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
13835

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

13834

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

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. 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
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
