J!) . We say that an interval structure is measurable if there is a measure for it. A comparative quality can thus be said to be measurable according to the liberalized theory if it can be identified with a measurable interval structure. The measurable interval structures can be characterized axiomatically. Let us define the 3-place relation }ffi (of 'betweenness') so that for each x, y, zEM, it holds that
JE(x,y,z) .. x
~
Y& Y
~
z or z
~
y& Y
~
x.
Let us also recursively define the 4-place relation e n, n = 1,2, ..., so that for each x,y,z, wEM, it holds that (1) (2)
C 1(x,y,z,w) .. y==z & lB5(x,y,w) & Jl(x,y,z,w) & Jl(z,w,x,y); cr: l(X,y,z, w) - there are u, vE M such that e n(x,y, u,v) & e 1(u, v.z. w) .
e
The meaning of, for example, 5(x,y,z,w) is that the interval between x and y is as wide as the interval between z and w, and that the interval between
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x and w can be divided into 5 + 1 subintervals which are all equal to the interval between x and y. One can prove the following representation theorem : An interval structure (M, == , ~ ,II) is measurable if and only if the following conditions hold for each .r.y.z, w,u,vEM: 9 (1) (2)
(3) (4)
== y - not x ~ y & not y ~ x. x ~ y -+ not y -e x. Not x ~ y & noty ~ Z -+ not x ~ z. !I(x,y,z,w) or !I(z,w,x,y).
x
!I(x,y,z,w) & !I(z, W,U, v) -+ !I(x,y,u, v). !I(x,y,y,x). (7) There is a z E M such that !I(x,z,z,y) & !I(z,y,x,z). (8) x == y & Kx.z,u. w) -+ !I(y,z,u,w). (9) JE(x,y,z) -+ not !I(x,z,x,y). (10) JE(x.y,z) & JE(u.w,v) & !I(x,y,u,w) & !I(y,z,w,v) -+ !I(x,z,u,v). (11) Not !I(u,v,x,y) -+ there is a zEM such that JE(u,z,v) & !I(x,y,u,z). (12) !I(x,y,u,v) & x =1= y -+ there are z,wEM and a number n=1,2, ... such that Cn(u,z,w,v) & !I(u,z,x,y). (5) (6)
One can also prove the following uniqueness theorem: If m and m' are two measures for an interval structure (M, ==, ~ ,II), then m' is a positive linear transformation of m. 10 Subjective loudness is an example of a comparative quality which (so far as can be judged) is measurable in the present, specified sense. Debreu 1960, Luce & Tukey 1964, and Luce 1966 study a multidimensional modification of intensities. The difference consists in the fact that the structure has more than one domain and the relation !I is limited. In the twodimensional case, i.e., the one with two domains, the limitation is such that !I(x,y,z,w) presupposes that x and z are in the first domain, and y and w in the second. The structures in these cases are well suited to be explicata for multidimensional comparative qualities like, for example, a person's multidimensional qualification for a certain task.
22. The liberalizedtheory of measurement: von Neumann & Morgenstern A primitive thermometer can be constructed thus: We want to find out the temperature of the water in a bowl. We assume that we can determine if the water in two bowls is equally hot simply by feeling the water. Then we mix icewater and boiling water in another bowl in such a way that the water in
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the two bowls are equally hot. If the mixture consists of 1/2 part ice-water and 1/2 part boiling water the temperature is 50 0 ( centigrade) . If the mixture consists of 3/5 icewater and 2/5 boiling water, then the temperature is 40°, etc. In von Neumann & Morgenstern 1947, this idea is used for the measurement of utility . If one wants to find out where the utility X lies in the interval between the utility Y and the utility Z, then one mixes Y and Z so that the mixture becomes as good as X. If the mixture consists of 3/5 Yand 2/5 Z, then X will be located 2/5 of the distance from Y towards Z. The mixture, of course, cannot be done literally, but it may be done probabilitywise like a ticket in a lottery where 3/5 of the tickets give Yand 2/5 of the tickets give Z as prize . Mixtures can, of course, also be mixed: for example, 3/ 10 of the mixture just described and 7/ 10 W. This mixture corresponds to a ticket in a lottery where 3/10 of the tickets give a ticket in the first lottery as prize, and 7/ 10 of the tickets give W. A mixture of a parts X and I-a parts Y, where 0< a < 1, can be denoted thus: (X[a]Y). By a mixture-structure, we shall understand a system (M, ==,
+ (l - a) . y.
A homomorphism from a mixture-structure onto a mixture-scale is an operation m defined for M, and with R as the class of all m(x) with xEM such that x == y - m(x) = m(y); x
for each a such that O
MEASUREMENT: AN ESSAY IN PHILOSOPHY OF SCIENCE
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theorem is proved: A mixture-structure (M, ==, ~ ,[a]) is measurable if and only if the following conditions hold for each x,y,z: (l) (2) (3) (4) (5)
(6) . (7) (8)
==
y - x .f€ y and y .f€ y. Y ~ y .f€ x. x.f€ y & Y .f€ z ~ x .f€ z. x[a]y = y[1-a]x . (x[a]y)[b]y = x[a' b]y. If x == y , then x[a]z == y[a]z . If x ~ y, then x ~ x[a]y ~ y. If x ~ Y ~ z, then there are a and b such that x[a]z ~ y ~ x[b]z. x
x
~
Also the following uniqueness theorem is proved: If m and m I are two measures for a mixture-structure, then m is a positive linear transformation I
ofm.
Modifications of von Neumann- Morgenstern's theory have been given by Herstein & Milnor 1953, Hausner 1954, Pfanzagl 1959, and Aumann 1962, among others. Excellent surveys can be found in Luce & Raiffa 1957, Adams 1960, Luce & Suppes 1965, and Fischburn 1968. We may obtain one of the possible modifications of this theory by limiting ourselves to mixtures with equal parts, i.e., mixtures of the type X[ll2]y . The limitation naturally is not very essential; for example, X[1/4]y can be constructed as (X[1/2]y) [1/2]y. Measurable mixture-structures of this kind are evidently closely related to dense pseudo-intensities.
23. Operationism and meaningfulness A common distinction is that between empirical and non-empirical structures. Length, conceived as a combination structure, is an example of an empirical structure; just as subjective loudness, conceived as a structural intensity (U, ==, ~ ,lDJ), is an empirical structure . A numerical structure, for example, the difference scale (R, =, < ,D) is, on the other hand, nonempirical. In our attempt above to fix a satisfactory measurement type for subjective loudness, a specification of (U, ==, ~ ,lDJ) was included as an essential step. The specification was both operational and axiomatic. The assertion that subjective loudness is empirical can now be understood to mean that the operational specification is relatively detailed. Operationism is a doctrine which (in this context) can be formulated thus: If a specification of an empirical structure is a step in the fixing of a type of
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measurement, then the specification shall be exclusively operational. An operational specification of subjective loudness (U, == , ~ ,V) may very well disconfirm the idea that (U, ==, ~ ,V) is a comparative aspect: the relation 'as strong as', for instance, may not tum out to be transitive. It is in line with operationism that one then tries to find some type of measurement for subjective loudness where the scale is not an ordinal scale. A scale to which one could then have recourse is the semi-ordinal scale (R, =', ~', ...) which was defined in § 4. A semi-ordinal scale (R, =', < ') can be conceived of as an ordinal scale with the margin of error 1. An aspect (U,==',~' , ...) such that there is a measure for it with respect to a semi-ordinal scale can be called a semi-comparative aspect. Structures (M, == " ~ '), which can be mapped homomorphically into a semi-ordinal scale, are usually called semi-orders . They were introduced, in measurement context, by Luce 1956 and Scott & Suppes 1958, where also a representation theorem was proved . A semi-ordinal scale (R,=',<',D', ... ), where for all x,y,z,wER, D'(x.y.z, w) .. x
+y-
2 :s;;
z+
w
+
2,
may be called a semi-difference scale. The aspect (U, ==', ~ ' ,D', ...) such that there is a measure for it with respect to a semi-difference scale can be called a semi-intensity. Structures related to semi-intensities were introduced in Gerlach 1957. It should be pointed out that the idea of such structures in essential respects, goes back to Wiener 1921. (See Luce & Suppes 1965.) Another extension of semi-orders to what perhaps could be called semicombination-structures has been studied by Adams 1965. If for operational reasons we should identify subjective loudness with a semi-intensity (U, == ', ~ ,,D'), then we should conceive of a judgement like 'X is as loud as Y' (where X, YE U) as meaningless if 'as loud as' refers to a transitive relation. The question what meaningfulness means in connection with measurement is important, and several attempts at an answer have been made. See, for example, Weitzenhoffer 1951, Suppes 1959, Adams, Fagot & Robinson 1965, and Pfanzagl 1968. Suppes' theory corresponds roughly to the following: With each type of measurement (M,A, S), let us associate a class of transformations, namely, a class T of transformations in A's domain M such that all measures in M for A with respect to S can be transformed into each other by means of transformations in T. Let us say that a transformation t is ' admissible' for m if m is a measure in M for A with respect to S, and if t is included in the class of transformations associated with (M,A,S). A relation between
MEASUREMENT: AN ESSAY IN PHILOSOPHY OF SCIENCE
269
measures can now be said to be meaningful if it is invariant with respect to admissible transformations of the measures. A 'numerical' statement, which expresses a relation between measures, is meaningful if the relation is meaningful. Example. Let m be a temperature measure with linear transformations admissible. Then the statement m(X)=m(Y) is meaningful since it is equivalent to t(m(X» =t(m(y). The statement m(X) =2 ' m(y) , on the other hand, is meaningless.
A statement can often concern both numerical relations between measures and other relations between objects. Example. If X is twice as warm as Y, and if m(X)=100, then m(Y)=50.
The problem how to define meaningfulness for statements of this more complex kind is open. 24. A terminological note 1. Quantity and magnitude. These two terms are often met with in the classical theory of measurement; frequently, no clear distinction is made between them. But when the distinction is made - as, for example, in Russell 1903 and Lenzen 1931 - then it can be explained thus: A quantity of a certain kind (or extensive quantity, according to the terminology in Suppes 1951) is an element in a fundamentally measurable combination structure which represents a comparative quality of corresponding kind. A measuring-rod is an example ofa quantity of the kind length. The measuring-rod 's length, on the other hand, is an example of a magnitude of the kind length. This magnitude can be identified with an equivalence class, namely, the class of all quantities of the kind length which are just as long as the measuring-rod . If one makes the distinction between quantity and magnitude in this way, then one can ask just what it is that is measured in case of length: is it the quantity of the kind length, or is it, as, for example, ~sell 1903 holds, magnitudes? Both of these standpoints are fully defensible: the former leads to the first version of the classical theory of measurement (where a measure is a homomorphism), and the latter leads to the second version (where a measure is an isomorphism). Another meaning of the term quantity goes back to Carnap 1926 and is adopted in Hempel 1952: A quantity, or a quantitative concept, is an operation which assigns numbers to things. A measure for length should
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then be an example of a quantity. Still another meaning, as far as the term quantity is concerned, was suggested in this essay: A quantity is a certain kind of aspect. 2. Scales. The concept of scale is often a bit vague in measurement theory. The most common conception can be specified thus: A scale for an empirical structure A is a measure for A with respect to some numerical structure S; a scale for A is thus a measurement (A,S,m) . A scale is a scale for some empirical structure. Scales are usually classified in some important classes: ordinal scales, interval scales, and quotient scales. The concept of interval scale has been defined in two ways: (1) a scale (A,S,m) is an interval scale if and only if for each m', it holds that m' is a positive linear transformation of m in A's domain if (A,S,m') is a scale; and (2) a scale (A,S,m) is an interval-scale if and only if for each m' it holds that m' is a positive linear transformation of m in A's domain if and only if (A,S,m') is a scale. The concepts of ordinal scale and quotient scale can be defined analogously. As far as quotient scale is concerned, we have similarity transformations instead of positive linear transformations, and as far as ordinal scale is concerned , we have monotone transformations. These definitions - or, in any case, the alternative (1) of the definitions - are given in Suppes & Zinnes 1963, and (in all essentials) in Pfanzagl 1968. They seem to adequately explicate the vaguer concept of scale given in Stevens 1946, 1951, and 1959. A discussion of Stevens' concept and a somewhat different explication of it is given in Wedberg 1968. Another concept of scale is given in Coombs 1950 and 1952. An interval scale (or "ordinal-interval" scale) in Coombs' sense can be defined as a measurement (A,S,m) where S is a numerical structure (R, =, < ,I) . Still another concept of scale occurs in Adams, Fagot & Robinson 1965: For example, an interval scale is a system (M,R, 'Nt,n such that (1) M is a non-empty domain, R is the class of all real numbers, 'Nt is a class of mappings m : M -+ R, and T is a class of transformations t: R -+ R; (2) 'Nt contains all transformations of mappings mE 'Nt by tE T; (3) T contains the identity transformation and all compositions of transformations in T; and, characteristic for interval scales, (4) T is the class of all positive linear transformations, i.e., all t such that t(x) = x' p + q for some real number q and some positive real number p. Finally, still another concept of scale has been used in this essay: A scale, in measurement context, is a certain numerical structure.
