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All the ideas for 'Confessions of a Philosopher', 'Intermediate Logic' and 'The Tarskian Turn'

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119 ideas

1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined
Philosophy is the most general intellectual discipline [Horsten]
2. Reason / D. Definition / 2. Aims of Definition
A definition should allow the defined term to be eliminated [Horsten]
3. Truth / A. Truth Problems / 1. Truth
Semantic theories of truth seek models; axiomatic (syntactic) theories seek logical principles [Horsten]
Truth is a property, because the truth predicate has an extension [Horsten]
3. Truth / A. Truth Problems / 2. Defining Truth
Truth has no 'nature', but we should try to describe its behaviour in inferences [Horsten]
3. Truth / A. Truth Problems / 5. Truth Bearers
Propositions have sentence-like structures, so it matters little which bears the truth [Horsten]
3. Truth / C. Correspondence Truth / 2. Correspondence to Facts
Modern correspondence is said to be with the facts, not with true propositions [Horsten]
3. Truth / C. Correspondence Truth / 3. Correspondence Truth critique
The correspondence 'theory' is too vague - about both 'correspondence' and 'facts' [Horsten]
3. Truth / D. Coherence Truth / 2. Coherence Truth Critique
The coherence theory allows multiple coherent wholes, which could contradict one another [Horsten]
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 [Horsten]
3. Truth / F. Semantic Truth / 1. Tarski's Truth / a. Tarski's truth definition
Tarski's hierarchy lacks uniform truth, and depends on contingent factors [Horsten]
Tarski Bi-conditional: if you'll assert φ you'll assert φ-is-true - and also vice versa [Horsten]
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 [Horsten]
3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
Axiomatic approaches avoid limiting definitions to avoid the truth predicate, and limited sizes of models [Horsten]
'Reflexive' truth theories allow iterations (it is T that it is T that p) [Horsten]
Axiomatic approaches to truth avoid the regress problem of semantic theories [Horsten]
A good theory of truth must be compositional (as well as deriving biconditionals) [Horsten]
An axiomatic theory needs to be of maximal strength, while being natural and sound [Horsten]
The Naďve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar [Horsten]
Axiomatic theories take truth as primitive, and propose some laws of truth as axioms [Horsten]
By adding truth to Peano Arithmetic we increase its power, so truth has mathematical content! [Horsten]
3. Truth / G. Axiomatic Truth / 2. FS Truth Axioms
Friedman-Sheard theory keeps classical logic and aims for maximum strength [Horsten]
3. Truth / G. Axiomatic Truth / 3. KF Truth Axioms
Kripke-Feferman has truth gaps, instead of classical logic, and aims for maximum strength [Horsten]
3. Truth / H. Deflationary Truth / 2. Deflationary Truth
Inferential deflationism says truth has no essence because no unrestricted logic governs the concept [Horsten]
Deflationism skips definitions and models, and offers just accounts of basic laws of truth [Horsten]
Deflationism concerns the nature and role of truth, but not its laws [Horsten]
This deflationary account says truth has a role in generality, and in inference [Horsten]
Deflationism says truth isn't a topic on its own - it just concerns what is true [Horsten]
Deflation: instead of asserting a sentence, we can treat it as an object with the truth-property [Horsten]
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
Venn Diagrams map three predicates into eight compartments, then look for the conclusion [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
'Disjunctive Normal Form' is ensuring that no conjunction has a disjunction within its scope [Bostock]
'Conjunctive Normal Form' is ensuring that no disjunction has a conjunction within its scope [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock]
'Negation' says that Γ,¬φ|= iff Γ|=φ [Bostock]
'Disjunction' says that Γ,φ∨ψ|= iff Γ,φ|= and Γ,ψ|= [Bostock]
'Conjunction' says that Γ|=φ∧ψ iff Γ|=φ and Γ|=ψ [Bostock]
'Assumptions' says that a formula entails itself (φ|=φ) [Bostock]
'Thinning' allows that if premisses entail a conclusion, then adding further premisses makes no difference [Bostock]
The 'conditional' is that Γ|=φ→ψ iff Γ,φ|=ψ [Bostock]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock]
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 [Horsten]
