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 | 15649 | In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage |
1 | p.2 | 15648 | Instead of a truth definition, add a primitive truth predicate, and axioms for how it works |
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 |
1.2 | p.3 | 15653 | We can add Reflexion Principles to Peano Arithmetic, which assert its consistency or soundness |
1.3 | p.4 | 15654 | If truth is defined it can be eliminated, whereas axiomatic truth has various commitments |
1.3 | p.4 | 15655 | Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms? |
1.3 | p.4 | 15656 | Deflationists say truth merely serves to express infinite conjunctions |
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.5 | 16296 | Tarski's Theorem renders any precise version of correspondence impossible |
1 | p.6 | 16297 | Semantic theories avoid Tarski's Theorem by sticking to a sublanguage |
11 | p.146 | 16323 | The object language/ metalanguage distinction is the basis of model theory |
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 that 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 | 16341 | Normally we only endorse a theory if we believe it to be sound |
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.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 recently 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 |
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' |