display all the ideas for this combination of philosophers
7 ideas
18953 | Modern notation frees us from Aristotle's restriction of only using two class-names in premises [Putnam] |
Full Idea: In modern notation we can consider potential logical principles that Aristotle never considered because of his general practice of looking at inferences each of whose premises involved exactly two class-names. | |
From: Hilary Putnam (Philosophy of Logic [1971], Ch.3) | |
A reaction: Presumably you can build up complex inferences from a pair of terms, just as you do with pairs in set theory. |
18949 | The universal syllogism is now expressed as the transitivity of subclasses [Putnam] |
Full Idea: On its modern interpretation, the validity of the inference 'All S are M; All M are P; so All S are P' just expresses the transitivity of the relation 'subclass of'. | |
From: Hilary Putnam (Philosophy of Logic [1971], Ch.1) | |
A reaction: A simple point I've never quite grasped. Since lots of syllogisms can be expressed as Venn Diagrams, in which the circles are just sets, it's kind of obvious really. So why does Sommers go back to 'terms'? See 'Term Logic'. |
18952 | '⊃' ('if...then') is used with the definition 'Px ⊃ Qx' is short for '¬(Px & ¬Qx)' [Putnam] |
Full Idea: The symbol '⊃' (read 'if...then') is used with the definition 'Px ⊃ Qx' ('if Px then Qx') is short for '¬(Px & ¬Qx)'. | |
From: Hilary Putnam (Philosophy of Logic [1971], Ch.3) | |
A reaction: So ⊃ and → are just abbreviations, and not really a proper part of the language. Notoriously, though, this is quite a long way from what 'if...then' means in ordinary English, and it leads to paradoxical oddities. |
18958 | In type theory, 'x ∈ y' is well defined only if x and y are of the appropriate type [Putnam] |
Full Idea: In the theory of types, 'x ∈ y' is well defined only if x and y are of the appropriate type, where individuals count as the zero type, sets of individuals as type one, sets of sets of individuals as type two. | |
From: Hilary Putnam (Philosophy of Logic [1971], Ch.6) |
9944 | We understand some statements about all sets [Putnam] |
Full Idea: We seem to understand some statements about all sets (e.g. 'for every set x and every set y, there is a set z which is the union of x and y'). | |
From: Hilary Putnam (Mathematics without Foundations [1967], p.308) | |
A reaction: His example is the Axiom of Choice. Presumably this is why the collection of all sets must be referred to as a 'class', since we can talk about it, but cannot define it. |
13655 | The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate [Putnam, by Shapiro] |
Full Idea: Putnam claims that the Löwenheim-Skolem theorems indicate that there is no 'fact of the matter' whether all sets are constructible. | |
From: report of Hilary Putnam (Models and Reality [1977]) by Stewart Shapiro - Foundations without Foundationalism | |
A reaction: [He refers to the 4th and 5th pages of Putnam's article] Shapiro offers (p.109) a critique of Putnam's proposal. |
9915 | V = L just says all sets are constructible [Putnam] |
Full Idea: V = L just says all sets are constructible. L is the class of all constructible sets, and V is the universe of all sets. | |
From: Hilary Putnam (Models and Reality [1977], p.425) |