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