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Ideas of Volker Halbach, by Text

[German, fl. 2010, Reader at the University of Oxford.]

2005 Axiomatic Theories of Truth (2005 ver)
1 p.2 15647 Truth definitions don't produce a good theory, because they go beyond your current language
1 p.2 15650 Axiomatic theories of truth need a weak logical framework, and not a strong metatheory
1 p.2 15648 Instead of a truth definition, add a primitive truth predicate, and axioms for how it works
1 p.2 15649 In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage
1.1 p.2 15651 Instead of saying x has a property, we can say a formula is true of x - as long as we have 'true'
1.1 p.3 15652 We can use truth instead of ontologically loaded second-order comprehension assumptions about properties
1.3 p.4 15656 Deflationists say truth merely serves to express infinite conjunctions
1.3 p.4 15655 Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms?
1.3 p.4 15654 If truth is defined it can be eliminated, whereas axiomatic truth has various commitments
2.1 p.5 15657 To prove the consistency of set theory, we must go beyond set theory
2011 Axiomatic Theories of Truth
1 p.3 16293 Traditional definitions of truth often make it more obscure, rather than less
1 p.3 16292 An explicit definition enables the elimination of what is defined
1 p.4 16294 Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage
1 p.6 16297 Semantic theories avoid Tarski's Theorem by sticking to a sublanguage
11 p.148 16324 Any definition of truth requires a metalanguage
12 p.150 16325 Analysis rests on natural language, but its ideal is a framework which revises language
14 p.163 16326 The main semantic theories of truth are Kripke's theory, and revisions semantics
15 p.195 16327 Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth
15.1 p.211 16329 Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts
15.2 p.212 16330 Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth
15.3 p.217 16331 The KF is much stronger deductively than FS, which relies on classical truth
16 p.229 16332 The KF theory is useful, but it is not a theory containing its own truth predicate
16.2 p.245 16333 The underestimated costs of giving up classical logic are found in mathematical reasoning
18 p.263 16335 In Strong Kleene logic a disjunction just needs one disjunct to be true
18 p.263 16334 In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value
19.3 p.275 16336 The liar paradox applies truth to a negated truth (but the conditional will serve equally)
19.5 p.280 16337 Disquotational truth theories are short of deductive power
2 p.11 16298 We need propositions to ascribe the same beliefs to people with different languages
2 p.13 16299 Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory
21 p.306 16338 Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge
21.2 p.314 16340 Truth axioms need a base theory, because that is where truth issues arise
21.2 p.314 16339 Truth axioms prove objects exist, so truth doesn't seem to be a logical notion
22.1 p.322 16342 You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system
22.1 p.322 16341 Normally we only endorse a theory if we believe it to be sound
22.1 p.323 16344 Soundness must involve truth; the soundness of PA certainly needs it
22.1 p.323 16343 The global reflection principle seems to express the soundness of Peano Arithmetic
23 p.330 16345 That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction
24.2 p.340 16346 Maybe necessity is a predicate, not the usual operator, to make it more like truth
24.2 p.341 16347 Many new paradoxes may await us when we study interactions between frameworks
3 p.15 16301 If people have big doubts about truth, a definition might give it more credibility
3 p.23 16305 We know a complete axiomatisation of truth is not feasible
4 p.25 16307 Don't trust analogies; they are no more than a guideline
4 p.25 16308 Set theory was liberated early from types, and recent truth-theories are exploring type-free
4.1 p.25 16309 Every attempt at formal rigour uses some set theory
5.1 p.29 16310 A theory is some formulae and all of their consequences
5.2 p.35 16311 To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction'
6 p.41 16312 To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals
6 Df 6.6 p.44 16313 A theory is 'conservative' if it adds no new theorems to its base theory [PG]
7 p.53 16315 The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals
7 p.56 16316 Deflationists say truth is just for expressing infinite conjunctions or generalisations
7 p.61 16317 The main problem for deflationists is they can express generalisations, but not prove them
8 p.66 16318 Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents
8 p.67 16319 Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism
8 p.67 16320 Some say deflationism is axioms which are conservative over the base theory
8.3 p.83 16321 The compactness theorem can prove nonstandard models of PA
8.6 p.106 16322 CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA
II Intro p.51 16314 Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free'