Ideas from 'The Tarskian Turn' by Leon Horsten [2011], by Theme Structure

[found in 'The Tarskian Turn' by Horsten,Leon [MIT 2011,978-0-262-01586-8]].

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1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined
Philosophy is the most general intellectual discipline
2. Reason / D. Definition / 2. Aims of Definition
A definition should allow the defined term to be eliminated
3. Truth / A. Truth Problems / 1. Truth
Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles
Truth is a property, because the truth predicate has an extension
3. Truth / A. Truth Problems / 2. Defining Truth
Tarski proved that any reasonably expressive language suffers from the liar paradox
Truth has no 'nature', but we should try to describe its behaviour in inferences
3. Truth / A. Truth Problems / 5. Truth Bearers
Propositions have sentence-like structures, so it matters little which bears the truth
3. Truth / C. Correspondence Truth / 2. Correspondence to Facts
Modern correspondence is said to be with the facts, not with true propositions
3. Truth / C. Correspondence Truth / 3. Correspondence Truth critique
The correspondence 'theory' is too vague - about both 'correspondence' and 'facts'
3. Truth / D. Coherence Truth / 2. Coherence Truth Critique
The coherence theory allows multiple coherent wholes, which could contradict one another
3. Truth / E. Pragmatic Truth / 1. Pragmatic Truth
The pragmatic theory of truth is relative; useful for group A can be useless for group B
3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa
Tarski's hierarchy lacks uniform truth, and depends on contingent factors
3. Truth / F. Semantic Truth / 1. Tarski's Truth / c. Meta-language for truth
Semantic theories have a regress problem in describing truth in the languages for the models
3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
'Reflexive' truth theories allow iterations (it is T that it is T that p)
Axiomatic approaches to truth avoid the regress problem of semantic theories
A good theory of truth must be compositional (as well as deriving biconditionals)
The Naďve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar
Axiomatic theories take truth as primitive, and propose some laws of truth as axioms
By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content!
An axiomatic theory needs to be of maximal strength, while being natural and sound
Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models
3. Truth / G. Axiomatic Truth / 2. FS Truth Axioms
Friedman-Sheard theory keeps classical logic and aims for maximum strength
3. Truth / G. Axiomatic Truth / 3. KF Truth Axioms
Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength
3. Truth / H. Deflationary Truth / 2. Deflationary Truth
Inferential deflationism says truth has no essence because no unrestricted logic governs the concept
Deflationism skips definitions and models, and offers just accounts of basic laws of truth
Deflationism concerns the nature and role of truth, but not its laws
Deflationism says truth isn't a topic on its own - it just concerns what is true
Deflation: instead of asserting a sentence, we can treat it as an object with the truth-property
This deflationary account says truth has a role in generality, and in inference
4. Formal Logic / E. Nonclassical Logics / 1. Nonclassical Logics
Nonclassical may accept T/F but deny applicability, or it may deny just T or F as well
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
Doubt is thrown on classical logic by the way it so easily produces the liar paradox
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
A theory is 'non-conservative' if it facilitates new mathematical proofs
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F)
5. Theory of Logic / I. Semantics of Logic / 5. Satisfaction
Satisfaction is a primitive notion, and very liable to semantical paradoxes
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
The first incompleteness theorem means that consistency does not entail soundness
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
Strengthened Liar: 'this sentence is not true in any context' - in no context can this be evaluated
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
ZFC showed that the concept of set is mathematical, not logical, because of its existence claims
Set theory is substantial over first-order arithmetic, because it enables new proofs
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
Predicativism says mathematical definitions must not include the thing being defined
7. Existence / D. Theories of Reality / 7. Facts / b. Types of fact
We may believe in atomic facts, but surely not complex disjunctive ones?
7. Existence / D. Theories of Reality / 9. Vagueness / e. Supervaluation for vagueness
In the supervaluationist account, disjunctions are not determined by their disjuncts
If 'Italy is large' lacks truth, so must 'Italy is not large'; but classical logic says it's large or it isn't
11. Knowledge Aims / A. Knowledge / 3. Belief / c. Aim of beliefs
Some claim that indicative conditionals are believed by people, even though they are not actually held true
19. Language / C. Semantics / 1. Semantics
A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding