51 ideas
22285 | Impredicative definitions are circular, but fine for picking out, rather than creating something [Potter] |
22301 | The Identity Theory says a proposition is true if it coincides with what makes it true [Potter] |
22324 | It has been unfortunate that externalism about truth is equated with correspondence [Potter] |
17749 | Post proved the consistency of propositional logic in 1921 [Walicki] |
17765 | Propositional language can only relate statements as the same or as different [Walicki] |
17764 | Boolean connectives are interpreted as functions on the set {1,0} [Walicki] |
10702 | Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter] |
10713 | Usually the only reason given for accepting the empty set is convenience [Potter] |
17752 | The empty set is useful for defining sets by properties, when the members are not yet known [Walicki] |
17753 | The empty set avoids having to take special precautions in case members vanish [Walicki] |
13044 | Infinity: There is at least one limit level [Potter] |
10708 | Nowadays we derive our conception of collections from the dependence between them [Potter] |
13546 | The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter] |
17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |
10707 | Mereology elides the distinction between the cards in a pack and the suits [Potter] |
17741 | To determine the patterns in logic, one must identify its 'building blocks' [Walicki] |
10704 | We can formalize second-order formation rules, but not inference rules [Potter] |
22279 | Frege's sign |--- meant judgements, but the modern |- turnstile means inference, with intecedents [Potter] |
22291 | Deductivism can't explain how the world supports unconditional conclusions [Potter] |
10703 | Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter] |
22295 | Modern logical truths are true under all interpretations of the non-logical words [Potter] |
17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki] |
17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki] |
17761 | A compact axiomatisation makes it possible to understand a field as a whole [Walicki] |
17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki] |
17757 | Members of ordinals are ordinals, and also subsets of ordinals [Walicki] |
17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki] |
17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki] |
17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki] |
17760 | Two infinite ordinals can represent a single infinite cardinal [Walicki] |
10712 | If set theory didn't found mathematics, it is still needed to count infinite sets [Potter] |
17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki] |
17882 | It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter] |
17754 | Inductive proof depends on the choice of the ordering [Walicki] |
22310 | The formalist defence against Gödel is to reject his metalinguistic concept of truth [Potter] |
22298 | Why is fictional arithmetic applicable to the real world? [Potter] |
22287 | If 'concrete' is the negative of 'abstract', that means desires and hallucinations are concrete [Potter] |
13043 | A relation is a set consisting entirely of ordered pairs [Potter] |
22284 | 'Greater than', which is the ancestral of 'successor', strictly orders the natural numbers [Potter] |
13042 | If dependence is well-founded, with no infinite backward chains, this implies substances [Potter] |
13041 | Collections have fixed members, but fusions can be carved in innumerable ways [Potter] |
12887 | A whole must have one characteristic, an internal relation, and a structure [Rescher/Oppenheim] |
10709 | Priority is a modality, arising from collections and members [Potter] |
17742 | Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki] |
22281 | A material conditional cannot capture counterfactual reasoning [Potter] |
22327 | Knowledge from a drunken schoolteacher is from a reliable and unreliable process [Potter] |
22273 | Traditionally there are twelve categories of judgement, in groups of three [Potter] |
22290 | The phrase 'the concept "horse"' can't refer to a concept, because it is saturated [Potter] |
22283 | Compositionality should rely on the parsing tree, which may contain more than sentence components [Potter] |
22282 | 'Direct compositonality' says the components wholly explain a sentence meaning [Potter] |
22296 | Compositionality is more welcome in logic than in linguistics (which is more contextual) [Potter] |