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NOTES This essay is an extended version of Kanger 1963 (cf. the references at the end of the paper). The translation from Swedish has been done by Wendy 10 Silver . The work was supported by a grant from the Wenner-Gren Foundation. 2 An excellent presentation of the distinction is given in Suppes & Zinnes 1963. 3 Sometimes the name "extensive system" is used for combination structure. 4 The proof is in Suppes 1951. 5 The proof is in Suppes 1951. 6 Holder 1901, and Huntington 1902 gave a characterization corresponding to the conditions (1) - (8) above for C-structures which have the C-scale of all positive real numbers as isomorphic image. 7 See Birkhoff 1948, p. 226. 8 Some data about the theory's origin can be found in Stevens 1959. 9 The proof is in Suppes & Winet 1955. 10 The proof is in Suppes & Winet 1955. REFERENCES Adams, E .W. Survey ofBernoullian utility theory. Included in: Solomon (ed.): Mathematical thinking in the measurement of behavior. Glencoe 1960. Adams, E.W. Elements of a theory of inexactmeasurement. Philosophy of science , vol. 32 (1965). Adams, E.W. On the nature and purpose of measurement. Synthese vol. 16 (1966). Adams, E.W., Fagot, R.F. & Robinson, R.E. A theory of appropriate statistics. Psychometrica, vol. 30 (1965). Aumann , R.I. Utility theory without the completeness axiom. Econometrica, vol. 30 (1962). Correction, vol. 32 (1964). Bettazzi, R. Teoria delle grandezze . Pisa, 1890. Birkhoff, G. Lattice theory. New York, 1948. Bridgman , P.W . The logic of modern physics. New York, 1927. Burali-Forti, C. Sulla teoria delle grandezze. Rivista di matematica, vol. 3 (1893). Campbell , N.R. Physics: the elements. Cambridge, 1920. Campbell, N.R. An account of the principles of measurement and calculation. London, 1928. Carnap, R. Physikalische Begriffsbildung. Karlsruhe, 1926. Coombs, C.H . Psychological scaling without a unit of measurement. Psychological review, vol. 57 (1950). Coombs, C.H. A theoryof psychological scaling. Engineering research bulletin, no. 34, Ann Arbor, Mich., 1952. Coombs, C.H. A theory of data. New York , 1964. Debreu, G. Topological methods in cardinalutility theory. Included in: Arrow , K.1., Karlin, S. & Suppes, P. (ed.): Mathematical methods in the social sciences 1959, Stanford 1960. Ellis, B. Basic concepts of measurement. Cambridge, 1966. Fischburn, P.C . Utility theory. Management science, vol. 14 (1968). Gerlach, M.W. Interval measurement of subjective magnitudes with subliminal differences. Dissertation, Stanford University , 1957.
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Hausner, M. Multidimensional utilities. Included in: Thrall, R.M., Coombs, C.H. & Davis, R.L. (ed.): Decision processes, New York, 1954. Helmholtz, H. von. Ziihlen und Messen, erkenntnistheoretisch betrachtet. Philosophische Aufslitze Eduard Zeller gewidmet. Leipzig, 1887. Hempel, C. Fundamentals of concept formation in empirical science. International encyclopedia of unified science, vol. 2, nr. 7. Chicago, 1952. Herstein, LN. & Milnor, J . An axiomatic approach to measurable utility. Econometrica, vol. 21 (1953). Huntington, E. V. A complete set of postulates for the theory of absolute continuous magnitude. Transactions of the American Mathematical Society, vol. 3 (1902). Holder, O. Die Axiome der Quantitiit und die Lehre vom Mass. Leipziger Berichte, Math.Phys. Klasse, vol. 53 (1901). Kanger, S. Mlitning: en vetenskapsteoretisk essay. Stockholm, 1963. Lenzen, V.F . The nature of physical theory. London, 1931. Luce, R.D . Semiorders and the theory of utility discrimination. Econometrica, vol. 24 (1956). Luce, R.D . Two extensions of conjoint measurement. Journal of mathematical psychology, vol. 3 (1966). Luce, R.D . & Raiffa, H. Games and decisions. New York, 1957. Luce, R.D. & Suppes, P. Preference, utility and subjective probability. Included in: Luce, R.D. , Bush, R.R. & Galanter, E. (ed.): Handbook of Mathematical Psychology, vol. 3, New York, 1965. Luce, R.D. & Tukey, J. W. Simultaneous conjoint measurement: A new type offundamental measurement. Journal of mathematical psychology, vol. 1 (1964). Nagel, E. On the logic of measurement. New York, 1930. Nagel, E. Measurement. Erkenntnis, vol. 2 (1931). Neumann, J. von & Morgenstern, O. Theory of games and economic behavior. Princeton, 1944. 2nd ed. 1947. Pfanzagl, J. A general theory of measurement: Applications to utility. Naval research logistics quarterly, vol. 6 (1959). Pfanzagl, J . Theory of measurement. Vienna, 1968. Russell, B. The principles of mathematics. London, 1903. Scott, D. Measurement structures and linear inequalities. Journal of mathematical psychology, vol. 1 (1964). Scott, D. & Suppes, P. Foundational aspects of theories of measurement. The journal of symbolic logic, vol. 23 (1958). Stevens, S.S . On the theory of scales and measurement. Science, vol. 103 (1946). Stevens, S .S. Mathematics, measurement and psychophysics. Included in: Stevens (ed.): Handbook of experimental psychology. New York, 1951. Stevens, S.S . Measurement, psychophysics and utility. Included in: Churchman, C.W. & Ratoosh, P. (ed.): Measurement: definitions and theories, New York, 1959. Suppes, P. A set of independent axioms for extensive quantities. Portugaliae mathematica, vol. 10 (1951). Suppes, P. Measurement, empirical meaningfulness and three-valued logic. Included in: Churchman, C.W. & Ratoosh, P. (ed.): Measurement: definitions and theories, New York, 1959. Suppes, P. Set-theoretical structures in science. Stanford, 1967.
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Suppes, P. & Winer, M. An axiomatization of utility based on the notion of utility differences. Management science, vol. 1 (1955). Suppes , P. & Zinnes, J .L. Basic measurement theory. Included in: Luce, R.D ., Bush, R.R. & Galanter, E. (ed.): Handbook of mathematical psychology, vol. 1, New York, 1963. Torgerson, W.S. Theory and methods of scaling. New York, 1958. Veronese, G. Il continuo rettilineo e l 'assioma v. d'Archimede. Atti della R. Accad. dei Lincei, ser. 4, vol. 6 (1889). Wedberg, A. Additive measures from an elementary logical point of view. Included in: Philosophical essays dedicated to Gunnar Aspelin, Lund, 1963. Wedberg, A. Om S.S. Stevens ' klassifikation av mattskalor. Included in: Sanning, dikt, tro: Till Ingemar Hedenius , Stockholm, 1968. Weitzenhoffer, A.M . Mathematical structures and psychological measurement. Psychometrica, vol. 16 (1951). Wiener , N. A new theory of measurement: A study in the logic of mathematics. Proceedings of the London Mathematical Society, ser. 2, vol. 19 (1919-1920).
THE NOTION OF A PHONEME
By a phonology (in the abstract sense) I shall understand an algebra < U,A,R,S> such that: (1) (2)
(3)
< U/' > is a cancellation semigroup,
R is a congruence relation on < U, A> , S is a nonempty subset of U that is closed under R.
To be more explicit: by a phonology we shall understand a structure A ,R,S > where U is a nonempty domain, A is a binary operation in U (i.e ., an operation we can perform on pairs of members of U and which then yields a member of U), R is a binary relation between members of U, and S is a set of elements in U such that the following conditions hold for all elements x, y and z in U;
< U,
(Ia) (Ib) (Ic) (2a) (2b) (2c) (2d) (2e) (3a) (3b)
(XAy)A Z = XA(yA Z) , if x""'z = yA z, then x=y, if ZAX = zAy, then x=y, xRx if x R y, then y R x, if x R y and y R z, then x R z, if x R y, then xAz R yAz, if x R y, then ZAX R zAy
S is nonempty, if x is in S and if x R y, then y is in S.
Let D be any spoken language or dialect. By the phonology for D I shall vaguely understand the phonology < U, ,R,S> in which A
(4)
(5)
(6)
(7)
U is the set of all types of utterances (expressionsy
is the operation of concatenation of expressions, (thus, if x and y are expressions, then xAy is the expression that consists of x immediately followed by y), R is the relation of synonymy in D between expressions, S is the set of expressions, the instances of which are occurrences of sentences in D. A
274
G. Holmstrom-Hintikka, S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. I, 274-278. © 2001 All Rights Reserved, Printed by Kluwer Academic Publishers , the Netherlands. Originally published in Statistical Methods in Linguistics 3 (1964), 43-48.
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THE NOTION OF A PHONEME
The aim of this brief note is to suggest a construction of a notion of a phoneme in D within the framework of the phonology for D. It may be regarded as an attempt at an exact explication of the phoneme notion in a manner independent of phonetics. I shall assume that the reader has some familiarity with phonemics .? I begin with some preliminary definitions. Let < V, ,R,S > be the phonology for D. Let the letters x, y, z, w be variables for members of V, and let X and Y be variables for subsets of U . A
Def. 1: By the concatenation set of X - in symbols C(X) - we shall understand the least subset Yof V such that X is a subset of Y and xAy is a member of Y whenever x and yare members of Y. Def. 2: We say that x is a part of y if there are z and w such that y=ZAX or y=xAw or y=ZAXA W , Def. 3: hold (i) (ii) (iii) (iv)
We say that x is a variant of y in D if the following four conditions for every z and w : if ZAXA W or ZAyA w are members of S, then ZAXA W R ZAyA w , if ZAX or zAy are members of S, then ZAX R zAy, if xAw or yAw are members of S, then xAw R yAw, if x or yare members of S, then x R y .
We easily prove that the variant relation thus defined is a congruence relation on the algebra < V , > and that the set S is closed under the variant relation . Note that the extension of the variant relation is independent of how we understand the synonymy relation x R y when x and y are not members of S. 3 A
Def. 4: We say that X is a base for D if there is a subset Y of V which fulfills the following three conditions: (i) Y is a subset of the concatenation set of X, (ii) no member of Y is a part of a member of X, and (iii) S is a subset of the concatenation set of the set of all variants of members of Y in D. Def. 5: We say that X is a phonematic base for D if X is a base for D and no base for D has less members than X. A phonematic base for D consists of a small number of simple expressions by means of which we can form the members of Y. These members we may regard as kinds of words with a strictly standardized pronunciation. By means of these words or pronunciation variants of them we shall be able to form every sentence in the language D.
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Among the various phonematic bases for D we now select - somewhat arbitrarily - one base which we denote by P. A phoneme in D can now be constructed simply as a member of P: Def. 6: x is a phoneme in D if and only if x is a member of the phonematic base P for D. This notion of a phoneme has the virtue of satisfying two desiderata: (a) (b)
The number of phonemes in D is small (less than 100 in any case when D is a natural language). Every sentence in D can be constructed as a variant of a sequence of phonemes in D. But in most cases it does not satisfy a stronger version of (b):
(c)
Every sentence in D can be constructed as a sequence of variants of phonemes in D.
In particular, (c) is not satisfied in case prosodic features - relative stress, for instance - affect meaning in D. . We may redefine the notion of a phoneme in order to save (C).4 But then we have to give up desideratum (a), as the following example will show: Let if be an o-expression with a stress of degree n (n= 1,2, ...). Similarly let ;n be an i-expression with stress n. Clearly 0 3 is not a variant of 0 1, since 0 3 cannot always replace 01 without affecting meaning: Pmp03rt and Pmpolrt, for instance, differ in meaning. Similarly 05 is not a variant of 03 or of 01, since (+mp05rt differs in meaning from i4mp03rt and l04mpO lrt. We may continue the argument to show that there is a large number of o-expressions, no two of which are variants of each other. Hence, if (c) shall be maintained there must be a large number of o-phonemes, and we cannot maintain (a). If we do not maintain (c) we need only (say) two o-phonemes, one with stress and one without, to provide for prosodic distinctions in D like the one between the two senses of import? A main problem in the study of the phonology for D is the problem of the uniqueness of the phoneme system for D. To formulate the problem in an exact manner we need two definitions: Def. 7: Let F be any phonematic base for D. By the phonology generated by F for D (or simply: the F-phonology for D) we shall understand the subalgebra
THE NOTION OF A PHONEME
277
We may get an example of a phonematic base for D that is equivalent with F if we modify F by increasing the stress or the pitch of the members of F in a manner that preserves relative stress and pitch in F. (We increase the volume or the speed of the grammophone, so to speak.) The problem of the uniqueness of the phoneme system for D can now be put this way: Is every phonematic base for D equivalent with the phonematic base P? An affirmative answer to this problem seems plausible in most cases when D is a natural language. But we may easily construct an artificial language which has non-equivalent phonematic bases. For instance, let the sentences of the language be the expressions da, dada, dadada, etc and the expressions ad, adad, adadad, etc. We have two phonematic bases for this language: the first consists of the expressions a and d, the second of da and ad, and the two bases are not equivalent. The study of the phonology for D mingles with semantics and syntax. In particular, these sciences may improve our understanding of the synonymy relation R and the sentence set S involved in the phonology. However, as long as our concern is the phonemes, we do not have to depend very much on semantics and syntax. We must agree on some fundamentals, but I believe we may disagree on many details in semantics and syntax and still have to agree on the phonematic bases for D.6 In this sense phonemics can be said to be almost independent of semantics and syntax. NOTES I Note that the expressions are not particular occurrences of sounds but types of sounds. Two sound-occurrences are said to be instances of the same sound-type if and only if they are phonetically very similar. Note also that an expression is not necessarily a type of simple sounds, it may be a type of sound sequences as well. 2 The linguistic background of the notion of a phoneme is given in standard manuals like Z.S. Harris : Methods in Structural Linguistics (1951), C.F. Hockett: A Manual of Phonology (1955), and H .A. Gleason: An Introduction to Descriptive Linguistics (1961). Among shorter papers of particular interest in connection with exact explications of the phoneme notion, we may mention: B. Bloch: A Set of Postulate for Phonemic Analysis, Language 24 (1948), A. Wedberg: Laran om uttrycks- oeh innehallsformer i glossematiken, Menneske og milj« (Copenhagen 1953), and G.E. Peterson & F. Harary : Foundations of Phonemic Theory, Proceedings of Symposia in Applied Mathematics, vol. 12 (providence, R. I. 1961). The algebra and logic required for exact explications of the phoneme notion is given, e.g., in R.R. Stoll: Introduction to Set Theory and Logic (1963). Phonemics is said to originate with the classical works of Bloomfield and Trubetskoi in the thirties. There were some forerunners , I would like to mention one: Adolf Noreen; who in his grand work Vart sprak (vol. I, Lund 1903) clearly anticipated the modern notion of a phoneme.
