Combining Philosophers

Ideas for Alfred Tarski, Max Weber and Ian Hacking

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4 ideas

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 [Hacking]
	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.

13834	Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking]
	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 [Hacking]
	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)

4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables

19195	Truth tables give prior conditions for logic, but are outside the system, and not definitions [Tarski]
	Full Idea: Logical sentences are often assigned preliminary conditions under which they are true or false (often given as truth tables). However, these are outside the system of logic, and should not be regarded as definitions of the terms involved.
	From: Alfred Tarski (The Semantic Conception of Truth [1944], 15)
	A reaction: Hence, presumably, the connectives are primitives (with no nature or meaning), and the truth tables are axioms for their use? This opinion of Tarski's may have helped shift the preference towards natural deduction introduction and elimination rules.