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Single Idea 13610
[filed under theme 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
]
Full Idea
For ¬,→ Schemas: (A1) |-φ→(ψ→φ), (A2) |-(φ→(ψ→ξ)) → ((φ→ψ)→(φ→ξ)), (A3) |-(¬φ→¬ψ) → (ψ→φ), Rule:DET:|-φ,|-φ→ψ then |-ψ
Gist of Idea
A logic with ¬ and → needs three axiom-schemas and one rule as foundation
Source
David Bostock (Intermediate Logic [1997], 5.2)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.194
A Reaction
A1 says everything implies a truth, A2 is conditional proof, and A3 is contraposition. DET is modus ponens. This is Bostock's compact near-minimal axiom system for proposition logic. He adds two axioms and another rule for predicate logic.
Related Idea
Idea 13619
Quantification adds two axiom-schemas and a new rule [Bostock]
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]
|
4631
|
In ideal circumstances, an axiom should be such that no rational agent could possibly object to its use
[Baggini /Fosl]
|
4632
|
'Natural' systems of deduction are based on normal rational practice, rather than on axioms
[Baggini /Fosl]
|