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5. Theory of Logic / H. Proof Systems / 2. Axiomatic Proof

[proofs built up from some initially accepted truths]

7 ideas

22277	Boole's method was axiomatic, achieving economy, plus multiple interpretations [Boole, by Potter]
	Full Idea: Boole's work was an early example of the axiomatic method, whereby intellectual economy is achieved by studying a set of axioms in which the primitive terms have multiple interpretations.
	From: report of George Boole (The Laws of Thought [1854]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 02 'Boole'
	A reaction: Unclear about this. I suppose the axioms are just syntactic, and a range of semantic interpretations can be applied. Are De Morgan's Laws interpretations, or implications of the syntactic axioms? The latter, I think.

13609	Frege produced axioms for logic, though that does not now seem the natural basis for logic [Frege, by Kaplan]
	Full Idea: Frege's work supplied a set of axioms for logic itself, at least partly because it was a well-known way of presenting the foundations in other disciplines, especially mathematics, but it does not nowadays strike us as natural for logic.
	From: report of Gottlob Frege (Begriffsschrift [1879]) by David Kaplan - Dthat 5.1
	A reaction: What Bostock has in mind is the so-called 'natural' deduction systems, which base logic on rules of entailment, rather than on a set of truths. The axiomatic approach uses a set of truths, plus the idea of possible contradictions.

13619	Quantification adds two axiom-schemas and a new rule [Bostock]
	Full Idea: New axiom-schemas for quantifiers: (A4) |-∀ξφ → φ(α/ξ), (A5) |-∀ξ(ψ→φ) → (ψ→∀ξφ), plus the rule GEN: If |-φ the |-∀ξφ(ξ/α).
	From: David Bostock (Intermediate Logic [1997], 5.6)
	A reaction: This follows on from Idea 13610, where he laid out his three axioms and one rule for propositional (truth-functional) logic. This Idea plus 13610 make Bostock's proposed axiomatisation of first-order logic.

13622	Axiom systems from Frege, Russell, Church, Lukasiewicz, Tarski, Nicod, Kleene, Quine... [Bostock]
	Full Idea: Notably axiomatisations of first-order logic are by Frege (1879), Russell and Whitehead (1910), Church (1956), Lukasiewicz and Tarski (1930), Lukasiewicz (1936), Nicod (1917), Kleene (1952) and Quine (1951). Also Bostock (1997).
	From: David Bostock (Intermediate Logic [1997], 5.8)
	A reaction: My summary, from Bostock's appendix 5.8, which gives details of all of these nine systems. This nicely illustrates the status and nature of axiom systems, which have lost the absolute status they seemed to have in Euclid.

13687	No assumptions in axiomatic proofs, so no conditional proof or reductio [Sider]
	Full Idea: Axiomatic systems do not allow reasoning with assumptions, and therefore do not allow conditional proof or reductio ad absurdum.
	From: Theodore Sider (Logic for Philosophy [2010], 2.6)
	A reaction: Since these are two of the most basic techniques of proof which I have learned (in Lemmon), I shall avoid axiomatic proof systems at all costs, despites their foundational and Ockhamist appeal.

13688	Good axioms should be indisputable logical truths [Sider]
	Full Idea: Since they are the foundations on which a proof rests, the axioms in a good axiomatic system ought to represent indisputable logical truths.
	From: Theodore Sider (Logic for Philosophy [2010], 2.6)

12198	Geometrical axioms in logic are nowadays replaced by inference rules (which imply the logical truths) [Rumfitt]
	Full Idea: The geometrical style of formalization of logic is now little more than a quaint anachronism, largely because it fails to show logical truths for what they are: simply by-products of rules of inference that are applicable to suppositions.
	From: Ian Rumfitt (Logical Necessity [2010], §1)
	A reaction: This is the rejection of Russell-style axiom systems in favour of Gentzen-style natural deduction systems (starting from rules). Rumfitt quotes Dummett in support.