77 ideas
13860 | We can only learn from philosophers of the past if we accept the risk of major misrepresentation [Wright,C] |
13883 | The best way to understand a philosophical idea is to defend it [Wright,C] |
10142 | The attempt to define numbers by contextual definition has been revived [Wright,C, by Fine,K] |
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] |
9868 | An expression refers if it is a singular term in some true sentences [Wright,C, by Dummett] |
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] |
13861 | Number theory aims at the essence of natural numbers, giving their nature, and the epistemology [Wright,C] |
13892 | One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C] |
13867 | Instances of a non-sortal concept can only be counted relative to a sortal concept [Wright,C] |
17441 | Wright thinks Hume's Principle is more fundamental to cardinals than the Peano Axioms are [Wright,C, by Heck] |
13862 | There are five Peano axioms, which can be expressed informally [Wright,C] |
17853 | Number truths are said to be the consequence of PA - but it needs semantic consequence [Wright,C] |
17854 | What facts underpin the truths of the Peano axioms? [Wright,C] |
13894 | Sameness of number is fundamental, not counting, despite children learning that first [Wright,C] |
10140 | We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Wright,C, by Fine,K] |
8692 | Frege has a good system if his 'number principle' replaces his basic law V [Wright,C, by Friend] |
17440 | Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Wright,C, by Heck] |
13893 | It is 1-1 correlation of concepts, and not progression, which distinguishes natural number [Wright,C] |
13888 | If numbers are extensions, Frege must first solve the Caesar problem for extensions [Wright,C] |
13869 | Number platonism says that natural number is a sortal concept [Wright,C] |
13870 | We can't use empiricism to dismiss numbers, if numbers are our main evidence against empiricism [Wright,C] |
13873 | Treating numbers adjectivally is treating them as quantifiers [Wright,C] |
13899 | The Peano Axioms, and infinity of cardinal numbers, are logical consequences of how we explain cardinals [Wright,C] |
13896 | The aim is to follow Frege's strategy to derive the Peano Axioms, but without invoking classes [Wright,C] |
7804 | Wright has revived Frege's discredited logicism [Wright,C, by Benardete,JA] |
13863 | Logicism seemed to fail by Russell's paradox, Gödel's theorems, and non-logical axioms [Wright,C] |
13895 | The standard objections are Russell's Paradox, non-logical axioms, and Gödel's theorems [Wright,C] |
13884 | The idea that 'exist' has multiple senses is not coherent [Wright,C] |
13877 | Singular terms in true sentences must refer to objects; there is no further question about their existence [Wright,C] |
9878 | Contextually defined abstract terms genuinely refer to objects [Wright,C, by Dummett] |
13868 | Sortal concepts cannot require that things don't survive their loss, because of phase sortals [Wright,C] |
12189 | Logical necessity involves a decision about usage, and is non-realist and non-cognitive [Wright,C, by McFetridge] |
9723 | Sentences with 'if' are only conditionals if they can read as A-implies-B [Enderton] |
3568 | Surely ALL truths are externally justified, by the facts? [Cross,A] |
13866 | A concept is only a sortal if it gives genuine identity [Wright,C] |
13865 | 'Sortal' concepts show kinds, use indefinite articles, and require grasping identities [Wright,C] |
13890 | Entities fall under a sortal concept if they can be used to explain identity statements concerning them [Wright,C] |
13898 | If we can establish directions from lines and parallelism, we were already committed to directions [Wright,C] |
13882 | A milder claim is that understanding requires some evidence of that understanding [Wright,C] |
7320 | Holism cannot give a coherent account of scientific methodology [Wright,C, by Miller,A] |
13885 | If apparent reference can mislead, then so can apparent lack of reference [Wright,C] |
17857 | We can accept Frege's idea of object without assuming that predicates have a reference [Wright,C] |