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The variant relation in the sense of Def. 3 corresponds to the relation called free variation in phonemics. If we change the defmition by replacing the occurrences of "or" with occurrences of "and" , we get a wider relation which includes also the relation called complementary distribution in phonemics. Two expressions x and y stand in the variant relation in this wider sense if and only if they stand in the relation of free variation or complementary distribution. Or, to use another term of phonemics : x and y stand in the wider variant relation if and only if they do not contrast. This relation plays an important role in phonemics. Very often a phoneme is constructed as a set of simple sounds representable in a phonetic alphabet . Two simple sounds of this kind are put into the same phoneme only if they stand in this wider variant relation. Thus the relation is conceived as a necessary condition for being members (or "allophones") of the same phoneme. (We shall note that it is not always a sufficient condition . If it were made sufficient also, we may run into contradictions because the wider variant relation is not transitive . Two simple sounds x and y may both be complementary to Z, and hence, by the sufficiency of the condition, they shall belong to the same phoneme as z. At the same time x and y may contrast, and hence, by the necessity of the condition, they shall belong to different phonemes.) 4 We can do this trivially by changing Def. 4 to: X is a base for D if and only if the following two conditions hold: (i) no member of S is a part of a member of X, and (ii) S is a subset of the concatenation set of the set of all variants of members of X. 5 In phonemics, (c), or ideas which amount to (c), are sometimes 'maintained in combination with (a). For instance, (c) is implicit in the characterization of phonemes in terms of a process of segmentation and classification . This process may roughly be described as follows: We cut the stream of speech into small expressions (segments) and classify them in such a way that two segments are put into the same class if and only if they are variants of each other. Among the possible processes of this kind we select one which yields a minimum number of classes. The classes of segments obtained in this process we identify with the phonemes. 6 For instance, if we maintain the odd view that a sentence x is synonymous with the sentence y if and only if x and y have the same truth value, then we probably get the same phonematic bases as those we get with a more normal view on synonymy . 3
PUBLISHED WRITINGS OF STIG KANGER
A. BOOKS AND ARTICLES
1.
2. 3. 4. 5.
6. 7. 8.
9.
10.
11.
"A Note on Partial Postulate Sets for Propositional Logic", Theoria 21 (1955), 99-104. Provability in Logic, Stockholm (Acta Universitatis Stockholmensis. Stockholm Studies in Philosophy 1) 1957,47 pages. "The Morning Star Paradox" , Theoria 23 (1957) , 1-11. "A Note on Quantification and Modalities", Theoria 23 (1957), 133134. "On the Characterization of Modalities ", Theoria 23 (1957), 152-153. New Foundations for Ethical Theory, Part l, Stockholm, 1957, 42 pages . Reprinted (with minor changes) in 20 . Handbok i logik. Dell. Logisk konsekvens [Handbook of Logic. Part 1. Logical Consequence], Stockholm (Filosofiska studier utgivna av Filosofiska institutionen vidStockholms universitet 4) 1959, 91 pages. Miitning : en vetenskapsteoretisk essay [Measurement: An Essay in Philosophy of Science], Stockholm (Filosofiska studier utgivna av Filosofiska institutionen vid Stockholms universitet 8) 1963, 45 pages . For an English translation see 21 . "Matning" [Measurement] . In Ann-Mari Henschen-Dahlquist (ed.), Filosofiska studiertilliignade KonradMarc-Wogau, Uppsala (Filosofiska studier utgivna av Filosofiska foreningen och Filosofiska institutionen vid Uppsala universitet 1), 1962, pp. 149-167. "Rattighetsbegreppet" [The Notion of a Right]. In Sjufilosofiska studier tilliignade Anders Wedberg , Stockholm (Filosofiska studier utgivna av Filosofiska institutionen vid Stockholms universitet 9) 1963, pp. 79-102 . An English translation of this paper is included (with minor changes) in 14. "A Simplified Proof Method for Elementary Logic". In P. Braffort & D. Hirschberg (eds.), Computer Programming and Formal Systems, Amsterdam (North -Holland Publishing Company , Studies in Logic and the Foundations of Mathematics), 1963, pp. 87-94. Reprinted in 31. For a Russian translation see 15. 279
G. Holmstrom-Hintikka, S. Lindstrom and R. Sliwinski (eds.), Collected Papers of Stig Kanger with Essays on his Life and Work, Vol. I, 279-283.
280
PUBLISHED WRITINGS OF STIG KANGER
12. "The Notion of Phoneme", Statistical Methods in Linguistics 3 (1964), 43-48. 13. "En algebraisk logikkalkyl" [An Algebraic Logic Calculus] . In Analyser och argument: filosofiska uppsatser tilliignade Andries MacLeod, Uppsala (Filosofiska studier utgivna av Filosofiska foreningen och Filosofiska institutionen vid Uppsala universitet 4), 1966, pp. 117-125. 14. (Together with Helle Kanger) "Rights and Parliamentarism", Theoria 32 (1966), 85-115 . Reprinted (with minor changes) in 23. 15. "Uproscennyj metod dokazatelstva dla elementarnoj logiki" [Russian translation of 11 by S. U. Maslov, with added footnotes by the translator] . In A.V. Idelson and G.E. Mine (eds.), Matematiceskaa teoria logiceskogo vyvoda , Matematiceskaa logika i osnovania matematiki, Izdatelstvo "Nauka," Moscow, 1967, pp. 200-207. 16. "Equivalent Theories", Theoria 34 (1968), 1-6. 17. "Preferenslogik" [Preference Logic]. In Hjalmar Wennerberg (ed.) , Nio filosofiska studier tilliignade Konrad Marc- Wogau, Uppsala (Filosofiska studier utgivna av Filosofiska foreningen och Filosofiska institutionen vid Uppsala universitet 6), 1968, pp. I-B. 18. "Beslut i demokratisk ordning" [Decision by Democratic Procedure] . In Sanning dikt tro, Stockholm (Bonniers), 1968, pp. 141-147 . 19. "Equational Calculi and Automatic Demonstration". In Tom Pauli (ed.), Logic and Value: Essays Dedicated to Thorild Dahlquist on His Fiftieth Birthday, Uppsala (Filosofiska studier utgivna av Filosofiska foreningen och Filosofiska institutionen vid Uppsala universitet 9), 1970, 220-226 . 20. "New Foundations for Ethical Theory". In Risto Hilpinen (ed.), Deontic Logic: Introductory and Systematic Readings, Dordrecht (Reidel, Synthese Library) 1971, pp. 36-58. Reprint of 6. 21. "Measurement: An Essay in Philosophy of Science " , Theoria 38 (1972), 1-44. English transI. of 8. 22. "Law and Logic", Theoria 38 (1972), 105-129 . 23. (Together with Helle Kanger) "Rights and Parliamentarism". In R.E. Olson and A.M. Paul (eds.), Contemporary Philosophy in Scandinavia, Baltimore (Johns Hopkins Press), 1972, pp. 213-236. Reprint of 14. 24. "Entailment" . In Modality, Morality and Other Problems of Sense and Nonsense : Essays Dedicated to Soren Hal/den, Lund 1973, pp. 168179. 25. (Editor) Proceedings of the 3rd Scandinavian Logic Symposium, Amsterdam (North-Holland Publishing Company, Studies in Logic and the Foundations of Mathematics 82), 1975.
PUBLISHED WRITINGS OF STIG KANGER
281
26. "The Paradox of the Unexpected Hanging , Regained Again". In Wright and Wrong: Mini-essays in Honor of Georg Henrik von Wright on His Sixtieth Birthday June 14, 1976, Publications of the Group in Logic and Methodology of Real Finland vol 3, 1976, pp . 19-23. 27. "Choice and Modality" . In Oand o: Mini-essays in Honor of Krister Segerberg on His Fortieth Birthday April 26, 1976, Publications of the Group in Logic and Methodology of Real Finland, vol 4, 1976, pp. 2532. 28. "Nagra synpunkter pa begreppet inflytande" [Some Aspects on the Concept of Influence]. In Filosofiska smulor tilliignade Konrad MarcWogau, 75 ar, Uppsala (Filosofiska studier utgivna av Filosofiska foreningen och Filosofiska institutionen vid Uppsala universitet 27), 1977, pp . 12-23. 29. "A Note on Preference Logic" . In ThD60: Philosophical Essays Dedicated to Thorild Dahlquist on His Sixtieth Birthday, Uppsala (Filosofiska studier utgivna av Filosofiska foreningen och Filosofiska institutionen vid Uppsala universitet 32), 1980, pp . 37-38. 30. (Editor together with Sven Ohman) Philosophy and Grammar, Papers from the Symposium on the Occasion of the Quincentennial of Uppsala University, Uppsala, 1978, D. Reidel Publishing Company, 1981. 31. "A Simplified Proof Method for Elementary Logic" . In Joerg Siekmann and Graham Wrightson (eds.) , Automation of Reasoning: Classical Papers on Computational Logic 1957-1966. Springer-Verlag, 1983. Reprint of 11. 32. "On Realization of Human Rights" . In Ghita Holmstrom and Andrew J.I . Jones (eds.), Action, Logic and Social Theory. Dedicated to Ingmar Porn on the Occasion ofHis 50th Birthday, Helsinki (Acta Philosophica Fennica 38) 1985, pp . 71-78. 33. "Unavoidability" . In Logic and Abstraction: Essays Dedicated to Per Lindstrom on His Fifthieth Birthday, Goteborg (Acta Philosophica Gothoburgensia 1), 1986, pp. 227-236 . B. REVIEWS
Review of A .N. Prior, Time and Modality . Being the John Locke Lectures for 1955-6 Delivered in the University of Oxford (Oxford University Press, London, Glasgow, New York, Toronto, 1957), The Journal of Symbolic Logic 25 (1960), 342-343.
282
PUBLISHED WRITINGS OF STIG KANGER
Review of A.N. Prior, "Diodorus and Modal Logic: A Correction" (The Philosophical Quarterly 8 (1958), 226-230), The Journal of Symbolic Logic 25 (1960), 343. Review of A.N . Prior, "Time After Time" (Mind, n.s . 67 (1958), 244-246) , The Journal of Symbolic Logic 25 (1960), 343. Review of A.N. Prior, "Thank Goodness that's Over" (Philosophy 34 (1959), 12-17), L. Jonathan Cohen, "Professor Prior on Thanking Goodness that's Over" (ibid., pp. 360-:362), A.N. Prior, "Mr . Cohen on Thanking Goodness that p and q" (ibid., pp. 362-363), L. Jonathan Cohen, "A Brief Rejoinder to Professor Prior" (ibid., pp. 363-364), The Journal of Symbolic Logic 25 (1960),343 . Review of A.N . Prior, "The Syntax of Time-distinctions" (Franciscan Studies 18:2 (1958), 105-120), The Journal of Symbolic Logic 27 (1962), 114-115. Review of Antonio Monteiro, "Matrices de Morgan caracteristiques pour Ie calcul propositionnel classique" (Anais de Academia Brasileira de Ciencas 32 (1960), 1-7), Mathematical Reviews 28 (#18) (1964), 3. Review of L. Jonathan Cohen, "Can the Logic of Indirect Discourse be Formalised" (The Journal of Symbolic Logic 22 (1957), 225-232), A.N. Prior, "Epimenides the Cretan" (ibid., 23 (1958), 261-266), R.L. Goodstein, "On the Formalisation of Indirect Discourse" (ibid., 32 (1958) ,417-419), L. Jonathan Cohen, "Professor Goodstein's Formalisation of the Policeman", (ibid., p. 420) , The Journal ofSymbolic Logic 32 (1967), 549-550. Review of L. Jonathan Cohen, "Why do Cretans Have to Say so Much?" (Philosophical Studies 12 (1961),72-78), A.N. Prior, "Indirect Speech Again" (ibid., 14 (1963), 12-15), L. Jonathan Cohen, "Indirect Speech: A Rejoinder to Prof. A.N. Prior" (ibid., pp . 15-18), A.N . Prior, "Indirect Speech and Extensionality" (ibid., 15 (1964), 35-38), L. Jonathan Cohen, "Indirect Speech: A Further Rejoinder to Professor Prior" (ibid., pp. 38-40), The Journal of Symbolic Logic 32 (1967), 550. Review of L. Jonathan Cohen, "A Formalisation of Referentially Opaque Contexts" (The Journal of Symbolic Logic 25 (1960), 193-202), The Journal of Symbolic Logic 32 (1967), 550. Review of Alan Ross Anderson, "Completeness Theorems for the Systems E of Entailment and EQ of Entailment with Quantification" tZeitschrift fUr mathematische Logik und Grundlagen der Mathematik 6 (1960), 201-216), The Journal of Symbolic Logic 36 (1971), 520.
