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

Eder, Günther

doi:10.1080/01445340.2015.1028610

Cited by 6 publications

(6 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

Section: Philosophical Backbone and Context Of This Workmentioning

confidence: 99%

“…As it is not exactly clear what formal theory Quine had in mind for his account-if he had any-the discussion of his account of logical validity is complicated and entangled with sophisticated historical points. Many have attacked him [15,4,8,13], and some have tried to defend him, or at least to soothe the critics [2,19,7].It is common to view the substitutional account as an alternative to the model theoretic account, and yet when it comes to make it precise, people have often used model theory. For example, [2] developed one of the best versions of the substitutional account, which however is framed in model theory.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Substitutional Validity for Modal Logic

Grossi

2023

Notre Dame J. Formal Logic

View full text Add to dashboard Cite

In the substitutional framework, validity is truth under all substitutions of the non logical vocabulary. I develop a theory where the □ is interpreted as substitutional validity. I show how to prove soundness and completeness for common modal calculi using this definition. Philosophical Backbone and Context of this Work1.1 Simple truth vs truth in a model According to the substitutional account of logical consequence, an argument is logically valid if it preserves truth under all uniform substitutions of the non logical vocabulary. The substitutional account has a longstanding pedigree. 1 Quine is probably the main modern proponent and the target of most of the contemporary discussion, especially in his 'Philosophy of Logic'. As it is not exactly clear what formal theory Quine had in mind for his account-if he had any-the discussion of his account of logical validity is complicated and entangled with sophisticated historical points. Many have attacked him [15,4,8,13], and some have tried to defend him, or at least to soothe the critics [2,19,7].It is common to view the substitutional account as an alternative to the model theoretic account, and yet when it comes to make it precise, people have often used model theory. For example, [2] developed one of the best versions of the substitutional account, which however is framed in model theory. First, they argue that Quine's definition of logical truth is incomplete 'since "truth" must be relative to an interpreted language'. Then, they consider one specific model-what they call a 'Quine model'-and define a sentence to be logically true when true in that model, under all substitutions. Whilst their account answers many questions others have raised, it neglects the fact that the substitutional account does not need the scaffolding of model theory to work: it is really a genuine alternative. In their system, on the other hand, the substitutional account is not very different from model theory: all the semantic work is done exactly in the same way, what changes is only the definition of validity.What is the point of developing an alternative to model theory to frame substitutional validity? In model theory we define the concept of truth in a model. However,

show abstract

Section: Philosophical Backbone and Context Of This Workmentioning

confidence: 99%

mentioning

confidence: 99%

Substitutional Validity for Modal Logic

Grossi

2023

Notre Dame J. Formal Logic

View full text Add to dashboard Cite

show abstract

“…Reductionism causes there problems similar to those discussed above. For a further discussion of Quine's theory see (Eder ).…”

mentioning

confidence: 99%

The Substitutional Analysis of Logical Consequence

Halbach

2018

Nous

View full text Add to dashboard Cite

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' interpretationthat 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. Consequentia 'formalis' vocatur quae in omnibus terminis valet retenta forma consimili. Vel si vis expresse loqui de vi sermonis, consequentia formalis est cui omnis propositio similis in forma quae formaretur esset bona consequentia [ . . . ] John Buridan, Tractatus de Consequentiis (Hubien 1976, I.3, p.22f) Formal Validity and Semantic Definitions of Logical ConsequenceAt the origin of logic is the observation that arguments sharing certain forms never have true premisses and a false conclusion. Similarly, all sentences of certain forms are always true. Traditionally, in order to show that an argument is not valid, logicians replaced nonlogical or categorematic terms in an argument with terms of the same respective grammatical category so that the premisses become true and the conclusion false. For instance, the following argument is not valid:Some horses are animals. Therefore all horses are animals. C 2018 Wiley Periodicals, Inc.

show abstract

“…Günther Eder objects to this predicate‐substitutional definition of satisfaction on the grounds that to prove it is co‐extensive with the model‐theoretic one, we need to prove a completeness result and the HB theorem (Eder , p. 191). This reasoning rests on a petitio .…”

mentioning

confidence: 99%

First‐order Logical Validity and the Hilbert‐bernays Theorem

Ebbs

Goldfarb

2018

Philosophical Issues

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 6 publications

References 16 publications

Substitutional Validity for Modal Logic

Substitutional Validity for Modal Logic

The Substitutional Analysis of Logical Consequence

First‐order Logical Validity and the Hilbert‐bernays Theorem

Contact Info

Product

Resources

About