On the Foundations of Logical Modality

On the Foundations of Logical Modality TUOMAS E. TAHKO

ABSTRACT Kit Fine has suggested that most types of modality, such as conceptual and logical modality, can be reduced to metaphysical modality. In this paper we will examine the grounds of logical modality and specifically how we could restrict logical modality to metaphysical modality, given the Finean understanding of metaphysical modality. It will be suggested that logical modality could only be independent of metaphysical modality if logic has modal content before it has been applied, that is, we can only make sense of logical concepts through their application to the world. An analysis of some key logical concepts and principles, such as the law of non-contradiction, will be offered in support of this view. Finally, we will consider why we take logical validity to be a proof of rationality – it appears that rationality as well is grounded in the world as opposed to language or grammar.

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What do we mean when we say that something is logically possible or logically necessary? Presumably, anything that does not violate the laws of logic is logically possible, whereas logical principles such as the law of identity are logically necessary. Accordingly, logical modality could be said to concern logical validity: anything that is logically valid is logically possible. Generally, logic is said to be a study which aims to distinguish between valid and invalid arguments. A logic for language L aims to prove all the valid arguments in L. It should be sound, namely all proved arguments should be valid, and it should be complete, namely all the valid arguments in L should have proofs. Logical necessity can then be defined with the help of a valuation function: A is necessarily true (i.e. valid) if and only if A is true in every logically possible world in which A exists. This is how one might start an inquiry into possible world semantics, but as our question concerns the foundations of logical modality, it will not do to give an explanation in modal terms, that is, to resort to the possible worlds jargon. We need to know what logically possible worlds in fact are. This is a question for the metaphysics of modality. Here our first concern is whether there are different fundamental kinds of modality and where we should place logical modality if there are. It was once common to think that logical modality in a very strict sense is the only type of modality – it concerns things that are true strictly in virtue of the laws of logic, such as the law of identity. However, it is difficult to account for the necessity of other than logical or mathematical truths with this type of modality, and with the emergence of a posteriori necessity it is now metaphysical modality which is widely considered to be the best candidate for a fundamental type of modality. 1 Thus, we should be able to ground logical modality in metaphysical modality, and as metaphysical modality (which is sometimes called broad logical modality) clearly has a wider scope than (strict) logical 1 See for instance Fine (2002: 255) for further discussion.

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modality, the most natural way to do this is by restriction. Let us examine how, exactly, might logical modality be restricted to metaphysical modality.2 Firstly, our preferred understanding of ‘metaphysical modality’ is that ‘[W]e should view metaphysical necessity as a special case of essence’ (Fine 1994: 8), rather than the other way around. The central idea of this conception of metaphysical modality is that necessities are necessary in virtue of the natures of whichever entities the modality concerns. This has some implications for the relationship between metaphysical and logical modality. To restrict logical modality to metaphysical modality, we might try something like this: ‘[T]he logical necessities can be taken to be the propositions which are true in virtue of the nature of all logical concepts’ (ibid.). More needs to be said though, for what does it mean that a proposition is true in virtue of the nature of logical concepts? Well, logical concepts are things like ‘negation’ (~), ‘implication’ (→), ‘conjunction’ (˄) and ‘disjunction’ (˅).3 Many things seem to be true in virtue of the nature of these concepts, such as transitivity: if A → B and B → C, then A → C. What we need to ask now is: what is this thing, namely transitivity, which is true in virtue of logical concepts? What does transitivity concern? Firstly, it should be noted here that the answers to these questions that will be defended are independent of Kit Fine’s account. We sympathise with Fine’s account of metaphysical modality, but Fine might very well resist a further premise that we wish to introduce, namely that the nature of logical concepts is such that they are not meaningful unless applied to the world, as we will proceed to argue. So, an account which agrees with Fine about logical necessities being true in virtue of the nature of

2 Fine (1994) discusses this in passing, but is primarily concerned with restricting conceptual modality to metaphysical modality. 3 We will omit discussion of quantifiers here, the point can be made without it.

