MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  meredith Unicode version

Theorem meredith 1472
Description: Carew Meredith's sole axiom for propositional calculus. This amazing formula is thought to be the shortest possible single axiom for propositional calculus with inference rule ax-mp 5, where negation and implication are primitive. Here we prove Meredith's axiom from ax-1 6, ax-2 7, and ax-3 8. Then from it we derive the Lukasiewicz axioms luk-1 1488, luk-2 1489, and luk-3 1490. Using these we finally re-derive our axioms as ax1 1499, ax2 1500, and ax3 1501, thus proving the equivalence of all three systems. C. A. Meredith, "Single Axioms for the Systems (C,N), (C,O) and (A,N) of the Two-Valued Propositional Calculus," The Journal of Computing Systems vol. 1 (1953), pp. 155-164. Meredith claimed to be close to a proof that this axiom is the shortest possible, but the proof was apparently never completed.

An obscure Irish lecturer, Meredith (1904-1976) became enamored with logic somewhat late in life after attending talks by Lukasiewicz and produced many remarkable results such as this axiom. From his obituary: "He did logic whenever time and opportunity presented themselves, and he did it on whatever materials came to hand: in a pub, his favored pint of porter within reach, he would use the inside of cigarette packs to write proofs for logical colleagues." (Contributed by NM, 14-Dec-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Wolf Lammen, 28-May-2013.)

Assertion
Ref Expression
meredith

Proof of Theorem meredith
StepHypRef Expression
1 pm2.21 108 . . . . . . 7
2 ax-3 8 . . . . . . 7
31, 2imim12i 57 . . . . . 6
43com13 80 . . . . 5
54con1d 124 . . . 4
65com12 31 . . 3
76a1d 25 . 2
8 ax-1 6 . . 3
98imim1d 75 . 2
107, 9ja 161 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4
This theorem is referenced by:  axmeredith  1473  merco1  1546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
  Copyright terms: Public domain W3C validator