71 ideas
| 7950 | Philosophy tries to explain how the actual is possible, given that it seems impossible [Macdonald,C] |
| 7923 | 'Did it for the sake of x' doesn't involve a sake, so how can ontological commitments be inferred? [Macdonald,C] |
| 7933 | Don't assume that a thing has all the properties of its parts [Macdonald,C] |
| 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] |
| 7944 | Reduce by bridge laws (plus property identities?), by elimination, or by reducing talk [Macdonald,C] |
| 7938 | Relational properties are clearly not essential to substances [Macdonald,C] |
| 7967 | Being taller is an external relation, but properties and substances have internal relations [Macdonald,C] |
| 7965 | Does the knowledge of each property require an infinity of accompanying knowledge? [Macdonald,C] |
| 7934 | Tropes are abstract (two can occupy the same place), but not universals (they have locations) [Macdonald,C] |
| 7958 | Properties are sets of exactly resembling property-particulars [Macdonald,C] |
| 7972 | Tropes are abstract particulars, not concrete particulars, so the theory is not nominalist [Macdonald,C] |
| 7959 | How do a group of resembling tropes all resemble one another in the same way? [Macdonald,C] |
| 7960 | Trope Nominalism is the only nominalism to introduce new entities, inviting Ockham's Razor [Macdonald,C] |
| 7951 | Numerical sameness is explained by theories of identity, but what explains qualitative identity? [Macdonald,C] |
| 7964 | How can universals connect instances, if they are nothing like them? [Macdonald,C] |
| 7971 | Real Nominalism is only committed to concrete particulars, word-tokens, and (possibly) sets [Macdonald,C] |
| 7955 | Resemblance Nominalism cannot explain either new resemblances, or absence of resemblances [Macdonald,C] |
| 7961 | A 'thing' cannot be in two places at once, and two things cannot be in the same place at once [Macdonald,C] |
| 7926 | We 'individuate' kinds of object, and 'identify' particular specimens [Macdonald,C] |
| 7936 | Unlike bundles of properties, substances have an intrinsic unity [Macdonald,C] |
| 7930 | The bundle theory of substance implies the identity of indiscernibles [Macdonald,C] |
| 7932 | A phenomenalist cannot distinguish substance from attribute, so must accept the bundle view [Macdonald,C] |
| 7937 | When we ascribe a property to a substance, the bundle theory will make that a tautology [Macdonald,C] |
| 7939 | Substances persist through change, but the bundle theory says they can't [Macdonald,C] |
| 7940 | A substance might be a sequence of bundles, rather than a single bundle [Macdonald,C] |
| 7948 | A statue and its matter have different persistence conditions, so they are not identical [Macdonald,C] |
| 7929 | A substance is either a bundle of properties, or a bare substratum, or an essence [Macdonald,C] |
| 7941 | Each substance contains a non-property, which is its substratum or bare particular [Macdonald,C] |
| 7942 | The substratum theory explains the unity of substances, and their survival through change [Macdonald,C] |
| 7943 | A substratum has the quality of being bare, and they are useless because indiscernible [Macdonald,C] |
| 12887 | A whole must have one characteristic, an internal relation, and a structure [Rescher/Oppenheim] |
| 7927 | At different times Leibniz articulated three different versions of his so-called Law [Macdonald,C] |
| 7928 | The Identity of Indiscernibles is false, because it is not necessarily true [Macdonald,C] |
| 9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
| 7947 | In continuity, what matters is not just the beginning and end states, but the process itself [Macdonald,C] |