Combining Texts

All the ideas for 'A Tour through Mathematical Logic', 'Two-Dimensional Semantics' and 'Second Philosophy'

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39 ideas

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
13520 A 'tautology' must include connectives [Wolf,RS]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
13524 Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
13521 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
13523 Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV
13529 Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
13526 Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
13534 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
13535 First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
10594 Henkin semantics is more plausible for plural logic than for second-order logic [Maddy]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13531 Model theory reveals the structures of mathematics [Wolf,RS]
13532 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
13519 Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
13533 First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
13537 An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
13539 The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 4. Completeness
13538 If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
13525 Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13530 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13518 Modern mathematics has unified all of its objects within set theory [Wolf,RS]
10. Modality / A. Necessity / 3. Types of Necessity
14703 Superficial necessity is true in all worlds; deep necessity is thus true, no matter which world is actual [Schroeter]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / b. Conceivable but impossible
14714 Contradictory claims about a necessary god both seem apriori coherent [Schroeter]
12. Knowledge Sources / A. A Priori Knowledge / 8. A Priori as Analytic
14704 2D semantics gives us apriori knowledge of our own meanings [Schroeter]
18. Thought / C. Content / 5. Twin Earth
14706 Your view of water depends on whether you start from the actual Earth or its counterfactual Twin [Schroeter]
18. Thought / C. Content / 7. Narrow Content
14711 Rationalists say knowing an expression is identifying its extension using an internal cognitive state [Schroeter]
19. Language / A. Nature of Meaning / 1. Meaning
14717 Internalist meaning is about understanding; externalist meaning is about embedding in a situation [Schroeter]
19. Language / C. Assigning Meanings / 2. Semantics
14720 Semantic theory assigns meanings to expressions, and metasemantics explains how this works [Schroeter]
19. Language / C. Assigning Meanings / 4. Compositionality
14695 Semantic theories show how truth of sentences depends on rules for interpreting and joining their parts [Schroeter]
19. Language / C. Assigning Meanings / 7. Extensional Semantics
14696 Simple semantics assigns extensions to names and to predicates [Schroeter]
14697 'Federer' and 'best tennis player' can't mean the same, despite having the same extension [Schroeter]
19. Language / C. Assigning Meanings / 8. Possible Worlds Semantics
14698 Possible worlds semantics uses 'intensions' - functions which assign extensions at each world [Schroeter]
14699 Possible worlds make 'I' and that person's name synonymous, but they have different meanings [Schroeter]
14709 Possible worlds semantics implies a constitutive connection between meanings and modal claims [Schroeter]
14719 In the possible worlds account all necessary truths are same (because they all map to the True) [Schroeter]
19. Language / C. Assigning Meanings / 10. Two-Dimensional Semantics
14701 Array worlds along the horizontal, and contexts (world,person,time) along the vertical [Schroeter]
14702 If we introduce 'actually' into modal talk, we need possible worlds twice to express this [Schroeter]
14705 Do we know apriori how we refer to names and natural kinds, but their modal profiles only a posteriori? [Schroeter]
14715 2D fans defend it for conceptual analysis, for meaning, and for internalist reference [Schroeter]
14716 2D semantics can't respond to contingent apriori claims, since there is no single proposition involved [Schroeter]