Comments on: On single and shortest axioms for Boolean logic http://www.mathrix.org/liquid/archives/on-single-and-shortest-axioms The blog of Hector Zenil Thu, 29 Dec 2011 20:29:03 +0000 hourly 1 http://wordpress.org/?v=3.3.1 By: John Halleck http://www.mathrix.org/liquid/archives/on-single-and-shortest-axioms/comment-page-1#comment-31805 John Halleck Thu, 29 Dec 2011 20:29:03 +0000 http://www.mathrix.org/liquid/?p=133#comment-31805 Opps... a English reference for Tarski's claim is: Alfred Tarski [1956] Logic, Semantics, Metamathematics, Papers from 1923 to 1938 (Translated by J. H. Woodger), Clarendon Press / Oxford UP, Oxford UK 1956. (A second revised edition has been issued by J. Corcoran (ed.), at Hackett Pub. Co., Indianapolis IN, in 19832 .) Opps… a English reference for Tarski’s claim is:

Alfred Tarski [1956] Logic, Semantics, Metamathematics, Papers from
1923 to 1938 (Translated by J. H. Woodger), Clarendon Press / Oxford UP,
Oxford UK 1956. (A second revised edition has been issued by J. Corcoran
(ed.), at Hackett Pub. Co., Indianapolis IN, in 19832 .)

]]> By: John Halleck http://www.mathrix.org/liquid/archives/on-single-and-shortest-axioms/comment-page-1#comment-31804 John Halleck Thu, 29 Dec 2011 20:24:49 +0000 http://www.mathrix.org/liquid/?p=133#comment-31804 If you have an interest in single axioms for systems, you might be interested in Tarsi's 1930 result, which is that if you have a logic defined by axioms, the rule of Modus Ponens, and the rule of Uniform Substitution, then is MUST have a single axiom basis if it can prove two specific theorems he gave. Giving names to the theorems for discussion: I: Cpp p -> p K: CpCqp p -> (q -> p) C: CCpCqrCqCpr (p -> (q -> r)) -> (q -> (p -> r)) D: CpCqCCpCqrr p -> (q -> ((p -> (q -> r)) -> r)) E: CpCqCCpCqrCsr p -> (q -> ((p -> (q -> r)) -> (s -> r))) C*: CpCqCCpCqrr p -> (q -> ((p -> (q -> r)) -> r)) Tarski proved the theorems K + D and the theorems K + E insured the existence of a single axiom for the system. Since then the provability of the following pairs proved the existence of a single axiom for the system. K+C, K+C*, I+E If you have an interest in single axioms for systems, you might be interested in Tarsi’s 1930 result, which is that if you have a logic defined by axioms, the rule of Modus Ponens, and the rule of Uniform Substitution, then is MUST have a single axiom basis if it can prove two specific theorems he gave.

Giving names to the theorems for discussion:
I: Cpp p -> p
K: CpCqp p -> (q -> p)
C: CCpCqrCqCpr (p -> (q -> r)) -> (q -> (p -> r))
D: CpCqCCpCqrr p -> (q -> ((p -> (q -> r)) -> r))
E: CpCqCCpCqrCsr p -> (q -> ((p -> (q -> r)) -> (s -> r)))
C*: CpCqCCpCqrr p -> (q -> ((p -> (q -> r)) -> r))

Tarski proved the theorems K + D and the theorems K + E insured the existence of a single axiom for the system.

Since then the provability of the following pairs proved the existence of a single axiom for the system.

K+C, K+C*, I+E

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