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Single Idea 15350

[filed under theme 3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth ]

Full Idea

The Naïve Theory of Truth collects all the Tarski bi-conditionals of a language and takes them as axioms. But no consistent theory extending Peano arithmetic can prove all of them. It is inconsistent, and even formalises the liar paradox.

Clarification

The bi-conditional is 'the sentence-is-true is equivalent to the sentence'

Gist of Idea

The Naïve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar

Source

Leon Horsten (The Tarskian Turn [2011], 03.5.2)

Book Ref

Horsten,Leon: 'The Tarskian Turn' [MIT 2011], p.38

A Reaction

[compressed] This looks to me like the account of truth that Davidson was working with, since he just seemed to be compiling bi-conditionals for tricky cases. (Wrong! He championed the Compositional Theory, Horsten p.71)

The 35 ideas with the same theme [theories of truth built from a set of axioms]:

16300 Ockham had an early axiomatic account of truth [William of Ockham, by Halbach]
15716 If axioms and their implications have no contradictions, they pass my criterion of truth and existence [Hilbert]
19190 We need an undefined term 'true' in the meta-language, specified by axioms [Tarski]
15322 Tarski's had the first axiomatic theory of truth that was minimally adequate [Tarski, by Horsten]
16306 Tarski defined truth, but an axiomatisation can be extracted from his inductive clauses [Tarski, by Halbach]
19141 Tarski thought axiomatic truth was too contingent, and in danger of inconsistencies [Tarski, by Davidson]
23296 We can elucidate indefinable truth, but showing its relation to other concepts [Davidson]
14967 Certain three-valued languages can contain their own truth predicates [Kripke, by Gupta]
14966 The Tarskian move to a metalanguage may not be essential for truth theories [Kripke, by Gupta]
19137 We can get a substantive account of Tarski's truth by adding primitive 'true' to the object language [Etchemendy]
15332 'Reflexive' truth theories allow iterations (it is T that it is T that p) [Horsten]
15346 Axiomatic approaches to truth avoid the regress problem of semantic theories [Horsten]
15350 The Naïve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar [Horsten]
15351 Axiomatic theories take truth as primitive, and propose some laws of truth as axioms [Horsten]
15361 A good theory of truth must be compositional (as well as deriving biconditionals) [Horsten]
15367 By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content! [Horsten]
15371 An axiomatic theory needs to be of maximal strength, while being natural and sound [Horsten]
15373 Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models [Horsten]
16294 Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage [Halbach]
16326 The main semantic theories of truth are Kripke's theory, and revisions semantics [Halbach]
16299 Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory [Halbach]
16340 Truth axioms need a base theory, because that is where truth issues arise [Halbach]
16305 We know a complete axiomatisation of truth is not feasible [Halbach]
16311 To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction' [Halbach]
16313 A theory is 'conservative' if it adds no new theorems to its base theory [Halbach, by PG]
16315 The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals [Halbach]
16318 Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents [Halbach]
16322 CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA [Halbach]
16314 Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free' [Halbach]
15648 Instead of a truth definition, add a primitive truth predicate, and axioms for how it works [Halbach]
15650 Axiomatic theories of truth need a weak logical framework, and not a strong metatheory [Halbach]
15655 Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms? [Halbach]
15654 If truth is defined it can be eliminated, whereas axiomatic truth has various commitments [Halbach]
19124 A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness [Halbach/Leigh]
19126 If deflationary truth is not explanatory, truth axioms should be 'conservative', proving nothing new [Halbach/Leigh]