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logical concepts, but disagrees with our suggestion concerning what the nature of logical concepts is would certainly be possible. What we hope to demonstrate is that if the Finean account of metaphysical modality is combined with our account of the nature of logical concepts, then we have some very interesting ramifications for logic as a discipline. At the same time we hope to defend the idea that logical modality can be reduced to metaphysical modality. When we formulate transitivity as we did above (if A → B and B → C, then A → C), it seems to express something about the relationship of A, B and C. Of course, we could just as well have used P, Q and R; these letters are merely placeholders, variables. The proposition only seems to become meaningful when we replace the placeholders with something in the world, i.e. if the white ball caused the red ball to move, and the red ball caused the black ball to move, then the white ball caused the black ball to move. There is a shift of interest here, as in this example transitivity concerns causation. If causation indeed is transitive, then we could replace A, B and C with any appropriate causal chain. We suspect that this reveals something about the nature of logical modality. Whenever we attach meaning to a law of logic, we do it by replacing the placeholders with something that is of interest to us. The order of explanation, however, seems to be from these objects of interest to laws of logic: causation is not transitive (if it is transitive) because we can formulate transitivity, rather, we have formulated transitivity because causation (and other things) seems to be subject to it. The same appears to be true of all laws of logic, such as the law of non-contradiction (LNC). We can formulate LNC as follows: ~(P ˄ ~P), but this formulation tells us nothing unless we add the further assumption that the placeholder P can be replaced with any proposition.4 This assumption is a crucial part of the meaning of any law of logic 4 Which is why Aristotle’s formulation of the law of non-contradiction is to be preferred: ‘[T]he same attribute cannot at the same time belong and not belong to the same subject in the same respect’

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because it fixes its application: the law becomes meaningful only with reference to the world. This would seem to be quite uncontroversial; logic becomes fully operational only when reference to the world is introduced, although this is generally assumed implicitly: ‘In logical semantics we take the standing for relations for granted, in the way that we assume grammatical rules given’ (Beall & van Fraassen 2003: 24). Here ‘standing for’ refers to the relationship between expressions and things in the world. The interesting questions about LNC, or any law of logic, concern its applicability in the world. Is LNC true of all propositions, is it necessary? Presumably it is, but can this be determined before we have applied the law? It would seem that this cannot be the case, as it was just said that laws of logic only become meaningful with reference to the world. This underlines the supposed distinction between logical and metaphysical modality. If the laws of logic have modal content before any application is introduced, then it seems that logical modality could be independent of metaphysical modality, but if modal content is only present after logic has been applied, then logical modality is just a species of metaphysical modality. At least, this is the case if our suggestion about the nature of logical concepts is correct. To settle this dilemma, we must further examine the nature of the laws of logic, namely, why we have the very laws of logic that we in fact have, and further, why do we consider logical validity to be a proof of rationality? If logic is a study of valid, rational argumentation, then there must be an answer to the latter question. A typical answer is that the structure of logic reflects language or grammar, but this is a very dangerous path to take, because language does not seem to be able to maintain rationality. Language is hardly as consistent and clearly structured as logic attempts to be. Indeed, the view that logic is somehow grounded in language has caused immense trouble due to all sorts of linguistic paradoxes, such as the Liar, that can be easily (Aristotle 1984: 1005b19-20).

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formulated. If we take this path, we also leave ourselves open to an attack from the likes of Graham Priest5, who has questioned one of our core logical principles, the law of non-contradiction. Even if we leave Priest’s attack aside, it would seem that logic has to be independent of language if we wish to insist that it has something to do with rationality, because rationality cannot be grounded in language. Let us motivate this claim. For the sake of simplicity, we will adopt a very loose definition of rationality here: rationality concerns reasoning from a set of premises to a conclusion, provided that certain conditions hold. A rational line of reasoning might go as follows.

P1: Objects with a higher density than water will sink in water. P2: This rock has a higher density than water. Conc.: This rock will sink in water.

This perfectly rational line of reasoning relies on a number of background conditions. We are interested in these conditions, as they are the bedrock of rationality. So, what are these conditions? Well, quite simply, in the above line of reasoning it is assumed that reality is consistent. It is assumed, for instance, that the rock will either sink or not sink. It is assumed that there is a consistent law which governs the behaviour of objects. We might ask, why do we assume such things? The answer is obvious: we assume that reality is consistent (i.e. subject to the law of non-contradiction) because it appears to be so. It should thus be evident that the order of explanation in regard to consistency is from how things appear to be in reality to a universal law concerning reality, namely LNC. This is the answer to the question why we have the very laws of logic that we in fact have – because those are the laws that reality appears to follow. If reality was not 5 See for instance his (2006).

