# 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
```

## 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]