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

			Full Idea: Today we expect that anything worth calling a definition should imply a semantics.
			From: Ian Hacking (What is Logic? [1979], §10)
			A reaction: He compares this with Gentzen 1935, who was attempting purely syntactic definitions of the logical connectives.

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'

			Full Idea: If A |- B and B |- C, then A |- C. This generalises to: If Γ|-A,Θ and Γ,A |- Θ, then Γ |- Θ. Gentzen called this 'cut'. It is the transitivity of a deduction.
			From: Ian Hacking (What is Logic? [1979], §06.3)
			A reaction: I read the generalisation as 'If A can be either a premise or a conclusion, you can bypass it'. The first version is just transitivity (which by-passes the middle step).

13835

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

			Full Idea: Only the cut rule can have a conclusion that is less complex than its premises. Hence when cut is not used, a derivation is quite literally constructive, building up from components. Any theorem obtained by cut can be obtained without it.
			From: Ian Hacking (What is Logic? [1979], §08)

13833

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

			Full Idea: 'Dilution' (or 'Thinning') provides an essential contrast between deductive and inductive reasoning; for the introduction of new premises may spoil an inductive inference.
			From: Ian Hacking (What is Logic? [1979], §06.2)
			A reaction: That is, inductive logic (if there is such a thing) is clearly non-monotonic, whereas classical inductive logic is monotonic.

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

			Full Idea: I don't believe English is by nature classical or intuitionistic etc. These are abstractions made by logicians. Logicians attend to numerous different objects that might be served by 'If...then', like material conditional, strict or relevant implication.
			From: Ian Hacking (What is Logic? [1979], §15)
			A reaction: The idea that they are 'abstractions' is close to my heart. Abstractions from what? Surely 'if...then' has a standard character when employed in normal conversation?

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

			Full Idea: First-order logic is the strongest complete compact theory with a Löwenheim-Skolem theorem.
			From: Ian Hacking (What is Logic? [1979], §13)

13844

A limitation of first-order logic is that it cannot handle branching quantifiers

			Full Idea: Henkin proved that there is no first-order treatment of branching quantifiers, which do not seem to involve any idea that is fundamentally different from ordinary quantification.
			From: Ian Hacking (What is Logic? [1979], §13)
			A reaction: See Hacking for an example of branching quantifiers. Hacking is impressed by this as a real limitation of the first-order logic which he generally favours.

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

13842

Second-order completeness seems to need intensional entities and possible worlds

			Full Idea: Second-order logic has no chance of a completeness theorem unless one ventures into intensional entities and possible worlds.
			From: Ian Hacking (What is Logic? [1979], §13)

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

			Full Idea: My doctrine is that the peculiarity of the logical constants resides precisely in that given a certain pure notion of truth and consequence, all the desirable semantic properties of the constants are determined by their syntactic properties.
			From: Ian Hacking (What is Logic? [1979], §09)
			A reaction: He opposes this to Peacocke 1976, who claims that the logical connectives are essentially semantic in character, concerned with the preservation of truth.

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

			Full Idea: For some purposes the variables of first-order logic can be regarded as prepositions and place-holders that could in principle be dispensed with, say by a system of arrows indicating what places fall in the scope of which quantifier.
			From: Ian Hacking (What is Logic? [1979], §11)
			A reaction: I tend to think of variables as either pronouns, or as definite descriptions, or as temporary names, but not as prepositions. Must address this new idea...

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

			Full Idea: A Löwenheim-Skolem theorem holds for anything which, on my delineation, is a logic.
			From: Ian Hacking (What is Logic? [1979], §13)
			A reaction: I take this to be an unusually conservative view. Shapiro is the chap who can give you an alternative view of these things, or Boolos.