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Ideas for 'fragments/reports', 'Russell's Metaphysical Logic' and 'W.V. Quine'

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6 ideas

4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
8472 Sentential logic is consistent (no contradictions) and complete (entirely provable) [Orenstein]
     Full Idea: Sentential logic has been proved consistent and complete; its consistency means that no contradictions can be derived, and its completeness assures us that every one of the logical truths can be proved.
     From: Alex Orenstein (W.V. Quine [2002], Ch.5)
     A reaction: The situation for quantificational logic is not quite so clear (Orenstein p.98). I do not presume that being consistent and complete makes it necessarily better as a tool in the real world.
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
8476 Axiomatization simply picks from among the true sentences a few to play a special role [Orenstein]
     Full Idea: In axiomatizing, we are merely sorting out among the truths of a science those which will play a special role, namely, serve as axioms from which we derive the others. The sentences are already true in a non-conventional or ordinary sense.
     From: Alex Orenstein (W.V. Quine [2002], Ch.5)
     A reaction: If you were starting from scratch, as Euclidean geometers may have felt they were doing, you might want to decide which are the simplest truths. Axiomatizing an established system is a more advanced activity.
4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic
8480 S4: 'poss that poss that p' implies 'poss that p'; S5: 'poss that nec that p' implies 'nec that p' [Orenstein]
     Full Idea: The five systems of propositional modal logic contain successively stronger conceptions of necessity. In S4 'it is poss that it is poss that p' implies 'it is poss that p'. In S5, 'it is poss that it is nec that p' implies 'it is nec that p'.
     From: Alex Orenstein (W.V. Quine [2002], Ch.7)
     A reaction: C.I. Lewis originated this stuff. Any serious student of modality is probably going to have to pick a system. E.g. Nathan Salmon says that the correct modal logic is even weaker than S4.
4. Formal Logic / F. Set Theory ST / 1. Set Theory
8474 Unlike elementary logic, set theory is not complete [Orenstein]
     Full Idea: The incompleteness of set theory contrasts sharply with the completeness of elementary logic.
     From: Alex Orenstein (W.V. Quine [2002], Ch.5)
     A reaction: This seems to be Quine's reason for abandoning the Frege-Russell logicist programme (quite apart from the problems raised by Gödel.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
21705 Reducibility says any impredicative function has an appropriate predicative replacement [Linsky,B]
     Full Idea: The Axiom of Reducibility avoids impredicativity, by asserting that for any predicate of given arguments defined by quantifying over higher-order functions or classes, there is another co-extensive but predicative function of the same type of arguments.
     From: Bernard Linsky (Russell's Metaphysical Logic [1999], 1)
     A reaction: Eventually the axiom seemed too arbitrary, and was dropped. Linsky's book explores it.
4. Formal Logic / G. Formal Mereology / 1. Mereology
8465 Mereology has been exploited by some nominalists to achieve the effects of set theory [Orenstein]
     Full Idea: The theory of mereology has had a history of being exploited by nominalists to achieve some of the effects of set theory.
     From: Alex Orenstein (W.V. Quine [2002], Ch.3)
     A reaction: Some writers refer to mereology as a 'theory', and others as an area of study. This appears to be an interesting line of investigation. Orenstein says Quine and Goodman showed its limitations.