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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 Truth definitions don't produce a good theory, because they go beyond your current language
1 p.2 Instead of a truth definition, add a primitive truth predicate, and axioms for how it works
1 p.2 Axiomatic theories of truth need a weak logical framework, and not a strong metatheory
1 p.2 In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage
1.1 p.2 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 We can use truth instead of ontologically loaded second-order comprehension assumptions about properties
1.2 p.3 We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness
1.3 p.4 Deflationists say truth merely serves to express infinite conjunctions
1.3 p.4 Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms?
1.3 p.4 If truth is defined it can be eliminated, whereas axiomatic truth has various commitments
2.1 p.5 To prove the consistency of set theory, we must go beyond set theory
2011 Axiomatic Theories of Truth
1 p.3 Traditional definitions of truth often make it more obscure, rather than less
1 p.3 An explicit definition enables the elimination of what is defined
1 p.4 Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage
1 p.5 Tarski's Theorem renders any precise version of correspondence impossible
1 p.6 Semantic theories avoid Tarski's Theorem by sticking to a sublanguage
11 p.146 The object language/ metalanguage distinction is the basis of model theory
11 p.148 Any definition of truth requires a metalanguage
12 p.150 Analysis rests on natural language, but its ideal is a framework which revises language
14 p.163 The main semantic theories of truth are Kripke's theory, and revisions semantics
15 p.195 Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth
15.1 p.211 Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts
15.2 p.212 Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth
15.3 p.217 The KF is much stronger deductively that FS, which relies on classical truth
16 p.229 The KF theory is useful, but it is not a theory containing its own truth predicate
16.2 p.245 The underestimated costs of giving up classical logic are found in mathematical reasoning
18 p.263 In Strong Kleene logic a disjunction just needs one disjunct to be true
18 p.263 In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value
19.3 p.275 The liar paradox applies truth to a negated truth (but the conditional will serve equally)
19.5 p.280 Disquotational truth theories are short of deductive power
2 p.11 We need propositions to ascribe the same beliefs to people with different languages
2 p.13 Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory
21 p.306 Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge
21.2 p.314 Truth axioms need a base theory, because that is where truth issues arise
21.2 p.314 Truth axioms prove objects exist, so truth doesn't seem to be a logical notion
22.1 p.322 You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system
22.1 p.322 Normally we only endorse a theory if we believe it to be sound
22.1 p.323 Soundness must involve truth; the soundness of PA certainly needs it
22.1 p.323 The global reflection principle seems to express the soundness of Peano Arithmetic
23 p.330 That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction
24.2 p.340 Maybe necessity is a predicate, not the usual operator, to make it more like truth
24.2 p.341 Many new paradoxes may await us when we study interactions between frameworks
3 p.15 If people have big doubts about truth, a definition might give it more credibility
3 p.23 We know a complete axiomatisation of truth is not feasible
4 p.25 Don't trust analogies; they are no more than a guideline
4 p.25 Set theory was liberated early from types, and recently truth-theories are exploring type-free
4.1 p.25 Every attempt at formal rigour uses some set theory
5.1 p.29 A theory is some formulae and all of their consequences
5.2 p.35 To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction'
6 p.41 To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals
6 Df 6.6 p.44 A theory is 'conservative' if it adds no new theorems to its base theory
7 p.53 The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals
7 p.56 Deflationists say truth is just for expressing infinite conjunctions or generalisations
7 p.61 The main problem for deflationists is they can express generalisations, but not prove them
8 p.66 Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents
8 p.67 Some say deflationism is axioms which are conservative over the base theory
8 p.67 Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism
8.3 p.83 The compactness theorem can prove nonstandard models of PA
8.6 p.106 CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA
II Intro p.51 Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free'