89 ideas
| 23917 | Contrary statements can both be reasonable, if they are meant in two different ways [Aristotle] |
| 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] |
| 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] |
| 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] |
| 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] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 5869 | The thesis of the Form of the Good (or of anything else) is verbal and vacuous [Aristotle] |
| 16640 | Form is the principle that connects a thing's constitution (rather than being operative) [Hill,N] |
| 22505 | The two right angles of a triangle necessitate that a quadrilateral has four [Aristotle] |
| 22513 | Knowing is having knowledge; understanding is using knowledge [Aristotle] |
| 23906 | Courage from spirit is natural and unconquerable, as seen in the young [Aristotle] |
| 20204 | Whether the mind has parts is irrelevant, since it obviously has distinct capacities [Aristotle] |
| 22506 | A man is the cause of what is within his power, and what he causes is in his power [Aristotle] |
| 22504 | Only a human being can be a starting point for an action [Aristotle] |
| 22510 | Some emotional states are too strong for human nature [Aristotle] |
| 23913 | Nearly all the good and bad states of character are concerned with feelings [Aristotle] |
| 23918 | Akrasia is the clash of two feelings - goodness and pleasure [Aristotle] |
| 22515 | Choice results when deliberation brings together an opinion with an inclination [Aristotle] |
| 22508 | Unlike in inanimate things, in animate things actions have more than one starting point [Aristotle] |
| 22514 | The deliberative part of the soul discerns explanatory causes [Aristotle] |
| 23319 | We are responsible if our actions reflect our motivation [Aristotle, by Frede,M] |
| 22507 | An action is voluntary when it is accompanied by thought of some kind [Aristotle] |
| 22512 | Acts are voluntary if done knowingly, by the agent, and in his power to avoid it [Aristotle] |
| 22509 | What is natural for us is either there at birth, or appears by normal processes [Aristotle] |
| 18230 | No one would choose life just for activities not done for their own sake [Aristotle] |
| 23909 | Wearing a shoe is its intrinsic use, and selling it (as a shoe) is its coincidental use [Aristotle] |
| 5870 | Everything seeks, not a single good, but its own separate good [Aristotle] |
| 5877 | We judge people from their deeds because we cannot see their choices (which matter more) [Aristotle] |
| 5868 | Horses, birds and fish are not happy, lacking a divine aspect to their natures [Aristotle] |
| 5865 | Happiness involves three things, of which the greatest is either wisdom, virtue, or pleasure [Aristotle] |
| 5876 | Virtue is different from continence [Aristotle] |
| 5872 | Excellence is the best state of anything (like a cloak) which has an employment or function [Aristotle] |
| 5874 | Character virtues (such as courage) are of the non-rational part, which follows the rational part [Aristotle] |
| 22516 | Character is shown by what is or is not enjoyed, and virtue chooses the mean among them [Aristotle] |
| 22517 | We judge character not by their actions, but by their reasons for actions [Aristotle] |
| 5875 | Character (éthos) is developed from habit (ethos) [Aristotle] |
| 23914 | People sometimes exhibit both extremes together, but the mean is contrary to both of them [Aristotle] |
| 23911 | Possessors of a virtue tend to despise what reason shows to be its opposite [Aristotle] |
| 23910 | Greatness of soul produces all the virtues - and vice versa [Aristotle] |
| 23908 | If someone just looks at or listens to beautiful things, they would not be thought intemperate [Aristotle] |
| 23905 | Courage follows reason, which tells us to choose what is noble [Aristotle] |
| 23912 | Honour depends on what it is for, and whether it is bestowed by worthy people [Aristotle] |
| 5871 | Goods in the soul are more worthy than those outside it, as everybody wants them [Aristotle] |
| 23920 | Decent people can be friends with base people [Aristotle] |
| 23919 | Friendship cannot be immediate; it takes time, and needs testing [Aristotle] |
| 23915 | The main function of politics is to produce friendship [Aristotle] |
| 23916 | The best cure for mutual injustice is friendship [Aristotle] |
| 5866 | It is folly not to order one's life around some end [Aristotle] |
| 5878 | Eyes could be used for a natural purpose, or for unnatural seeing, or for a non-seeing activity [Aristotle] |
| 5873 | Each thing's function is its end [Aristotle] |