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Boolos and the Metamathematics of Quine's Definitions of Logical Truth and Consequence
Abstract
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.
... The alleged co-extensionality of Quine's substitutional account with the model-theoretic is discussed and put into question inEder (2016) following and expanding an argument developed inBoolos (1975). 5 See, for example Corcoran's 1972 paper "Conceptual Structure of Classical Logic", where the author already cast doubts on the material adequacy of Tarski's model-theoretic concept of logical consequence. ...
I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic conception of the working mathematician.
... The alleged co-extensionality of Quine's substitutional account with the model-theoretic is discussed and put into question inEder (2016) following and expanding an argument developed inBoolos (1975). 5 See, for example Corcoran's 1972 paper "Conceptual Structure of Classical Logic", where the author already cast doubts on the material adequacy of Tarski's model-theoretic concept of logical consequence. ...
I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic conception of the working mathematician.
... Reductionism causes there problems similar to those discussed above. For a further discussion of Quine's theory see (Eder 2016). 11 Under Linnebo's (2012) definition logical validity is also obviously truth preserving. ...
A substitutional account of logical validity for formal first‐order languages is developed and defended against competing accounts such as the model‐theoretic definition of validity. Roughly, a substitution instance of a sentence is defined as the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category and possibly relativizing quantifiers. In particular, predicate symbols can be replaced with formulae possibly containing additional free variables. A sentence is defined to be logically true iff all its substitution instances are satisfied by all variable assignments. Logical consequence is defined analogously. Satisfaction is taken to be a primitive notion and axiomatized.
For every set‐theoretic model in the sense of model theory there exists a corresponding substitutional interpretation in a sense to be specified. Conversely, however, there are substitutional interpretations – in particular the ‘intended’ interpretation – that lack a model‐theoretic counterpart. The substitutional definition of logical validity overcomes the weaknesses of more restrictive accounts of substitutional validity; unlike model‐theoretic logical consequence, the substitutional notion is trivially and provably truth preserving. In Kreisel's squeezing argument the formal notion of substitutional validity naturally slots into the place of intuitive validity.
In this paper, we give a groundwork for the foundations of the semantic concept of logical consequence. We first give an opinionated survey of recent discussions on the model-theoretic concept, in particular Etchemendy’s criticisms and responses, alluding to Kreisel’s squeezing argument. We then present a view that in a sense the semantic concept of logical consequence irreducibly depends on the meaning of logical expressions but in another sense the extensional adequacy of the semantic account of first-order logical consequence is also of fundamental importance. We further point out a connection with proof-theoretic semantics.
LENLS is an annual international workshop on formal syntax, semantics and pragmatics. It will be held as one of the workshops of the JSAI International Symposia on AI (JSAI-isAI2019) sponsored by the Japan Society for Artificial Intelligence (JSAI).
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.
What we call the Hilbert‐Bernays (HB) Theorem establishes that for any satisfiable first‐order quantificational schema S, there are expressions of elementary arithmetic that yield a true sentence of arithmetic when they are substituted for the predicate letters in S. Our goals here are, first, to explain and defend W. V. Quine's claim that the HB theorem licenses us to define the first‐order logical validity of a schema in terms of predicate substitution; second, to clarify the theorem by sketching an accessible and illuminating new proof of it; and, third, to explain how Quine's substitutional definition of logical notions can be modified and extended in ways that make it more attractive to contemporary logicians.
Did Tarski Commit Tarski's Fallacy ? Author(s): G. Y. Sher Source: The Journal of Symbolic Logic, Vol. 61, No. 2 (Jun., 1996), pp. 653-686 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2275681 Accessed: 16/06/2009 13:06 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=asl. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact support@jstor.org. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org
In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.
According to Quine, logic is first-order. For him, two underlying logic of all rational thought is a standard first-order logic without identity, without individual constants and without function constants, a logic which is two-valued, tenseless, extensional, non-modal, non-intuitionistic and one-sorted, a logic which is (in a sense) a proper sublogic of virtually every logic used as an underlying logic in tha literature of classical mathematics and science, a logic which merits being called conservative. His conception of the logical properties gives priority to logical truth. Roughly, Quine holds that in arder for a sentence to be logically trae it is necessary and sufficient for it to be true and to remain true under any uniform lexical substitution of its content-terms. Derivatively, in order for a sentence to be logically implied by a set of sentences F it is necessary and sufficient for there to be no single uniform lexical substitution of content-terms that makes every member of F true and false. Since, intuitively speaking, the relation of a sentence to its lexical subctitutions is a matter of grammar, logical truth and logical implication are thus a matter, a Quine omphasizec, of grammar and truth. My parpase ic to diccuss Quinas distintive view of logical truth and logical implication and to analyze the main features and philosophical import of his conception. This paper has five sections. The first fixes the terminology and stresses the fact that Quine takes logical truth to be prior to logical implication. Section two identifies three core features of the Quinean conception; his fixed-universe, fixed-content, non-modal conception is discussed in the light of Tarcki's distinctive contribution to these perennial issues. Section three analyses the grounds for Quine`s view and his claim that his account is co-extensional with the current model-theoretic conception. Two necessary conditions for this alleged adequacy are considered: (a) the first-order language must contain elementary arithmetic, (b) identity must be nonlogical. In section four the role that Quines "parsimonious' ontology plays in his conception is briefly discussed in historical perspective. Section five conludes with a brief historical survey of the main achievements of Bolzano, Russell, Tarski and Quine logical truth and consequence.
