# Pledger's S3 extension 12p

[Pledger, 1972]

## Notes

"The 12p semantic condition is that every normal world y is either related
to a nonnormal world (in which case LLp is always false at y, or related
only to itself (in which case q => Lq is always true at y"
- [Pledger,
1980, p683"

This system has 12 distinct proper affirmative modalities (plus the
improper modality p) and is contained in neither
K2 nor S5
[Pledger, 1972, p270-271]

The distinct affirmative modalities of the syatem are:

p implied by Lp it implies Mp
Lp implied by LLp it implies p, LMLp
Mp it implies MMp implied by p, MLMp
LLp it implies Lp, LMLLp
MMp implied by Mp, MLMMp
LMp implied by LMLp, LMLLp it implies LMMp, MLMp
MLp it implies MLMp, MLMMp implied by LMLp, MLLp
LMLp implied by Lp, LMLLp it implies LMp, MLp, LMMp
MLMp it implies Mp, MLMMp implied by LMp, MLp, MLLp
LMMp implied by LMp, LMLp it implies MLMMp
MLLp it implies MLp, MLMp implied by LMLLp
LMLLp implied by LLp it implies LMp, LMLp, MLLp
MLMMp it implies MMp implied by MLp, LMMp, MLMp

[Pledger, 1972, p270-271]

## Based on

The system 12p is: the system S3 plus any
one of the following: (Arranged as axiom, dual of axiom)

LLMp => p, p => MMLp
LLMp => Lp, Mp => MMLp
LLMp => LLp, MMP => MMLp

[Pledger, 1972, p275]

# Basis for

12pa = 12p + Axiom S
(MMp)

[Pledger,
1972, p279]

The system 12pb is the system 12p plus
MLMMp

[Pledger,
1972, p279]

# Other Containment

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© Copyright 2000, by John Halleck, All Rights Reserved.

This page is http://www.cc.utah.edu/~nahaj/logic/structures/systems/12p.html

This page was last modified on September 13th, 2006