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Single Idea 16310
[filed under theme 5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
]
Full Idea
A theory is a set of formulae closed under first-order logical consequence.
Gist of Idea
A theory is some formulae and all of their consequences
Source
Volker Halbach (Axiomatic Theories of Truth [2011], 5.1)
Book Ref
Halbach,Volker: 'Axiomatic Theories of Truth' [CUP 2011], p.29
The
47 ideas
from 'Axiomatic Theories of Truth'
16292
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An explicit definition enables the elimination of what is defined
[Halbach]
|
16293
|
Traditional definitions of truth often make it more obscure, rather than less
[Halbach]
|
16294
|
Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage
[Halbach]
|
16297
|
Semantic theories avoid Tarski's Theorem by sticking to a sublanguage
[Halbach]
|
16324
|
Any definition of truth requires a metalanguage
[Halbach]
|
16325
|
Analysis rests on natural language, but its ideal is a framework which revises language
[Halbach]
|
16326
|
The main semantic theories of truth are Kripke's theory, and revisions semantics
[Halbach]
|
16327
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Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth
[Halbach]
|
16329
|
Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts
[Halbach]
|
16330
|
Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth
[Halbach]
|
16331
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The KF is much stronger deductively than FS, which relies on classical truth
[Halbach]
|
16332
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The KF theory is useful, but it is not a theory containing its own truth predicate
[Halbach]
|
16333
|
The underestimated costs of giving up classical logic are found in mathematical reasoning
[Halbach]
|
16335
|
In Strong Kleene logic a disjunction just needs one disjunct to be true
[Halbach]
|
16334
|
In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value
[Halbach]
|
16336
|
The liar paradox applies truth to a negated truth (but the conditional will serve equally)
[Halbach]
|
16337
|
Disquotational truth theories are short of deductive power
[Halbach]
|
16299
|
Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory
[Halbach]
|
16298
|
We need propositions to ascribe the same beliefs to people with different languages
[Halbach]
|
16338
|
Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge
[Halbach]
|
16340
|
Truth axioms need a base theory, because that is where truth issues arise
[Halbach]
|
16339
|
Truth axioms prove objects exist, so truth doesn't seem to be a logical notion
[Halbach]
|
16341
|
Normally we only endorse a theory if we believe it to be sound
[Halbach]
|
16344
|
Soundness must involve truth; the soundness of PA certainly needs it
[Halbach]
|
16342
|
You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system
[Halbach]
|
16343
|
The global reflection principle seems to express the soundness of Peano Arithmetic
[Halbach]
|
16345
|
That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction
[Halbach]
|
16346
|
Maybe necessity is a predicate, not the usual operator, to make it more like truth
[Halbach]
|
16347
|
Many new paradoxes may await us when we study interactions between frameworks
[Halbach]
|
16301
|
If people have big doubts about truth, a definition might give it more credibility
[Halbach]
|
16305
|
We know a complete axiomatisation of truth is not feasible
[Halbach]
|
16307
|
Don't trust analogies; they are no more than a guideline
[Halbach]
|
16308
|
Set theory was liberated early from types, and recent truth-theories are exploring type-free
[Halbach]
|
16309
|
Every attempt at formal rigour uses some set theory
[Halbach]
|
16310
|
A theory is some formulae and all of their consequences
[Halbach]
|
16311
|
To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction'
[Halbach]
|
16312
|
To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals
[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]
|
16316
|
Deflationists say truth is just for expressing infinite conjunctions or generalisations
[Halbach]
|
16317
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The main problem for deflationists is they can express generalisations, but not prove them
[Halbach]
|
16319
|
Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism
[Halbach]
|
16320
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Some say deflationism is axioms which are conservative over the base theory
[Halbach]
|
16318
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Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents
[Halbach]
|
16321
|
The compactness theorem can prove nonstandard models of PA
[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]
|