81 ideas
| 17663 | If you know what it is, investigation is pointless. If you don't, investigation is impossible [Armstrong] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [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] |
| 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] |
| 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] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [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] |
| 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] |
| 10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
| 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [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] |
| 17688 | Negative facts are supervenient on positive facts, suggesting they are positive facts [Armstrong] |
| 17691 | Nothing is genuinely related to itself [Armstrong] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 17679 | All instances of some property are strictly identical [Armstrong] |
| 12677 | Armstrong holds that all basic properties are categorical [Armstrong, by Ellis] |
| 17666 | Actualism means that ontology cannot contain what is merely physically possible [Armstrong] |
| 17667 | Dispositions exist, but their truth-makers are actual or categorical properties [Armstrong] |
| 17687 | If everything is powers there is a vicious regress, as powers are defined by more powers [Armstrong] |
| 17678 | Universals are just the repeatable features of a world [Armstrong] |
| 17669 | Realist regularity theories of laws need universals, to pick out the same phenomena [Armstrong] |
| 17677 | Past, present and future must be equally real if universals are instantiated [Armstrong] |
| 15442 | Universals are abstractions from their particular instances [Armstrong, by Lewis] |
| 17686 | Universals are abstractions from states of affairs [Armstrong] |
| 17668 | It is likely that particulars can be individuated by unique conjunctions of properties [Armstrong] |
| 13197 | The notion of substance is one of the keys to true philosophy [Leibniz] |
| 17680 | The identity of a thing with itself can be ruled out as a pseudo-property [Armstrong] |
| 17693 | The necessary/contingent distinction may need to recognise possibilities as real [Armstrong] |
| 17685 | Induction aims at 'all Fs', but abduction aims at hidden or theoretical entities [Armstrong] |
| 17683 | Science suggests that the predicate 'grue' is not a genuine single universal [Armstrong] |
| 17675 | Unlike 'green', the 'grue' predicate involves a time and a change [Armstrong] |
| 17674 | The raven paradox has three disjuncts, confirmed by confirming any one of them [Armstrong] |
| 17672 | A good reason for something (the smoke) is not an explanation of it (the fire) [Armstrong] |
| 17684 | To explain observations by a regular law is to explain the observations by the observations [Armstrong] |
| 17676 | Best explanations explain the most by means of the least [Armstrong] |
| 17664 | Each subject has an appropriate level of abstraction [Armstrong] |
| 17692 | We can't deduce the phenomena from the One [Armstrong] |
| 17689 | Absences might be effects, but surely not causes? [Armstrong] |
| 17682 | A universe couldn't consist of mere laws [Armstrong] |
| 17662 | Science depends on laws of nature to study unobserved times and spaces [Armstrong] |
| 17690 | Oaken conditional laws, Iron universal laws, and Steel necessary laws [Armstrong, by PG] |
| 17670 | Newton's First Law refers to bodies not acted upon by a force, but there may be no such body [Armstrong] |
| 8582 | Regularities are lawful if a second-order universal unites two first-order universals [Armstrong, by Lewis] |
| 17671 | A naive regularity view says if it never occurs then it is impossible [Armstrong] |
| 17681 | The laws of nature link properties with properties [Armstrong] |
| 16246 | Rather than take necessitation between universals as primitive, just make laws primitive [Maudlin on Armstrong] |
| 9480 | Armstrong has an unclear notion of contingent necessitation, which can't necessitate anything [Bird on Armstrong] |
| 13198 | Gravity is within matter because of its structure, and it can be explained. [Leibniz] |