49 ideas
10633 | 'Some critics admire only one another' cannot be paraphrased in singular first-order [Linnebo] |
8472 | Sentential logic is consistent (no contradictions) and complete (entirely provable) [Orenstein] |
8476 | Axiomatization simply picks from among the true sentences a few to play a special role [Orenstein] |
8480 | S4: 'poss that poss that p' implies 'poss that p'; S5: 'poss that nec that p' implies 'nec that p' [Orenstein] |
8474 | Unlike elementary logic, set theory is not complete [Orenstein] |
10779 | A comprehension axiom is 'predicative' if the formula has no bound second-order variables [Linnebo] |
23445 | Naïve set theory says any formula defines a set, and coextensive sets are identical [Linnebo] |
8465 | Mereology has been exploited by some nominalists to achieve the effects of set theory [Orenstein] |
10781 | A 'pure logic' must be ontologically innocent, universal, and without presuppositions [Linnebo] |
10638 | A pure logic is wholly general, purely formal, and directly known [Linnebo] |
8452 | Traditionally, universal sentences had existential import, but were later treated as conditional claims [Orenstein] |
8475 | The substitution view of quantification says a sentence is true when there is a substitution instance [Orenstein] |
10778 | Can second-order logic be ontologically first-order, with all the benefits of second-order? [Linnebo] |
10636 | Plural plurals are unnatural and need a first-level ontology [Linnebo] |
10639 | Plural quantification may allow a monadic second-order theory with first-order ontology [Linnebo] |
10783 | Plural quantification depends too heavily on combinatorial and set-theoretic considerations [Linnebo] |
10635 | Second-order quantification and plural quantification are different [Linnebo] |
10641 | Traditionally we eliminate plurals by quantifying over sets [Linnebo] |
10640 | Instead of complex objects like tables, plurally quantify over mereological atoms tablewise [Linnebo] |
23447 | In classical semantics singular terms refer, and quantifiers range over domains [Linnebo] |
23443 | The axioms of group theory are not assertions, but a definition of a structure [Linnebo] |
23444 | To investigate axiomatic theories, mathematics needs its own foundational axioms [Linnebo] |
8454 | The whole numbers are 'natural'; 'rational' numbers include fractions; the 'reals' include root-2 etc. [Orenstein] |
23446 | You can't prove consistency using a weaker theory, but you can use a consistent theory [Linnebo] |
23448 | Mathematics is the study of all possible patterns, and is thus bound to describe the world [Linnebo] |
14085 | 'Deductivist' structuralism is just theories, with no commitment to objects, or modality [Linnebo] |
14084 | Non-eliminative structuralism treats mathematical objects as positions in real abstract structures [Linnebo] |
14086 | 'Modal' structuralism studies all possible concrete models for various mathematical theories [Linnebo] |
14087 | 'Set-theoretic' structuralism treats mathematics as various structures realised among the sets [Linnebo] |
14089 | Structuralism differs from traditional Platonism, because the objects depend ontologically on their structure [Linnebo] |
14083 | Structuralism is right about algebra, but wrong about sets [Linnebo] |
14090 | In mathematical structuralism the small depends on the large, which is the opposite of physical structures [Linnebo] |
8473 | The logicists held that is-a-member-of is a logical constant, making set theory part of logic [Orenstein] |
23441 | Logical truth is true in all models, so mathematical objects can't be purely logical [Linnebo] |
23442 | Game Formalism has no semantics, and Term Formalism reduces the semantics [Linnebo] |
14091 | There may be a one-way direction of dependence among sets, and among natural numbers [Linnebo] |
10643 | We speak of a theory's 'ideological commitments' as well as its 'ontological commitments' [Linnebo] |
10637 | Ordinary speakers posit objects without concern for ontology [Linnebo] |
8458 | Just individuals in Nominalism; add sets for Extensionalism; add properties, concepts etc for Intensionalism [Orenstein] |
14088 | An 'intrinsic' property is either found in every duplicate, or exists independent of all externals [Linnebo] |
10782 | The modern concept of an object is rooted in quantificational logic [Linnebo] |
8457 | The Principle of Conservatism says we should violate the minimum number of background beliefs [Orenstein] |
8477 | People presume meanings exist because they confuse meaning and reference [Orenstein] |
8471 | Three ways for 'Socrates is human' to be true are nominalist, platonist, or Montague's way [Orenstein] |
10634 | Predicates are 'distributive' or 'non-distributive'; do individuals do what the group does? [Linnebo] |
8484 | If two people believe the same proposition, this implies the existence of propositions [Orenstein] |
6000 | The goal is rationality in the selection of things according to nature [Diogenes of Babylon, by Blank] |
5999 | The good is what is perfect by nature [Diogenes of Babylon, by Blank] |
6001 | Justice is a disposition to distribute according to desert [Diogenes of Babylon, by Blank] |