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Single Idea 16314
[filed under theme 3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
]
Full Idea
I sort theories of truth into the large families of 'typed' and 'type-free'. Roughly, typed theories prohibit a truth predicate's application to sentences with occurrences of that predicate, and one cannot prove the truth of sentences containing 'true'.
Gist of Idea
Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free'
Source
Volker Halbach (Axiomatic Theories of Truth [2011], II Intro)
Book Ref
Halbach,Volker: 'Axiomatic Theories of Truth' [CUP 2011], p.51
A Reaction
The problem sentence the typed theories are terrified of is the Liar Sentence. Typing produces a hierarchy of languages, referring down to the languages below them.
Related Idea
Idea 6006
If you say truly that you are lying, you are lying [Eubulides, by Dancy,R]
The
35 ideas
with the same theme
[theories of truth built from a set of axioms]:
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16300
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Ockham had an early axiomatic account of truth
[William of Ockham, by Halbach]
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15716
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If axioms and their implications have no contradictions, they pass my criterion of truth and existence
[Hilbert]
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19190
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We need an undefined term 'true' in the meta-language, specified by axioms
[Tarski]
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15322
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Tarski's had the first axiomatic theory of truth that was minimally adequate
[Tarski, by Horsten]
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16306
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Tarski defined truth, but an axiomatisation can be extracted from his inductive clauses
[Tarski, by Halbach]
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19141
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Tarski thought axiomatic truth was too contingent, and in danger of inconsistencies
[Tarski, by Davidson]
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23296
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We can elucidate indefinable truth, but showing its relation to other concepts
[Davidson]
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14967
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Certain three-valued languages can contain their own truth predicates
[Kripke, by Gupta]
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14966
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The Tarskian move to a metalanguage may not be essential for truth theories
[Kripke, by Gupta]
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19137
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We can get a substantive account of Tarski's truth by adding primitive 'true' to the object language
[Etchemendy]
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15332
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'Reflexive' truth theories allow iterations (it is T that it is T that p)
[Horsten]
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15346
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Axiomatic approaches to truth avoid the regress problem of semantic theories
[Horsten]
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15350
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The Naďve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar
[Horsten]
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15351
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Axiomatic theories take truth as primitive, and propose some laws of truth as axioms
[Horsten]
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15361
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A good theory of truth must be compositional (as well as deriving biconditionals)
[Horsten]
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15367
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By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content!
[Horsten]
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15371
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An axiomatic theory needs to be of maximal strength, while being natural and sound
[Horsten]
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15373
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Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models
[Horsten]
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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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16326
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The main semantic theories of truth are Kripke's theory, and revisions semantics
[Halbach]
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16299
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Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory
[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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16305
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We know a complete axiomatisation of truth is not feasible
[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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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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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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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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15648
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Instead of a truth definition, add a primitive truth predicate, and axioms for how it works
[Halbach]
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15650
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Axiomatic theories of truth need a weak logical framework, and not a strong metatheory
[Halbach]
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15655
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Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms?
[Halbach]
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15654
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If truth is defined it can be eliminated, whereas axiomatic truth has various commitments
[Halbach]
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19124
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A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness
[Halbach/Leigh]
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19126
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If deflationary truth is not explanatory, truth axioms should be 'conservative', proving nothing new
[Halbach/Leigh]
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