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Ideas for 'Mahaprajnaparamitashastra', 'A Slim Book about Narrow Content' and 'Philosophy of Logic'

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4 ideas

4. Formal Logic / A. Syllogistic Logic / 1. Aristotelian Logic
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.
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
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'.
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / a. Symbols of PC
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.
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
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)