A Defense of the Kripkean Account of Logical Truth in First-Order Modal Logic

This paper responds to criticism of the Kripkean account of logical truth in first-order modal logic. The criticism, largely ignored in the literature, claims that when the box and diamond are interpreted as the logical modality operators, the Kripkean account is extensionally incorrect because it fails to reflect the fact that all sentences stating truths about what is logically possible are themselves logically necessary. I defend the Kripkean account by arguing that some true sentences about logical possibility are not logically necessary.     

Actuality, Necessity, and Logical Truth

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. Earlier versions of this paper were read at the universities of Graz, Maribor, and Salzburg, and at a workshop on the philosophy of logic at the National Autonomous University of Mexico (UNAM) in Mexico City. My thanks to those present at these events for many helpful suggestions. Thanks are also due to an anonymous referee for Philosophical Studies.     

On the Substitutional Characterization of First-Order Logical Truth

I consider the well-known criticism of Quine's characterization of first-order logical truth that it expands the class of logical truths beyond what is sanctioned by the model-theoretic account. Briefly, I argue that at best the criticism is shallow and can be answered with slight alterations in Quine's account. At worse the criticism is defective because, in part, it is based on a misrepresentation of Quine. This serves not only to clarify Quine's position, but also to crystallize what is and what is not at issue in choosing the model-theoretic account of first-order logical truth over one in terms of substitutions. I conclude by highlighting the need for justifying the belief that the definition of first-order logical truth in terms of models is superior to its definition in terms of substitutions.     

A Counterfactual Analysis of the Concepts of Logical Truth and Necessity

This paper analyzes the logical truths as (very roughly) those truths that would still have been true under a certain range of counterfactual perturbations.What’s nice is that the relevant range is characterized without relying (overtly, at least) upon the notion of logical truth. This approach suggests a conception of necessity that explains what the different varieties of necessity (logical, physical, etc.) have in common, in virtue of which they are all varieties of necessity. However, this approach places the counterfactual conditionals in an unfamiliar foundational role.     

Zitierte zeichenreihen

We use quotation marks when we wish to refer to an expression. We can and do so refer even when this expression is composed of characters which do not occur in our alphabet. That's why Tarski's, Quine's, and Geach's theories of quotation don't work. The proposals of Davidson, Frege, and C. Washington, however, do not provide a plausible account of quotation either. The problem is to construct a Tarskian theory of truth for an object language which contains quotation marks, without appealing to quotation marks in the metalanguage. I propose to supply Tarski's truth definition with an axiom which determines the denotation of all expressions containing quotation marks. According to this axiom, quotation marks create a non-extensional context. Since our admitting such contexts does not lead to any difficulties in our recursive truth characterization, we may indeed dispense with extensionalism. Finally, I argue that we classify and denote expressions in the very same way that we classify and denote extralinguistic entities.     

The Trouble with Harrison's 'The Trouble with Tarski'

In 'The Trouble with Tarski', The Philosophical Quarterly , 48 (1998), pp. 1–22, Jonathan Harrison attacks 'Tarski-style' truth theories for both formalized and natural languages, on the grounds that (1) truth cannot be a property of sentences; (2) if it could be, T-sentences would have to be necessary truths, which they are not; and (3) T-sentences are not necessarily true and can even can be false. I reply that (1) cannot be an objection to Tarskian truth theories, since these can be formulated in terms of whatever truth bearers might be. Thesis (2) is unjustified: Harrison's argument for it depends on an equivocation. Thesis (3) is false, since the right-hand side of a T-sentence is a meta-language translation of the object-language sentence described on the left-hand side, and this guarantees its truth.
email: boisvert@phil.ufl.edu     

Ray on Tarski on Logical Consequence

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.     

On Mixed Inferences and Pluralism about Truth Predicates

A familiar view concerning sentences about the ethical or comical is that they are not truth-apt – not capable of being true or false. Against this view Peter Geach famously noted that such sentences may figure in true (or false) conditionals and so must thereby be truth-apt. In response to Geach's argument, Crispin Wright proposed a pluralism about truth predicates, a pluralism which sees Geach's conditional clauses as being true in a sense that avoids realism about the entities involved. I defend Wright's proposal against an interesting recent attack by Christine Tappolet, and show that, pace Tappolet, Wright's proposed pluralism is compatible with the Tarskian idea that validity is necessary truth-preservation.     

