87 ideas
| 8060 | In the 17th-18th centuries morality offered a cure for egoism, through altruism [MacIntyre] |
| 8053 | Twentieth century social life is re-enacting eighteenth century philosophy [MacIntyre] |
| 8047 | Philosophy has been marginalised by its failure in the Enlightenment to replace religion [MacIntyre] |
| 6161 | Structuralism is neo-Kantian idealism, with language playing the role of categories of understanding [Rowlands] |
| 8062 | Proof is a barren idea in philosophy, and the best philosophy never involves proof [MacIntyre] |
| 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] |
| 6163 | If bivalence is rejected, then excluded middle must also be rejected [Rowlands] |
| 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] |
| 6155 | Supervenience is a one-way relation of dependence or determination between properties [Rowlands] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 6154 | It is argued that wholes possess modal and counterfactual properties that parts lack [Rowlands] |
| 6157 | Tokens are dated, concrete particulars; types are their general properties or kinds [Rowlands] |
| 6159 | Strong idealism is the sort of mess produced by a Cartesian separation of mind and world [Rowlands] |
| 8052 | To find empiricism and science in the same culture is surprising, as they are really incompatible [MacIntyre] |
| 8057 | Unpredictability doesn't entail inexplicability, and predictability doesn't entail explicability [MacIntyre] |
| 8054 | Social sciences discover no law-like generalisations, and tend to ignore counterexamples [MacIntyre] |
| 6152 | Minds are rational, conscious, subjective, self-knowing, free, meaningful and self-aware [Rowlands] |
| 6173 | Content externalism implies that we do not have privileged access to our own minds [Rowlands] |
| 6174 | If someone is secretly transported to Twin Earth, others know their thoughts better than they do [Rowlands] |
| 21050 | I can only make decisions if I see myself as part of a story [MacIntyre] |
| 6158 | Supervenience of mental and physical properties often comes with token-identity of mental and physical particulars [Rowlands] |
| 8056 | AI can't predict innovation, or consequences, or external relations, or external events [MacIntyre] |
| 6168 | The content of a thought is just the meaning of a sentence [Rowlands] |
| 6167 | Action is bodily movement caused by intentional states [Rowlands] |
| 8059 | The good life for man is the life spent seeking the good life for man [MacIntyre] |
| 8034 | We still have the appearance and language of morality, but we no longer understand it [MacIntyre] |
| 8036 | Unlike expressions of personal preference, evaluative expressions do not depend on context [MacIntyre] |
| 8049 | Moral judgements now are anachronisms from a theistic age [MacIntyre] |
| 8045 | The failure of Enlightenment attempts to justify morality will explain our own culture [MacIntyre] |
| 8051 | Mention of 'intuition' in morality means something has gone wrong with the argument [MacIntyre] |
| 6177 | Moral intuition seems unevenly distributed between people [Rowlands] |
| 8048 | When 'man' is thought of individually, apart from all roles, it ceases to be a functional concept [MacIntyre] |
| 8035 | In trying to explain the type of approval involved, emotivists are either silent, or viciously circular [MacIntyre] |
| 8037 | The expression of feeling in a sentence is in its use, not in its meaning [MacIntyre] |
| 8040 | Emotivism cannot explain the logical terms in moral discourse ('therefore', 'if..then') [MacIntyre] |
| 8042 | Nowadays most people are emotivists, and it is embodied in our culture [MacIntyre] |
| 8058 | Maybe we can only understand rules if we first understand the virtues [MacIntyre] |
| 7097 | Virtue is secondary to a role-figure, defined within a culture [MacIntyre, by Statman] |
| 8043 | Characters are the masks worn by moral philosophies [MacIntyre] |
| 8061 | If morality just is emotion, there are no external criteria for judging emotions [MacIntyre] |
| 8038 | Since Moore thinks the right action produces the most good, he is a utilitarian [MacIntyre] |
| 8050 | There are no natural or human rights, and belief in them is nonsense [MacIntyre] |
| 6156 | The 17th century reintroduced atoms as mathematical modes of Euclidean space [Rowlands] |
| 6170 | Natural kinds are defined by their real essence, as in gold having atomic number 79 [Rowlands] |
| 6178 | It is common to see the value of nature in one feature, such as life, diversity, or integrity [Rowlands] |
| 8055 | If God is omniscient, he confronts no as yet unmade decisions, so decisions are impossible [MacIntyre] |