Ideas from 'What is Logic?' by Ian Hacking [1979], by Theme Structure
[found in 'A Philosophical Companion to FirstOrder Logic' (ed/tr Hughes,R.I.G.) [Hackett 1993,0872201813]].
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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. FirstOrder Logic
13840

Firstorder logic is the strongest complete compact theory with LöwenheimSkolem

13844

A limitation of firstorder logic is that it cannot handle branching quantifiers

5. Theory of Logic / A. Overview of Logic / 7. SecondOrder Logic
13842

Secondorder completeness seems to need intensional entities and possible worlds

5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
13829

If logical truths essentially depend on logical constants, we had better define the latter

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. LöwenheimSkolem Theorems
13843

If it is a logic, the LöwenheimSkolem theorem holds for it