How to Think about Fallibilism

Almost every contemporary theory of knowledge is a version of fallibilism, yet an adequate statement of fallibilism has not yet been provided. Standard definitions cannot account for fallibilistic knowledge of necessary truths. I consider and reject several attempts to resolve this difficulty before arguing that a belief is an instance of fallibilistic knowledge when it could have failed to be knowledge. This is a fully general account of fallibilism that applies to knowledge of necessary truths. Moreover, it reveals, not only the connection between fallibility and error, but the connection between fallibility and accidental truth as well.     

Necessary Moral Truths and Theistic Metaethics

Theistic metaethics usually places one key restriction on the explanation of moral facts, namely: every moral fact must ultimately be explained by some fact about God. But the widely held belief that moral truths are necessary truths seems to undermine this claim. If a moral truth is necessary, then it seems like it neither needs nor has an explanation. Or so the objection typically goes. Recently, two proponents of theistic metaethics — William Lane Craig and Mark Murphy — have argued that this objection is flawed. They claim that even if a truth is necessary, it does not follow that it neither needs nor has an explanation. In this article, I challenge Craig and Murphy’s reasoning on three main grounds. First, I argue that the counterexamples they use to undermine the necessary truth objection to theistic metaethics are flawed. While they may provide some support for the notion that necessary truths can be explained, they do not provide support for the notion that necessary moral truths can be explained. Second, I argue that the principles of explanation that Murphy and Craig use to support theistic metaethics are either question-begging (in the case of Murphy) or improperly motivated (in the case of Craig). And third, I provide a general defence of the claim that necessary moral truths neither need nor have an explanation.     

A Note on Formality and Logical Consequence

Logic is formal in the sense that all arguments of the same form as logically valid arguments are also logically valid and hence truth-preserving. However, it is not known whether all arguments that are valid in the usual model-theoretic sense are truth-preserving. Tarski claimed that it could be proved that all arguments that are valid (in the sense of validity he contemplated in his 1936 paper on logical consequence) are truth-preserving. But he did not offer the proof. The question arises whether the usual model-theoretic sense of validity and Tarski's 1936 sense are the same. I argue in this paper that they probably are not, and that the proof Tarski had in mind, although unusable to prove that model-theoretically valid arguments are truth-preserving, can be used to prove that arguments valid in Tarski's 1936 sense are truth-preserving.     

Remarks on Bolzano's Conception of Necessary Truth

This essay presents a new interpretation of Bolzano's account of necessary truth as set out in §182 of the Theory of Science. According to this interpretation, Bolzano's conception is closely related to that of Leibniz, with some important differences. In the first place, Bolzano's conception of necessary truth embraces not only what Leibniz called metaphysical or brute necessities but also moral necessities (truths grounded in God's choice of the best among all metaphysical possibilities). Second, in marked contrast to Leibniz, Bolzano maintains that there is still plenty of room for contingency even on this broader conception of necessity.     

塔斯基：语义性真理论与符合论

塔斯基语义性真理论是否为符合论是一个争论的热点。目前有两种观点认为它是符合论,其一是根据对象语言与元语言的区分,到目前为止此观点已受到激烈批评,其二是根据“满足”概念的递归定义,此观点到目前为止还没有被人详细地阐明。本文认为塔斯基的语义性真理论是符合论,为此将首先对第一种观点及其反对意见进行深入分析,并得出结论认为该观点不能成立,继而详细阐明第二种观点,表明它是塔斯基本人的意图。在讨论中本文将对一系列的逻辑哲学问题进行探讨。     

Necessity and Triviality

In this paper I argue that there are some sentences whose truth makes no demands on the world, being trivially true in that their truth-conditions are trivially met. I argue that this does not amount to their truth-conditions being met necessarily: we need a non-modal understanding of the notion of the demands the truth of a sentence makes, lest we be blinded to certain conceptual possibilities. I defend the claim that the truths of pure mathematics and set theory are trivially true, and hence accepting their truth brings no ontological commitment; I further defend the claim that the truths of applied mathematics and set theory do not demand the existence of numbers or sets. While the notion of a demand must not be reduced to anything modal, I nonetheless argue that sentences that are trivially true must also be necessary, lest we violate a very weak version of the principle that truth depends on the world. I further argue that all necessary truths are trivially true, lest we admit unexplained necessities. I end by showing one important consequence of this: I argue that if there are truthmakers for intrinsic predications, they must be states of affairs rather than tropes.     

