70 ideas
| 9023 | If you say that a contradiction is true, you change the meaning of 'not', and so change the subject [Quine] |
| 9012 | Talk of 'truth' when sentences are mentioned; it reminds us that reality is the point of sentences [Quine] |
| 9011 | Truth is redundant for single sentences; we do better to simply speak the sentence [Quine] |
| 9013 | We can eliminate 'or' from our basic theory, by paraphrasing 'p or q' as 'not(not-p and not-q)' [Quine] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 9020 | My logical grammar has sentences by predication, then negation, conjunction, and existential quantification [Quine] |
| 9028 | Maybe logical truth reflects reality, but in different ways in different languages [Quine] |
| 10014 | Quine rejects second-order logic, saying that predicates refer to multiple objects [Quine, by Hodes] |
| 10828 | Quantifying over predicates is treating them as names of entities [Quine] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 9024 | Excluded middle has three different definitions [Quine] |
| 10012 | Quantification theory can still be proved complete if we add identity [Quine] |
| 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] |
| 9016 | Names are not essential, because naming can be turned into predication [Quine] |
| 9015 | Universal quantification is widespread, but it is definable in terms of existential quantification [Quine] |
| 9025 | You can't base quantification on substituting names for variables, if the irrationals cannot all be named [Quine] |
| 9026 | Some quantifications could be false substitutionally and true objectually, because of nameless objects [Quine] |
| 10705 | Putting a predicate letter in a quantifier is to make it the name of an entity [Quine] |
| 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] |
| 9027 | A sentence is logically true if all sentences with that grammatical structure are true [Quine] |
| 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] |
| 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] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 9017 | Predicates are not names; predicates are the other parties to predication [Quine] |
| 9018 | A physical object is the four-dimensional material content of a portion of space-time [Quine] |
| 8249 | Class membership is not transitive, unlike being part of a part of the whole [Lesniewski, by George/Van Evra] |
| 9019 | Four-d objects helps predication of what no longer exists, and quantification over items from different times [Quine] |
| 9014 | Some conditionals can be explained just by negation and conjunction: not(p and not-q) [Quine] |
| 9009 | Single words are strongly synonymous if their interchange preserves truth [Quine] |
| 9007 | It makes no sense to say that two sentences express the same proposition [Quine] |
| 9008 | There is no rule for separating the information from other features of sentences [Quine] |
| 9010 | We can abandon propositions, and just talk of sentences and equivalence [Quine] |
| 9021 | A good way of explaining an expression is saying what conditions make its contexts true [Quine] |