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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]
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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
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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
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Axiomatization simply picks from among the true sentences a few to play a special role
[Orenstein]
|
7797
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Axiom systems of logic contain axioms, inference rules, and definitions of proof and theorems
[Girle]
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4631
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In ideal circumstances, an axiom should be such that no rational agent could possibly object to its use
[Baggini /Fosl]
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4632
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'Natural' systems of deduction are based on normal rational practice, rather than on axioms
[Baggini /Fosl]
|