Satisfying Predicates: Kleene's Proof of the Hilbert–Bernays Theorem

The Hilbert–Bernays Theorem establishes that for any satisfiable first-order quantificational schema S, one can write out linguistic expressions that are guaranteed to yield a true sentence of elementary arithmetic when they are substituted for the predicate letters in S. The theorem implies that if L is a consistent, fully interpreted language rich enough to express elementary arithmetic, then a schema S is valid if and only if every sentence of L that can be obtained by substituting predicates of L for predicate letters in S is true. The theorem therefore licenses us to define validity substitutionally in languages rich enough to express arithmetic. The heart of the theorem is an arithmetization of Gödel's completeness proof for first-order predicate logic. Hilbert and Bernays were the first to prove that there is such an arithmetization. Kleene established a strengthened version of it, and Kreisel, Mostowski, and Putnam refined Kleene's result. Despite the later refinements, Kleene's presentation of the arithmetization is still regarded as the standard one. It is highly compressed, however, and very difficult to read. My goals in this paper are expository: to present the basics of Kleene's arithmetization in a less compressed, more easily readable form, in a setting that highlights its relevance to issues in the philosophy of logic, especially to Quine's substitutional definition of logical truth, and to formulate the Hilbert–Bernays Theorem in a way that incorporates Kreisel's, Mostowski's, and Putnam's refinements of Kleene's result.     

Quine’s Substitutional Definition of Logical Truth and the Philosophical Significance of the Löwenheim-Hilbert-Bernays Theorem

The Löwenheim-Hilbert-Bernays theorem states that, for an arithmetical first-order language L, if S is a satisfiable schema, then substitution of open sentences of L for the predicate letters of S results in true sentences of L. For two reasons, this theorem is relevant to issues relative to Quine’s substitutional definition of logical truth. First, it makes it possible for Quine to reply to widespread objections raised against his account (the lexicon-dependence problem and the cardinality-dependence problem). These objections purport to show that Quine’s account overgenerates: it would count as logically true sentences which intuitively or model-theoretically are not so. Second, since this theorem is a crucial premise in Quine’s proof of the equivalence between his substitutional account and the model-theoretic one, it enables him to show that, from a metamathematical point of view, there is no need to favour the model-theoretic account over one in terms of substitutions. The purpose of that essay is thus to explore the philosophical bearings of the Löwenheim-Hilbert-Bernays theorem on Quine’s definition of logical truth. This neglected aspect of Quine’s argumentation in favour of a substitutional definition is shown to be part of a struggle against the model-theoretic prejudice in logic. Such an exploration leads to reassess Quine’s peculiar position in the history of logic.     

A CRITIQUE OF FREGE ON COMMON NOUNS

Frege analyzed the grammatical subject‐term ‘S’ in quantified subject‐predicate sentences, ‘q S are P’, as being logically predicative. This is in contrast to Aristotelian Logic, according to which it is a logical subject‐term, like the proper name ‘a’ in ‘a is P’– albeit a plural one, designating many particulars. I show that Frege’s arguments for his analysis are unsound, and explain how he was misled to his position by the mathematical concept of function. If common nouns in this grammatical subject position are indeed logical subject‐terms, this should require a thorough reevaluation of the adequacy of Frege’s predicate calculus as a tool for the analysis of the logic and semantics of natural language.     

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.     

Can First-Order Logical Truth be Defined in Purely Extensional Terms?

W. V. Quine thinks logical truth can be defined in purely extensional terms, as follows: a logical truth is a true sentence that exemplifies a logical form all of whose instances are true. P. F. Strawson objects that one cannot say what it is for a particular use of a sentence to exemplify a logical form without appealing to intensional notions, and hence that Quine's efforts to define logical truth in purely extensional terms cannot succeed. Quine's reply to this criticism is confused in ways that have not yet been noticed in the literature. This may seem to favour Strawson's side of the debate. In fact, however, a proper analysis of the difficulties that Quine's reply faces suggests a new way to clarify and defend the view that logical truth can be defined in purely extensional terms.     

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.     