Reinflating Logical Consequence

Shapiro (Philos Q 61:320–342, 2011) argues that, if we are deflationists about truth, we should be deflationists about logical consequence. Like the truth predicate, he claims, the logical consequence predicate is merely a device of generalisation and more substantial characterisation, e.g. proof- or model-theoretic, is mistaken. I reject his analogy between truth and logical consequence and argue that, by appreciating how the logical consequence predicate is used as well as the goals of proof theory and model theory, we can be deflationists about truth but not logical consequence.     

Boolos and the Metamathematics of Quine's Definitions of Logical Truth and Consequence

The paper is concerned with Quine's substitutional account of logical truth. The critique of Quine's definition tends to focus on miscellaneous odds and ends, such as problems with identity. However, in an appendix to his influential article On Second Order Logic, George Boolos offered an ingenious argument that seems to diminish Quine's account of logical truth on a deeper level. In the article he shows that Quine's substitutional account of logical truth cannot be generalized properly to the general concept of logical consequence. The purpose of this paper is threefold: first, to introduce the reader to the metamathematics of Quine's substitutional definition of logical truth; second, to make Boolos' result accessible to a broader audience by giving a detailed and self-contained presentation of his proof; and, finally, to discuss some of the possible implications and how a defender of the Quinean concepts might react to the challenge posed by Boolos' result.     

Truth and epistemic value

The notion of more truth, or of more truth and less falsehood, is central to epistemology. Yet, I argue, we have no idea what this consists in, as the most natural or obvious thing to say—that more truth is a matter of a greater number of truths, and less falsehood is a matter of a lesser number of falsehoods—is ultimately implausible. The issue is important not merely because the notion of more truth and less falsehood is central to epistemology, but because an implicit, false picture of what this consists in underpins and gives shape to much contemporary epistemology.     

LOGICAL TRUTH IN MODAL LOGIC

In this paper, I consider the criticism due to Hartry Field, John Pollack, William Hanson and James Hawthorne that the Kripkean requirement that a logical truth in modal logic be true at all possible worlds in all quantified model structures is unmotivated and misses some logical truths. These authors do not see the basis for making the logical truth of a modal sentence turn on more than the model structure given by one reading of the modal operator(s) which occur in the sentence. The primary goal here is to motivate the Kripkean requirement.     

Deutscher's problem is not Popper's problem

The so-called knowability paradox results from Fitch's argument that if there are any unknown truths, then there are unknowable truths. This threatens recent versions of semantical antirealism, the central thesis of which is that truth is epistemic. When this is taken to mean that all truths are knowable, antirealism is thus committed to the conclusion that no truths are unknown. The correct antirealistic response to the paradox should be to deny that the fundamental thesis of the epistemic nature of truth entails the knowability of all truths. Correctly understood, the antirealistic conditions on a proposition's truth do not require that the proposition possess a verification-procedure which, when executed under the given conditions, issues in an agent's recognition of truth, but merely that there be a verification-procedure which, under these conditions, takes the value true. The knowability paradox and the related idealism problem (that antirealism seems, but is not, committed to the necessary existence of an epistemic agent) draw attention to the fact that certain propositions, those that are about verification-procedures themselves, may under certain conditions take the value true despite their unperformability under these circumstances. Thus these propositions' procedures can only be performed when the propositions are false, and they gain the appearance of antirealistic impossibility (e.g., that there is an unknown truth). This differs from the unperformability that antirealists object to, pertaining merely to matters of execution rather than to the logical structure of the procedures themselves. The force of antirealism's notion of epistemic truth is piecemeal, rather than consisting in a blanket characterization of truth as knowable.     

Generalized Logical Consequence: Making Room for Induction in the Logic of Science

We present a framework that provides a logic for science by generalizing the notion of logical (Tarskian) consequence. This framework will introduce hierarchies of logical consequences, the first level of each of which is identified with deduction. We argue for identification of the second level of the hierarchies with inductive inference. The notion of induction presented here has some resonance with Popper's notion of scientific discovery by refutation. Our framework rests on the assumption of a restricted class of structures in contrast to the permissibility of classical first-order logic. We make a distinction between deductive and inductive inference via the notions of compactness and weak compactness. Connections with the arithmetical hierarchy and formal learning theory are explored. For the latter, we argue against the identification of inductive inference with the notion of learnable in the limit. Several results highlighting desirable properties of these hierarchies of generalized logical consequence are also presented.