21 ideas
10301 | The axiom of choice is controversial, but it could be replaced [Shapiro] |
8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
10588 | First-order logic is Complete, and Compact, with the Löwenheim-Skolem Theorems [Shapiro] |
10298 | Some say that second-order logic is mathematics, not logic [Shapiro] |
10299 | If the aim of logic is to codify inferences, second-order logic is useless [Shapiro] |
10300 | Logical consequence can be defined in terms of the logical terminology [Shapiro] |
12219 | Whether a modal claim is true depends on how the object is described [Quine, by Fine,K] |
10922 | Objects are the values of variables, so a referentially opaque context cannot be quantified into [Quine] |
10290 | Second-order variables also range over properties, sets, relations or functions [Shapiro] |
10590 | Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro] |
10296 | The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro] |
10297 | The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro] |
10292 | Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro] |
10294 | Second-order logic has the expressive power for mathematics, but an unworkable model theory [Shapiro] |
10718 | A natural number is a property of sets [Maddy, by Oliver] |
8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
10591 | Logicians use 'property' and 'set' interchangeably, with little hanging on it [Shapiro] |
10923 | Aristotelian essentialism says a thing has some necessary and some non-necessary properties [Quine] |
10921 | Necessity can attach to statement-names, to statements, and to open sentences [Quine] |
10924 | Necessity is in the way in which we say things, and not things themselves [Quine] |