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Single Idea 16333

[filed under theme 5. Theory of Logic / A. Overview of Logic / 6. Classical Logic ]

Full Idea

The costs of giving up classical logic are easily underestimated, …the price being paid in terms of mathematical reasoning.

Gist of Idea

The underestimated costs of giving up classical logic are found in mathematical reasoning

Source

Volker Halbach (Axiomatic Theories of Truth [2011], 16.2)

Book Ref

Halbach,Volker: 'Axiomatic Theories of Truth' [CUP 2011], p.245

A Reaction

No one cares much about such costs, until you say they are 'mathematical'. Presumably this is a message to Graham Priest and his pals.

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]
16342 You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system [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]
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]
16320 Some say deflationism is axioms which are conservative over the base theory [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]
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]
15650 Axiomatic theories of truth need a weak logical framework, and not a strong metatheory [Halbach]
15648 Instead of a truth definition, add a primitive truth predicate, and axioms for how it works [Halbach]
15649 In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage [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]
15656 Deflationists say truth merely serves to express infinite conjunctions [Halbach]
15654 If truth is defined it can be eliminated, whereas axiomatic truth has various commitments [Halbach]
15655 Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms? [Halbach]
15657 To prove the consistency of set theory, we must go beyond set theory [Halbach]