88 ideas
| 18495 | The best philosophers I know are the best people I know [Heil] |
| 18494 | Using a technical vocabulary actually prevents discussion of the presuppositions [Heil] |
| 18506 | Questions of explanation should not be confused with metaphyics [Heil] |
| 18535 | Without abstraction we couldn't think systematically [Heil] |
| 18534 | Truth relates truthbearers to truthmakers [Heil] |
| 18531 | Philosophers of the past took the truthmaking idea for granted [Heil] |
| 18509 | Not all truths need truthmakers - mathematics and logic seem to be just true [Heil] |
| 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] |
| 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] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [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] |
| 18518 | Infinite numbers are qualitatively different - they are not just very large numbers [Heil] |
| 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] |
| 18500 | How could structures be mathematical truthmakers? Maths is just true, without truthmakers [Heil] |
| 18539 | Our categories lack the neat arrangement needed for reduction [Heil] |
| 18505 | Fundamental ontology aims at the preconditions for any true theory [Heil] |
| 18499 | Our quantifications only reveal the truths we accept; the ontology and truthmakers are another matter [Heil] |
| 18512 | Ontology aims to give the fundamental categories of being [Heil] |
| 18508 | Most philosophers now (absurdly) believe that relations fully exist [Heil] |
| 18532 | If causal relations are power manifestations, that makes them internal relations [Heil] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 18510 | We need properties to explain how the world works [Heil] |
| 18522 | Categorical properties were introduced by philosophers as actual properties, not if-then properties [Heil] |
| 18513 | Emergent properties will need emergent substances to bear them [Heil] |
| 18540 | Predicates only match properties at the level of fundamentals [Heil] |
| 18533 | In Fa, F may not be a property of a, but a determinable, satisfied by some determinate [Heil] |
| 18511 | Properties have causal roles which sets can't possibly have [Heil] |
| 18523 | Are all properties powers, or are there also qualities, or do qualities have the powers? [Heil] |
| 18524 | Properties are both qualitative and dispositional - they are powerful qualities [Heil] |
| 18498 | Abstract objects wouldn't be very popular without the implicit idea of truthmakers [Heil] |
| 18507 | Substances bear properties, so must be simple, and not consist of further substances [Heil] |
| 18515 | Spatial parts are just regions, but objects depend on and are made up of substantial parts [Heil] |
| 18516 | A 'gunky' universe would literally have no parts at all [Heil] |
| 18514 | Many wholes can survive replacement of their parts [Heil] |
| 18517 | Dunes depend on sand grains, but line segments depend on the whole line [Heil] |
| 18502 | If basic physics has natures, then why not reality itself? That would then found the deepest necessities [Heil] |
| 18496 | If possible worlds are just fictions, they can't be truthmakers for modal judgements [Heil] |
| 18525 | Mental abstraction does not make what is abstracted mind-dependent [Heil] |
| 18504 | Only particulars exist, and generality is our mode of presentation [Heil] |
| 18503 | You can think of tomatoes without grasping what they are [Heil] |
| 18537 | Linguistic thought is just as imagistic as non-linguistic thought [Heil] |
| 18538 | Non-conscious thought may be unlike conscious thought [Heil] |
| 16383 | Puzzled Pierre has two mental files about the same object [Recanati on Kripke] |
| 18536 | The subject-predicate form reflects reality [Heil] |
| 18497 | Many reject 'moral realism' because they can't see any truthmakers for normative judgements [Heil] |
| 18519 | If there were infinite electrons, they could vanish without affecting total mass-energy [Heil] |
| 18526 | We should focus on actual causings, rather than on laws and causal sequences [Heil] |
| 18527 | Probabilistic causation is not a weak type of cause; it is just a probability of there being a cause [Heil] |
| 18520 | Electrons are treated as particles, but they lose their individuality in relations [Heil] |
| 18501 | Maybe the universe is fine-tuned because it had to be, despite plans by God or Nature? [Heil] |