Ideas of Volker Halbach, by Theme

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

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