66 ideas
13520 | A 'tautology' must include connectives [Wolf,RS] |
13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
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] |
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] |
14549 | Maybe truths are necessitated by the facts which are their truthmakers [Mumford/Anjum] |
14585 | We have more than five senses; balance and proprioception, for example [Mumford/Anjum] |
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] |
7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
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] |