# 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