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
A 'free' logic can have empty names, and a 'universally free' logic can have empty domains [Bostock]
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 [Horsten]
Truth is the basic notion in classical logic [Bostock]
Elementary logic cannot distinguish clearly between the finite and the infinite [Bostock]
Fictional characters wreck elementary logic, as they have contradictions and no excluded middle [Bostock]
5. Theory of Logic / B. Logical Consequence / 3. Deductive Consequence |-
The syntactic turnstile |- φ means 'there is a proof of φ' or 'φ is a theorem' [Bostock]
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid' [Bostock]
Validity is a conclusion following for premises, even if there is no proof [Bostock]
It seems more natural to express |= as 'therefore', rather than 'entails' [Bostock]
5. Theory of Logic / B. Logical Consequence / 5. Modus Ponens
MPP is a converse of Deduction: If Γ |- φ→ψ then Γ,φ|-ψ [Bostock]
MPP: 'If Γ|=φ and Γ|=φ→ψ then Γ|=ψ' (omit Γs for Detachment) [Bostock]
Deduction Theorem: ψ only derivable from φ iff φ→ψ are axioms [Horsten]
5. Theory of Logic / D. Assumptions for Logic / 4. Identity in Logic
The sign '=' is a two-place predicate expressing that 'a is the same thing as b' (a=b) [Bostock]
If we are to express that there at least two things, we need identity [Bostock]
|= α=α and α=β |= φ(α/ξ ↔ φ(β/ξ) fix identity [Bostock]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Truth-functors are usually held to be defined by their truth-tables [Bostock]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'zero-place' function just has a single value, so it is a name [Bostock]
A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock]
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
A theory is 'non-conservative' if it facilitates new mathematical proofs [Horsten]
5. Theory of Logic / F. Referring in Logic / 1. Naming / a. Names
In logic, a name is just any expression which refers to a particular single object [Bostock]
5. Theory of Logic / F. Referring in Logic / 1. Naming / e. Empty names
An expression is only a name if it succeeds in referring to a real object [Bostock]
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / b. Definite descriptions
Definite descriptions don't always pick out one thing, as in denials of existence, or errors [Bostock]
We are only obliged to treat definite descriptions as non-names if only the former have scope [Bostock]
Definite desciptions resemble names, but can't actually be names, if they don't always refer [Bostock]
Because of scope problems, definite descriptions are best treated as quantifiers [Bostock]
Definite descriptions are usually treated like names, and are just like them if they uniquely refer [Bostock]
5. Theory of Logic / F. Referring in Logic / 2. Descriptions / c. Theory of definite descriptions
Names do not have scope problems (e.g. in placing negation), but Russell's account does have that problem [Bostock]
5. Theory of Logic / G. Quantification / 1. Quantification
'Prenex normal form' is all quantifiers at the beginning, out of the scope of truth-functors [Bostock]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
If we allow empty domains, we must allow empty names [Bostock]
5. Theory of Logic / H. Proof Systems / 1. Proof Systems
An 'informal proof' is in no particular system, and uses obvious steps and some ordinary English [Bostock]
5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof
Quantification adds two axiom-schemas and a new rule [Bostock]
Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock]
5. Theory of Logic / H. Proof Systems / 3. Proof from Assumptions
'Conditonalised' inferences point to the Deduction Theorem: If Γ,φ|-ψ then Γ|-φ→ψ [Bostock]
The Deduction Theorem greatly simplifies the search for proof [Bostock]
Proof by Assumptions can always be reduced to Proof by Axioms, using the Deduction Theorem [Bostock]
The Deduction Theorem and Reductio can 'discharge' assumptions - they aren't needed for the new truth [Bostock]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock]
Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it [Bostock]
In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock]
Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E) [Bostock]
5. Theory of Logic / H. Proof Systems / 5. Tableau Proof
Tableau proofs use reduction - seeking an impossible consequence from an assumption [Bostock]
