System S3.5 (Åqvist)

[Åqvist, 1964]

[Hughes and Cresswell, 1996]

[Pledger, 1972]

Notes

[Hughes and Cresswell, 1996] mention a system S3.3 (originally investigated by Feys, which was later shown to be system S3.5 itself. [Pledger 1972, sec 6, p276] [Pledger 2001]

This system has also been called S3(S) [Pledger 2000]

This system is system 12r (Pledger) enumeration of systems containing S3 [Pledger, 1972]

This system has 12 distinct proper postive modalities (and the improper modality p). They are:

p      implied by Lp          it implies Mp

Lp     implied by LLp         it implies p, LMp, MLp, LMMLp
Mp     it implies MMp         implied by p, LMp, MLp, MLLMp

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

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

LLMp   implied by LLp         it implies LMp, MLLMp
MMLp   it implies MMp         implied by MLp, LMMLp

LMMp   implied by LMp, LMMLp  it implies MMp
MLLp   it implies MLp, MLLMp  implied by LLp

LMMLp  implied by Lp          it implies LMMp, MMLp
MLLMp  it implies Mp          implied by LLMp, MLLp

Based on

The system S3.5 is:

S3 + [~Lp > L~Lp] [Åqvist, 1964, p82]

OR

The system S3.5 is:

S3 + axiom 5 [Mp>LMp] [Hughes and Cresswell, 1996, p364]

[Pledger, 1972, p270-271]

OR

The system S3.5 is:

S3 + B [p>LMp] [Hughes and Cresswell, 1996, p208]

OR

The system S3.5 is the system S3 plus any of the following: (Arranged as axiom, dual of axiom.)

LML => p,   p  => MLMp
LML => Lp,  Mp => MLMp

[Pledger, 1972, p275]

OR

The system S3.5 is the system S3 plus the axiom LMLL(true). [Pledger, 1972, sec 8, p282]

Basis for

S3.5 + MMp [Axiom-S] = S9 [Hughes and Cresswell, 1996, p208, p364]


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This page was last modified on October 11th, 2005