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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2. Reason / D. Definition / 3. Types of Definition
A decent modern definition should always imply a semantics
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction
Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C'
Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with
5. Theory of Logic / A. Overview of Logic / 4. Pure Logic
The various logics are abstractions made from terms like 'if...then' in English
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic is the strongest complete compact theory with L÷wenheim-Skolem
A limitation of first-order logic is that it cannot handle branching quantifiers
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order completeness seems to need intensional entities and possible worlds
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically
If logical truths essentially depend on logical constants, we had better define the latter
5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic
Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
If it is a logic, the L÷wenheim-Skolem theorem holds for it