117 ideas
| 9414 | Metaphysics is the mapping of possibilities [Lowe, by Mumford] |
| 16414 | Science needs metaphysics to weed out its presuppositions [Lowe, by Hofweber] |
| 8282 | Only metaphysics can decide whether identity survives through change [Lowe] |
| 16127 | Metaphysics tells us what there could be, rather than what there is [Lowe] |
| 8262 | How can a theory of meaning show the ontological commitments of two paraphrases of one idea? [Lowe] |
| 19463 | Induction assumes some uniformity in nature, or that in some respects the future is like the past [Ayer] |
| 8315 | Maybe facts are just true propositions [Lowe] |
| 8319 | One-to-one correspondence would need countable, individuable items [Lowe] |
| 8309 | A set is a 'number of things', not a 'collection', because nothing actually collects the members [Lowe] |
| 8322 | I don't believe in the empty set, because (lacking members) it lacks identity-conditions [Lowe] |
| 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] |
| 8312 | It is better if the existential quantifier refers to 'something', rather than a 'thing' which needs individuation [Lowe] |
| 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] |
| 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] |
| 8297 | Numbers are universals, being sets whose instances are sets of appropriate cardinality [Lowe] |
| 8266 | Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous [Lowe] |
| 8302 | Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe] |
| 8298 | Sets are instances of numbers (rather than 'collections'); numbers explain sets, not vice versa [Lowe] |
| 8311 | If 2 is a particular, then adding particulars to themselves does nothing, and 2+2=2 [Lowe] |
| 8310 | Does the existence of numbers matter, in the way space, time and persons do? [Lowe] |
| 8321 | All possible worlds contain abstracta (e.g. numbers), which means they contain concrete objects [Lowe] |
| 8300 | Perhaps possession of causal power is the hallmark of existence (and a reason to deny the void) [Lowe] |
| 8281 | Heraclitus says change is new creation, and Spinoza that it is just phases of the one substance [Lowe] |
| 8270 | Events are changes or non-changes in properties and relations of persisting objects [Lowe] |
| 8308 | Events are ontologically indispensable for singular causal explanations [Lowe] |
| 8314 | Are facts wholly abstract, or can they contain some concrete constituents? [Lowe] |
| 8316 | Facts cannot be wholly abstract if they enter into causal relations [Lowe] |
| 8318 | The problem with the structured complex view of facts is what binds the constituents [Lowe] |
| 8323 | It is whimsical to try to count facts - how many facts did I learn before breakfast? [Lowe] |
| 8313 | Facts are needed for truth-making and causation, but they seem to lack identity criteria [Lowe] |
| 8258 | Two of the main rivals for the foundations of ontology are substances, and facts or states-of-affairs [Lowe] |
| 8301 | Some abstractions exist despite lacking causal powers, because explanation needs them [Lowe] |
| 8283 | Ontological categories are not natural kinds: the latter can only be distinguished using the former [Lowe] |
| 8284 | The top division of categories is either abstract/concrete, or universal/particular, or necessary/contingent [Lowe] |
| 13122 | Lowe divides things into universals and particulars, then kinds and properties, and abstract/concrete [Lowe, by Westerhoff] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 8273 | Is 'the Thames is broad in London' relational, or adverbial, or segmental? [Lowe] |
| 8285 | I prefer 'modes' to 'tropes', because it emphasises their dependence [Lowe] |
| 8295 | Why cannot a trope float off and join another bundle? [Lowe] |
| 8286 | Tropes cannot have clear identity-conditions, so they are not objects [Lowe] |
| 8294 | How can tropes depend on objects for their identity, if objects are just bundles of tropes? [Lowe] |
| 8296 | Does a ball snug in plaster have one trope, or two which coincide? [Lowe] |
| 8288 | Sortal terms for universals involve a substance, whereas adjectival terms do not [Lowe] |
| 8293 | Real universals are needed to explain laws of nature [Lowe] |
| 8307 | Particulars are instantiations, and universals are instantiables [Lowe] |
| 16130 | To be an object at all requires identity-conditions [Lowe] |
| 8265 | Our commitment to the existence of objects should depend on their explanatory value [Lowe] |
| 8275 | Objects are entities with full identity-conditions, but there are entities other than objects [Lowe] |
| 8267 | Perhaps concrete objects are entities which are in space-time and subject to causality [Lowe] |
| 8263 | An object is an entity which has identity-conditions [Lowe] |
| 8268 | Some things (such as electrons) can be countable, while lacking proper identity [Lowe] |
| 8303 | Criteria of identity cannot individuate objects, because they are shared among different types [Lowe] |
| 8292 | Diversity of two tigers is their difference in space-time; difference of matter is a consequence [Lowe] |
| 8291 | Individuation principles identify what kind it is; identity criteria distinguish items of the same kind [Lowe] |
| 16128 | A 'substance' is an object which doesn't depend for existence on other objects [Lowe] |
| 8279 | The identity of composite objects isn't fixed by original composition, because how do you identify the origin? [Lowe] |
| 8271 | An object 'endures' if it is always wholly present, and 'perdures' if different parts exist at different times [Lowe] |
| 8272 | How can you identify temporal parts of tomatoes without referring to tomatoes? [Lowe] |
| 8305 | A clear idea of the kind of an object must precede a criterion of identity for it [Lowe] |
| 8290 | One view is that two objects of the same type are only distinguished by differing in matter [Lowe] |
| 15079 | 'Conceptual' necessity is narrow logical necessity, true because of concepts and logical laws [Lowe] |
| 16063 | Metaphysical necessity is logical necessity 'broadly construed' [Lowe, by Lynch/Glasgow] |
| 8260 | Logical necessity can be 'strict' (laws), or 'narrow' (laws and definitions), or 'broad' (all logical worlds) [Lowe] |
| 16131 | The metaphysically possible is what acceptable principles and categories will permit [Lowe] |
| 8320 | Does every abstract possible world exist in every possible world? [Lowe] |
| 19461 | Knowing I exist reveals nothing at all about my nature [Ayer] |
| 19459 | To say 'I am not thinking' must be false, but it might have been true, so it isn't self-contradictory [Ayer] |
| 19460 | 'I know I exist' has no counterevidence, so it may be meaningless [Ayer] |
| 8280 | While space may just be appearance, time and change can't be, because the appearances change [Lowe] |
| 8276 | Properties or qualities are essentially adjectival, not objectual [Lowe] |
| 19464 | We only discard a hypothesis after one failure if it appears likely to keep on failing [Ayer] |
| 19462 | Induction passes from particular facts to other particulars, or to general laws, non-deductively [Ayer] |
| 8289 | The idea that Cartesian souls are made of some ghostly 'immaterial' stuff is quite unwarranted [Lowe] |
| 8299 | Abstractions are non-spatial, or dependent, or derived from concepts [Lowe] |
| 8306 | You can think of a direction without a line, but a direction existing with no lines is inconceivable [Lowe] |
| 8317 | To cite facts as the elements in causation is to confuse states of affairs with states of objects [Lowe] |
| 8269 | Points are limits of parts of space, so parts of space cannot be aggregates of them [Lowe] |