74 ideas
| 22140 | The greatest philosophers are methodical; it is what makes them great [Grice] |
| 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] |
| 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] |
| 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] |
| 10076 | The 'range' of a function is the set of elements in the output set created by the function [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] |
| 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] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 1212 | Replacing timbers on Theseus' ship was the classic illustration of the problem of growth and change [Plutarch] |
| 13856 | Conditionals are truth-functional, but we must take care with misleading ones [Grice, by Edgington] |
| 8948 | The odd truth table for material conditionals is explained by conversational conventions [Grice, by Fisher] |
| 13767 | Conditionals might remain truth-functional, despite inappropriate conversational remarks [Edgington on Grice] |
| 10990 | Conditionals are truth-functional, but unassertable in tricky cases? [Grice, by Read] |
| 14277 | A person can be justified in believing a proposition, though it is unreasonable to actually say it [Grice, by Edgington] |
| 5958 | The sun is always bright; it doesn't become bright when it emerges [Plutarch] |
| 5959 | Some philosophers say the soul is light [Plutarch] |
| 5960 | When the soul is intelligent and harmonious, it is part of god and derives from god [Plutarch] |
| 5952 | Rather than being the whole soul, maybe I am its chief part? [Plutarch] |
| 5951 | If atoms have no qualities, they cannot possibly produce a mind [Plutarch] |
| 5963 | Some say emotion is a sort of reason, and others say virtue concerns emotion [Plutarch] |
| 7752 | Only the utterer's primary intention is relevant to the meaning [Grice] |
| 7751 | Meaning needs an intention to induce a belief, and a recognition that this is the speaker's intention [Grice] |
| 7753 | We judge linguistic intentions rather as we judge non-linguistic intentions, so they are alike [Grice] |
| 22330 | Grice said patterns of use are often semantically irrelevant, because it is a pragmatic matter [Grice, by Glock] |
| 18045 | Grice's maxim of quality says do not assert what you believe to be false [Grice, by Magidor] |
| 18044 | Grice's maxim of manner requires one to be as brief as possible [Grice, by Magidor] |
| 10991 | Key conversational maxims are 'quality' (assert truth) and 'quantity' (leave nothing out) [Grice, by Read] |
| 18046 | Grice's maxim of quantity says be sufficiently informative [Grice, by Magidor] |
| 20796 | Action needs an affinity for a presentation, and an impulse toward the affinity [Plutarch] |
| 1477 | Being manly and brave is the result of convention, not of human nature [Plutarch] |
| 1478 | Animals don't value pleasure, as they cease sexual intercourse after impregnation [Plutarch] |
| 5948 | The good life involves social participation, loyalty, temperance and honesty [Plutarch] |
| 1479 | Animals have not been led into homosexuality, because they value pleasure very little [Plutarch] |
| 5950 | If only atoms exist, how do qualities arise when the atoms come together? [Plutarch] |
| 5974 | People report seeing through rocks, or over the horizon, or impossibly small works [Plutarch] |
| 5957 | Absurd superstitions make people atheist, not disharmony in nature [Plutarch] |
| 5955 | No one will ever find a city that lacks religious practices [Plutarch] |