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Single Idea 16322
[filed under theme 3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
]
Full Idea
Compositional Truth CT proves the consistency of Peano arithmetic, which is not provable in Peano arithmetic by Gödel's second incompleteness theorem. Hence the theory CT is not conservative over Peano arithmetic.
Gist of Idea
CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA
Source
Volker Halbach (Axiomatic Theories of Truth [2011], 8.6)
Book Ref
Halbach,Volker: 'Axiomatic Theories of Truth' [CUP 2011], p.106
Related Ideas
Idea 16318
Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents [Halbach]
Idea 10122
Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman]
Idea 16313
A theory is 'conservative' if it adds no new theorems to its base theory [Halbach, by PG]
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
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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
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Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge
[Halbach]
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16340
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Truth axioms need a base theory, because that is where truth issues arise
[Halbach]
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16339
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Truth axioms prove objects exist, so truth doesn't seem to be a logical notion
[Halbach]
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16341
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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
|
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
|
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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