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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

15647

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

1

p.2

15648

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

1

p.2

15649

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

1

p.2

15650

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

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 secondorder comprehension assumptions about properties

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 truthdefinition, though it could admit it at a later stage

1

p.6

16297

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

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

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

15.1

p.211

16329

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

15.2

p.212

16330

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

15.3

p.217

16331

The KF is much stronger deductively than 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 recent truththeories are exploring typefree

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 nonnegative 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 [PG]

7

p.53

16315

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

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 'typefree'
