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
16325

Analysis rests on natural language, but its ideal is a framework which revises language

2. Reason / D. Definition / 2. Aims of Definition
16292

An explicit definition enables the elimination of what is defined

2. Reason / E. Argument / 3. Analogy
16307

Don't trust analogies; they are no more than a guideline

3. Truth / A. Truth Problems / 1. Truth
16330

Truthvalue 'gluts' allow two truth values together; 'gaps' give a partial conception of truth

16339

Truth axioms prove objects exist, so truth doesn't seem to be a logical notion

3. Truth / A. Truth Problems / 2. Defining Truth
15647

Truth definitions don't produce a good theory, because they go beyond your current language

16293

Traditional definitions of truth often make it more obscure, rather than less

16301

If people have big doubts about truth, a definition might give it more credibility

16324

Any definition of truth requires a metalanguage

3. Truth / C. Correspondence Truth / 3. Correspondence Truth critique
16296

Tarski's Theorem renders any precise version of correspondence impossible

3. Truth / F. Semantic Truth / 1. Tarski's Truth / c. Metalanguage for truth
15649

In semantic theories of truth, the predicate is in an objectlanguage, and the definition in a metalanguage

16297

Semantic theories avoid Tarski's Theorem by sticking to a sublanguage

3. Truth / F. Semantic Truth / 2. Semantic Truth
16337

Disquotational truth theories are short of deductive power

3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
15650

Axiomatic theories of truth need a weak logical framework, and not a strong metatheory

15655

Should axiomatic truth be 'conservative'  not proving anything apart from implications of the axioms?

15654

If truth is defined it can be eliminated, whereas axiomatic truth has various commitments

15648

Instead of a truth definition, add a primitive truth predicate, and axioms for how it works

16294

Axiomatic truth doesn't presuppose a truthdefinition, though it could admit it at a later stage

16326

The main semantic theories of truth are Kripke's theory, and revisions semantics

16322

CT proves PA consistent, which PA can't do on its own, so CT is not conservative over PA

16318

Compositional Truth CT has the truth of a sentence depending of the semantic values of its constituents

16315

The Tarski Biconditional theory TB is Peano Arithmetic, plus truth, plus all Tarski biconditionals

16299

Gödel numbering means a theory of truth can use Peano Arithmetic as its base theory

16340

Truth axioms need a base theory, because that is where truth issues arise

16305

We know a complete axiomatisation of truth is not feasible

16314

Theories of truth are 'typed' (truth can't apply to sentences containing 'true'), or 'typefree'

16311

To axiomatise Tarski's truth definition, we need a binary predicate for his 'satisfaction'

16313

A theory is 'conservative' if it adds no new theorems to its base theory [PG]

3. Truth / G. Axiomatic Truth / 2. FS Truth Axioms
16327

FriedmanSheard is typefree Compositional Truth, with two inference rules for truth

3. Truth / G. Axiomatic Truth / 3. KF Truth Axioms
16331

The KF is much stronger deductively that FS, which relies on classical truth

16332

The KF theory is useful, but it is not a theory containing its own truth predicate

16329

KripkeFeferman theory KF axiomatises Kripke fixedpoints, with Strong Kleene logic with gluts

3. Truth / H. Deflationary Truth / 2. Deflationary Truth
15656

Deflationists say truth merely serves to express infinite conjunctions

16316

Deflationists say truth is just for expressing infinite conjunctions or generalisations

16338

Deflationism says truth is a disquotation device to express generalisations, adding no new knowledge

16317

The main problem for deflationists is they can express generalisations, but not prove them

16320

Some say deflationism is axioms which are conservative over the base theory

16319

Compositional Truth CT proves generalisations, so is preferred in discussions of deflationism

4. Formal Logic / E. Nonclassical Logics / 3. ManyValued Logic
16335

In Strong Kleene logic a disjunction just needs one disjunct to be true

16334

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
15657

To prove the consistency of set theory, we must go beyond set theory

16309

Every attempt at formal rigour uses some set theory

5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
16333

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
15652

We can use truth instead of ontologically loaded secondorder comprehension assumptions about properties

5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
15651

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
16310

A theory is some formulae and all of their consequences

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
16323

The object language/ metalanguage distinction is the basis of model theory

5. Theory of Logic / K. Features of Logics / 3. Soundness
16342

You cannot just say all of Peano arithmetic is true, as 'true' isn't part of the system

16341

Normally we only endorse a theory if we believe it to be sound

16344

Soundness must involve truth; the soundness of PA certainly needs it

5. Theory of Logic / L. Paradox / 1. Paradox
16347

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
16336

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
16321

The compactness theorem can prove nonstandard models of PA

16343

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
15653

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
16312

To reduce PA to ZF, we represent the nonnegative integers with von Neumann ordinals

6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
16308

Set theory was liberated early from types, and recently truththeories are exploring typefree

7. Existence / C. Structure of Existence / 2. Reduction
16345

That Peano arithmetic is interpretable in ZF set theory is taken by philosophers as a reduction

10. Modality / A. Necessity / 2. Nature of Necessity
16346

Maybe necessity is a predicate, not the usual operator, to make it more like truth

19. Language / D. Propositions / 4. Mental Propositions
16298

We need propositions to ascribe the same beliefs to people with different languages
