71 ideas
| 11103 | We aren't stuck with our native conceptual scheme; we can gradually change it [Quine] |
| 16477 | Asserting not-p is saying p is false [Russell] |
| 16484 | There are four experiences that lead us to talk of 'some' things [Russell] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 16486 | The physical world doesn't need logic, but the mental world does [Russell] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 2947 | Questions wouldn't lead anywhere without the law of excluded middle [Russell] |
| 16480 | A disjunction expresses indecision [Russell] |
| 16483 | Disjunction may also arise in practice if there is imperfect memory. [Russell] |
| 16479 | 'Or' expresses hesitation, in a dog at a crossroads, or birds risking grabbing crumbs [Russell] |
| 16481 | 'Or' expresses a mental state, not something about the world [Russell] |
| 16487 | Maybe the 'or' used to describe mental states is not the 'or' of logic [Russell] |
| 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
| 10074 | A 'total function' maps every element to one element in 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] |
| 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] |
| 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] |
| 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] |
| 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] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [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] |
| 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] |
| 16475 | A 'heterological' predicate can't be predicated of itself; so is 'heterological' heterological? Yes=no! [Russell] |
| 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] |
| 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] |
| 11092 | A river is a process, with stages; if we consider it as one thing, we are considering a process [Quine] |
| 11093 | We don't say 'red' is abstract, unlike a river, just because it has discontinuous shape [Quine] |
| 11101 | General terms don't commit us ontologically, but singular terms with substitution do [Quine] |
| 11096 | Discourse generally departmentalizes itself to some degree [Quine] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 11099 | Understanding 'is square' is knowing when to apply it, not knowing some object [Quine] |
| 11094 | 'Red' is a single concrete object in space-time; 'red' and 'drop' are parts of a red drop [Quine] |
| 11097 | Red is the largest red thing in the universe [Quine] |
| 17595 | To unite a sequence of ostensions to make one object, a prior concept of identity is needed [Quine] |
| 11095 | We should just identify any items which are indiscernible within a given discourse [Quine] |
| 16482 | All our knowledge (if verbal) is general, because all sentences contain general words [Russell] |
| 4758 | Naïve realism leads to physics, but physics then shows that naïve realism is false [Russell] |
| 16476 | For simple words, a single experience can show that they are true [Russell] |
| 16485 | Perception can't prove universal generalisations, so abandon them, or abandon empiricism? [Russell] |
| 11104 | Concepts are language [Quine] |
| 11102 | Apply '-ness' or 'class of' to abstract general terms, to get second-level abstract singular terms [Quine] |
| 16478 | A mother cat is paralysed if equidistant between two needy kittens [Russell] |