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consistent we would not have formulated LNC in the first place. How does this imply that logical truths are meaningful only if they are applied? Well, if the source of logical truths, the reason why we have the very logical laws that we have, is that those are the laws that can be applied, then surely there is a connection here to the meaning of these laws. If we say that the law of non-contradiction is true, it means that we believe the world to be subject to this law. At any rate, this is the metaphysical understanding of truth that we are presently interested in. Given this conception of the notion of truth, the truth or falsity of logical laws should be settled by considering their applications in the world. Accordingly, logical validity does not have a bearing on truth understood as a metaphysical notion. A logic for language L may have an internal test for validity, that is, an argument may be said to be valid and to express a logical truth if there is a proof for it in L, but this is not enough to make the argument meaningful, or so we wish to suggest. Thus, the notion of logical truth does not help us to make sense of the foundations of logical modality, if modality is understood in the Finean fashion. This is of utmost importance for the task at hand: the Finean conception of metaphysical modality, which grounds it in essences, fixes modality to the world. This just means that we have to analyse modality with reference to how things stand in the world. Now, if logical modality is grounded in metaphysical modality, if it is simply a sub-species of metaphysical modality, then the same applies to logical modality. The notion of logical truth is clearly not sufficient to uphold this reference to the world, for simply the fact that there is a proof for a certain proposition in a given language L does not fix the truth of the proposition in regard to the actual world, unless L can be shown to correspond accurately with the actual world. Consequently, if the essences of logical concepts include their applicability to the world, then it seems quite sensible to

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determine the meaning of logical concepts via this relation, call it ‘standing for’ or something else. What we wish to suggest is that the metaphysical notion of truth is intimately connected with meaning and thus if we say that LNC is true in a metaphysical sense, then we are also saying something about the meaning of LNC. It is implicit in what has been said above that we can have only one true logic; this is one of the interesting ramifications of the account at hand. If what has been said is correct, then reality, whether it is consistent or not, can only be arranged in one way, and there can only be one logic that corresponds with it. We should, perhaps, consider how this fits in with recent discussion about logical pluralism (cf. Beall and Restall 2006). In a somewhat trivial sense, we should have no objections to the idea that we could be pluralists about logical truth. This is the sense in which we can have quite different, even incompatible logical systems, as long as they are consistent within a given framework. These may be useful because they have interesting applications, or they may be rival systems and claim to reach a more accurate correspondence with reality. However, only one of these logical systems can be true in a deeper, metaphysical sense, insofar as they are incompatible. If language L accurately corresponds with the actual world, then any language that is incompatible with L will surely fail to maintain correspondence with the actual world. Alternative logics can be true only in the sense that, say, classical mechanics is true, that is, within a given framework. We have no quarrel with logic done within a framework like this, but the logical systems most interesting from a metaphysical point of view are certainly the ones which claim universal application, correspondence with reality. Thus, we should be careful with the use of the notion of ‘logical truth’, for if it is taken to imply truth in a logical system, any logical system, then it has little bearing on truth in a metaphysically deep sense, and it will certainly not help us in regard to our inquiry into the foundations

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of logical modality (cf. Beall and Restall 2006, p. 100-102). We are now in a position to settle the dilemma that we faced earlier, i.e. whether logical modality is independent of metaphysical modality or just a species of metaphysical modality. As we saw, logical modality could only be independent of metaphysical modality if logic has modal content before it has been applied. It appears now that logic is applied even before it is formulated – the laws of logic are based on the structure of reality. This suggests that there is no place for logical modality without reference to the world. If LNC is necessary, it is necessary because the world makes it so. This analysis applies to all logical principles. Consider the law of identity, i.e. ‘A is A’. It might seem that the law of identity is necessary before we give an interpretation for A, as it strikes us as self-evident that anything is identical with itself. But we must ask: why does is strike us as self-evident that anything is identical with itself? Furthermore, if it is the case that the law of identity is necessary, in virtue of what is this so? Here is a thought: the law of identity strikes us as self-evident because it seems that entities are essentially the very entities they are and not other entities. If this is true, then we have also located the source for the necessity of the law, namely, the law of identity is necessary in virtue of the essences of entities, vindicating the idea that logical modality should be treated in the same manner as metaphysical modality. The law was formulated because it appeared to be true – in the metaphysical sense of truth as opposed to truth in a logical system. Finally, consider how we understand logical operators, such as negation. A negation must be a negation of something and this once again points towards it being applied; we cannot understand negation independently of the world. However, instead of repeating what was said above, we could examine the relationship between logical operators and propositions. It might appear that turning our attention to propositions offers an