Quine on the Nature of Naturalism

Quine's metaphilosophical naturalism is often dismissed as overly “scientistic.” Many contemporary naturalists reject Quine's idea that epistemology should become a “chapter of psychology” (1969a, 83) and urge for a more “liberal,” “pluralistic,” and/or “open‐minded” naturalism instead. Still, whenever Quine explicitly reflects on the nature of his naturalism, he always insists that his position is modest and that he does not “think of philosophy as part of natural science” (1993, 10). Analyzing this tension, Susan Haack has argued that Quine's naturalism contains a “deep‐seated and significant ambivalence” (1993a, 353). In this paper, I argue that a more charitable interpretation is possible—a reading that does justice to Quine's own pronouncements on the issue. I reconstruct Quine's position and argue (i) that Haack and Quine, in their exchanges, have been talking past each other and (ii) that once this mutual misunderstanding is cleared up, Quine's naturalism turns out to be more modest, and hence less scientistic, than many contemporary naturalists have presupposed. I show that Quine's naturalism is first and foremost a rejection of the transcendental. It is only after adopting a broadly science‐immanent perspective that Quine, in regimenting our language, starts making choices that many contemporary philosophers have argued to be unduly restrictive.     

紧缩论,证明与保守性

紧缩论者主张真谓词表达了一种逻辑概念,它的全部意义都体现在所有塔斯基式的T-语句中。Shapiro近来论证说,将紧缩论的公理添加到一阶皮亚诺算术公理系统(PA)中,在该扩张理论中能够证明PA的可靠性,并在此基础上证明PA的一致性,这表明紧缩论不具有保守性,因此真谓词不是紧缩的。本文论证,扩张理论预设了反射原则,这导致它推出了更多的东西,而反射原则是可证性谓词定义的推论,这才是造成扩张理论非保守性的真正根源。针对紧缩论的非保守性论证因此失效了。     

Non‐Classical Knowledge

The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely‐held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single‐premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent theory validating all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker‐than‐classical K₃ logic. I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to—whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non‐classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities.     

Two Dogmatists

Grice and Strawson's ‘In Defense of a Dogma’ is admired even by revisionist Quineans such as Putnam (1962) who should know better. The analytic/synthetic distinction they defend is distinct from that which Putnam successfully rehabilitates. Theirs is the post‐positivist distinction bounding a grossly enlarged analytic. It is not, as they claim, the sanctified product of a long philosophic tradition, but the cast‐off of a defunct philosophy ‐ logical positivism. The fact that the distinction can be communally drawn does not show that it is based on a real difference. Sub‐categories that can be grouped together by enumeration will do the trick. Quine's polemical tactic (against which Grice and Strawson protest) of questioning the intelligibility of the distinction is indeed objectionable, but his argument can be revived once it is realized that ‘analytic’ et al. are theoretic terms, and there is no extant theory to make sense of them. Grice and Strawson's paradigm of logical impossibility is, in fact, possible. Their attempt to define synonymy in Quinean terms is a failure, nor can they retain analyticity along with the Quinean thesis of universal revisability. The dogma, in short, is indefensible.     

On Value‐Attributions: Semantics and Beyond

This paper is driven by the idea that the contextualism‐relativism debate regarding the semantics of value‐attributions turns on certain extra‐semantic assumptions that are unwarranted. One is the assumption that the many‐place predicate of truth, deployed by compositional semantics, cannot be directly appealed to in theorizing about people's assessments of truth value but must be supplemented (if not replaced) by a different truth‐predicate, obtained through certain “postsemantic” principles. Another is the assumption that semantics assigns to sentences not only truth values (as a function of various parameters, such as contexts, worlds and times), but also semantic contents, and that what context‐sensitive expressions contribute to content are contextually determined elements. My first aim in this paper will be to show how the two assumptions have shaped two ways of understanding the debate between contextualism and relativism. My second aim will be to show that both assumptions belong outside semantics and are, moreover, questionable.     

PERSPECTIVE‐NEUTRAL INTRINSIC VALUE

Is it possible to do a good thing, or to make the world a better place? Some argue that it is not possible, because perspective‐neutral value does not exist. Some argue that ‘good’ does not play the right grammatical role; or that all good things are good ‘in a way’; or that goodness is inherently perspective‐dependent. I argue that the logical and semantic properties of ‘good’ are what we should expect of an evaluative predicate; that the many ways of being good don't threaten the thesis that some ways are perspective‐independent; and that there are clear examples of perspective‐independent goodness.     

