Combining Texts

Ideas for 'On the Question of Absolute Undecidability', 'Modern Philosophy:introduction and survey' and 'Thinking About Logic'

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7 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 / 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 / 1. Set Theory
17884 Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner]
     Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory.
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro)
17893 'Reflection principles' say the whole truth about sets can't be captured [Koellner]
     Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level.
     From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1)
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.