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alternative analysis of the status of the laws of logic. For instance, we can analyse negation as follows: negation (~P) states that the proposition P is false, where P is a placeholder for any proposition. We can thus account for the necessary application strictly in terms of propositions, at least seemingly avoiding reference to the world. One should expect that we will not be quite satisfied with this. There is nothing wrong with this analysis as such, but it indeed avoids reference to the world only seemingly, as presumably we will have to offer some sort of an analysis of propositions. We do not wish to dwell on the matter here, but it seems reasonably clear that propositions are mind-independent (cf. e.g. Bealer 1998). Accordingly, propositions only introduce a further iteration to the story: yes, we can analyse logical modality in terms of propositions, but propositions themselves need to be applied as well, they are only interesting to us as statements concerning mind-independent reality. The upshot of this discussion is that logical modality is not a distinct kind of modality, but rather a sub-species of metaphysical modality. It is true that we can analyse the relationships between logical concepts strictly in logical terms, without any direct reference to the world, and we can further distance the logical analysis from the world by resorting to a talk of propositions. However, the question about the modal content of logical possibilities and necessities remains unanswered: something that is ‘necessary’ in a model does not necessarily hold genuine modal content. It can only do so if it is still necessary after the appropriate placeholders have been replaced. For instance, as we saw above, LNC, which is presumably necessary in the latter, genuine sense, only has this modal content because it is a true, universal law concerning mindindependent reality. It was also suggested that the overlap between logical and metaphysical modality is due to the fact that the laws of logic have been formulated with reference to the world, i.e. they are formal representations about the governing

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laws of reality. LNC for instance is derived from the apparent consistency in the world.6 All this has some very interesting ramifications. One major question concerns alternative logics, such as paraconsistent logics. For the sake of fallibilism, we should always leave room for the possibility that reality is not consistent after all and that one of these alternative logics is true in the metaphysically substantial sense noted above, albeit this seems highly unlikely. Nevertheless, we should not undermine the value of such alternative logics – they can certainly be useful in other regards. For a metaphysician, however, only a logic that truly reflects the laws of reality will be interesting in the sense that it tells us something about genuine, logical modality understood as a sub-species of metaphysical modality.

References Aristotle (1984) Metaphysics, trans. W. D. Ross, revised by J. Barnes (Princeton, NJ: Princeton University Press). Bealer, G. (1998) ‘Propositions’, Mind 107: 1-32. Beall, J. C. and van Fraassen, B. C. (2003) Possibilities and Paradox: an introduction to modal and many-valued logic (Oxford: Oxford University Press). Beall, J. C. and Restall, G. (2006) Logical Pluralism (Oxford: Clarendon Press). Fine, K. (1994) ‘Essence and Modality’, J. E. Tomberlin (ed.), Philosophical Perspectives 8: Logic and Language (Atascadero, CA: Ridgeview), pp. 1-16. Fine, K. (2002) ‘The Varieties of Necessity’, in Gendler, T. S. & Hawthorne J. (eds.), Conceivability and Possibility (Oxford: Oxford University Press), pp. 253-281. 6 Admittedly, if Priest (2006) is right, reality might not be consistent, but even if this were the case (which we seriously doubt), it does not undermine the argument at hand: our judgements about the world are always fallible and revisable. In fact, this only corroborates the claim that the laws of logic concern mind-independent reality.

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Priest, G. (2006) In Contradiction: A Study of the Transconsistent, 2nd expanded ed. (Oxford: Clarendon Press).

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