Quine's Naturalized Epistemology and the Third Dogma of Empiricism

This essay reconsiders Davidson's critical attribution of the scheme‐content distinction to Quine's naturalized epistemology. It focuses on Davidson's complaint that the presence of this distinction leads Quine to mistakenly construe neural input as evidence. While committed to this distinction, Quine's epistemology does not attempt to locate a justificatory foundation in sensory experience and does not then equate neural intake with evidence. Quine's central epistemological task is an explanatory one that attempts to scientifically clarify the route from stimulus to science. Davidson's critical remarks wrongly assign concerns to Quine's view that it does not have and further obscures the status of his naturalized conception of epistemology.     

Quine's Naturalism and Behaviorisms

This paper investigates the complicated relations between various versions of naturalism, behaviorism, and mentalism within the framework of W. V. O. Quine's thinking. It begins with Roger Gibson's reconstruction of Quine's behaviorisms and argues that it lacks a crucial ontological element and misconstrues the relation between philosophy and science. After getting clear of Quine's naturalism, the paper distinguishes between evidential, methodological, and ontological behaviorisms. The evidential and methodological versions are often conflated, but they need to be clearly distinguished in order to see whether Quine's argument against mentalism is cogent. The paper argues that Quine's naturalism supports only the weakest version of behaviorism, that is, the evidential one, but this version is compatible with mentalistic semantics. Quine's opposition to mentalism is an overreaction against the behaviorist camp. By contrast, Jerry Fodor's objection to José Luis Bermúdez is an overreaction from the opposite direction.     

Open‐endedness,Schemas and Ontological Commitment

Second‐order axiomatizations of certain important mathematical theories—such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth‐value. Second‐order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth‐values. The status of second‐order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one can get some of the technical advantages of second‐order axiomatizations—categoricity, in particular—while walking free of worries about ontological commitment. In so arguing he appeals to the notion of an open‐ended schema—a schema that holds no matter how the language of the relevant theory is extended. Contra McGee, we argue that second‐order quantification and open‐ended schemas are on a par when it comes to ontological commitment.     

The Principle of Sufficient Reason,the Ontological Argument and the Is/Ought Distinction

Kant's attack on metaphysics consists in large part in his attack on a principle that he names the Supreme Principle of Pure Reason. This principle, it is not often noticed, is the Principle of Sufficient Reason [PSR]. In interpreting this principle as such, I argue that Kant's attack on the PSR (and thereby his attack on dogmatic metaphysics as a whole) depends on Kant's claim that existence is not a first‐order predicate. If existence isn't what Kant calls a real predicate, the PSR is false. While this constitutes a powerful Kantian argument against dogmatic rationalism, it also poses a problem for Kant. For, as I argue, if the PSR is true, Kant's argument that existence isn't a first‐order predicate is false. In this sense, Kant's attack on the PSR is begging the question vis‐á‐vis radical metaphysicians (e.g. Spinoza). Concluding the paper I suggest relying on Kant's 'is'/'ought' distinction in avoiding this circularity, thereby reinforcing the Kantian critique.     

(ANTI)‐Anti‐Intellectualism and the Sufficiency Thesis

Anti‐intellectualists about knowledge‐how insist that, when an agent S knows how to φ, it is in virtue of some ability, rather than in virtue of any propositional attitudpaes, S has. Recently, a popular strategy for attacking the anti‐intellectualist position proceeds by appealing to cases where an agent is claimed to possess a reliable ability to φ while nonetheless intuitively lacking knowledge‐how to φ. John Bengson and Marc Moffett and Carlotta Pavese have embraced precisely this strategy and have thus claimed, for different reasons, that anti‐intellectualism is defective on the grounds that possessing the ability to φ is not sufficient for knowing how to φ. We investigate this strategy of argument‐by‐counterexample to the anti‐intellectualist's sufficiency thesis and show that, at the end of the day, anti‐intellectualism remains unscathed.