51 ideas
5750 | Consistency is modal, saying propositions are consistent if they could be true together [Melia] |
5737 | Predicate logic has connectives, quantifiers, variables, predicates, equality, names and brackets [Melia] |
5744 | First-order predicate calculus is extensional logic, but quantified modal logic is intensional (hence dubious) [Melia] |
17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
5740 | Second-order logic needs second-order variables and quantification into predicate position [Melia] |
5741 | If every model that makes premises true also makes conclusion true, the argument is valid [Melia] |
17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
17928 | Ordinal numbers represent order relations [Colyvan] |
17923 | Intuitionists only accept a few safe infinities [Colyvan] |
17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
16554 | Activities have place, rate, duration, entities, properties, modes, direction, polarity, energy and range [Machamer/Darden/Craver] |
5736 | No sort of plain language or levels of logic can express modal facts properly [Melia] |
5735 | Maybe names and predicates can capture any fact [Melia] |
16556 | Penicillin causes nothing; the cause is what penicillin does [Machamer/Darden/Craver] |
5746 | The Identity of Indiscernibles is contentious for qualities, and trivial for non-qualities [Melia] |
5738 | We may be sure that P is necessary, but is it necessarily necessary? [Melia] |
5732 | 'De re' modality is about things themselves, 'de dicto' modality is about propositions [Melia] |
5739 | Sometimes we want to specify in what ways a thing is possible [Melia] |
5734 | Possible worlds make it possible to define necessity and counterfactuals without new primitives [Melia] |
5742 | In possible worlds semantics the modal operators are treated as quantifiers [Melia] |
5743 | If possible worlds semantics is not realist about possible worlds, logic becomes merely formal [Melia] |
5749 | Possible worlds could be real as mathematics, propositions, properties, or like books [Melia] |
5751 | The truth of propositions at possible worlds are implied by the world, just as in books [Melia] |
16562 | We understand something by presenting its low-level entities and activities [Machamer/Darden/Craver] |
17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
16563 | The explanation is not the regularity, but the activity sustaining it [Machamer/Darden/Craver] |
17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
16555 | Functions are not properties of objects, they are activities contributing to mechanisms [Machamer/Darden/Craver] |
16528 | Mechanisms are not just push-pull systems [Machamer/Darden/Craver] |
16529 | Mechanisms are systems organised to produce regular change [Machamer/Darden/Craver] |
16530 | A mechanism explains a phenomenon by showing how it was produced [Machamer/Darden/Craver] |
16553 | Our account of mechanism combines both entities and activities [Machamer/Darden/Craver] |
16559 | Descriptions of explanatory mechanisms have a bottom level, where going further is irrelevant [Machamer/Darden/Craver] |
17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
16564 | There are four types of bottom-level activities which will explain phenomena [Machamer/Darden/Craver] |
16561 | We can abstract by taking an exemplary case and ignoring the detail [Machamer/Darden/Craver] |
17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
5748 | We accept unverifiable propositions because of simplicity, utility, explanation and plausibility [Melia] |
16558 | Laws of nature have very little application in biology [Machamer/Darden/Craver] |