Ideas from 'What is Logic?' by Ian Hacking [1979], by Theme Structure
[found in 'A Philosophical Companion to First-Order Logic' (ed/tr Hughes,R.I.G.) [Hackett 1993,0-87220-181-3]].
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expand these ideas
2. Reason / D. Definition / 3. Types of Definition
13838
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A decent modern definition should always imply a semantics
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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
13833
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'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction
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13834
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Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C'
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13835
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Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with
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5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
13845
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The various logics are abstractions made from terms like 'if...then' in English
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5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
13844
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A limitation of first-order logic is that it cannot handle branching quantifiers
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13840
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First-order logic is the strongest complete compact theory with Löwenheim-Skolem
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5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
13842
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Second-order completeness seems to need intensional entities and possible worlds
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5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
13837
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With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically
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5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
13839
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Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
13843
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If it is a logic, the Löwenheim-Skolem theorem holds for it
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