Does general validity or real world validity better represent the intuitive notion of logical truth for sentential modal languages with an actuality connective? In (Philosophical Studies 130:436–459, 2006) I argued in favor of general validity, and I criticized the arguments of Zalta (Journal of Philosophy 85:57–74, 1988) for real world validity. But in Nelson and Zalta (Philosophical Studies 157:153–162, 2012) Michael Nelson and Edward Zalta criticize my arguments and claim to have established the superiority of real world validity. Section 1 (...) of the present paper introduces the problem and sets out the basic issues. In Sect. 2 I consider three of Nelson and Zalta’s arguments and find all of them deficient. In Sect. 3 I note that Nelson and Zalta direct much of their criticism at a phrase (‘true at a world from the point of view of some distinct world as actual’) I used only inessentially in Hanson (Philosophical Studies 130:436–459, 2006), and that their account of the philosophical foundations of modal semantics leaves them ill equipped to account for the plausibility of modal logics weaker than S5. Along the way I make several general suggestions for ways in which philosophical discussions of logical matters–especially, but not limited to, discussions of truth and logical truth for languages containing modal and indexical terms–might be facilitated and made more productive. (shrink)
The traditional view that all logical truths are metaphysically necessary has come under attack in recent years. The contrary claim is prominent in David Kaplan’s work on demonstratives, and Edward Zalta has argued that logical truths that are not necessary appear in modal languages supplemented only with some device for making reference to the actual world (and thus independently of whether demonstratives like ‘I’, ‘here’, and ‘now’ are present). If this latter claim can be sustained, it strikes close to the (...) heart of the traditional view. I begin this paper by discussing and refuting Zalta’s argument in the context of a language for propositional modal logic with an actuality connective (section 1). This involves showing that his argument in favor of real world validity his preferred explication of logical truth, is fallacious. Next (section 2) I argue for an alternative explication of logical truth called general validity. Since the rule of necessitation preserves general validity, the argument of section 2 provides a reason for affirming the traditional view. Finally (section 3) I show that the intuitive idea behind the discredited notion of real world validity finds legitimate expression in an object language connective for deep necessity. (shrink)
The time-honored view that logic is a non-empirical enterprise is still widely accepted, but it is not always recognized that there are (at least) two distinct ways in which this view can be made precise. One way focuses on the knowledge we can have of logical matters, the other on the nature of the logical consequence relation itself. More specifically; the first way embodies the claim that knowledge of whether the logical consequence relation holds in a particular case is knowledge (...) that can be had a priori (if at all). The second way presupposes a distinction between structural and non-structural properties and relations, and it holds that logical consequence is to be defined exdusively in terms of the former. It is shown that the two ways are not coextensive by giving an example of a logic that is non-empirical in the second way but not in the first. (shrink)
In "Logical consequence: A defense of Tarski" (Journal of Philosophical Logic, vol. 25, 1996, pp. 617-677), Greg Ray defends Tarski's account of logical consequence against the criticisms of John Etchemendy. While Ray's defense of Tarski is largely successful, his attempt to give a general proof that Tarskian consequence preserves truth fails. Analysis of this failure shows that de facto truth preservation is a very weak criterion of adequacy for a theory of logical consequence and should be replaced by a stronger (...) absence-of-counterexamples criterion. It is argued that the latter criterion reflects the modal character of our intuitive concept of logical consequence, and it is shown that Tarskian consequence can be proved to satisfy this criterion for certain choices of logical constants. Finally, an apparent inconsistency in Ray's interpretation of Tarski's position on the modal status of the consequence relation is noted. (shrink)
Some widely accepted arguments in the philosophy of mathematics are fallacious because they rest on results that are provable only by using assumptions that the con- clusions of these arguments seek to undercut. These results take the form of bicon- ditionals linking statements of logic with statements of mathematics. George Boolos has given an argument of this kind in support of the claim that certain facts about second-order logic support logicism, the view that mathematics—or at least part of it—reduces to (...) logic. Hilary Putnam has offered a similar argument for the view that it is indifferent whether we take mathematics to be about objects or about what follows from certain postulates. In this paper I present and rebut these arguments. (shrink)
In this paper I argue that there can be genuine (as opposed to merely verbal) disputes about whether a sentence form is logically true or an argument form is valid. I call such disputes ?cases of deviance?, of which I distinguish a weak and a strong form. Weak deviance holds if one disputant is right and the other wrong, but the available evidence is insufficient to determine which is which. Strong deviance holds if there is no fact of the matter. (...) In section 2 I argue that weak deviance need not be trivial and may even be interesting. Section 3 considers what it could mean to say that logic is determined by a theory, especially a theory of meaning, an idea that arises in section 2. In section 4 I discuss the dispute between classical and relevance logicians over entailment and argue that it is a case of strong deviance. Finally, in section 5 I show that the result of the previous section is not absolute but relative to the background logic used in reaching it. (shrink)
Although the use of possible worlds in semantics has been very fruitful and is now widely accepted, there is a puzzle about the standard definition of validity in possible-worlds semantics that has received little notice and virtually no comment. A sentence of an intensional language is typically said to be valid just in case it is true at every world under every model on every model structure of the language. Each model structure contains a set of possible worlds, and models (...) are defined relative to model structures, assigning truth-values to sentences at each world countenanced by the model structure. The puzzle is why more than one model structure is used in the definition of validity. There is presumably just one class of all possible worlds and just one model structure defined on this class that does correctly the things that model structures are supposed to do. (These include, but need not be limited to, specifying the set of individuals in each world as well as various accessibility relations between worlds.) Why not define validity simply as truth at every world under every model on this one model structure? What is the point of bringing in more model structures than just this one?
We investigate these questions in some detail and conclude that for many intensional languages the puzzle points to a genuine difficulty: the standard definition of validity is insufficiently motivated. We begin (Section 1) by showing that a plausible and natural account of validity for intensional languages can be based on a single model structure, and that validity so defined is analogous in important respects to the standard account of validity for extensional languages. We call this notion of validity "validity!", and in Section 2 we contrast it with the standard notion, which we call "validity2". Several attempts are made to discover a rationale for the almost universal acceptance of validity2, but in most of these attempts we come up empty-handed. So in Section 3 we investigate validity! for some intensional languages. Our investigation includes providing axiomatizations for several propositional and predicate logics, most of which are provably complete. The completeness proofs are given in the Appendix, which also contains a sketch of a compactness proof for one of the predicate logics. (shrink)