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Single Idea 19124
[filed under theme 3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
]
Full Idea
If a natural theory of truth is added to Peano Arithmetic, it is not necessary to add explicity global reflection principles to assert soundness, as the truth theory proves them. Truth theories thus prove soundess, and allows its expression.
Gist of Idea
A natural theory of truth plays the role of reflection principles, establishing arithmetic's soundness
Source
Halbach,V/Leigh,G.E. (Axiomatic Theories of Truth (2013 ver) [2013], 1.2)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.3
A Reaction
This seems like a big attraction of axiomatic theories of truth for students of metamathematics.
Related Idea
Idea 19123
If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh]
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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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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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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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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15650
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Axiomatic theories of truth need a weak logical framework, and not a strong metatheory
[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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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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