83 ideas
| 18996 | A statement S is 'partly true' if it has some wholly true parts [Yablo] |
| 19006 | An 'enthymeme' is an argument with an indispensable unstated assumption [Yablo] |
| 8859 | The main modal logics disagree over three key formulae [Yablo] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 18999 | y is only a proper part of x if there is a z which 'makes up the difference' between them [Yablo] |
| 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] |
| 19001 | 'Pegasus doesn't exist' is false without Pegasus, yet the absence of Pegasus is its truthmaker [Yablo] |
| 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] |
| 9138 | An infinite series of sentences asserting falsehood produces the paradox without self-reference [Yablo, by Sorensen] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 8865 | If 'the number of Democrats is on the rise', does that mean that 50 million is on the rise? [Yablo] |
| 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] |
| 19002 | A nominalist can assert statements about mathematical objects, as being partly true [Yablo] |
| 8863 | We must treat numbers as existing in order to express ourselves about the arrangement of planets [Yablo] |
| 10580 | Mathematics is both necessary and a priori because it really consists of logical truths [Yablo] |
| 8862 | Platonic objects are really created as existential metaphors [Yablo] |
| 10579 | Putting numbers in quantifiable position (rather than many quantifiers) makes expression easier [Yablo] |
| 10577 | Concrete objects have few essential properties, but properties of abstractions are mostly essential [Yablo] |
| 10578 | We are thought to know concreta a posteriori, and many abstracta a priori [Yablo] |
| 19489 | For me, fictions are internally true, without a significant internal or external truth-value [Yablo] |
| 19490 | Make-believe can help us to reason about facts and scientific procedures [Yablo] |
| 19491 | 'The clouds are angry' can only mean '...if one were attributing emotions to clouds' [Yablo] |
| 8864 | We quantify over events, worlds, etc. in order to make logical possibilities clearer [Yablo] |
| 19494 | Fictionalism allows that simulated beliefs may be tracking real facts [Yablo] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 8858 | Philosophers keep finding unexpected objects, like models, worlds, functions, numbers, events, sets, properties [Yablo] |
| 14381 | A statue is essentially the statue, but its lump is not essentially a statue, so statue isn't lump [Yablo, by Rocca] |
| 18998 | Parthood lacks the restriction of kind which most relations have [Yablo] |
| 19493 | Governing possible worlds theory is the fiction that if something is possible, it happens in a world [Yablo] |
| 19542 | It is nonsense that understanding does not involve knowledge; to understand, you must know [Dougherty/Rysiew] |
| 19543 | To grasp understanding, we should be more explicit about what needs to be known [Dougherty/Rysiew] |
| 19541 | Rather than knowledge, our epistemic aim may be mere true belief, or else understanding and wisdom [Dougherty/Rysiew] |
| 19540 | Don't confuse justified belief with justified believers [Dougherty/Rysiew] |
| 19539 | If knowledge is unanalysable, that makes justification more important [Dougherty/Rysiew] |
| 19004 | Gettier says you don't know if you are confused about how it is true [Yablo] |
| 19007 | A theory need not be true to be good; it should just be true about its physical aspects [Yablo] |
| 18993 | If sentences point to different evidence, they must have different subject-matter [Yablo] |
| 19003 | Most people say nonblack nonravens do confirm 'all ravens are black', but only a tiny bit [Yablo] |
| 10805 | A sentence should be recarved to reveal its content or implication relations [Yablo] |
| 18992 | Sentence-meaning is the truth-conditions - plus factors responsible for them [Yablo] |
| 19538 | Entailment is modelled in formal semantics as set inclusion (where 'mammals' contains 'cats') [Dougherty/Rysiew] |
| 18994 | The content of an assertion can be quite different from compositional content [Yablo] |
| 18997 | Truth-conditions as subject-matter has problems of relevance, short cut, and reversal [Yablo] |
| 19005 | Not-A is too strong to just erase an improper assertion, because it actually reverses A [Yablo] |
| 8861 | Hardly a word in the language is devoid of metaphorical potential [Yablo] |