PUBLISHED WRITINGS OF STIG KANGER
283
Review of Nuel D. Belnap, Jr, "EQ and the First Order Functional Calculus" (Zeitschrift fUr mathematische Logik und Grundlagen der Mathematik 6 (1960), 217-218), The Journal of Symbolic Logic 36 (1971),520. Review of Alan Ross Andersson and Nuel D. Belnap, Jr , "First Degree Entailments" (Mathematische Annalen 149 (1963),302-319), The Journal of Symbolic Logic 36 (1971), 520-521. Review of David Makinson, "An Alternative Characterisation of First Degree Entailment" (Logique et Analyse, n.s. 8 (1965) 308-311), The Journal of Symbolic Logic 36 (1971) , 521. Review of Patrick Suppes, "Finite Equal-interval Measurement Structures" (Theorla 38 (1972), 45-63) , Mathematical Reviews 51 (#9863) (1976), 1375.
INDEX OF NAMES
Ackermann, W. 39, 83, 84 Adams, E. W. 260,267,268,270,271 Andersson, A. R. 39, 83, 84, 115, 134, 151,167,168 Aristotle 49, 106, 109 Arrow , K. J. 168,231-235 Aumann, R. J. 267,271 Austin , J. 132 Ayer, A . 1. 118 Belnap , N. D. 83 Bernays, P. 7,38,39,40 Beth, E. W. 39,59,64, 111 Bettazi, R. 260, 271 Birkhoff, G. 271 Bloch, B. 277 Bolzano, B. 110, 111 Bouvere, K. L. de 69 Braffort, P. 58 Bridgman, P . W . 271 Burali -Forti, C. 260,271 Bush , R. R. 168, 272, 273 Campbell, N. R. 260,271 Carnap, R . 39, 40, 42, 45, 46-47 , 49, 50, 151, 153-158, 166,260,269,271 Cartwright, D. 164, 168 Chellas, B. 151, 156-158 , 166, 168 Church, A. 40,42,49,50,82, 109, 151 Churchman, C. W . 272 Commons, J. R. 105, 134 Comte, A. G . 166 Coombs , C. H. 208,263,270,271 ,272 Corbin, A. L. 134, 145 Curry, H . B. 7, 39,40 Dahlquist, T . 39 Danielsson, S. 168 Davidson, D . 168 Davis, R . L. 272 Debreu, G. 264, 265, 271 Dreben, B. 38,39,40
Eckhoff, T . 134 Enger , G. 231 Ellis , B. 260, 271 Fagot, R . F . 268, 270, 271 Feys, R. 34-35,39,40 Fishburn, P. C . 230,267,271 Fitting, M . 167 Frege, G. 49,50, 110, 111 Fusilier, R. 145 Galanter, E. 168,272,273 Geach, P. 109 Gelernter, H. 58,64 Gentzen, G. 5,39,40 Gerlach, M. W. 268,271 Gilmore, P. C . 59, 64 Gleason, H. A. 277 GOdel, K. 38, 39,40 Hagerstrom, A . 118 Hall, J. 145 Hallden, S. 168, 199 Hansen, K. B. 70 Hansson, B. 151, 167, 168, 200-201, 204 Harary, F . 277 Hare, R. M. 118 Harris, Z. S. 277 Hausner, M . 208,267,272 Hedenius, I. 118 Helmholtz, H . von 260, 272 Hempel, C. 260, 269, 272 Henkin , L. 38, 40 Henschen-Dahlquist, A. 231 Herbrand, 1. 39, 40 Hermes, H. 7,39,40 Herstein, I. N. 267,272 Heyting, A. 3 Hilbert, D. 7,38,39,40 Hilpinen, R . 119, 167, 168 Hintikka, J. 151, 154-158, 166, 167
286
INDEX OF NAMES
Hirschberg , D. 58 Hockett, C. F. 277 Hofstadter , A. 102, 109, III Hohfeld, W. N. 105, 120, 133-135 , 145, 168, 177 Holder, O. 260,271 ,272 Holmstrom, G. 179 Hospers, J. 119 Hume, D. 119 Huntington, E. V. 260,271,272 Johansson, I. 3, 7 Jones, A. J. I. 179 Jonsson, B. 39, 40, 212, 230 Kalinowski, J. 166 Kanger, H. 120-145 ,151,164,168,177 Kanger, S. 38, 39, 40, 50, 59, 64, 81, 109, 115, 145, 151, 153-158 , 164, 166-167 , 168, 170-171 , 177, 271, 272 Kant, I. III Kemeny, J. G. 39,40 Kenny, A. 168 Ketonen, O. 39,40 Kleene, S. C. 7,39,40 Klibansky, R. 167 Klug, U. 166 Kocourek, A. 134, 145 Kripke, S. 83, 155-158, 167
Morgenstern, O. 206,265-267,272 Moritz, M. 134, 168 Nagel, E. 260, 272 Neumann, J. von 206,265-267,272 Newell, A. 58, 64 Noreen, A. 277 Ogden, C. K. 118 Olson, R. E. 120 Oppenheim, F. 164, 168, 177 Paul, A. M. 120 Pauli, T. 76, 169 Perry, R. B. 119 Peterson, G. E. 277 Pfanzagl, J. 260,267,268,270,272 Plott, C. R. 212, 224, 230 Porn, I. lSI, 164, 167, 168, 169, 177, 184-185, 190 Prawitz, D. 59,64 Prawitz , H. 64 Prior, A. N. 109, 115 Quine, W. V. O. 38,39,40-41,45-46, 49, 50, 111
Langford, C. H. 39, 40 Lenzen, V. F. 269,272 Lewis, C. I. 34-35, 39, 40 , 42, 50 Lindahl, L. 177, 184-185 Lindstrom , P. 69 Luce, R. D. 151, 168,231, 264, 265, 267,268,272,273 Lukasiewicz, J. 7
Rabinowicz, W. 190 Raiffa, H. 151, 168,231,267,272 Ratoosh, P. 272 Rescher, N. 168, 169,200 Richards, I. A. 118 Rider , S. 170, 199 Robinson, R. E. 268,270,271 Rochester, N. 64 Ross, A. 134, 166 Ross, W. D. lOS , 119 Russell, B. 46, 49, 50, 118, 119, 260, 269,272
Mally, E. 102 March, J. G. 168 Mates, B. 49,50, 110 McKinsey, J. J. C. 102,109 ,111 Miller, R. 8 Milnor, J. 267,272 Moore, G. E. lOS, 119 Moore , O. K. 115
Salmond, J. 133 Scholz, H. 7, 39,40 Schonfinkel, M. 39 Schreiber, R. 166 Schutte, K. 39,41 Scott, D. 156-158, 167, 264, 268, 272 Segerberg , K. 157, 167, 169,211 ,212 Sellars, W . 119
INDEX OF NAMES
Sen, A. 212, 213, 215, 219, 222, 230 Shen, Y. 39,41 Silver, W. 1. 271 Simon, H. 58, 64 Skolem, T. 38,41 Smullyan, R. 49,50 Stevens, S. S. 263,270,271,272 Stevenson, C. 118 Stoll, R. R. 277 Suppes, P. 168,260,263-264,267,268, 269,270,271,272-273 Tammelo, I. 166 Tarski, A. 39, 40, 41, 49, 50, 69, 71, 108,110, 152-153, 167,212,230 Terry , H. T. 133 Thrall, R. M. 272 Torgerson, W. S. 260,273 Tukey, J. W. 264,265 ,272
287
Urquhart, A. 93 Vaught, R. L. 41 Verney, D. V. '145 Veronese, G. 260, 273 Vogera, N. 59, 64 Wang, H. 59, 64 Wedberg, A. 8, 260, 270, 273, 277 Weinberger, O. 166 Weitzenhoffer, A. M. 268,273 Wennerberg, H. 168, 199 Wiener, N. 268,273 Winet, M 263,271,273 Wright, G.H. von 94, 102, 111, 151, 167, 168, 200-201, 208 Zinnes, 1. L. 260, 270, 271, 273
SUBJECT INDEX
Ability 173-174 Absolute modality 29 Action 160-162 Algebraic logic 65-69,70-75 Algebra ic logic calculus 70, 71-75 Analys is, paradox of 48-49 Analytic judgement 110 Analytic necessity 30 Analytic proposition 44, 111 As good as 199 Aspect 239 Aspects, classification of 242-247 At least as good as 199 Automated theorem proving 58-64, 7681 Axiomatic calculus 70 Better than 199 Blameworthiness 105-106, 165 C-scale 261-263 C-structure 261-262 Calculus LC 15-18 Calculus 54/54* 35 Calculus 55/55* 35-38 Calculus tlt* 35 Choice 211-212,214-230 Choice function 214, 218 Claim 106, 121-122 Classical logic 5-6 Comparative quality 247-248 Completeness 22,24,74, 78-81 Condition 186 Congruence 67 Consistency axiom 229 Contingency 110 Contradictory judgement 110 Contravalidity 110 Control parliamentarism 140-141 Conversion 128-129 Conviction, logic of 94-95
Conviction operator 149-151 Coordination 128-129 Correctness 107, 113 Counter-ability 173-174 Counter-claim 121-122 Counter-freedom 121-122 Counter-immunity 121-122 Counter-power 121-122 Counter-security 173-174 Crucial system 21,23-24 Cylindrification 202 Decision method 27-28,37,39,77-81 Decision operator 149-151 , 158-166 Decision problem 27-28, 37, 39 Decision theory 211-212,214-230,231236 Deduction branch 18 Deduction branch, explicit 18 Delegation parliamentarism 141 Democratic decision-making 231-236 Denotation 113 Deontic logic 99-119 Deontic operator 101-102, 151 Deontic statement 10I Difference scale 256-257 Disability 106 Disutility 202 Do-operator 149-151 ,158-160,179-181 Do-predicate 187-188 Dummy 62 Duty 106 Effective proof procedure 26-28, 58-64 , 77-81 Elementary net 22-23 Emotivism 116 Entailment 82-93 Entailment, strong 82-83 Entailment, ultraintensional 82, 88-93 Entailment, weak 82-83, 88-93
290
SUBJECT INDEX
Epistemic operators 149-151 Equational calculus 76-81 Equational logic 71, 73-74, 76-81 , 8393 Equivalence 66-67 Equivalent theories 65-69 Ethical theory 99-1()() Explicit deduction branch 18 Exposure 106 Extensional net 22-23 Forbidden action 102 Frame 13 Freedom 121-122 Frege-Hilbert calculus 70 Full normal quasi-deduction 26 Government parliamentarism 141 Grounded partial ordering 218, 224-225 Gentzen Hauptsatz 5, 25 Gentzen system 7, 15-17,35-38,58-61, 70,88-91 Heuristics 58 Homogeneously spotted formula 36 Homomorphism 240 Human right 172-184 Immunity 106, 121-122 Imperative operator 101 Impossibility theorems 233-236 Incorrectness 107, 113 Influence 164,170-177 Influence of higher order 175-177 Influence, types of 172-174 Interpretation, quantificational 46-47 Interpretation, syntactic 45 Interval scale 264 Interval structure 264 Intuitionistic logic 3-6 Inversion 128-129 Irreflexive relations 217, 221-223 Irrevocability 189-190 Is-Ought 118 Iterative modality 30 Jaskowski-Gentzen calculus 70 Jurisprudence 146-169
L-Ianguage 147-148 Law 146-169 Laws, system of 146 Liability 106 Logic of conviction 94-95 Logical necessity 30 Logical truth 15, 24 Lowenheim-Skolem theorem 25 Magnitude 269-270 Main structure 22-23 May-operator 122, 179-181 Meaningfulness 251,260,267-269 Measure 239,241 Measurement 241 Measurement theory 239-273 Measurement theory, liberalized 263-267 Measurement type 249-250 Metrical sentence 250-251 Minimal calculus 3-6 Minister parliamentarism 141 Mixture scale 266-267 Mixture structure 266-267 Modal logic 28-38, 42-50, 52-53, 5457,211-212 Modalities, classification of 29-30 Modality 28,54-57 ,211-212 Modality, absolute 29 Modality, iterative 30 Model 107-108 Morning Star paradox 42-50 Natural deduction calculus 70 Naturalistic fallacy 117-118,119 Neat partial order 218. 223-224 Net 22-23 Norm 108 Normal deduction branch 19-20 Normal demonstration 77 Normal proof 77 Normal system 15 Normal valuation 14 Normally intensional entailment 82, 8688 Normative logic 113-115 Normative sentence 101-102 Numerical structure 240
SUBJECT INDEX Obligatory action 102 Operationism 267-268 Ought-operator 101,103 ,104,147,149151, 158-160 Paradox of analysis 48-49 Paradox of unexpected hanging 94-96 Parliamentarism 140-142,231-236 Permitted action 102 Phoneme 274-278 Phonology 274,276 Position structure 135 Position structure, reduced 141-144 Predicative net 22-23 Preference 165-166, 199-208,209-210, 214-230 Preference logic 199-208, 209-210 Privilege 106 Proof 17,37 Proposition 44 Propositional logic 3-7 Power 106, 121-122 Praiseworthiness 105-106 , 165 Pseudo-parliamentarism 141 Quantification 52-53 Quantificational interpretation 46-47 Quantity 269-270 Quasi-deduction 17, 36-37 Quasi-deduction, full normal 26 Quasi-sequent 11 Relevant implication 82, 88-93 Representation theorem 259-263 , 265267 Responsibility 165 Right, concept of 130-131 , 162-164, 179-184 Right, scope of 131 Right-operator 102-104 Right +-operator 103-104 Rights, atomic types of 126-127 Rights, rule of 131-132, 181-184 Rights , simple types of 120-131 , 162164 Rights-type, concept of 130-131
291
Scale 241,270 Scale-determination 248-249 Security 173-174 Seeing-to- it-that-operator 104, 122-123, 148,151,171-172,179-181 Semantic tableaux 39, 59 Semantics 13-15,29,31-34,43-44,5457, 106-115, 151-160 Semi-orderings 217,226-227 Sequent 11 Sequent calculus 15-17 , 70, 88-91 Set structure 23 Set-theoretical necessity 30 Set-theoretical principle 23 Set theory 22-23 Shall-operator 122-123, 147, 149-151, 158-160, 179-181 Soundness 24 Stability axiom 228-229 Stoic-Megaric logic 49, 110 Strong entailment 82-83 Structure 239 Substructure 240 Synonymous 110 Synonymous theories 66-67 Syntactic interpretation 45 Synthetic judgement 110 System 15, 44, 107 Tarski calculus 70 Thinning 90 Transitive closure 218 Ultraintensional entailment 82, 88-93 Unavoidability 147, 149-151, 158-160, 186-196 Unavoidability-operator 147, 149-151, 158-160 Unavoidability-predicate 187-189 Unexpected hanging, paradox of 94-96 Uniqueness 243-246,260-263,265 ,267, 277 Unit 239 Utilitarianism 103 Utility 202
292
SUBJECT INDEX
Validity 15, 22, 32-33, 44, 5, 110 Valuation 13-15, 29, 43-44 , 106-110, 153-154 Value-logic 202-203 Value plane 201-204
Weak entailment 82-83 ,88-93 Weak ordering 217,227-228 Welfare progr am 103 Wrong-operator 102
SYNTHESE LIBRARY 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14.
