# Pledger's S3 extension 16s

[Pledger, 1972]

Pledger's 16s is Sobociński's 3.01
[Sobociński,
1976a]

## Notes

This system has 16 distinct proper affirmative modalities,
(Plus the improper modality p) and is
contained in both S5 amd K2.
[Pledger, 1972, p270-271]

The distinct affirmative modalities of the system 16s 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, LLMLp/LMLLp
MMp implied by Mp, MLMMp/MMLMp
LMp implied by LLMp, LMLp it implies LMMp, MLMp
MLp it implies MLMp, MMLp implied by LMLp, MLLp
LLMp implied by LLMLp/LMLLp it implies LMp, MLLMp
MMLp it implies MLMMp/MMLMp implied by MLp, LMMLp
LMLp implied by Lp, LLMLp/LMLLp it implies LMp, MLp, LMMLp
MLMp it implies Mp, MLMMp/MMLMp implied by LMp, MLp, MLLMp
LMMp implied by LMp, LMMLp it implies MLMMp/MMLMp
MLLp it implies MLp, MLLMp implied by LLMLp/LMLLp
LLMLp/LMLLp implied by LLp it implies LLMp, LMLp, MLLp
MLMMp/MMLMp it implies MMp implied by LMMp, MLMp, MMLp
LMMLp implied by LMLp it implies LMMp, MMLp
MLLMp it implies MLMp implied by LLMp, MLLp

[Pledger, 1972, p270-271]

## Based on

The system 16s is: the system S3 plus any
one of the following:

- Necessity based
- LMLLp => LLMLp
- LMLLp => LLMp
- LMLLM => LLMp

- Possibility based
- MMLMp => MLMMp
- MMLp => MLMMLp
- MMLp => MLMMp

[Pledger, 1972, p275]

## Basis for

16sb = 16s plus either of

[Pledger, 1972, p279]

## Based on

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

This page is http://www.cc.utah.edu/~nahaj/logic/structures/systems/16s.html

This page was last modified on October 11th, 2005