# System C4

[Hughes and Cresswell,
1968]

[Anderson and Belnap,
1962]

[Ulrich,
1981]

## Notes

There are at least 6, and no more than 8 two axiom bases for C4
[EFHW,
2002, pg. 170]

## Based on

Originally from
[Anderson and Belnap,
1962,
p31-32]

## Based on

- C1: p => p
- C2:
(p=>(q=>r)) => ((p=>q)=>(p=>r))
- C4: a => (p=>a)
[Where a is strict (I.E. of the form X => y)]

[Hughes and Cresswell,
1968,
p295]

## Based on

Note that this is the shortest possible two axiom basis for C4

- p=>(q=>q)
- (p=>(q=>r)) => ( (p=>q) => (s => (p => r)))

[EFHW,
2002, pg. 170]

## Based on

Note that this is the shortest possible single axiom basis for C4
[EFHW,
2002, pg. 171]

- (p=>((q=>(r=>r)))=>(p=>s)) => ((s=>t)=>(u=>(p=>t)))

[EFHW,
2002, pg. 171]

## Basis for

# Go to ...

## Notes

My thanks to Dolph Ulrich [Ulrich,
email, 2005] for
straightening me out and documenting that the two C4's that used to be on
this page were the same system.

© Copyright 2006,2007,2008, by John Halleck, All Rights Reserved.

This page is http://www.cc.utah.edu/~nahaj/logic/structures/systems/c4.html

This page was last modified on July 17th, 2008