15. 16. 17. 18.
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20. 21. 22. 23.
J. M. Bochenski, A Precis ofMathematical Logic . Translated from French and German by O. Bird. 1959 ISBN 90-277-0073-7 P. Guiraud, Problemes et methodes de la statistique linguistique. 1959 ISBN 90-277-0025-7 H. Freudenthal (ed.), The Concept and the Role ofthe Model in Mathematics and Natural and ISBN 90-277 -0017-6 Social Sciences. 1961 E. W. Beth, Formal Methods. An Introduction to Symbolic Logic and to the Study of Effective Operations in Arithmetic and Logic. 1962 ISBN 90-277-0069-9 B. H. Kazemier and D. Vuysje (eds.), Logic and Language. Studies dedicated to Professor Rudolf Carnap on the Occasion of His 70th Birthday. 1962 ISBN 90-277-0019-2 M. W. Wartofsky (ed.), Proceedings of the Boston Colloquium for the Philosophy of Science , 1961-1962. [Boston Studies in the Philosophy of Science, Vol. 1]1963 ISBN 90-277-0021 -4 A. A. Zinov'ev, Philosophical Problems ofMany-valued Logic. A revised edition, edited and translated (from Russian) by G. Kiing and D.D. Corney. 1963 ISBN 90-277-0091-5 G. Gurvitch, The Spectrum ofSocial TIme. Translated from French and edited by M. Korenbaum and P. Bosserman. 1964 ISBN 90-277-0006-0 P. Lorenzen, Formal Logic . Translated from German by FJ. Crosson. 1965 ISBN 9O-277-OO80-X R. S. Cohen and M. W. Wartofsky (eds .), Proceedings ofthe Boston Colloquiumfor the Philosophy of Science , 1962-1964. In Honor of Philipp Frank. [Boston Studies in the Philosophy of Science, Vol. 11]1965 ISBN 90-277-9004-0 E. W. Beth, Mathematical Thought. An Introduction to the Philosophy of Mathematics. 1965 ISBN 90-277-0070-2 E. W. Beth and J. Piaget, Mathematical Epistemology and Psychology. Translated from French by W. Mays . 1966 ISBN 90-277-0071-0 G. Kiing, Ontology and the Logistic Analysis ofLanguage. An Enqu iry into the Contemporary Views on Universals. Revised ed., translated from German. 1967 ISBN 90-277-0028-1 R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the Philosophy ofSciences , 1964-1966. In Memory of Norwood Russell Hanson. [Boston Studies ISBN 90-277-0013-3 in the Philosophy of Science, Vol. III] 1967 C. D. Broad, Induction, Probability, and Causation . Selected Papers . 1968 ISBN 90-277 -0012-5 G. Patzig, Aristotle's Theory ofthe Syllogism. A Logical -philosophical Study of Book A of the Prior Analytics. Translated from German by J. Barnes . 1968 ISBN 90-277 -0030-3 ISBN 90-277 -0084-2 N. Rescher, Topics in Philosophical Logic. 1968 R. S. Cohen and M. W. Wartofsky (eds.) , Proceedings of the Boston Colloquium for the Philosophy of Science, 1966-1968, Part I. [Boston Studies in the Philosophy of Science, Vol. IV] 1969 ISBN 90-277-0014-1 R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the Philosophy of Science, 1966-1968, Part 11. [Boston Studies in the Philosophy of Science, Vol. V] 1969 ISBN 90-277-OO15-X J. W. Davis, D. J. Hockney and W. K. Wilson (eds.) , Philosophical Logic. 1969 ISBN 90-277-0075-3 D. Davidson and J. Hintikka (eds.), Words and Objections. Essays on the Work ofW. V. Quine. 1969, rev. ed. 1975 ISBN 90-277-0074-5; Pb 90-277-0602-6 P. Suppes, Studies in the Methodology and Foundations ofScience. Selected Papers from 1951 to 1969. 1969 ISBN 90-277-0020-6 J. Hintikka, Models for Modalities. Selected Essays. 1969 ISBN 90-277-0078-8; Pb 90-277-0598-4
SYNTHESE LIBRARY 24. 25. 26. 27. 28. 29. 30. 3 I. 32. 33. 34. 35. 36. 37. 38. 39.
40. 41. 42. 43. 44. 45. 46.
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N. Rescher et al. (eds .), Essays in Honor ofCarl G. Hempel. A Tribute on the Occasion of His 65th Birthday. 1969 ISBN 90-277-0085-0 P. V. Tavanec (ed .), Problems of the Logic of Scientific Knowledge. Translated from Russian. 1970 ISBN 90-277-0087-7 M. Swain (ed .), Induction, Acceptance, and Rational Belief. 1970 ISBN 90-277-0086-9 R. S. Cohen and R. J. Seeger (eds.), Ernst Mach: Physicist and Philosopher. [Boston Studies in the Philosophy of Science, Vol. VI]. 1970 ISBN 90-277-0016-8 J. Hintikka and P. Suppes, Information and Inference. 1970 ISBN 90-277-0155-5 K. Lambert, Philosophical Problems in Logic. Some Recent Developments. 1970 ISBN 90-277-0079-6 ISBN 9O-277-0161-X R. A. Eberle, Nominalistic Systems. 1970 P.Weingartner and G. Zecha (eds.),Induction, Physics, and Ethics. 1970 ISBN 90-277-0158-X E. W. Beth, Aspects ofModern Logic. Translated from Dutch. 1970 ISBN 90-277-0173-3 R. Hilpinen (ed .), Deontic Logic. Introductory and Systematic Readings. 1971 See also No. 152. ISBN Pb (1981 rev.) 90-277-1302-2 J.-L. Krivine, Introduction to Axiomatic Set Theory. Translated from French. 1971 ISBN 90-277-0169-5; Pb 90-277-0411 -2 J. D. Sneed, The Logical Structure ofMathematical Physics. 2nd rev. ed., 1979 . ISBN 90-277-1056-2; Pb 90-277-1059-7 C. R. Kordig, The Justification ofScientific Change. 1971 ISBN 90-277-0181-4; Pb 90-277-0475-9 M. Capek, Bergson and Modern Physics. A Reinterpretation and Re-evaluation. [Boston ISBN 90-277-0186-5 Studies in the Philosophy of Science, Vol. VII] 1971 N. R. Hanson, What I Do Not Believe , and Other Essays . Ed. by S. Toulmin and H. Woolf. 1971 ISBN 90-277-0191 -1 R. C. Buck and R. S. Cohen (eds.), PSA 1970. Proceedings of the Second Biennial Meeting of the Philosophy of Science Association, Boston, Fall 1970. In Memory of Rudolf Carnap. [Boston Studies in the Philosophy of Science, Vol. VIII] 1971 ISBN 90-277-0187-3; Pb 90-277-0309-4 D. Davidson and G. Harman (eds.), Semantics ofNatural Language. 1972 ISBN 90-277-0304-3; Pb 90-277-0310-8 Y. Bar-Hillel (ed.), Pragmatics ofNatural Languages. 1971 ISBN 90-277-0194-6; Pb 90-277-0599-2 S. Stenlund, Combinators, Terms and Proof Theory. 1972 ISBN 90-277-0305-1 M. Strauss, Modern Physics and Its Philosophy. Selected Paper in the Logic, History, and Philosophy of Science. 1972 ISBN 90-277-0230-6 M. Bunge, Method , Model and Matter. 1973 ISBN 90-277-0252-7 M. Bunge, Philosophy ofPhysics. 1973 ISBN 90-277-0253-5 A. A. Zinov'ev, Foundations of the Logical Theory ofScientific Knowledge (Complex Logic). Revised and enlarged English edition with an appendix by G. A. Smirnov, E. A. Sidorenka, A. M. Fedina and L. A. Bobrova. [Boston Studies in the Philosophy of Science, Vol.IX] 1973 ISBN 90-277-0193-8; Pb 90-277-0324-8 L. Tondl, Scientific Procedures. A Contribution concerning the Methodological Problems of Scientific Concepts and Scientific Explanation. Translated from Czech by D. Short. Edited by R.S. Cohen and M.W. Wartofsky. [Boston Studies in the Philosophy of Science, Vol. X] 1973 ISBN 90-277-0147-4; Pb 9O-277-0323-X N. R. Hanson, Constellations and Conjectures. 1973 ISBN 90-277-0192-X
SYNTHESE LIBRARY 49. 50. 51. 52. 53. 54.
55. 56. 57. 58. 59. 60.
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K. J. J. Hintikka, J. M. E. Moravcsik and P. Suppes (eds.), Approaches to Natural Language. 1973 ISBN 90-277-0220-9; Pb 90-277-0233-0 M. Bunge (ed.), Exact Philosophy. Problems, Tools and Goals. 1973 ISBN 90-277-0251-9 R. J. Bogdan and I. Niiniluoto (eds.), Logic, Language and Probability. 1973 ISBN 90-277-0312-4 G. Pearce and P. Maynard (eds.), Conceptual Change. 1973 ISBN 9O-277-0287-X; Pb 90-277-0339-6 I. Niiniluoto and R. Tuomela, Theoretical Concepts and Hypothetico-inductive Inference. 1973 ISBN 90-277-0343-4 R. Fraisse, Course ofMathematical Logic - Volume I: Relation and Logical Formula. Translated from French. 1973 ISBN 90-277-0268-3; Pb 90-277-0403-1 (For Volume 2 see under No. 69). A. Griinbaum, Philosophical Problems of Space and Time. Edited by R.S. Cohen and M.W. Wartofsky. 2nd enlarged ed. [Boston Studies in the Philosophy of Science, Vol. XII] 1973 ISBN 90-277-0357-4; Pb 90-277-0358-2 P. Suppes (ed.), Space, Time and Geometry. 1973 ISBN 90-277-0386-8; Pb 90-277-0442-2 H. Kelsen, Essays in Legal and Moral Philosophy. Selected and introduced by O. Weinberger. Translated from German by P. Heath. 1973 ISBN 90-277-0388-4 R. J. Seeger and R. S. Cohen (eds.), Philosophical Foundations of Science. [Boston Studies in the Philosophy of Science, Vol. XIj1974 ISBN 90-277-0390-6; Pb 90-277-0376-0 R. S. Cohen and M. W. Wartofsky(eds.), Logical and Epistemological Studies in Contemporary Physics . [Boston Studies in the Philosophy of Science, Vol. XIIIj1973 ISBN 90-277-0391-4; Pb 90-277-0377-9 R. S. Cohen and M. W. Wartofsky (eds.), Methodological and Historical Essays in the Natural and Social Sciences. Proceedings of the Boston Colloquium for the Philosophy of Science, 1969-1972. [Boston Studies in the Philosophy of Science, Vol. XIVj1974 ISBN 90-277-0392-2; Pb 90-277-0378-7 R. S. Cohen, J. J. Stachel and M. W. Wartofsky (eds.), For Dirk Struik . Scientific , Historical and Political Essays. [Boston Studies in the Philosophy of Science, Vol. XVj 1974 ISBN 90-277-0393-0; Pb 90-277-0379-5 K. Ajdukiewicz, Pragmatic Logic . Translated from Polish by O. Wojtasiewicz. 1974 ISBN 90-277-0326-4 S. Stenlund (ed.), Logical Theory and Semantic Analysis. Essays dedicated to Stig Kanger on His 50th Birthday. 1974 ISBN 90-277-0438-4 K. F. Schaffner and R. S. Cohen (eds.), PSA 1972. Proceedings ofthe Third Biennial Meeting of the Philosophy ofScience Association. [Boston Studies in the Philosophy of Science, Vol.XXj 1974 ISBN 90-277-0408-2; Pb 90-277-0409-0 H. E. Kyburg, Jr., The Logical Foundations ofStatistical Inference. 1974 ISBN 90-277-0330-2; Pb 90-277-0430-9 M. Grene, The Understanding ofNature. Essays in the Philosophy of Biology. [Boston Studies in the Philosophy of Science, Vol. XXIIIj1974 ISBN 90-277-0462-7; Pb 90-277-0463-5 J. M. Broekman, Structuralism: Moscow, Prague, Paris. Translated from German, 1974 ISBN 90-277-0478-3 N. Geschwind, Selected Papers on Language and the Brain. [Boston Studies in the Philosophy of Science, Vol.XVIj1974 ISBN 90-277-0262-4; Pb 90-277-0263-2 R. Fraisse, Course ofMathematical Logic - Volume2: Model Theory. Translated from French. 1974 ISBN 90-277-0269-1; Pb 90-277-0510-0 (For Volume 1 see under No. 54)
SYNTHESE LIBRARY 70. 71. 72. 73. 74.