Gary Kemp: Quine's Relationship with Analytic Philosophy: I try to explain why Quine, for all his fame amongst analytic philosophers, has so few explicit followers within analytic philosophy of the past fifty years – despite the fact that naturalism, at least as broadly conceived, is so popular. Partly it's because Quine's particular version of naturalism is so demanding, partly it's because the nature and seriousness of his commitment to extensionalism are not well recognized, and partly because his views were formulated from an extremely general and abstract point of view in the philosophy of science. I focus on the second and especially on the third. Quine is more radical and more alien than is customarily appreciated.
Some philosophers, notably Professors Quine and Geach, have stressed the analogies they see between pronouns of the vernacular and the bound variables of quantification theory. Geach, indeed, once maintained that ‘for a philosophical theory of reference, then, it is all one whether we consider bound variables or pronouns of the vernacular'. This slightly overstates Geach's positition since he recognizes that some pronouns of ordinary language do function differently from bound variables; he calls such pronouns ‘pronouns of laziness'. Geach's characterisation of pronouns of laziness has varied from time to time, but the general idea should be clear from a paradigm example:
(1) A man who sometimes beats his wife has more sense than one who always gives in to her.
The pronouns ‘one’ and ‘her’ go proxy for a noun or a noun phrase (here: ‘a man’ and ‘his wife’) in the sense that the pronoun is replaceable in paraphrase by simple repetition of its antecedent.
PART I: ORIENTATION Terms and questions Foundationalism and foundations of mathematics PART II: LOGIC AND MATHEMATICS Theory Metatheory Second-order logic and mathematics Advanced metatheory PART III: HISTORY AND PHILOSOPHY The historical triumph of first-order languages Second-order logic and rule-following The competition Index.
Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics. In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second-order arithmetic. Additional results are presented in an appendix. © The Association for Symbolic Logic 2009 and Cambridge University Press, 2009.
In recent work on Frege, one of the most salient issues has been whether he was prepared to make serious use of semantical notions such as reference and truth. I argue here Frege did make very serious use of semantical concepts. I argue, first, that Frege had reason to be interested in the question how the axioms and rules of his formal theory might be justified and, second, that he explicitly commits himself to offering a justification that appeals to the notion of reference. I then discuss the justifications Frege offered, focusing on his discussion of inferences involving free variables, in section 17 of Grundgesetze, and his argument, in sections 29-32, that every well-formed expression of his formal language has a unique reference.
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.
The standard approach to what I call “proof-theoretic semantics”, which is mainly due to Dummett and Prawitz, attempts to
give a semantics of proofs by defining what counts as a valid proof. After a discussion of the general aims of proof-theoretic
semantics, this paper investigates in detail various notions of proof-theoretic validity and offers certain improvements of
the definitions given by Prawitz. Particular emphasis is placed on the relationship between semantic validity concepts and
validity concepts used in normalization theory. It is argued that these two sorts of concepts must be kept strictly apart.
The celebrated theorem of Löwenheim and Skolem tells us that every consistent set S of quantificational schemata (i.e., every set of well-formed formulas of the lower predicate calculus admitting of a true interpretation in some non-empty universe) admits of a true numerical interpretation (i.e., an interpretation of predicate letters such that all schemata of S come out true when the variables of quantification are construed as ranging over just the positive integers).
Later literature goes farther, and shows how, given S , actually to produce a numerical interpretation which will fit S in case S is consistent. The general case is covered by Kleene (see Bibliography). The special case where S contains just one schema (or any finite number, since we can form their conjunction) had been dealt with by Hilbert and Bernays. Certain extensions, along lines not to be embarked on here, have been made by Kleene, Kreisel, Hasenjäger, and Wang.
My present purpose is expository: to make the construction of the numerical interpretation, and the proof of its adequacy, more easily intelligible than they hitherto have been. The reasoning is mainly Kleene's, though closer in some ways to earlier reasoning of Gödel.
‘The consistency of Frege's foundations of arithmetic On Being and Saying: Essays in Honor of Richard Cartwright
- G Boolos
Introduction to Metamathematics
- S C Kleene
- P Noordhoff