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Ideas for 'fragments/reports', 'Thinking About Logic' and 'Introduction to Mathematical Logic'

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5 ideas

5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
     Full Idea: In order to construct precise and valid patterns of arguments one has to determine their 'building blocks'. One has to identify the basic terms, their kinds and means of combination.
     From: Michal Walicki (Introduction to Mathematical Logic [2012], History Intro)
     A reaction: A deceptively simple and important idea. All explanation requires patterns and levels, and it is the idea of building blocks which makes such things possible. It is right at the centre of our grasp of everything.
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
Not all validity is captured in first-order logic [Read]
     Full Idea: We must recognise that first-order classical logic is inadequate to describe all valid consequences, that is, all cases in which it is impossible for the premisses to be true and the conclusion false.
     From: Stephen Read (Thinking About Logic [1995], Ch.2)
     A reaction: This is despite the fact that first-order logic is 'complete', in the sense that its own truths are all provable.
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
The non-emptiness of the domain is characteristic of classical logic [Read]
     Full Idea: The non-emptiness of the domain is characteristic of classical logic.
     From: Stephen Read (Thinking About Logic [1995], Ch.2)
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Semantics must precede proof in higher-order logics, since they are incomplete [Read]
     Full Idea: For the realist, study of semantic structures comes before study of proofs. In higher-order logic is has to, for the logics are incomplete.
     From: Stephen Read (Thinking About Logic [1995], Ch.9)
     A reaction: This seems to be an important general observation about any incomplete system, such as Peano arithmetic. You may dream the old rationalist dream of starting from the beginning and proving everything, but you can't. Start with truth and meaning.
5. Theory of Logic / A. Overview of Logic / 8. Logic of Mathematics
We should exclude second-order logic, precisely because it captures arithmetic [Read]
     Full Idea: Those who believe mathematics goes beyond logic use that fact to argue that classical logic is right to exclude second-order logic.
     From: Stephen Read (Thinking About Logic [1995], Ch.2)