69 ideas
| 9724 | Until the 1960s the only semantics was truth-tables [Enderton] |
| 9703 | 'dom R' indicates the 'domain' of objects having a relation [Enderton] |
| 9705 | 'fld R' indicates the 'field' of all objects in the relation [Enderton] |
| 9704 | 'ran R' indicates the 'range' of objects being related to [Enderton] |
| 9710 | We write F:A→B to indicate that A maps into B (the output of F on A is in B) [Enderton] |
| 9707 | 'F(x)' is the unique value which F assumes for a value of x [Enderton] |
| 13201 | ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton] |
| 9712 | A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton] |
| 9713 | A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton] |
| 13204 | The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton] |
| 13206 | A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton] |
| 9699 | The 'powerset' of a set is all the subsets of a given set [Enderton] |
| 9700 | Two sets are 'disjoint' iff their intersection is empty [Enderton] |
| 9702 | A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton] |
| 9701 | A 'relation' is a set of ordered pairs [Enderton] |
| 9706 | A 'function' is a relation in which each object is related to just one other object [Enderton] |
| 9708 | A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton] |
| 9709 | A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton] |
| 9711 | A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton] |
| 9714 | A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton] |
| 9717 | A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton] |
| 13200 | Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton] |
| 13199 | The empty set may look pointless, but many sets can be constructed from it [Enderton] |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| 9715 | An 'equivalence relation' is a reflexive, symmetric and transitive binary relation [Enderton] |
| 9716 | We 'partition' a set into distinct subsets, according to each relation on its objects [Enderton] |
| 13202 | Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton] |
| 13205 | We can only define functions if Choice tells us which items are involved [Enderton] |
| 9722 | Inference not from content, but from the fact that it was said, is 'conversational implicature' [Enderton] |
| 9718 | Validity is either semantic (what preserves truth), or proof-theoretic (following procedures) [Enderton] |
| 9721 | A logical truth or tautology is a logical consequence of the empty set [Enderton] |
| 9994 | A truth assignment to the components of a wff 'satisfy' it if the wff is then True [Enderton] |
| 9719 | A proof theory is 'sound' if its valid inferences entail semantic validity [Enderton] |
| 9720 | A proof theory is 'complete' if semantically valid inferences entail proof-theoretic validity [Enderton] |
| 9995 | Proof in finite subsets is sufficient for proof in an infinite set [Enderton] |
| 9996 | Expressions are 'decidable' if inclusion in them (or not) can be proved [Enderton] |
| 9997 | For a reasonable language, the set of valid wff's can always be enumerated [Enderton] |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| 2730 | Because 'gold is malleable' is necessary does not mean that it is analytic [Audi,R] |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| 2715 | Beliefs are based on perception, memory, introspection or reason [Audi,R] |
| 2735 | Could you have a single belief on its own? [Audi,R] |
| 2736 | We can make certain of what we know, so knowing does not entail certainty [Audi,R] |
| 2721 | If you gradually remove a book's sensory properties, what is left at the end? [Audi,R] |
| 2722 | Sense-data theory is indirect realism, but phenomenalism is direct irrealism [Audi,R] |
| 2728 | The concepts needed for a priori thought may come from experience [Audi,R] |
| 2727 | Red and green being exclusive colours seems to be rationally graspable but not analytic [Audi,R] |
| 2717 | How could I see a field and believe nothing regarding it? [Audi,R] |
| 2716 | To see something as a field, I obviously need the concept of a field [Audi,R] |
| 2719 | Sense data imply representative realism, possibly only representing primary qualities [Audi,R] |
| 2720 | Sense-data (and the rival 'adverbial' theory) are to explain illusions and hallucinations [Audi,R] |
| 2718 | Perception is first simple, then objectual (with concepts) and then propositional [Audi,R] |
| 2729 | Virtually all rationalists assert that we can have knowledge of synthetic a priori truths [Audi,R] |
| 2741 | The principles of justification have to be a priori [Audi,R] |
| 2725 | To remember something is to know it [Audi,R] |
| 2724 | I might remember someone I can't recall or image, by recognising them on meeting [Audi,R] |
| 2731 | Justification is either unanchored (infinite or circular), or anchored (in knowledge or non-knowledge) [Audi,R] |
| 2739 | Internalism about justification implies that there is a right to believe something [Audi,R] |
| 2732 | Maths may be consistent with observations, but not coherent [Audi,R] |
| 2733 | It is very hard to show how much coherence is needed for justification [Audi,R] |
| 2734 | A consistent madman could have a very coherent belief system [Audi,R] |
| 2738 | Consistent accurate prediction looks like knowledge without justified belief [Audi,R] |
| 2740 | A reliability theory of knowledge seems to involve truth as correspondence [Audi,R] |
| 2737 | 'Reliable' is a very imprecise term, and may even mean 'justified' [Audi,R] |
| 2726 | We can be ignorant about ourselves, for example, our desires and motives [Audi,R] |
| 20064 | Actions are not mere effects of reasons, but are under their control [Audi,R] |