103 ideas
| 19504 | My modus ponens might be your modus tollens [Pritchard,D] |
| 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] |
| 14562 | A process is unified as an expression of a collection of causal powers [Mumford/Anjum] |
| 14541 | Events are essentially changes; property exemplifications are just states of affairs [Mumford/Anjum] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 14553 | Weak emergence is just unexpected, and strong emergence is beyond all deduction [Mumford/Anjum] |
| 14538 | Powers explain properties, causes, modality, events, and perhaps even particulars [Mumford/Anjum] |
| 14555 | Powers offer no more explanation of nature than laws do [Mumford/Anjum] |
| 14557 | Powers are not just basic forces, since they combine to make new powers [Mumford/Anjum] |
| 14583 | Dispositionality is a natural selection function, picking outcomes from the range of possibilities [Mumford/Anjum] |
| 14536 | We say 'power' and 'disposition' are equivalent, but some say dispositions are manifestable [Mumford/Anjum] |
| 14584 | The simple conditional analysis of dispositions doesn't allow for possible prevention [Mumford/Anjum] |
| 14582 | Might dispositions be reduced to normativity, or to intentionality? [Mumford/Anjum] |
| 14542 | If statue and clay fall and crush someone, the event is not overdetermined [Mumford/Anjum] |
| 14535 | Pandispositionalists say structures are clusters of causal powers [Mumford/Anjum] |
| 14561 | Perdurantism imposes no order on temporal parts, so sequences of events are contingent [Mumford/Anjum] |
| 14579 | Dispositionality is the core modality, with possibility and necessity as its extreme cases [Mumford/Anjum] |
| 14580 | Dispositions may suggest modality to us - as what might not have been, and what could have been [Mumford/Anjum] |
| 14552 | Relations are naturally necessary when they are generated by the essential mechanisms of the world [Mumford/Anjum] |
| 14578 | Possibility might be non-contradiction, or recombinations of the actual, or truth in possible worlds [Mumford/Anjum] |
| 19503 | An improbable lottery win can occur in a nearby possible world [Pritchard,D] |
| 14549 | Maybe truths are necessitated by the facts which are their truthmakers [Mumford/Anjum] |
| 19505 | Moore begs the question, or just offers another view, or uses 'know' wrongly [Pritchard,D, by PG] |
| 14585 | We have more than five senses; balance and proprioception, for example [Mumford/Anjum] |
| 19499 | We can have evidence for seeing a zebra, but no evidence for what is entailed by that [Pritchard,D] |
| 19500 | Favouring: an entailment will give better support for the first belief than reason to deny the second [Pritchard,D] |
| 19502 | Maybe knowledge just needs relevant discriminations among contrasting cases [Pritchard,D] |
| 19498 | Epistemic internalism usually says justification must be accessible by reflection [Pritchard,D] |
| 19506 | Externalism is better than internalism in dealing with radical scepticism [Pritchard,D] |
| 19496 | Disjunctivism says perceptual justification must be both factual and known by the agent [Pritchard,D] |
| 19497 | Metaphysical disjunctivism says normal perceptions and hallucinations are different experiences [Pritchard,D] |
| 19495 | Epistemic externalism struggles to capture the idea of epistemic responsibility [Pritchard,D] |
| 19501 | We assess error against background knowledge, but that is just what radical scepticism challenges [Pritchard,D] |
| 19507 | Radical scepticism is merely raised, and is not a response to worrying evidence [Pritchard,D] |
| 14576 | Smoking disposes towards cancer; smokers without cancer do not falsify this claim [Mumford/Anjum] |
| 14551 | If causation were necessary, the past would fix the future, and induction would be simple [Mumford/Anjum] |
| 14571 | The only full uniformities in nature occur from the essences of fundamental things [Mumford/Anjum] |
| 14570 | Nature is not completely uniform, and some regular causes sometimes fail to produce their effects [Mumford/Anjum] |
| 14569 | It is tempting to think that only entailment provides a full explanation [Mumford/Anjum] |
| 14568 | A structure won't give a causal explanation unless we know the powers of the structure [Mumford/Anjum] |
| 14556 | Strong emergence seems to imply top-down causation, originating in consciousness [Mumford/Anjum] |
| 14566 | Causation by absence is not real causation, but part of our explanatory practices [Mumford/Anjum] |
| 14577 | Causation may not be transitive. Does a fire cause itself to be extinguished by the sprinklers? [Mumford/Anjum] |
| 14563 | Causation is the passing around of powers [Mumford/Anjum] |
| 14587 | We take causation to be primitive, as it is hard to see how it could be further reduced [Mumford/Anjum] |
| 14533 | Causation doesn't have two distinct relata; it is a single unfolding process [Mumford/Anjum] |
| 14558 | A collision is a process, which involves simultaneous happenings, but not instantaneous ones [Mumford/Anjum] |
| 14559 | Does causation need a third tying ingredient, or just two that meet, or might there be a single process? [Mumford/Anjum] |
| 14565 | Sugar dissolving is a process taking time, not one event and then another [Mumford/Anjum] |
| 14567 | Privileging one cause is just an epistemic or pragmatic matter, not an ontological one [Mumford/Anjum] |
| 14537 | Coincidence is conjunction without causation; smoking causing cancer is the reverse [Mumford/Anjum] |
| 14573 | Occasionally a cause makes no difference (pre-emption, perhaps) so the counterfactual is false [Mumford/Anjum] |
| 14572 | Is a cause because of counterfactual dependence, or is the dependence because there is a cause? [Mumford/Anjum] |
| 14574 | Cases of preventing a prevention may give counterfactual dependence without causation [Mumford/Anjum] |
| 14539 | Nature can be interfered with, so a cause never necessitates its effects [Mumford/Anjum] |
| 14550 | We assert causes without asserting that they necessitate their effects [Mumford/Anjum] |
| 14546 | Necessary causation should survive antecedent strengthening, but no cause can always survive that [Mumford/Anjum] |
| 14575 | A 'ceteris paribus' clause implies that a conditional only has dispositional force [Mumford/Anjum] |
| 14548 | There may be necessitation in the world, but causation does not supply it [Mumford/Anjum] |
| 14554 | Laws are nothing more than descriptions of the behaviour of powers [Mumford/Anjum] |
| 14564 | If laws are equations, cause and effect must be simultaneous (or the law would be falsified)! [Mumford/Anjum] |