Ideas from 'Axiomatic Theories of Truth' by Volker Halbach [2011], by Theme Structure
[found in 'Axiomatic Theories of Truth' by Halbach,Volker [CUP 2011,9780521115810]].
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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
16339

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

16330

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

3. Truth / A. Truth Problems / 2. Defining Truth
16293

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

16324

Any definition of truth requires a metalanguage

16301

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

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

16322

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

16314

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

16294

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

16318

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

16326

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

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]

16315

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

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
16329

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

16332

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

16331

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

3. Truth / H. Deflationary Truth / 2. Deflationary Truth
16316

Deflationists say truth is just for expressing infinite conjunctions or generalisations

16317

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

16338

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

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
16334

In Weak Kleene logic there are 'gaps', neither true nor false if one component lacks a truth value

16335

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

4. Formal Logic / F. Set Theory ST / 1. 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 / E. Structures of Logic / 8. Theories in Logic
16310

A theory is some formulae and all of their consequences

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

16342

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

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 / 4. 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 / 6. 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
