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Single Idea 16308

[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory ]

Full Idea

While set theory was liberated much earlier from type restrictions, interest in type-free theories of truth only developed more recently.

Gist of Idea

Set theory was liberated early from types, and recent truth-theories are exploring type-free

Source

Volker Halbach (Axiomatic Theories of Truth [2011], 4)

Book Ref

Halbach,Volker: 'Axiomatic Theories of Truth' [CUP 2011], p.25

A Reaction

Tarski's theory of truth involves types (or hierarchies).

The 57 ideas from Volker Halbach

16292 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]
16297 Semantic theories avoid Tarski's Theorem by sticking to a sublanguage [Halbach]
16294 Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage [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 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 The KF is much stronger deductively than FS, which relies on classical truth [Halbach]
16332 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]
16343 The global reflection principle seems to express the soundness of Peano Arithmetic [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]
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]
16317 The main problem for deflationists is they can express generalisations, but not prove them [Halbach]
16316 Deflationists say truth is just for expressing infinite conjunctions or generalisations [Halbach]
16318 Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents [Halbach]
16319 Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism [Halbach]
16320 Some say deflationism is axioms which are conservative over the base theory [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]
15649 In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage [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]
15647 Truth definitions don't produce a good theory, because they go beyond your current language [Halbach]
15652 We can use truth instead of ontologically loaded second-order comprehension assumptions about properties [Halbach]
15651 Instead of saying x has a property, we can say a formula is true of x - as long as we have 'true' [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]
15656 Deflationists say truth merely serves to express infinite conjunctions [Halbach]
15657 To prove the consistency of set theory, we must go beyond set theory [Halbach]