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A. Grzegorczyk, An Outline ofMathematical Logic. Fundamental Results and Notions explained with all Details. Translated from Polish. 1974 ISBN 90-277-0359-0; Pb 90-277-0447-3 F. von Kutschera, Philosophy ofLanguage. 1975 ISBN 90-277-0591-7 J. Manninen and R. Tuomela (eds.), Essays on Explanation and Understanding. Studies in the Foundations of Humanities and Social Sciences . 1976 ISBN 90-277-0592-5 J. Hintikka (ed.), RudolfCarnap, Logical Empiricist. Materials and Perspectives. 1975 ISBN 90-277-0583-6 M. Capek (ed.), The Concepts of Space and Time. Their Structure and Their Development. [Boston Studies in the Philosophy of Science, Vol. XXII] 1976 ISBN 90-277-0355-8; Pb 90-277-0375-2 J. Hintikka and U. Remes, The Method of Analysis . Its Geometrical Origin and Its General Significance. [Boston Studies in the Philosophy of Science, Vol. XXV] 1974 ISBN 90-277-0532-1; Pb 90-277-0543-7 J. E. Murdoch and E. D. Sylla (eds.), The Cultural Context of Medieval Learning. [Boston Studies in the Philosophy of Science, Vol. XXVI] 1975 ISBN 90-277-0560-7; Pb 90-277-0587-9 S. Amsterdamski, Between Experience and Metaphysics . Philosophical Problems of the Evolution of Science. [Boston Studies in the Philosophy of Science, Vol. XXXV] 1975 ISBN 90-277-0568-2; Pb 90-277-0580-1 P. Suppes (ed.), Logic and Probability in Quantum Mechanics. 1976 ISBN 90-277-0570-4; Pb 9O-277-1200-X H. von Helmholtz: Epistemological Writings . The Paul Hertz I Moritz Schlick Centenary Edition of 1921 with Notes and Commentary by the Editors. Newly translated from German by M. F. Lowe. Edited, with an Introduction and Bibliography, by R. S. Cohen and Y. Elkana. [Boston Studies in the Philosophy of Science, Vol. XXXVII] 1975 ISBN 9O-277-029O-X; Pb 90-277-0582-8 J. Agassi, Science in Flux. [Boston Studies in the Philosophy of Science, Vol. XXVIII] 1975 ISBN 90-277-0584-4; Pb 90-277-0612-2 S. G. Harding (ed.), Can Theories Be Refuted? Essays on the Duhem-Quine Thesis. 1976 ISBN 90-277-0629-8; Pb 90-277-0630-1 S. Nowak, Methodology ofSociological Research. General Problems. 1977 ISBN 90-277-0486-4 J. Piaget, J.-B. Grize, A. Szeminsska and V. Bang, Epistemology and Psychology ofFunctions. Translated from French. 1977 ISBN 90-277-0804-5 M. Grene and E. Mendelsohn (eds.), Topics in the Philosophy ofBiology. [Boston Studies in the Philosophy of Science, Vol. XXVII] 1976 ISBN 9O-277-0595-X; Pb 90-277-0596-8 E. Fischbein, The Intuitive Sources ofProbabilistic Thinking in Children. 1975 ISBN 90-277-0626-3; Pb 90-277-1190-9 E. W. Adams, The Logic of Conditionals. An Application of Probability to Deductive Logic. 1975 ISBN 9O-277-0631-X M. Przelecki and R. Wojcicki (eds.), Twenty-Five Years of Logical Methodology in Poland. Translated from Polish. 1976 ISBN 90-277-0601-8 J. Topolski, The Methodology ofHistory. Translated from Polish by O. Wojtasiewicz. 1976 ISBN 9O-277-0550-X A. Kasher (ed.), Language in Focus: Foundations , Methods and Systems. Essays dedicated to Yehoshua Bar-Hillel. [Boston Studies in the Philosophy of Science, Vol. XLIII] 1976 ISBN 90-277-0644-1; Pb 90-277-0645-X
SYNTHESE LIBRARY 90. 91. 92. 93. 94. 95. 96. 97. 98. 99.
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J. Hintikka, The Intentions ofIntentionality and Other New Models for Modalities. 1975 ISBN 90-277-0633-6; Pb 90-277-0634-4 W. Stegmiiller, Collected Papers on Epistemology, Philosophy of Science and History of Philosophy. 2 Volumes. 1977 Set ISBN 90-277-0767-7 D. M. Gabbay, Investigations in Modal and Tense Logics with Applications to Problems in ISBN 90-277-0656-5 Philosophy and Linguistics. 1976 R. J. Bogdan, Local Induction . 1976 ISBN 90-277-0649-2 S. Nowak, Understanding and Prediction. Essays in the Methodology of Social and Behavioral Theories. 1976 ISBN 90-277-0558-5; Pb 90-277-1199-2 P.Mittelstaedt, Philosophical Problems ofModern Physics. [Boston Studies in the Philosophy of Science, Vol. XVIII] 1976 ISBN 90-277-0285-3; Pb 90-277-0506-2 G. Holton and W. A. Blanpied (eds .), Science and Its Public: The Changing Relationship. [Boston Studies in the Philosophy of Science, Vol. XXXIII] 1976 ISBN 90-277-0657-3; Pb 90-277-0658-1 ISBN 90-277-0671-9 M. Brand and D. Walton (eds.), Action Theory. 1976 P. Gochet, Outline ofa Nominalist Theory ofPropositions. An Essay in the Theory of Meaning and in the Philosophy of Logic. 1980 ISBN 90-277-1031-7 R. S. Cohen, P. K. Feyerabend, and M. W. Wartofsky (eds.), Essays in Memory ofImre Lakatos. [Boston Studies in the Philosophy of Science, Vol. XXXIX] 1976 ISBN 90-277-0654-9; Pb 90-277-0655-7 R. S. Cohen and J. J. Stachel (eds .), Selected Papers of Leon Rosenfield. [Boston Studies in the Philosophy of Science, Vol.XXI] 1979 ISBN 90-277-0651-4; Pb 90-277-0652-2 R. S. Cohen, C. A. Hooker, A. C. Michalos and J. W. van Evra (eds.), PSA 1974. Proceedings ofthe 1974 Biennial Meeting ofthe Philosophy ofScience Association. [Boston Studies in the Philosophy of Science, Vol. XXXII] 1976 ISBN 90-277-0647-6; Pb 90-277-0648-4 Y. Fried and J. Agassi, Paranoia. A Study in Diagnosis. [Boston Studies in the Philosophy of Science, Vol. L] 1976 ISBN 90-277-0704-9; Pb 90-277-0705-7 M. Przelecki, K. Szaniawski and R. Wojcicki (eds.), Formal Methods in the Methodology of Empirical Sciences. 1976 ISBN 90-277-0698-0 J. M. Vickers, Beliefand Probability. 1976 ISBN 90-277-0744-8 K. H. Wolff, Surrender and Catch. Experience and Inquiry Today. [Boston Studies in the Philosophy of Science, Vol.LI] 1976 ISBN 90-277-0758-8; Pb 90-277-0765-0 K. Kosik, Dialectics ofthe Concrete. A Study on Problems of Man and World. [Boston Studies in the Philosophy of Science, Vol.LII] 1976 ISBN 90-277-0761-8; Pb 90-277-0764-2 N. Goodman, The Structure of Appearance. 3rd ed. with an Introduction by G. Hellman. [Boston Studies in the Philosophy of Science, Vol. LIII] 1977 ISBN 90-277-0773-1; Pb 9O-277-0774-X K. Ajdukiewicz, The Scientific World-Perspective and Other Essays. 1931-1963. Translated from Polish. Edited and with an Introduction by J. Giedymin. 1978 ISBN 90-277-0527-5 R. L. Causey, Unity ofScience. 1977 ISBN 90-277-0779-0 R. E. Grandy, Advanced Logicfor Applications. 1977 ISBN 90-277-0781-2 R. P. McArthur, Tense Logic. 1976 ISBN 90-277-0697-2 L. Lindahl, Position and Change. A Study in Law and Logic. Translated from Swedish by P. Needham. 1977 ISBN 90-277-0787-1 ISBN 90-277-0810-X R. Tuomela, Dispositions. 1978 H. A. Simon, Models ofDiscovery and Other Topics in the Methods ofScience. [Boston Studies in the Philosophy of Science, Vol.LIV] 1977 ISBN 90-277-0812-6; Pb 90-277-0858-4
SYNTHESE LIBRARY 115. R. D. Rosenkrantz, Inference, Method and Decision. Towards a Bayesian Philosophy of Science. 1977 ISBN 90-277-0817-7; Pb 90-277-0818-5 116. R. Tuomela, Human Action and Its Explanation . A Study on the Philosophical Foundationsof Psychology. 1977 ISBN 9O-277-0824-X 117. M. Lazerowitz, The Language ofPhilosophy. Freud and Wittgenstein. [Boston Studies in the Philosophy of Science, Vol. LVjI977 ISBN 90-277-0826-6; Pb 90-277-0862-2 118. Not published 119. J. Pelc (ed.), Semiotics in Poland, 1894-1969. Translated from Polish. 1979 ISBN 90-277-0811 -8 120. 1. Porn, Action Theory and Social Science. Some Formal Models. 1977 ISBN 90-277-0846-0 121. J. Margolis, Persons and Mind . The Prospects of Nonreductive Materialism. [Boston Studies in the Philosophy of Science, Vol. LVIIjI977 ISBN 90-277-0854-1; Pb 90-277-0863-0 122. J. Hintikka, 1. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical ISBN 90-277-0879-7 Logic. 1979 123. T. A. F. Kuipers, Studies in Inductive Probability and Rational Expectation. 1978 ISBN 90-277-0882-7 124. E. Saarinen, R. Hilpinen, 1. Niiniluoto and M. P. Hintikka (eds.), Essays in Honour ofJaakko ISBN 90-277-0916-5 Hintikka on the Occasion ofHis 50th Birthday. 1979 125. G. Radnitzky and G. Andersson (eds.), Progress and Rationality in Science. [Boston Studies in the Philosophy of Science, Vol.LVIII] 1978 ISBN 90-277-0921-1; Pb 9O-277-0922-X 126. P. Mittelstaedt, Quantum Logic. 1978 ISBN 90-277-0925-4 127. K. A. Bowen, Model Theory for Modal Logic. Kripke Models for Modal Predicate Calculi. 1979 ISBN 90-277-0929-7 128. H. A. Bursen, Dismantling the Memory Machine. A Philosophical Investigation of Machine Theories of Memory. 1978 ISBN 90-277-0933-5 129. M. W. Wartofsky, Models. Representation and the Scientific Understanding. [Boston Studies ISBN 90-277-0736-7; Pb 90-277-0947-5 in the Philosophy of Science, Vol. XLVIII] 1979 130. D. Ihde, Technics and Praxis . A Philosophy of Technology. [Boston Studies in the Philosophy of Science, Vol. XXIV] 1979 ISBN 90-277-0953-X; Pb 90-277-0954-8 131. J. J. Wiatr (ed.), Polish Essays in the Methodology of the Social Sciences. [Boston Studies in the Philosophy of Science, Vol. XXIX] 1979 ISBN 90-277-0723-5; Pb 90-277-0956-4 ISBN 90-277-0958-0 132. W. C. Salmon (ed.), Hans Reichenbach : Logical Empiricist. 1979 133. P. Bieri, R.-P. Horstmann and L. Kriiger (eds.), Transcendental Arguments in Science. Essays in Epistemology. 1979 ISBN 90-277-0963-7; Pb 90-277-0964-5 134. M. Markovic and G. Petrovic (eds.), Praxis. Yugoslav Essays in the Philosophy and Methodology of the Social Sciences. [Boston Studies in the Philosophy of Science, Vol.XXXVI] 1979 ISBN 90-277-0727-8; Pb 90-277-0968-8 135. R. W6jcicki, Topics in the Formal Methodology ofEmpirical Sciences . Translated from Polish. 1979 ISBN 9O-277-1004-X 136. G. Radnitzky and G. Andersson (eds.), The Structure and Development of Science. [Boston Studies in the Philosophy of Science, Vol. LIX] 1979 ISBN 90-277-0994-7; Pb 90-277-0995-5 137. J. C. Webb, Mechan ism, Mentalism and Metamathematics. An Essay on Finitism. 1980 ISBN 90-277-1046-5 138. D. F. Gustafson and B. L. Tapscott (eds.), Body, Mind and Method. Essays in Honor of Virgil C. Aldrich. 1979 ISBN 90-277-1013-9 139. L. Nowak, The Structure ofIdealization. Towards a Systematic Interpretation of the Marxian Idea of Science. 1980 ISBN 90-277-1014-7
