71 ideas
1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
10354 | Correspondence could be with other beliefs, rather than external facts [Kusch] |
10353 | Tarskians distinguish truth from falsehood by relations between members of sets [Kusch] |
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] |
9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
10337 | We can have knowledge without belief, if others credit us with knowledge [Kusch] |
10357 | Methodological Solipsism assumes all ideas could be derived from one mind [Kusch] |
10339 | Foundations seem utterly private, even from oneself at a later time [Kusch] |
10331 | Testimony is reliable if it coheres with evidence for a belief, and with other beliefs [Kusch] |
10338 | The coherentist restricts the space of reasons to the realm of beliefs [Kusch] |
10340 | Individualistic coherentism lacks access to all of my beliefs, or critical judgement of my assessment [Kusch] |
10345 | Individual coherentism cannot generate the necessary normativity [Kusch] |
10350 | Cultures decide causal routes, and they can be critically assessed [Kusch] |
10343 | Process reliabilism has been called 'virtue epistemology', resting on perception, memory, reason [Kusch] |
10341 | Justification depends on the audience and one's social role [Kusch] |
10334 | Testimony is an area in which epistemology meets ethics [Kusch] |
10336 | Powerless people are assumed to be unreliable, even about their own lives [Kusch] |
10324 | Testimony does not just transmit knowledge between individuals - it actually generates knowledge [Kusch] |
10327 | Some want to reduce testimony to foundations of perceptions, memories and inferences [Kusch] |
10329 | Testimony won't reduce to perception, if perception depends on social concepts and categories [Kusch] |
10330 | A foundation is what is intelligible, hence from a rational source, and tending towards truth [Kusch] |
10325 | Vindicating testimony is an expression of individualism [Kusch] |
10335 | Myths about lonely genius are based on epistemological individualism [Kusch] |
10323 | Communitarian Epistemology says 'knowledge' is a social status granted to groups of people [Kusch] |
10348 | Private justification is justification to imagined other people [Kusch] |
10349 | To be considered 'an individual' is performed by a society [Kusch] |
10344 | Our experience may be conceptual, but surely not the world itself? [Kusch] |
10358 | Often socialising people is the only way to persuade them [Kusch] |
10333 | Communitarianism in epistemology sees the community as the primary knower [Kusch] |
1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
10351 | Natural kinds are social institutions [Kusch] |
1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
10332 | Omniscience is incoherent, since knowledge is a social concept [Kusch] |
1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |