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Single Idea 15326
[filed under theme 5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
]
Full Idea
Aside from logic, so little is needed to generate the liar paradox that one wonders whether the laws of classical logic are unrestrictedly valid after all. (Many theories of truth have therefore been formulated in nonclassical logic.)
Gist of Idea
Doubt is thrown on classical logic by the way it so easily produces the liar paradox
Source
Leon Horsten (The Tarskian Turn [2011], 01.2)
Book Ref
Horsten,Leon: 'The Tarskian Turn' [MIT 2011], p.5
A Reaction
Kripke uses Strong Kleene logic for his theory. The implication is that debates discussed by Horsten actually have the status of classical logic at stake, as well as the nature of truth.
Related Idea
Idea 6006
If you say truly that you are lying, you are lying [Eubulides, by Dancy,R]
The
50 ideas
from Leon Horsten
10881
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The concept of 'ordinal number' is set-theoretic, not arithmetical
[Horsten]
|
10882
|
Predicative definitions only refer to entities outside the defined collection
[Horsten]
|
10884
|
A theory is 'categorical' if it has just one model up to isomorphism
[Horsten]
|
10885
|
Computer proofs don't provide explanations
[Horsten]
|
15324
|
Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles
[Horsten]
|
15323
|
Truth is a property, because the truth predicate has an extension
[Horsten]
|
15325
|
Inferential deflationism says truth has no essence because no unrestricted logic governs the concept
[Horsten]
|
15329
|
Nonclassical may accept T/F but deny applicability, or it may deny just T or F as well
[Horsten]
|
15326
|
Doubt is thrown on classical logic by the way it so easily produces the liar paradox
[Horsten]
|
15331
|
Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength
[Horsten]
|
15328
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A theory is 'non-conservative' if it facilitates new mathematical proofs
[Horsten]
|
15330
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Friedman-Sheard theory keeps classical logic and aims for maximum strength
[Horsten]
|
15332
|
'Reflexive' truth theories allow iterations (it is T that it is T that p)
[Horsten]
|
15334
|
The coherence theory allows multiple coherent wholes, which could contradict one another
[Horsten]
|
15336
|
The pragmatic theory of truth is relative; useful for group A can be useless for group B
[Horsten]
|
15333
|
Modern correspondence is said to be with the facts, not with true propositions
[Horsten]
|
15337
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The correspondence 'theory' is too vague - about both 'correspondence' and 'facts'
[Horsten]
|
15338
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We may believe in atomic facts, but surely not complex disjunctive ones?
[Horsten]
|
15340
|
Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa
[Horsten]
|
15341
|
Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms
[Horsten]
|
15344
|
Deflationism skips definitions and models, and offers just accounts of basic laws of truth
[Horsten]
|
15345
|
Semantic theories have a regress problem in describing truth in the languages for the models
[Horsten]
|
15346
|
Axiomatic approaches to truth avoid the regress problem of semantic theories
[Horsten]
|
15348
|
Propositions have sentence-like structures, so it matters little which bears the truth
[Horsten]
|
15347
|
A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding
[Horsten]
|
15349
|
It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F)
[Horsten]
|
15350
|
The Naďve 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]
|
15352
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A definition should allow the defined term to be eliminated
[Horsten]
|
15353
|
The first incompleteness theorem means that consistency does not entail soundness
[Horsten]
|
15354
|
Tarski's hierarchy lacks uniform truth, and depends on contingent factors
[Horsten]
|
15355
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Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated
[Horsten]
|
15356
|
Deflationism concerns the nature and role of truth, but not its laws
[Horsten]
|
15358
|
Deflationism says truth isn't a topic on its own - it just concerns what is true
[Horsten]
|
15357
|
Philosophy is the most general intellectual discipline
[Horsten]
|
15359
|
Deflation: instead of asserting a sentence, we can treat it as an object with the truth-property
[Horsten]
|
15360
|
ZFC showed that the concept of set is mathematical, not logical, because of its existence claims
[Horsten]
|
15361
|
A good theory of truth must be compositional (as well as deriving biconditionals)
[Horsten]
|
15363
|
In the supervaluationist account, disjunctions are not determined by their disjuncts
[Horsten]
|
15362
|
If 'Italy is large' lacks truth, so must 'Italy is not large'; but classical logic says it's large or it isn't
[Horsten]
|
15364
|
English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable
[Horsten]
|
15366
|
Satisfaction is a primitive notion, and very liable to semantical paradoxes
[Horsten]
|
15367
|
By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content!
[Horsten]
|
15368
|
This deflationary account says truth has a role in generality, and in inference
[Horsten]
|
15369
|
Set theory is substantial over first-order arithmetic, because it enables new proofs
[Horsten]
|
15370
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Predicativism says mathematical definitions must not include the thing being defined
[Horsten]
|
15371
|
An axiomatic theory needs to be of maximal strength, while being natural and sound
[Horsten]
|
15372
|
Some claim that indicative conditionals are believed by people, even though they are not actually held true
[Horsten]
|
15373
|
Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models
[Horsten]
|
15374
|
Truth has no 'nature', but we should try to describe its behaviour in inferences
[Horsten]
|