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Single Idea 16339
[filed under theme 3. Truth / A. Truth Problems / 1. Truth
]
Full Idea
Two typed disquotation sentences, truth axioms of TB, suffice for proving that there at least two objects. Hence truth is not a logical notion if one expects logical notions to be ontologically neutral.
Gist of Idea
Truth axioms prove objects exist, so truth doesn't seem to be a logical notion
Source
Volker Halbach (Axiomatic Theories of Truth [2011], 21.2)
Book Ref
Halbach,Volker: 'Axiomatic Theories of Truth' [CUP 2011], p.314
Related Idea
Idea 16315
The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals [Halbach]
The
47 ideas
from 'Axiomatic Theories of Truth'
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16292
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An explicit definition enables the elimination of what is defined
[Halbach]
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16293
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Traditional definitions of truth often make it more obscure, rather than less
[Halbach]
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16297
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Semantic theories avoid Tarski's Theorem by sticking to a sublanguage
[Halbach]
|
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16294
|
Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage
[Halbach]
|
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16324
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Any definition of truth requires a metalanguage
[Halbach]
|
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16325
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Analysis rests on natural language, but its ideal is a framework which revises language
[Halbach]
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16326
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The main semantic theories of truth are Kripke's theory, and revisions semantics
[Halbach]
|
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16327
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Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth
[Halbach]
|
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16329
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Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts
[Halbach]
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16330
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Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth
[Halbach]
|
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16331
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The KF is much stronger deductively than FS, which relies on classical truth
[Halbach]
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16332
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The KF theory is useful, but it is not a theory containing its own truth predicate
[Halbach]
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16333
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The underestimated costs of giving up classical logic are found in mathematical reasoning
[Halbach]
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16335
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In Strong Kleene logic a disjunction just needs one disjunct to be true
[Halbach]
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16334
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In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value
[Halbach]
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16336
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The liar paradox applies truth to a negated truth (but the conditional will serve equally)
[Halbach]
|
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16337
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Disquotational truth theories are short of deductive power
[Halbach]
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16299
|
Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory
[Halbach]
|
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16298
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We need propositions to ascribe the same beliefs to people with different languages
[Halbach]
|
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16338
|
Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge
[Halbach]
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16340
|
Truth axioms need a base theory, because that is where truth issues arise
[Halbach]
|
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16339
|
Truth axioms prove objects exist, so truth doesn't seem to be a logical notion
[Halbach]
|
|
16341
|
Normally we only endorse a theory if we believe it to be sound
[Halbach]
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16344
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Soundness must involve truth; the soundness of PA certainly needs it
[Halbach]
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16342
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You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system
[Halbach]
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16343
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The global reflection principle seems to express the soundness of Peano Arithmetic
[Halbach]
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16345
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That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction
[Halbach]
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16346
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Maybe necessity is a predicate, not the usual operator, to make it more like truth
[Halbach]
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16347
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Many new paradoxes may await us when we study interactions between frameworks
[Halbach]
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16305
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We know a complete axiomatisation of truth is not feasible
[Halbach]
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16301
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If people have big doubts about truth, a definition might give it more credibility
[Halbach]
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16307
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Don't trust analogies; they are no more than a guideline
[Halbach]
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16308
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Set theory was liberated early from types, and recent truth-theories are exploring type-free
[Halbach]
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16309
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Every attempt at formal rigour uses some set theory
[Halbach]
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16310
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A theory is some formulae and all of their consequences
[Halbach]
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16311
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To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction'
[Halbach]
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16312
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To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals
[Halbach]
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16313
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A theory is 'conservative' if it adds no new theorems to its base theory
[Halbach, by PG]
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16315
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The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals
[Halbach]
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16317
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The main problem for deflationists is they can express generalisations, but not prove them
[Halbach]
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16316
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Deflationists say truth is just for expressing infinite conjunctions or generalisations
[Halbach]
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16318
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Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents
[Halbach]
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16319
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Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism
[Halbach]
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16320
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Some say deflationism is axioms which are conservative over the base theory
[Halbach]
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16321
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The compactness theorem can prove nonstandard models of PA
[Halbach]
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16322
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CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA
[Halbach]
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16314
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Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free'
[Halbach]
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