101 ideas
| 14227 | We could refer to tables as 'xs that are arranged tablewise' [Inwagen] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 8972 | What in the real world could ground the distinction between the sets {A,{A,B}} and {B,{A,B}}? [Inwagen] |
| 10662 | Mereology is 'nihilistic' (just atoms) or 'universal' (no restrictions on what is 'whole') [Inwagen, by Varzi] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 17587 | The 'Law' of Excluded Middle needs all propositions to be definitely true or definitely false [Inwagen] |
| 17558 | Variables are just like pronouns; syntactic explanations get muddled over dummy letters [Inwagen] |
| 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] |
| 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] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [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] |
| 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] |
| 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [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] |
| 17583 | There are no heaps [Inwagen] |
| 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] |
| 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] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 22180 | Multiple realisability is said to make reduction impossible [Okasha] |
| 17578 | I reject talk of 'stuff', and treat it in terms of particles [Inwagen] |
| 17582 | Singular terms can be vague, because they can contain predicates, which can be vague [Inwagen] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 17556 | Material objects are in space and time, move, have a surface and mass, and are made of some stuff [Inwagen] |
| 8264 | Maybe table-shaped particles exist, but not tables [Inwagen, by Lowe] |
| 17565 | Nihilism says composition between single things is impossible [Inwagen] |
| 14228 | If there are no tables, but tables are things arranged tablewise, the denial of tables is a contradiction [Liggins on Inwagen] |
| 14468 | Actions by artefacts and natural bodies are disguised cooperations, so we don't need them [Inwagen] |
| 17571 | Every physical thing is either a living organism or a simple [Inwagen] |
| 17562 | The statue and lump seem to share parts, but the statue is not part of the lump [Inwagen] |
| 17574 | If you knead clay you make an infinite series of objects, but they are rearrangements, not creations [Inwagen] |
| 17531 | I assume matter is particulate, made up of 'simples' [Inwagen] |
| 17560 | If contact causes composition, do two colliding balls briefly make one object? [Inwagen] |
| 17561 | If bricks compose a house, that is at least one thing, but it might be many things [Inwagen] |
| 17566 | I think parthood involves causation, and not just a reasonably stable spatial relationship [Inwagen] |
| 14230 | We can deny whole objects but accept parts, by referring to them as plurals within things [Inwagen, by Liggins] |
| 17557 | Special Composition Question: when is a thing part of something? [Inwagen] |
| 17564 | The essence of a star includes the released binding energy which keeps it from collapse [Inwagen] |
| 17575 | The persistence of artifacts always covertly involves intelligent beings [Inwagen] |
| 17577 | When an electron 'leaps' to another orbit, is the new one the same electron? [Inwagen] |
| 17589 | If you reject transitivity of vague identity, there is no Ship of Theseus problem [Inwagen] |
| 17588 | We should talk of the transitivity of 'identity', and of 'definite identity' [Inwagen] |
| 17572 | Actuality proves possibility, but that doesn't explain how it is possible [Inwagen] |
| 17579 | Counterparts reduce counterfactual identity to problems about similarity relations [Inwagen] |
| 17590 | A merely possible object clearly isn't there, so that is a defective notion [Inwagen] |
| 17591 | Merely possible objects must be consistent properties, or haecceities [Inwagen] |
| 22177 | Randomised Control Trials have a treatment and a control group, chosen at random [Okasha] |
| 22172 | Not all sciences are experimental; astronomy relies on careful observation [Okasha] |
| 22174 | The discoverers of Neptune didn't change their theory because of an anomaly [Okasha] |
| 22175 | Science mostly aims at confirming theories, rather than falsifying them [Okasha] |
| 22182 | Theories with unobservables are underdetermined by the evidence [Okasha] |
| 22185 | Two things can't be incompatible if they are incommensurable [Okasha] |
| 22176 | Induction is inferences from examined to unexamined instances of a given kind [Okasha] |
| 22178 | If the rules only concern changes of belief, and not the starting point, absurd views can look ratiional [Okasha] |
| 6981 | Determinism clashes with free will, as the past determines action, and is beyond our control [Inwagen, by Jackson] |
| 7101 | Virtue theory needs an external standard to judge behaviour and character [Inwagen, by Statman] |
| 22173 | Galileo refuted the Aristotelian theory that heavier objects fall faster [Okasha] |
| 17563 | The strong force pulls, but also pushes apart if nucleons get too close together [Inwagen] |
| 17559 | Is one atom a piece of gold, or is a sizable group of atoms required? [Inwagen] |
| 17580 | One's mental and other life is centred on the brain, unlike any other part of the body [Inwagen] |
| 17586 | At the lower level, life trails off into mere molecular interaction [Inwagen] |
| 17581 | Being part of an organism's life is a matter of degree, and vague [Inwagen] |
| 17567 | A flame is like a life, but not nearly so well individuated [Inwagen] |
| 17568 | A tumour may spread a sort of life, but it is not a life, or an organism [Inwagen] |
| 17569 | Unlike waves, lives are 'jealous'; it is almost impossible for them to overlap [Inwagen] |
| 17570 | The chemical reactions in a human life involve about sixteen elements [Inwagen] |
| 17584 | Some events are only borderline cases of lives [Inwagen] |
| 17576 | If God were to 'reassemble' my atoms of ten years ago, the result would certainly not be me [Inwagen] |
| 17585 | Life is vague at both ends, but could it be totally vague? [Inwagen] |
| 17366 | Virtually all modern views of speciation rest on relational rather than intrinsic features [Okasha] |
| 17573 | There is no reason to think that mere existence is a valuable thing [Inwagen] |