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Ideas for 'fragments/reports', 'Thinking About Logic' and 'Principia Mathematica'

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9 ideas

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
10987 Three traditional names of rules are 'Simplification', 'Addition' and 'Disjunctive Syllogism' [Read]
     Full Idea: Three traditional names for rules are 'Simplification' (P from 'P and Q'), 'Addition' ('P or Q' from P), and 'Disjunctive Syllogism' (Q from 'P or Q' and 'not-P').
     From: Stephen Read (Thinking About Logic [1995], Ch.2)
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
9542 The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules [Russell/Whitehead, by Hughes/Cresswell]
     Full Idea: The best known axiomatization of PL is Whitehead/Russell. There are four axioms: (p∨p)→p, q→(p∨q), (p→q)→(q∨p), and (q→r)→((p∨q)→(p∨r)), plus Substitution and Modus Ponens rules.
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by GE Hughes/M Cresswell - An Introduction to Modal Logic Ch.1
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / a. Systems of modal logic
11004 Necessity is provability in S4, and true in all worlds in S5 [Read]
     Full Idea: In S4 necessity is said to be informal 'provability', and in S5 it is said to be 'true in every possible world'.
     From: Stephen Read (Thinking About Logic [1995], Ch.4)
     A reaction: It seems that the S4 version is proof-theoretic, and the S5 version is semantic.
4. Formal Logic / E. Nonclassical Logics / 4. Fuzzy Logic
11018 There are fuzzy predicates (and sets), and fuzzy quantifiers and modifiers [Read]
     Full Idea: In fuzzy logic, besides fuzzy predicates, which define fuzzy sets, there are also fuzzy quantifiers (such as 'most' and 'few') and fuzzy modifiers (such as 'usually').
     From: Stephen Read (Thinking About Logic [1995], Ch.7)
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
11011 Same say there are positive, negative and neuter free logics [Read]
     Full Idea: It is normal to classify free logics into three sorts; positive free logics (some propositions with empty terms are true), negative free logics (they are false), and neuter free logics (they lack truth-value), though I find this unhelpful and superficial.
     From: Stephen Read (Thinking About Logic [1995], Ch.5)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
21720 Russell saw Reducibility as legitimate for reducing classes to logic [Linsky,B on Russell/Whitehead]
     Full Idea: The axiom of Reducibility ...is crucial in the reduction of classes to logic, ...and seems to be a quite legitimate logical notion for Russell.
     From: comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 6.4
     A reaction: This is an unusual defence of the axiom, which is usually presumed to have been kicked into the long grass by Quine. If one could reduce classes to logic, that would destroy the opposition to logicism in a single neat coup.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
11020 Realisms like the full Comprehension Principle, that all good concepts determine sets [Read]
     Full Idea: Hard-headed realism tends to embrace the full Comprehension Principle, that every well-defined concept determines a set.
     From: Stephen Read (Thinking About Logic [1995], Ch.8)
     A reaction: This sort of thing gets you into trouble with Russell's paradox (though that is presumably meant to be excluded somehow by 'well-defined'). There are lots of diluted Comprehension Principles.
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
10044 Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Russell/Whitehead, by Shapiro]
     Full Idea: Russell adduces two reasons against the extensional view of classes, namely the existence of the null class (which cannot very well be a collection), and the unit classes (which would have to be identical with their single elements).
     From: report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Structure and Ontology p.459
     A reaction: Gödel believes in the reality of classes. I have great sympathy with Russell, when people start to claim that sets are not just conveniences to help us think about things, but actual abstract entities. Is the singleton of my pencil is on this table?
18208 We regard classes as mere symbolic or linguistic conveniences [Russell/Whitehead]
     Full Idea: Classes, so far as we introduce them, are merely symbolic or linguistic conveniences, not genuine objects.
     From: B Russell/AN Whitehead (Principia Mathematica [1913], p.72), quoted by Penelope Maddy - Naturalism in Mathematics III.2