# Axiom 5

## Notes

[Hughes and Cresswell, 1996]

## Axiom 5

Mp>LMp

It is the defining axiom of the system S5

This axiom is also called axiom "E" by Hughes and Cresswell.
This leads to confusion when dealing with the "E" modal logic systems,
and when dealing with the propositional logic system E, so
I've chosen to follow the naming conventions of Chellas and call
it axiom 5.
The strict form of this axiom is called M10 by Zeman.

## Systems

S5 = Axiom K: L(p>q) > (Lp>Lq)
+ Lp>p
+ Axiom 5 [Mp>LMp]
[Hughes and Cresswell, 1996, p58]

S9 = S3
+ Axiom 5 [Mp>LMp]
+ MMp [Axiom S]
[Hughes and Cresswell, 1996, p364]

S3.5 = S3
+ Axiom 5 [Mp>LMp]
[Hughes and Cresswell, 1996, p364]

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This page was last modified on December 9th, 2006.