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Single Idea 9013

[filed under theme 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL ]

Full Idea

The construction of 'alternation' (using 'or') is useful in practice, but superfluous in theory. It can be paraphrased using only negation and conjunction. We say that 'p or q' is paraphrased as 'not(not-p and not-q)'.

Gist of Idea

We can eliminate 'or' from our basic theory, by paraphrasing 'p or q' as 'not(not-p and not-q)'

Source

Willard Quine (Philosophy of Logic [1970], Ch.2)

Book Ref

Quine,Willard: 'Philosophy of Logic' [Prentice-Hall 1970], p.24

A Reaction

Quine treats 'not' and 'and' as the axiomatic logical connectives, and builds the others from those, presumably because that is the smallest number he could get it down to. I quite like it, because it seems to mesh with basic thought procedures.

The 17 ideas with the same theme [statements treated as true without question]:

2145 In mathematics certain things have to be accepted without further explanation [Plato]
562 Axioms are the underlying principles of everything, and who but the philosopher can assess their truth? [Aristotle]
573 The axioms of mathematics are part of philosophy [Aristotle]
1668 An axiom is a principle which must be understood if one is to learn anything [Aristotle]
20791 Chrysippus has five obvious 'indemonstrables' of reasoning [Chrysippus, by Diog. Laertius]
5620 Philosophy has no axioms, as it is just rational cognition of concepts [Kant]
9154 Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence [Frege, by Burge]
9585 Since every definition is an equation, one cannot define equality itself [Frege]
9542 The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules [Russell/Whitehead, by Hughes/Cresswell]
9013 We can eliminate 'or' from our basic theory, by paraphrasing 'p or q' as 'not(not-p and not-q)' [Quine]
13610 A logic with ¬ and → needs three axiom-schemas and one rule as foundation [Bostock]
8086 Predicate logic retains the axioms of propositional logic [Devlin]
6299 Axioms are often affirmed simply because they produce results which have been accepted [Resnik]
8476 Axiomatization simply picks from among the true sentences a few to play a special role [Orenstein]
7797 Axiom systems of logic contain axioms, inference rules, and definitions of proof and theorems [Girle]
4632 'Natural' systems of deduction are based on normal rational practice, rather than on axioms [Baggini /Fosl]
4631 In ideal circumstances, an axiom should be such that no rational agent could possibly object to its use [Baggini /Fosl]