SYNTHESE LIBRARY 140. C. Perelman, The New Rhetoric and the Human ities. Essays on Rhetoric and Its Applications. Translated from French and German. With an Introduction by H. Zyskind. 1979 ISBN 90-277-1018-X; Pb 90-277-1019-8 141. W. Rabinowicz, Universalizability. A Study in Morals and Metaphysics. 1979 ISBN 90-277-1020-2 142. C. Perelman, Justice, Law and Argument. Essays on Moral and Legal Reasoning. Translated from French and German. With an Introduction by H.J. Berman. 1980 ISBN 90-277-1089-9; Pb 90-277-1090-2 143. S. Kanger and S. Ohman (eds.), Philosophy and Grammar. Papers on the Occasion of the Quincentennial of Uppsala University. 1981 ISBN 90-277-1091-0 144. T. Pawlowski, Concept Formation in the Humanities and the Social Sciences. 1980 ISBN 90-277-1096-1 145. J. Hintikka, D. Gruender and E. Agazzi (eds.), Theory Change, Ancient Axiomatics and Galileo 's Methodology. Proceedings of the 1978Pisa Conference on the History and Philosophy of Science, Volume I. 1981 ISBN 90-277-1126-7 146. J. Hintikka, D. Gruender and E. Agazzi (eds.), Probabilistic Thinking, Thermodynamics, and the Interaction of the History and Philosophy of Science. Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science, Volume 11.1981 ISBN 90-277-1127-5 147. U. Monnich (ed.), Aspects ofPhilosophical Logic. Some Logical Forays into Central Notions of Linguistics and Philosophy. 1981 ISBN 90-277-1201-8 148. D. M. Gabbay, Semanticallnvestigations in Heyting's Intuitionistic Logi c. 1981 ISBN 90-277-1202-6 149. E. Agazzi (ed.), Modern Logic -A Survey. Historical, Philosophical, and Mathematical Aspects of Modem Logic and Its Applications. 1981 ISBN 90-277-1137-2 150. A. F. Parker-Rhodes, The Theory ofIndistinguishables. A Search for Explanatory Principles below the Level of Physics. 1981 ISBN 90-277-1214-X 151. J. C. Pitt, Pictures. Images, and Conceptual Change. An Analysis of Wilfrid Sellars' Philosophy of Science. 1981 ISBN 90-277-1276-X; Pb 90-277-1277-8 152. R. Hilpinen (ed.), New Studies in Deontic Logic. Norms, Actions, and the Foundations of Ethics. 1981 ISBN 90-277-1278-6; Pb 90-277-1346-4 153. C. Dilworth, Scientific Progress. A Study Concerning the Nature of the Relation between Successive Scientific Theories. 3rd rev. ed., 1994 ISBN 0-7923-2487-0; Pb 0-7923-2488-9 154. D. Woodruff Smith and R. McIntyre, Husserl and Intentionality. A Study of Mind, Meaning, and Language. 1982 ISBN 90-277-1392-8; Pb 90-277-1730-3 155. R. J. Nelson, The Logic ofMind. 2nd. ed., 1989 ISBN 90-277-2819-4; Pb 90-277-2822-4 156. J. F. A. K. van Benthem, The Logic ofTime. A Model-Theoretic Investigation into the Varieties of Temporal Ontology, and Temporal Discourse. 1983; 2nd ed., 1991 ISBN 0-7923-1081-0 157. R. Swinburne (ed.), Space, Time and Causality. 1983 ISBN 90-277-1437-1 158. E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics. Ed. by R. D. Rozenkrantz. 1983 ISBN 90-277-1448-7; Pb (1989) 0-7923-0213-3 ISBN 90-277-1465-7 159. T. Chapman, Time: A Philosophical Analysis. 1982 160. E. N. Zalta, Abstract Objects: An Introduction to Axiomatic Metaphysics. 1983 ISBN 90-277-1474-6 161. S. Harding and M. B. Hintikka (eds.), Discovering Reality. Feminist Perspectives on Epistemology, Metaphysics, Methodology, and Philosophy of Science. 1983 ISBN 90-277-1496-7; Pb 90-277-1538-6 162. M. A. Stewart (ed.), Law, Morality and Rights. 1983 ISBN 90-277-1519-X
SYNTHESE LIBRARY 163. D. Mayr and G. Siissmann (eds .), Space. Time. and Mechanics. Basic Structures of a Physical Theory. 1983 ISBN 90-277-1525-4 164. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. I: Elements of Classical Logic. 1983 ISBN 90-277-1542-4 165. D. Gabbay and F. Guenthner (eds.), Handbook ofPhilosophical Logic . Vol. II: Extensions of Classical Logic. 1984 ISBN 90-277-1604-8 166. D. Gabbay and F. Guenthner (eds.), Handbook ofPhilosophical Logic. Vol.III: Alternativeto Classical Logic. 1986 ISBN 90-277-1605-6 167. D. Gabbay and F. Guenthner (eds .), Handbook ofPhilosophical Logic. Vol. IV: Topics in the Philosophy of Language. 1989 ISBN 90-277-1606-4 168. A. J. I. Jones, Communication and Meaning. An Essay in Applied Modal Logic. 1983 ISBN 90-277-1543-2 169. M. Fitting, ProofMethods for Modal and Intuitionistic Logics. 1983 ISBN 90-277-1573-4 170. J. Margolis, Culture and Cultural Entities. Towarda New Unity of Science. 1984 ISBN 90-277-1574-2 171. R. Tuomela, A Theory ofSocial Action. 1984 ISBN 90-277-1703-6 172. J. J. E. Gracia, E. Rabossi, E. Villanueva and M. Dascal (eds.), Philosophical Analysis in Latin ISBN 90-277-1749-4 America. 1984 173. P. Ziff, Epistemic Analysis. A Coherence Theory of Knowledge. 1984 ISBN 90-277-1751-7 174. P. Ziff, Antiaesthetics. An Appreciation of the Cow with the Subtile Nose. 1984 ISBN 90-277-1773-7 175. W. Balzer, D. A. Pearce, and H.-J. Schmidt (eds.), Reduction in Science. Structure, Examples. Philosophical Problems. 1984 ISBN 90-277-1811-3 176. A. Peczenik, L. Lindahl and B. van Roennund (eds .), Theory ofLegal Science . Proceedingsof the Conference on Legal Theory and Philosophy of Science (Lund, Sweden, December 1983). 1984 ISBN 90-277-1834-2 177. I. Niiniluoto, Is Science Progressive? 1984 ISBN 90-277-1835-0 178. B. K. Matilal and J. L. Shaw (eds.), Analytical Philosophy in Comparative Perspective. Exploratory Essays in Current Theories and Classical Indian Theories of Meaning and Reference. 1985 ISBN 90-277-1870-9 179. P. Kroes, Time: Its Structure and Role in Physical Theories. 1985 ISBN 90-277-1894-6 180. J. H. Fetzer, Sociobiology and Epistemology. 1985 ISBN 90-277-2005-3; Pb 90-277-2006-1 181. L. Haaparanta and J. Hintikka (eds.), Frege Synthesized. Essays on the Philosophical and Foundational Work of Gottlob Frege. 1986 ISBN 90-277-2126-2 182. M. Detlefsen, Hilbert's Program. An Essay on Mathematical Instrumentalism. 1986 ISBN 90-277-2151-3 183. J. L. Golden and J. J. Pilotta (eds.), Practical Reasoning in Human Affairs. Studies in Honor of Chaim Perelman. 1986 ISBN 90-277-2255-2 184. H. Zandvoort, Models ofScientific Development and the Case ofNuclear Magnetic Resonance. 1986 ISBN 90-277-2351-6 ISBN 90-277-2354-0 185. I. Niiniluoto, Truthlikeness. 1987 186. W. Balzer, C. U. Moulines and J. D. Sneed, An Architectonic for Science. The Structuralist Program. 1987 ISBN 90-277-2403-2 187. D. Pearce, Roads to Commensurability. 1987 ISBN 90-277-2414-8 188. L. M. Vaina (ed.), Matters of Intelligence. Conceptual Structures in Cognitive Neuroscience. 1987 ISBN 90-277-2460-1
SYNTHESE LIBRARY 189. H. Siegel, Relativism Refuted. A Critique of Contemporary Epistemological Relativism. 1987 ISBN 90-277-2469-5 190. W. Callebaut and R. Pinxten, Evolutionary Epistemology. A Multiparadigm Program, with a Complete Evolutionary Epistemology Bibliograph. 1987 ISBN 90-277-2582-9 191. J. Kmita, Problems in Historical Epistemology. 1988 ISBN 90-277-2199-8 192. J. H. Fetzer (ed.), Probability and Causality. Essays in Honor of Wesley C. Salmon, with an Annotated Bibliography. 1988 ISBN 90-277-2607-8; Pb 1-5560-8052-2 193. A. Donovan, L. Laudan and R. Laudan (eds.), Scrutinizing Science . Empirical Studies of Scientific Change. 1988 ISBN 90-277-2608-6 194. H.R. Otto and I .A. Tuedio (eds.), Perspectives on Mind. 1988 ISBN 90-277-264O-X 195. D. Batens and J.P. van Bendegem (eds.), Theory and Experiment. Recent Insights and New Perspectives on Their Relation. 1988 ISBN 90-277-2645-0 ISBN 90-277-2648-5 196. J. Osterberg, Selfand Others. A Study of Ethical Egoism. 1988 197. D.H. Helman (ed.), Analogical Reasoning. Perspectives of Artificial Intelligence, Cognitive Science, and Philosophy. 1988 ISBN 90-277-2711-2 198. J. Wolenski, Logic and Philosophy in the Lvov-Warsaw School. 1989 ISBN 90-277-2749-X 199. R. Wojcicki, Theory ofLogical Calculi. Basic Theory of Consequence Operations. 1988 ISBN 90-277-2785-6 200. J. Hintikka and M.B. Hintikka, The Logic of Epistemology and the Epistemology of Logic. Selected Essays. 1989 ISBN 0-7923-0040-8; Pb 0-7923-0041-6 201. E. Agazzi (ed.), Probability in the Sciences. 1988 ISBN 90-277-2808-9 ISBN 90-277-2814-3 202. M. Meyer (ed.), From Metaphysics to Rhetoric. 1989 203. R.L. Tieszen, Mathematical Intuition. Phenomenology and Mathematical Knowledge. 1989 ISBN 0-7923-0131-5 204. A. Melnick, Space. Time. and Thought in Kant. 1989 ISBN 0-7923-0135-8 205. D.W. Smith, The Circle ofAcquaintance. Perception, Consciousness, and Empathy. 1989 ISBN 0-7923-0252-4 206. M.H. Salmon (ed.), The Philosophy of Logical Mechanism . Essays in Honor of Arthur W. Burks. With his Responses, and with a Bibliography of Burk's Work. 1990 ISBN 0-7923-0325-3 207. M. Kusch, Language as Calculus vs. Language as Universal Med ium. A Study in Husserl, Heidegger, and Gadamer. 1989 ISBN 0-7923-0333-4 208. T.C. Meyering, Historical Roots of Cognitive Science. The Rise of a Cognitive Theory of Perception from Antiquity to the Nineteenth Century. 1989 ISBN 0-7923-0349-0 209. P. Kosso, Observability and Observation in Physical Science. 1989 ISBN 0-7923-0389-X 210. J. Krnita, Essays on the Theory ofScientific Cognition . 1990 ISBN 0-7923-0441-1 211. W. Sieg (ed .), Acting and Reflecting. The Interdisciplinary Tum in Philosophy. 1990 ISBN 0-7923-0512-4 212. J. Karpinski, Causality in Sociological Research. 1990 ISBN 0-7923-0546-9 213. H.A. Lewis (ed.), Peter Geach : Philosophical Encounters. 1991 ISBN 0-7923-0823-9 214. M. Ter Hark, Beyond the Inner and the Outer. Wittgenstein's Philosophy of Psychology. 1990 ISBN 0-7923-0850-6 215. M. Gosselin, Nominalism and Contemporary Nominalism. Ontological and Epistemological Implications of the Work ofW.V.O. Quine and ofN. Goodman. 1990 ISBN 0-7923-0904-9 216. J.H. Fetzer, D. Shatz and G. Schlesinger (eds.), Definitions and Definability. Philosophical Perspectives. 1991 ISBN 0-7923-1046-2 217. E. Agazzi and A. Cordero (eds.), Philosophy and the Origin and Evolution of the Universe. 1991 ISBN 0-7923-1322-4
