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

13833

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

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

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