94 ideas
18390 | All metaphysical discussion should be guided by a quest for truthmakers [Armstrong] |
16539 | A definition of a circle will show what it is, and show its generating principle [Lowe] |
16540 | Defining an ellipse by conic sections reveals necessities, but not the essence of an ellipse [Lowe] |
16548 | An essence is what an entity is, revealed by a real definition; this is not an entity in its own right [Lowe] |
16549 | Simple things like 'red' can be given real ostensive definitions [Lowe] |
18467 | Truth-making can't be entailment, because truthmakers are portions of reality [Armstrong] |
18468 | Armstrong says truthmakers necessitate their truth, where 'necessitate' is a primitive relation [Armstrong, by MacBride] |
18377 | Negative truths have as truthmakers all states of affairs relevant to the truth [Armstrong] |
18382 | The nature of arctic animals is truthmaker for the absence of penguins there [Armstrong] |
18394 | In mathematics, truthmakers are possible instantiations of structures [Armstrong] |
18384 | One truthmaker will do for a contingent truth and for its contradictory [Armstrong] |
18387 | The truthmakers for possible unicorns are the elements in their combination [Armstrong] |
18386 | What is the truthmaker for 'it is possible that there could have been nothing'? [Armstrong] |
18381 | Necessitating general truthmakers must also specify their limits [Armstrong] |
18396 | The set theory brackets { } assert that the member is a unit [Armstrong] |
18393 | For 'there is a class with no members' we don't need the null set as truthmaker [Armstrong] |
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] |
18392 | Classes have cardinalities, so their members must all be treated as units [Armstrong] |
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] |
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] |
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] |
18385 | Logical atomism builds on the simple properties, but are they the only possible properties? [Armstrong] |
18391 | 'Naturalism' says only the world of space-time exists [Armstrong] |
18374 | Truthmaking needs states of affairs, to unite particulars with tropes or universals. [Armstrong] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
18372 | We need properties, as minimal truthmakers for the truths about objects [Armstrong] |
18379 | The determinates of a determinable must be incompatible with each other [Armstrong] |
18378 | Length is a 'determinable' property, and one mile is one its 'determinates' [Armstrong] |
18373 | If tropes are non-transferable, then they necessarily belong to their particular substance [Armstrong] |
18400 | Properties are not powers - they just have powers [Armstrong] |
18397 | Powers must result in some non-powers, or there would only be potential without result [Armstrong] |
18399 | How does the power of gravity know the distance it acts over? [Armstrong] |
18371 | The class of similar things is much too big a truthmaker for the feature of a particular [Armstrong] |
16545 | The essence of lumps and statues shows that two objects coincide but are numerically distinct [Lowe] |
16546 | The essence of a bronze statue shows that it could be made of different bronze [Lowe] |
16551 | Grasping an essence is just grasping a real definition [Lowe] |
16542 | Explanation can't give an account of essence, because it is too multi-faceted [Lowe] |
16552 | If we must know some entity to know an essence, we lack a faculty to do that [Lowe] |
18389 | When entities contain entities, or overlap with them, there is 'partial' identity [Armstrong] |
16533 | Logical necessities, based on laws of logic, are a proper sub-class of metaphysical necessities [Lowe] |
16531 | 'Metaphysical' necessity is absolute and objective - the strongest kind of necessity [Lowe] |
16532 | 'Epistemic' necessity is better called 'certainty' [Lowe] |
16543 | If an essence implies p, then p is an essential truth, and hence metaphysically necessary [Lowe] |
16544 | Metaphysical necessity is either an essential truth, or rests on essential truths [Lowe] |
18388 | Possible worlds don't fix necessities; intrinsic necessities imply the extension in worlds [Armstrong] |
16538 | We could give up possible worlds if we based necessity on essences [Lowe] |
16534 | 'Intuitions' are just unreliable 'hunches'; over centuries intuitions change enormously [Lowe] |
18375 | General truths are a type of negative truth, saying there are no more ravens than black ones [Armstrong] |
16535 | A concept is a way of thinking of things or kinds, whether or not they exist [Lowe] |
16550 | Direct reference doesn't seem to require that thinkers know what it is they are thinking about [Lowe] |
18368 | For all being, there is a potential proposition which expresses its existence and nature [Armstrong] |
18370 | A realm of abstract propositions is causally inert, so has no explanatory value [Armstrong] |
18380 | Negative causations supervene on positive causations plus their laws? [Armstrong] |
16547 | H2O isn't necessary, because different laws of nature might affect how O and H combine [Lowe] |
18401 | The pure present moment is too brief to be experienced [Armstrong] |