Pledger's S3 extension 8q

[Pledger, 1972]


This system has 8 distinct proper affirmative modalities (plus the improper modality p) and is contained in K2 but not in 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, LMp
Mp    it implies MMp       implied by p, MLp

LLp                        it implies Lp, LLMp
MMp                        implied by Mp, MMLp

LMp   implied by Lp, LLMp  it implies MLp
MLp   it implies Mp, MMLp  implied by LMp

LLMp  implied by LLp       it implies LMp
MMLp  it implies MMp       implied by MLp

[Pledger, 1972, p270-271]

Based on

The system 8q is: the system S3 plus any one of the following: (Arranged as axiom, dual of axiom)

LMp => LMLLp  MLMMp => LMp
LMp => LMLp   MLMp  => LMp
LMp => MLLp   LMMp  => MLp
LMp => MLp    self dual
LMM => LMLp   MLMp  => MLLp
LMM => MLLp   LMMp  => MLLp
LMM => MLp    LMp   => MLLp

[Pledger, 1972, p275]

Basis for

8qa = 8q + MMp (Axiom S)

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