99 ideas
| 13734 | Modern Quinean metaphysics is about what exists, but Aristotelian metaphysics asks about grounding [Schaffer,J] |
| 13751 | If you tore the metaphysics out of philosophy, the whole enterprise would collapse [Schaffer,J] |
| 14600 | Analysis aims at secure necessary and sufficient conditions [Schaffer,J] |
| 13743 | We should not multiply basic entities, but we can have as many derivative entities as we like [Schaffer,J] |
| 14603 | 'Reification' occurs if we mistake a concept for a thing [Schaffer,J] |
| 14607 | T adds □p→p for reflexivity, and is ideal for modeling lawhood [Schaffer,J] |
| 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] |
| 10373 | Logical form can't dictate metaphysics, as it may propose an undesirable property [Schaffer,J] |
| 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] |
| 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] |
| 13741 | If 'there are red roses' implies 'there are roses', then 'there are prime numbers' implies 'there are numbers' [Schaffer,J] |
| 14604 | If a notion is ontologically basic, it should be needed in our best attempt at science [Schaffer,J] |
| 13748 | Grounding is unanalysable and primitive, and is the basic structuring concept in metaphysics [Schaffer,J] |
| 17304 | As causation links across time, grounding links the world across levels [Schaffer,J] |
| 17306 | If ground is transitive and irreflexive, it has a strict partial ordering, giving structure [Schaffer,J] |
| 14599 | Three types of reduction: Theoretical (of terms), Definitional (of concepts), Ontological (of reality) [Schaffer,J] |
| 13747 | Supervenience is just modal correlation [Schaffer,J] |
| 13744 | The cosmos is the only fundamental entity, from which all else exists by abstraction [Schaffer,J] |
| 10367 | There is only one fact - the True [Schaffer,J] |
| 13739 | Maybe categories are just the different ways that things depend on basic substances [Schaffer,J] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 14605 | Tropes are the same as events [Schaffer,J] |
| 14601 | Individuation aims to count entities, by saying when there is one [Schaffer,J] |
| 14082 | No sortal could ever exactly pin down which set of particles count as this 'cup' [Schaffer,J] |
| 13742 | There exist heaps with no integral unity, so we should accept arbitrary composites in the same way [Schaffer,J] |
| 13752 | The notion of 'grounding' can explain integrated wholes in a way that mere aggregates can't [Schaffer,J] |
| 14081 | Identities can be true despite indeterminate reference, if true under all interpretations [Schaffer,J] |
| 14606 | Only ideal conceivability could indicate what is possible [Schaffer,J] |
| 13749 | Belief in impossible worlds may require dialetheism [Schaffer,J] |
| 13740 | 'Moorean certainties' are more credible than any sceptical argument [Schaffer,J] |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| 17308 | Explaining 'Adam ate the apple' depends on emphasis, and thus implies a contrast [Schaffer,J] |
| 17305 | I take what is fundamental to be the whole spatiotemporal manifold and its fields [Schaffer,J] |
| 10359 | In causation there are three problems of relata, and three metaphysical problems [Schaffer,J] |
| 10372 | Causation may not be transitive; the last event may follow from the first, but not be caused by it [Schaffer,J] |
| 10374 | There are at least ten theories about causal connections [Schaffer,J] |
| 17307 | Nowadays causation is usually understood in terms of equations and variable ranges [Schaffer,J] |
| 10366 | Causation transcends nature, because absences can cause things [Schaffer,J] |
| 10377 | Causation may not be a process, if a crucial part of the process is 'disconnected' [Schaffer,J] |
| 10378 | A causal process needs to be connected to the effect in the right way [Schaffer,J] |
| 10382 | Causation can't be a process, because a process needs causation as a primitive [Schaffer,J] |
| 10375 | At least four rivals have challenged the view that causal direction is time direction [Schaffer,J] |
| 10389 | Causal order must be temporal, or else causes could be blocked, and time couldn't be explained [Schaffer,J] |
| 10390 | Causal order is not temporal, because of time travel, and simultanous, joint or backward causes [Schaffer,J] |
| 10380 | Causation is primitive; it is too intractable and central to be reduced; all explanations require it [Schaffer,J] |
| 10385 | If causation is just observables, or part of common sense, or vacuous, it can't be primitive [Schaffer,J] |
| 10387 | The notion of causation allows understanding of science, without appearing in equations [Schaffer,J] |
| 10388 | Causation is utterly essential for numerous philosophical explanations [Schaffer,J] |
| 10384 | If two different causes are possible in one set of circumstances, causation is primitive [Schaffer,J] |
| 10386 | If causation is primitive, it can be experienced in ourselves, or inferred as best explanation [Schaffer,J] |
| 10361 | Events are fairly course-grained (just saying 'hello'), unlike facts (like saying 'hello' loudly) [Schaffer,J] |
| 10360 | Causal relata are events - or facts, features, tropes, states, situations or aspects [Schaffer,J] |
| 10362 | One may defend three or four causal relata, as in 'c causes e rather than e*' [Schaffer,J] |
| 10368 | If causal relata must be in nature and fine-grained, neither facts nor events will do [Schaffer,J] |
| 10383 | The relata of causation (such as events) need properties as explanation, which need causation! [Schaffer,J] |
| 10393 | Our selection of 'the' cause is very predictable, so must have a basis [Schaffer,J] |
| 10394 | Selecting 'the' cause must have a basis; there is no causation without such a selection [Schaffer,J] |
| 10376 | The actual cause may make an event less likely than a possible more effective cause [Schaffer,J] |
| 10381 | All four probability versions of causation may need causation to be primitive [Schaffer,J] |