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Single Idea 15328
[filed under theme 5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
]
Full Idea
A theory is 'non-conservative' if it allows us to prove mathematical facts that go beyond what the background mathematical theory can prove on its own.
Gist of Idea
A theory is 'non-conservative' if it facilitates new mathematical proofs
Source
Leon Horsten (The Tarskian Turn [2011], 01.4)
Book Ref
Horsten,Leon: 'The Tarskian Turn' [MIT 2011], p.7
A Reaction
This is an instance of the relationship with mathematics being used as the test case for explorations of logic. It is a standard research method, because it is so precise, but should not be mistaken for the last word about a theory.
The
46 ideas
from 'The Tarskian Turn'
15323
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Truth is a property, because the truth predicate has an extension
[Horsten]
|
15324
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Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles
[Horsten]
|
15325
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Inferential deflationism says truth has no essence because no unrestricted logic governs the concept
[Horsten]
|
15329
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Nonclassical may accept T/F but deny applicability, or it may deny just T or F as well
[Horsten]
|
15326
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Doubt is thrown on classical logic by the way it so easily produces the liar paradox
[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
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'Reflexive' truth theories allow iterations (it is T that it is T that p)
[Horsten]
|
15331
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Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength
[Horsten]
|
15338
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We may believe in atomic facts, but surely not complex disjunctive ones?
[Horsten]
|
15333
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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]
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15334
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The coherence theory allows multiple coherent wholes, which could contradict one another
[Horsten]
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15336
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The pragmatic theory of truth is relative; useful for group A can be useless for group B
[Horsten]
|
15340
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Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa
[Horsten]
|
15341
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Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms
[Horsten]
|
15345
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Semantic theories have a regress problem in describing truth in the languages for the models
[Horsten]
|
15344
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Deflationism skips definitions and models, and offers just accounts of basic laws of truth
[Horsten]
|
15346
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Axiomatic approaches to truth avoid the regress problem of semantic theories
[Horsten]
|
15348
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Propositions have sentence-like structures, so it matters little which bears the truth
[Horsten]
|
15347
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A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding
[Horsten]
|
15349
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It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F)
[Horsten]
|
15350
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The Naďve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar
[Horsten]
|
15351
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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
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The first incompleteness theorem means that consistency does not entail soundness
[Horsten]
|
15354
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Tarski's hierarchy lacks uniform truth, and depends on contingent factors
[Horsten]
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15355
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Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated
[Horsten]
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15356
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Deflationism concerns the nature and role of truth, but not its laws
[Horsten]
|
15357
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Philosophy is the most general intellectual discipline
[Horsten]
|
15358
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Deflationism says truth isn't a topic on its own - it just concerns what is true
[Horsten]
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15359
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Deflation: instead of asserting a sentence, we can treat it as an object with the truth-property
[Horsten]
|
15360
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ZFC showed that the concept of set is mathematical, not logical, because of its existence claims
[Horsten]
|
15361
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A good theory of truth must be compositional (as well as deriving biconditionals)
[Horsten]
|
15362
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If 'Italy is large' lacks truth, so must 'Italy is not large'; but classical logic says it's large or it isn't
[Horsten]
|
15363
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In the supervaluationist account, disjunctions are not determined by their disjuncts
[Horsten]
|
15364
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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
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By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content!
[Horsten]
|
15368
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This deflationary account says truth has a role in generality, and in inference
[Horsten]
|
15369
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Set theory is substantial over first-order arithmetic, because it enables new proofs
[Horsten]
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15370
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Predicativism says mathematical definitions must not include the thing being defined
[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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15372
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Some claim that indicative conditionals are believed by people, even though they are not actually held true
[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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15374
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Truth has no 'nature', but we should try to describe its behaviour in inferences
[Horsten]
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