Combining Texts

Ideas for 'Identity and Spatio-Temporal Continuity', 'Intro to I: Classical Logic' and 'Introduction to the Philosophy of Mathematics'

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4 ideas

5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
9463 Classical logic is bivalent, has excluded middle, and only quantifies over existent objects [Jacquette]
     Full Idea: Classical logic (of Whitehead, Russell, Gödel, Church) is a two-valued system of propositional and predicate logic, in which all propositions are exclusively true or false, and quantification and predication are over existent objects only.
     From: Dale Jacquette (Intro to I: Classical Logic [2002], p.9)
     A reaction: All of these get challenged at some point, though the existence requirement is the one I find dubious.
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
17924 Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan]
     Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false.
     From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3)
     A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa.
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17929 Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan]
     Full Idea: Löwenheim proved that if a first-order sentence has a model at all, it has a countable model. ...Skolem generalised this result to systems of first-order sentences.
     From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2)
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17930 Axioms are 'categorical' if all of their models are isomorphic [Colyvan]
     Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic.
     From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2)
     A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'.