Ideas of Volker Halbach, by Theme
[German, fl. 2010, Reader at the University of Oxford.]
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1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
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16325
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Analysis rests on natural language, but its ideal is a framework which revises language
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2. Reason / D. Definition / 2. Aims of Definition
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16292
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An explicit definition enables the elimination of what is defined
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2. Reason / E. Argument / 3. Analogy
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16307
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Don't trust analogies; they are no more than a guideline
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3. Truth / A. Truth Problems / 1. Truth
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16330
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Truth-value 'gluts' allow two truth values together; 'gaps' give a partial conception of truth
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16339
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Truth axioms prove objects exist, so truth doesn't seem to be a logical notion
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3. Truth / A. Truth Problems / 2. Defining Truth
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15647
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Truth definitions don't produce a good theory, because they go beyond your current language
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16293
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Traditional definitions of truth often make it more obscure, rather than less
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16301
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If people have big doubts about truth, a definition might give it more credibility
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16324
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Any definition of truth requires a metalanguage
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3. Truth / F. Semantic Truth / 1. Tarski's Truth / c. Meta-language for truth
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15649
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In semantic theories of truth, the predicate is in an object-language, and the definition in a metalanguage
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16297
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Semantic theories avoid Tarski's Theorem by sticking to a sublanguage
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3. Truth / F. Semantic Truth / 2. Semantic Truth
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16337
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Disquotational truth theories are short of deductive power
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3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
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15655
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Should axiomatic truth be 'conservative' - not proving anything apart from implications of the axioms?
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15654
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If truth is defined it can be eliminated, whereas axiomatic truth has various commitments
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15650
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Axiomatic theories of truth need a weak logical framework, and not a strong metatheory
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15648
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Instead of a truth definition, add a primitive truth predicate, and axioms for how it works
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16322
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CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA
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16294
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Axiomatic truth doesn't presuppose a truth-definition, though it could admit it at a later stage
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16311
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To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction'
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16318
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Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents
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16326
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The main semantic theories of truth are Kripke's theory, and revisions semantics
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16299
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Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory
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16340
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Truth axioms need a base theory, because that is where truth issues arise
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16305
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We know a complete axiomatisation of truth is not feasible
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16313
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A theory is 'conservative' if it adds no new theorems to its base theory [PG]
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16315
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The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski bi-conditionals
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16314
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Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'type-free'
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3. Truth / G. Axiomatic Truth / 2. FS Truth Axioms
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16327
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Friedman-Sheard is type-free Compositional Truth, with two inference rules for truth
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3. Truth / G. Axiomatic Truth / 3. KF Truth Axioms
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16329
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Kripke-Feferman theory KF axiomatises Kripke fixed-points, with Strong Kleene logic with gluts
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16331
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The KF is much stronger deductively than FS, which relies on classical truth
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16332
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The KF theory is useful, but it is not a theory containing its own truth predicate
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3. Truth / H. Deflationary Truth / 2. Deflationary Truth
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15656
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Deflationists say truth merely serves to express infinite conjunctions
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16338
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Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge
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16317
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The main problem for deflationists is they can express generalisations, but not prove them
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16316
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Deflationists say truth is just for expressing infinite conjunctions or generalisations
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16320
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Some say deflationism is axioms which are conservative over the base theory
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16319
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Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism
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4. Formal Logic / E. Nonclassical Logics / 3. Many-Valued Logic
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16335
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In Strong Kleene logic a disjunction just needs one disjunct to be true
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16334
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In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
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15657
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To prove the consistency of set theory, we must go beyond set theory
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16309
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Every attempt at formal rigour uses some set theory
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5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
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16333
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The underestimated costs of giving up classical logic are found in mathematical reasoning
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5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
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15652
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We can use truth instead of ontologically loaded second-order comprehension assumptions about properties
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5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
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15651
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Instead of saying x has a property, we can say a formula is true of x - as long as we have 'true'
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5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
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16310
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A theory is some formulae and all of their consequences
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5. Theory of Logic / K. Features of Logics / 3. Soundness
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16341
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Normally we only endorse a theory if we believe it to be sound
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16344
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Soundness must involve truth; the soundness of PA certainly needs it
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16342
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You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system
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5. Theory of Logic / L. Paradox / 1. Paradox
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16347
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Many new paradoxes may await us when we study interactions between frameworks
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5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
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16336
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The liar paradox applies truth to a negated truth (but the conditional will serve equally)
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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16321
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The compactness theorem can prove nonstandard models of PA
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16343
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The global reflection principle seems to express the soundness of Peano Arithmetic
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6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
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16312
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To reduce PA to ZF, we represent the non-negative integers with von Neumann ordinals
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
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16308
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Set theory was liberated early from types, and recent truth-theories are exploring type-free
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7. Existence / C. Structure of Existence / 2. Reduction
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16345
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That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction
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10. Modality / A. Necessity / 2. Nature of Necessity
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16346
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Maybe necessity is a predicate, not the usual operator, to make it more like truth
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19. Language / D. Propositions / 4. Mental Propositions
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16298
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We need propositions to ascribe the same beliefs to people with different languages
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