SYNTHESE LIBRARY 218. M. Kusch, Foucault's Strata and Fields. An Investigation into Archaeological and Genealogical Science Studies. 1991 ISBN 0-7923-1462-X ISBN 0-7923-1495-6 219. C.J. Posy, Kant's Philosophy ofMathematics. Modem Essays. 1992 220. G. Van de Vijver, New Perspectives on Cybernetics. Self-Organization, Autonomy and Connectionism.1992 ISBN 0-7923-1519-7 ISBN 0-7923-1566-9 221. J.C. Nyfri, Tradition and Individuality. Essays. 1992 222. R. Howell, Kant 's Transcendental Deduction . An Analysis of Main Themes in His Critical Philosophy. 1992 ISBN 0-7923-1571-5 223. A. Garda de la Sienra, The Logical Foundations of the Marxian Theory ofValue. 1992 ISBN 0-7923-1778-5 224. D.S. Shwayder, Statement and Referent. An Inquiry into the Foundations of Our Conceptual Order. 1992 ISBN 0-7923-1803-X 225. M. Rosen, Problems of the Hegelian Dialectic. Dialectic Reconstructed as a Logic of Human Reality. 1993 ISBN 0-7923-2047-6 226. P. Suppes, Models and Methods in the Philosophy ofScience: Selected Essays . 1993 ISBN 0-7923-2211-8 227. R. M. Dancy (ed.), Kant and Critique : New Essays in Honor ofW. H. Werkmeister. 1993 ISBN 0-7923-2244-4 228. J. Wolenski (ed.), Philosophical Logic in Poland. 1993 ISBN 0-7923-2293-2 229. M. De Rijke (ed.), Diamonds and Defaults. Studies in Pure and Applied Intensional Logic. 1993 ISBN 0-7923-2342-4 230. B.K. Matilal and A. Chakrabarti (eds.), Knowingfrom Words. Western and Indian Philosophical Analysis of Understanding and Testimony. 1994 ISBN 0-7923-2345-9 231. S.A. Kleiner, The Logic ofDiscovery. A Theory of the Rationality of Scientific Research. 1993 ISBN 0-7923-2371-8 232. R. Festa, Optimum Inductive Methods. A Study in Inductive Probability, Bayesian Statistics, and Verisimilitude. 1993 ISBN 0-7923-2460-9 233. P.Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 1: Probability and Probabilistic Causality. 1994 ISBN 0-7923-2552-4 234. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 2: Philosophy of Physics, Theory Structure, and Measurement Theory. 1994 ISBN 0-7923-2553-2 235. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 3: Language, Logic, and Psychology. 1994 ISBN 0-7923-2862-0 Set ISBN (Vols 233-235) 0-7923-2554-0 236. D. Prawitz and D. WesterstlThl (eds.), Logic and Philosophy of Science in Uppsala. Papers from the 9th International Congress of Logic, Methodology, and Philosophy of Science. 1994 ISBN 0-7923-2702-0 237. L. Haaparanta (ed.), Mind . Meaning and Mathematics. Essays on the Philosophical Views of Husserl and Frege. 1994 ISBN 0-7923-2703-9 238. J. Hintikka (ed.), Aspects ofMetaphor. 1994 ISBN 0-7923-2786-1 239. B. McGuinness and G. Oliveri (eds.), The Philosophy ofMichael Dummett. With Replies from Michael Dummett. 1994 ISBN 0-7923-2804-3 240. D. Jamieson (ed.), Language. Mind. and Art. Essays in Appreciation and Analysis, In Honor of Paul Ziff. 1994 ISBN 0-7923-2810-8 241. G. Preyer, F. Siebelt and A. U1fig (eds.), Language, Mind and Epistemology. On Donald Davidson's Philosophy. 1994 ISBN 0-7923-2811-6 242. P. Ehrlich (ed.), Real Numbers. Generalizations ofthe Reals, and Theories ofContinua. 1994 ISBN 0-7923-2689-X
SYNTHESE LIBRARY 243. G. Debrock and M. Hulswit (eds.), Living Doubt. Essays concerning the epistemology of Charles Sanders Peirce. 1994 ISBN 0-7923-2898-1 244. J. Srzednicki, To Know or Not to Know. Beyond Realism and Anti-Realism. 1994 ISBN 0-7923-2909-0 245. R. Egidi (ed.), Wittgenstein: Mind and Language. 1995 ISBN 0-7923-3171-0 246. A. Hyslop, Other Minds . 1995 ISBN 0-7923-3245-8 247. L. Palos and M. Masuch (eds .), Applied Logic: How, What and Why. Logical Approaches to Natural Language. 1995 ISBN 0-7923-3432-9 248. M. Krynicki, M. Mostowski and L.M. Szczerba (eds.), Quantifiers: Logics , Models and ComISBN 0-7923-3448-5 putation . VolumeOne: Surveys. 1995 249. M. Krynicki, M. Mostowski and L.M. Szczerba (eds.), Quantifiers: Logics , Models and ComISBN 0-7923-3449-3 putation. VolumeTwo: Contributions. 1995 Set ISBN (Vols 248 + 249) 0-7923-3450-7 250. R.A. Watson, Representational Ideas from Plato to Patricia Churchland. 1995 ISBN 0-7923-3453-1 251. J. Hintikka (ed.), From Dedekind to Giidel. Essays on the Development of the Foundations of Mathematics. 1995 ISBN 0-7923-3484-1 252. A. Wisniewski, The Posing ofQuestions. Logical Foundations of Erotetic Inferences. 1995 ISBN 0-7923-3637-2 253. J. Peregrin, Doing Worlds with Words. Formal Semantics without Formal Metaphysics. 1995 ISBN 0-7923-3742-5 254. LA. Kieseppa, Truthlikenessfor Multidimensional, Quantitative Cognitive Problems . 1996 ISBN 0-7923-4005-1 255. P. Hugly and C. Sayward: Intensionality and Truth. An Essay on the Philosophy of A.N. Prior. 1996 ISBN 0-7923-4119-8 256. L. Hankinson Nelson and J. Nelson (eds.): Feminism, Science, and the Philosophy ofScience. 1997 ISBN 0-7923-4162-7 257. P.1. Bystrov and V.N.Sadovsky (eds.): Philosophical Logic and Logical Philosophy. Essays in Honour of Vladimir A. Smirnov. 1996 ISBN 0-7923-4270-4 258. A.E. Andersson and N-E. Sahlin (eds.): The Complexity of Creativity. 1996 ISBN 0-7923-4346-8 259. M.L. Dalla Chiara, K. Doets, D. Mundici and J. van Benthem (eds.): Logic and Scientific Methods. VolumeOne of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995. 1997 ISBN 0-7923-4383-2 260. M.L. Dalla Chiara, K. Doets, D. Mundici and J. van Benthem (eds.): Structures and Norms in Science. Volume Two of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence, August 1995. 1997 ISBN 0-7923-4384-0 Set ISBN (Vols 259 + 260) 0-7923-4385-9 261. A. Chakrabarti: Denying Existence. The Logic, Epistemology and Pragmatics of Negative Existentials and Fictional Discourse. 1997 ISBN 0-7923-4388-3 262. A. Biletzki: Talking Wolves. Thomas Hobbes on the Language of Politics and the Politics of Language. 1997 ISBN 0-7923-4425-1 263. D. Nute (ed.): Defeasible Deontic Logic . 1997 ISBN 0-7923-4630-0 ISBN 0-7923-4747-X 264. U. Meixner: Axiomatic Formal Ontology. 1997 265. I. Brinck: The Indexical 'I' , The First Person in Thought and Language. 1997 ISBN 0-7923-4741-2 266. G. Holmstrom-Hintikka and R. Tuomela (OOs.): Contemporary Action Theory. Volume I : Individual Action. 1997 ISBN 0-7923-4753-6; Set: 0-7923-4754-4
SYNTHESE LIBRARY 267. G. Holmstrom-Hintikka and R. Tuomela (OOs.): Contemporary Action Theory. Volume 2: Social Action. 1997 ISBN 0-7923-4752-8; Set: 0-7923-4754-4 268. B.-C. Park: Phenomenological Aspects ofWittgenstein's Philosophy. 1998 ISBN 0-7923-4813-3 269. J. Pasniczek: The Logic ofIntentional Objects. A Meinongian Versionof Classical Logic. 1998 Hb ISBN 0-7923-4880-X; Pb ISBN 0-7923-5578-4 270. P.W. Humphreys and J.H. Fetzer (OOs.): The New Theory of Reference. Kripke, Marcus, and Its Origins. 1998 ISBN 0-7923-4898-2 271. K. Szaniawski, A. Chmielewski and J. Wolenski (008.): On Science, Inference, Information and Decision Making . Selected Essays in the Philosophy of Science. 1998 ISBN 0-7923-4922-9 272. G.H. von Wright: In the Shadow ofDescartes. Essays in the Philosophy of Mind. 1998 ISBN 0-7923-4992-X 273. K. Kijania-Placek and J. Wolenski (OOs.): The Lvov-Warsaw School and Contemporary Philosophy. 1998 ISBN 0-7923-5105-3 274. D. Dedrick: Naming the Rainbow. Colour Language, Colour Science, and Culture. 1998 ISBN 0-7923-5239-4 275. L. Albertazzi (00.): Shapes ofForms. From Gestalt Psychology and Phenomenology to Ontology and Mathematics. 1999 ISBN 0-7923-5246-7 276. P. Fletcher: Truth, Proofand Infinity. A Theory of Constructions and Constructive Reasoning. 1998 ISBN 0-7923-5262-9 277. M. Fitting and R.L. Mendelsohn (OOs.): First -Order Modal Logic. 1998 Hb ISBN 0-7923-5334-X; Pb ISBN 0-7923-5335-8 278. J.N. Mohanty: Logic, Truth and the Modalitiesfrom a Phenomenological Perspective. 1999 ISBN 0-7923-5550-4 279. T. Placek: Mathematical lntiutionism and lntersubjectivity. A Critical Exposition of Arguments for Intuitionism. 1999 ISBN 0-7923-5630-6 280. A. Cantini, E. Casari and P. Minari (eds.): Logic and Foundations ofMathematics. 1999 ISBN 0-7923-5659-4 set ISBN 0-7923-5867-8 281. M.L. Dalla Chiara, R. Giuntini and F. Laudisa (OOs.): Language, Quantum, Music. 1999 ISBN 0-7923-5727-2; set ISBN 0-7923-5867-8 282. R. Egidi (00.): In Search ofa New Humanism . The Philosophy of Georg Hendrik von Wright. 1999 ISBN 0-7923-5810-4 283. F. Vollmer: Agent Causality. 1999 ISBN 0-7923-5848-1 284. J. Peregrin (00.): Truth and Its Nature (if Any). 1999 ISBN 0-7923-5865-1 285. M. De Caro (ed.): Interpretations and Causes. New Perspectives on Donald Davidson's Philosophy. 1999 ISBN 0-7923-5869-4 286. R. Murawski: Recursive Functions and Metamathematics. Problems of Completeness and Decidability, GOdel's Theorems. 1999 ISBN 0-7923-5904-6 287. T.A.F. Kuipers: From Instrumentalism to Constructive Realism. On Some Relations between Confirmation, Empirical Progress, and Truth Approximation. 2000 ISBN 0-7923-6086-9 288. G. Holmstrorn-Hintikka (00.): Medieval Philosoph y and Modern Times. 2000 ISBN 0-7923-6102-4 289. E. Grosholz and H. Breger (eds.): The Growth ofMathematical Knowledge. 2000 ISBN 0-7923-6151-2
SYNTHESE LIBRARY 290. 291. 292. 293. 294. 295. 296. 297. 298. 299. 300. 301. 302. 303.
G. Sommaruga: History and Philosophy of Constructive Type Theory . 2000 ISBN 0-7923-6180-6 J. Gasser (ed.) : A Boole Anthology. Recent and Classical Studies in the Logic of George Boole . 2000 ISBN 0-7923-6380-9 V.F. Hendricks, S.A . Pedersen and K.F. Jergensen (eds.): Proof Theory. History and Philosophical Significance. 2000 ISBN 0-7923-6544-5 W.L. Craig: The Tensed Theory of Time. A Critical Examination. 2000 ISBN 0-7923-6634-4 W.L. Craig : The Tenseless Theory of Time. A Critical Examination. 2000 ISBN 0-7923-6635-2 L. Albertazzi (ed.): The Dawn ofCognitive Science. Early European Contributors. 2001 ISBN 0-7923-6799-5 G. Forrai: Reference. Truth and Conceptual Schemes. A Defense of Internal Realism. 2001 ISBN 0-7923-6885-1 V.F. Hendricks, S.A. Pedersen and K.F. Jergensen (eds.): Probability Theory. Philosophy, Recent History and Relations to Science. 2001 ISBN 0-7923-6952-1 M. Esfeld: Holism in Philosophy ofMind and Philosophy ofPhysics. 2001 ISBN 0-7923-7003-1 E.C. Steinhart: The Logic ofMetaphor. Analogous Parts of Possible Worlds. 2001 ISBN 0-7923-7OO4-X To be published. T.A.F. Kuipers: Structures in Science Heuristic Patterns Based on Cognitive Structures. An Advanced Textbook in Neo-Classical Philosophy of Science. 2001 ISBN 0-7923-7117-8 G. Ho~ and S.S. Rakover (eds.): Explanation. Theoretical Approaches and Applications. 2001 ISBN 1-4020-0017-0 G. Holmstrom-Hintikka, S. Lindstrom and R. Sliwinski (eds.): Collected Papers ofStig Kanger with Essays on his Life and Work. Vol. I. 2001 ISBN 1-4020-0021-9 ; Pb ISBN 1-4020 -0022-7
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