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A Defense of Contingent Logical Truths∗ Agree That There Are Contingent Instances of LA1

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A Defense of Contingent Logical Truths∗ Agree That There Are Contingent Instances of LA1 Michael Nelson and Edward N. Zalta 2 then, at w1, the left side of our conditional is true and the right side false, making the conditional itself false at w1 and so only contingently true at the actual world. All sides of the debate on which we are about to embark A Defense of Contingent Logical Truths∗ agree that there are contingent instances of LA1. What is controversial is whether or not all instances of LA1 are an- alytic and logically true. Zalta argues that they are (and hence that we Michael Nelson need to learn to live with contingent logical and analytic truths). In his University of California/Riverside 2006, William Hanson argues to the contrary, defending the traditional and connection between logical truth, analyticity, and necessity. In this paper Edward N. Zalta we consider Hanson's case and argue that LA1 is indeed a logical and Stanford University analytic truth. 1. Formal Preliminaries and Zalta's Argument Abstract Whether or not all instances of LA1 are logically true depends on the A formula is a contingent logical truth when it is true in every notion of logical truth one adopts. In their pioneering work on the logic model M but, for some model M, false at some world of M. We of the actuality operator, Crossley and Humberstone (1977) distinguish argue that there are such truths, given the logic of actuality. Our two distinct notions of validity, which they dub general and real-world argument turns on defending Tarski's definition of truth and logical validity. We shall first characterize these competing notions, relate the truth, extended so as to apply to modal languages with an actuality choice between these notions to the logical status of LA1, and then turn to operator. We argue that this extension is the philosophically proper the matter of defending a choice of real-world validity over general validity. account of validity. We counter recent arguments to the contrary Both definitions assume that a model M of modal logic is a structure presented in William Hanson's `Actuality, Necessity, and Logical of the form hW; w ;V i, where W is a set of possible worlds, w is a Truth' (Philosophical Studies, 130 (2006): 437{459). α α distinguished world in W , and V is a valuation function that assigns a In a paper of 1988, Edward Zalta argues that there are logical truths set of worlds to each atomic sentence of the language.1 In what follows, and analytic truths that are not necessary. Primary examples are in- we shall assume an S5 propositional modal logic and so we need not stances of the schema include an accessibility relation, which will play no role in what follows. Moreover, we shall use `model' and `interpretation' interchangeably, since LA1: Aφ ! φ some authors use the former while others use the latter in talking about where φ is some contingent claim, and A is the actuality operator. Take as the same set-theoretic structure. an example the sentence `If it is actually the case that Obama is president, Let us assume that the definition of truthM of a formula φ at a world then Obama is president'. This sentence is contingent. The justifica- w (i.e., M; w j= φ) has been given recursively in the standard way.2 Then tion for this is straightforward. Given the contingency of the proposition 1We shall not be discussing those modal logicians who eliminate altogether the [Obama is president], there is a world at which it is false; i.e., a world, say actual world from the models of modal logic. Those logicians face two further problems, w1, where Obama is not president. At w1, the proposition [It is actually namely, that there seems to be no way for them to introduce an actuality operator and the case that Obama is president] is true, as that proposition is a propo- there is no way to define the notion of truth in a model if no reference to a distinguished actual world can be made (as done below). sition not about w1 but the actual world, where Obama is president. But 2I.e., as follows: ∗ This paper is published in Philosophical Studies, 157/1 (2012): 153{162. 1. When φ is an atomic sentence letter p, M; w j= φ iff w 2 VM (p) 3 A Defense of Contingent Logical Truths Michael Nelson and Edward N. Zalta 4 the notion of real-world validity is defined in terms of the notion of truth an intimate connection between truth in a model and logical truth. But in a model: φ is trueM (M j= φ) just in case φ is true at the distinguished the notion of truth in a model does not play this foundational role in world wα in M (i.e., M j= φ iff M; wα j= φ). We now define real-world the definition of G-validity. A formula is G-valid, recall, just in case it validity as: a formula φ is R-valid just in case φ is true in every model is true for every model M and world w. The notion of G-validity thus M, i.e., 8M(M j= φ). by-passes the notion of truth in a model and thus does not respect the By contrast, the definition of general validity bypasses the notion of foundational role this notion plays. So G-validity does not characterize a truth in a model and defines validity directly from the notion of truth at kind of genuine logical truth. a world in the model, as follows: φ is G-valid just in case, for every model M and every world w 2 WM , φ is trueM at w, i.e., iff 8M8w(M; w j= φ). 2. Hanson's First Two Problems Thus, G-validity is not defined in terms of the notion of truth in a model (M j= φ). Hanson raises two problems with Zalta's argument for the primacy of the Intuitively and informally, a formula is R-valid just in case, for every notion of R-validity. First, he challenges the claim that Tarski's (1936) model, it is true at the distinguished world of the model. A formula is definition of logical truth, which is in terms of an extensional (i.e., non- G-valid, on the other hand, just in case, for every model, it is true in every modal) language, carries over to a nonextensional (i.e., modal) language world of the model. So, whereas the G-validity of a formula involves its (Hanson 2006, 442).4 Second, he claims that, with a minor modification, truth value at counterfactual worlds of the models, its R-validity does not. the proponent of G-validity can explicate the notion of logical truth in The choice between these competing definitions of validity determines terms of truth in a model. We consider each claim in turn. the logical status of instances of LA1. While every instance of LA1 is It is true that Tarski's work on logical truth was carried out in the true in every model M, for every contingent instance of LA1, there is context of a nonmodal language and so when we move to a modal language a model M and world w 2 WM , where w is not the distinguished world certain changes and additions are necessary. But we claim that Tarski of M, such that is falseM at w. Hence, while every instance of LA1 provided us with a proper understanding of the notion of logical truth is R-valid, there are instances of LA1 that are not G-valid. This follows as truth under every interpretation.5 This basic insight should not be from the definition of the two notions of validity and the fact that there altered, no matter what new languages we go on to consider. What will are contingent instances of LA1. change is what constitutes an interpretation of the language in question. Zalta argues that the proper way to understand the notion of logical The basic conception of logical truth as truth across all interpretations truth excludes G-validity as a genuine kind of logical truth. The argument should remain the same, even when we move from a nonmodal to a modal turns on the role that the notion of truth in a model plays in our intuitive language. Our response to Hanson's first claim, then, is simply this. conception of logical truth. It is argued that only the notion of R-validity We agree that one should expect there to be changes in the semantic respects this role.3 The idea is that the most fundamental semantic notion definitions when we move from nonmodal to modal languages, but we is that of truth in a model. Logical truth is then explicated as truth in think those changes should take place in the definition of an interpretation every model. Truth invariance under permutation of the interpretation of 4For the purposes of this paper, we will consider the language of propositional the nonlogical vocabulary is what it is to be logically true. This establishes logic to be extensional and the language of propositional modal logic to be nonexten- sional. But see Zalta 1993, where it is argued that when propositions are assigned as 2. When φ has the form : , M; w j= φ iff it is not the case that M; w j= the denotations of the sentence letters in propositional modal logic, the language of 3. When φ has the form ! χ, M; w j= φ iff either it is not the case that M; w j= propositional modal logic becomes extensional.
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