Non-branching rules add lines, and branching rules need a split; a branch with a contradiction is 'closed' [Bostock]
A completed open branch gives an interpretation which verifies those formulae [Bostock]
In a tableau proof no sequence is established until the final branch is closed; hypotheses are explored [Bostock]
Tableau rules are all elimination rules, gradually shortening formulae [Bostock]
Unlike natural deduction, semantic tableaux have recipes for proving things [Bostock]
A tree proof becomes too broad if its only rule is Modus Ponens [Bostock]
5. Theory of Logic / H. Proof Systems / 6. Sequent Calculi
Each line of a sequent calculus is a conclusion of previous lines, each one explicitly recorded [Bostock]
A sequent calculus is good for comparing proof systems [Bostock]
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
Interpretation by assigning objects to names, or assigning them to variables first [Bostock, by PG]
It is easier to imagine truth-value gaps (for the Liar, say) than for truth-value gluts (both T and F) [Horsten]
5. Theory of Logic / I. Semantics of Logic / 4. Satisfaction
Satisfaction is a primitive notion, and very liable to semantical paradoxes [Horsten]
5. Theory of Logic / I. Semantics of Logic / 5. Extensionalism
Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects [Bostock]
If an object has two names, truth is undisturbed if the names are swapped; this is Extensionality [Bostock]
5. Theory of Logic / K. Features of Logics / 2. Consistency
A set of formulae is 'inconsistent' when there is no interpretation which can make them all true [Bostock]
A proof-system is 'absolutely consistent' iff we don't have |-(S)φ for every formula [Bostock]
For 'negation-consistent', there is never |-(S)φ and |-(S)¬φ [Bostock]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
The first incompleteness theorem means that consistency does not entail soundness [Horsten]
5. Theory of Logic / K. Features of Logics / 6. Compactness
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact [Bostock]
Compactness means an infinity of sequents on the left will add nothing new [Bostock]
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 [Horsten]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable [Horsten]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Ordinary or mathematical induction assumes for the first, then always for the next, and hence for all [Bostock]
Complete induction assumes for all numbers less than n, then also for n, and hence for all numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 6. 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 [Horsten]
Set theory is substantial over first-order arithmetic, because it enables new proofs [Horsten]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
Predicativism says mathematical definitions must not include the thing being defined [Horsten]
7. Existence / D. Theories of Reality / 8. Facts / b. Types of fact
We may believe in atomic facts, but surely not complex disjunctive ones? [Horsten]
7. Existence / D. Theories of Reality / 10. Vagueness / f. Supervaluation for vagueness
If 'Italy is large' lacks truth, so must 'Italy is not large'; but classical logic says it's large or it isn't [Horsten]
In the supervaluationist account, disjunctions are not determined by their disjuncts [Horsten]
8. Modes of Existence / A. Relations / 4. Formal Relations / a. Types of relation
A relation is not reflexive, just because it is transitive and symmetrical [Bostock]
Relations can be one-many (at most one on the left) or many-one (at most one on the right) [Bostock]
9. Objects / F. Identity among Objects / 5. Self-Identity
If non-existent things are self-identical, they are just one thing - so call it the 'null object' [Bostock]
10. Modality / A. Necessity / 6. Logical Necessity
The idea that anything which can be proved is necessary has a problem with empty names [Bostock]
11. Knowledge Aims / A. Knowledge / 4. Belief / c. Aim of beliefs
Some claim that indicative conditionals are believed by people, even though they are not actually held true [Horsten]
16. Persons / C. Self-Awareness / 3. Limits of Introspection
Why don't we experience or remember going to sleep at night? [Magee]
19. Language / C. Assigning Meanings / 1. Syntax
A theory of syntax can be based on Peano arithmetic, thanks to the translation by Gödel coding [Horsten]
19. Language / C. Assigning Meanings / 3. Predicates
A (modern) predicate is the result of leaving a gap for the name in a sentence [Bostock]