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

13838

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

13834

Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C'

13835

Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with

13833

'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction

5. Theory of Logic / A. Overview of Logic / 4. Pure Logic

13845

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

13840

First-order logic is the strongest complete compact theory with Löwenheim-Skolem

13844

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

13842

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

13837

With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically

5. Theory of Logic / E. Structures of Logic / 4. Variables in Logic

13839

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. Löwenheim-Skolem Theorems

13843

If it is a logic, the Löwenheim-Skolem theorem holds for it