47 ideas
| 7786 | Propositional logic handles negation, disjunction, conjunction; predicate logic adds quantifiers, predicates, relations [Girle] |
| 7798 | There are three axiom schemas for propositional logic [Girle] |
| 7799 | Proposition logic has definitions for its three operators: or, and, and identical [Girle] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 7797 | Axiom systems of logic contain axioms, inference rules, and definitions of proof and theorems [Girle] |
| 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] |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| 7794 | There are seven modalities in S4, each with its negation [Girle] |
| 7793 | ◊p → □◊p is the hallmark of S5 [Girle] |
| 7795 | S5 has just six modalities, and all strings can be reduced to those [Girle] |
| 7787 | Possible worlds logics use true-in-a-world rather than true [Girle] |
| 7788 | Modal logic has four basic modal negation equivalences [Girle] |
| 7796 | Modal logics were studied in terms of axioms, but now possible worlds semantics is added [Girle] |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| 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] |
| 7789 | Necessary implication is called 'strict implication'; if successful, it is called 'entailment' [Girle] |
| 4568 | If 'Queen of England' does not refer if there is no queen, its meaning can't refer if there is one [Cooper,DE] |
| 7790 | If an argument is invalid, a truth tree will indicate a counter-example [Girle] |
| 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] |
| 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] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 4574 | If some peoples do not have categories like time or cause, they can't be essential features of rationality [Cooper,DE] |
| 7800 | Analytic truths are divided into logically and conceptually necessary [Girle] |
| 7801 | Possibilities can be logical, theoretical, physical, economic or human [Girle] |
| 7792 | A world has 'access' to a world it generates, which is important in possible worlds semantics [Girle] |
| 4573 | If it is claimed that language correlates with culture, we must be able to identify the two independently [Cooper,DE] |
| 4575 | A person's language doesn't prove their concepts, but how are concepts deduced apart from language? [Cooper,DE] |
| 4561 | Many sentences set up dispositions which are irrelevant to the meanings of the sentences [Cooper,DE] |
| 4564 | I can meaningfully speculate that humans may have experiences currently impossible for us [Cooper,DE] |
| 4565 | The verification principle itself seems neither analytic nor verifiable [Cooper,DE] |
| 4563 | 'How now brown cow?' is used for elocution, but this says nothing about its meaning [Cooper,DE] |
| 4562 | Most people know how to use the word "Amen", but they do not know what it means [Cooper,DE] |
| 4571 | Reference need not be a hit-or-miss affair [Cooper,DE] |
| 4566 | Any thesis about reference is also a thesis about what exists to be referred to [Cooper,DE] |
| 4572 | If predicates name things, that reduces every sentence to a mere list of names [Cooper,DE] |
| 4576 | An analytic truth is one which becomes a logical truth when some synonyms have been replaced [Cooper,DE] |