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Single Idea 9154

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

Full Idea

Frege maintained a sophisticated version of the Euclidean position that knowledge of the axioms and theorems of logic, geometry, and arithmetic rests on the self-evidence of the axioms, definitions, and rules of inference.

Gist of Idea

Frege agreed with Euclid that the axioms of logic and mathematics are known through self-evidence

Source

report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Tyler Burge - Frege on Apriority Intro

Book Ref

'New Essays on the A Priori', ed/tr. Boghossian,P /Peacocke,C [OUP 2000], p.11

A Reaction

I am inclined to agree that they are indeed self-evident, but not in a purely a priori way. They are self-evident general facts about how reality is and how (it seems) that it must be. It seems to me closer to a perception than an insight.

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]