69 ideas
| 8797 | The negation of all my beliefs about my current headache would be fully coherent [Sosa] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10405 | In the iterative conception of sets, they form a natural hierarchy [Swoyer] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 10407 | Logical Form explains differing logical behaviour of similar sentences [Swoyer] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
| 10605 | Two functions are the same if they have the same extension [Smith,P] |
| 10075 | A 'partial function' maps only some elements to another set [Smith,P] |
| 10074 | A 'total function' maps every element to one element in another set [Smith,P] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 10598 | A theory is 'negation complete' if it proves all sentences or their negation [Smith,P] |
| 10597 | 'Complete' applies both to whole logics, and to theories within them [Smith,P] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P] |
| 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P] |
| 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P] |
| 10081 | A set is 'enumerable' is all of its elements can result from a natural number function [Smith,P] |
| 10083 | A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P] |
| 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) [Smith,P] |
| 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable [Smith,P] |
| 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P] |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [Smith,P] |
| 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers [Smith,P] |
| 10850 | Baby Arithmetic is complete, but not very expressive [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [Smith,P] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10421 | Supervenience is nowadays seen as between properties, rather than linguistic [Swoyer] |
| 10410 | Anti-realists can't explain different methods to measure distance [Swoyer] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 10399 | If a property such as self-identity can only be in one thing, it can't be a universal [Swoyer] |
| 10416 | Can properties have parts? [Swoyer] |
| 10417 | There are only first-order properties ('red'), and none of higher-order ('coloured') [Swoyer] |
| 10413 | The best-known candidate for an identity condition for properties is necessary coextensiveness [Swoyer] |
| 10402 | Various attempts are made to evade universals being wholly present in different places [Swoyer] |
| 10400 | Conceptualism says words like 'honesty' refer to concepts, not to properties [Swoyer] |
| 10403 | If properties are abstract objects, then their being abstract exemplifies being abstract [Swoyer] |
| 10406 | One might hope to reduce possible worlds to properties [Swoyer] |
| 10404 | Extreme empiricists can hardly explain anything [Swoyer] |
| 8794 | There are very few really obvious truths, and not much can be proved from them [Sosa] |
| 8796 | A single belief can trail two regresses, one terminating and one not [Sosa] |
| 8799 | If mental states are not propositional, they are logically dumb, and cannot be foundations [Sosa] |
| 8795 | Mental states cannot be foundational if they are not immune to error [Sosa] |
| 8798 | Vision causes and justifies beliefs; but to some extent the cause is the justification [Sosa] |
| 10408 | Intensions are functions which map possible worlds to sets of things denoted by an expression [Swoyer] |
| 10409 | Research suggests that concepts rely on typical examples [Swoyer] |
| 10401 | The F and G of logic cover a huge range of natural language combinations [Swoyer] |
| 10420 | Maybe a proposition is just a property with all its places filled [Swoyer] |
| 10412 | If laws are mere regularities, they give no grounds for future prediction [Swoyer] |
| 10411 | Two properties can have one power, and one property can have two powers [Swoyer] |