# System TRIVIAL

## Notes

TRIV is the trivial modal logic system.

TRIV has the axioms La==a and Ma==a.

The side effect of these axioms is that you can add or subtract modal operators to your hearts content without changing the truth of a statement.

This *very* degenerate system is only of theoretical interest.

Every consistent extension of system K which retains the rules US, MP, and Necessitation is contained in either the Trivial system or System Ver, or both. [Hughes and Cresswell, 1996, p67]

It is not possible to add any axioms to the system, that aren't already theorems, without making it inconsistent. [Hughes and Cresswell, 1996, p67]

If you add the system ver to this system, the result is inconsistent. [Hughes and Cresswell, 1996, p67]

Any consistent system that contains the axiom D (Lp>Mp) is contained in the system TRIV. [Hughes and Cresswell, 1996, p67-68]

It is characterized by frames with one reflexive world. [Hughes and Cresswell, 1996, p362]

The paper "An Adaptive Logic Based on Jaśkowski's D2", by Joke Meheus (2001) has a section that covers the semantics of Triv.

## Based on

TRIV = D + Axiom TRIV [ p==Lp ] [Hughes and Cresswell, 1996, p65]

TRIV = S3 (Or any system that contains it) plus any of the following

```
{ MLLMp, MLMp, MLMMp, MMLMp} => { p, Lp, LLp }
{ MMp, Mp, p} => { LMMLp, LMLp, LMLLp, LLMLp }

{ LMp, LMMp } => { p, Lp, LLp }
{ p, Mp, MMp } => { MLp, MLLp }

MMp => LLp

{ MMp, Mp } => { p, Lp }
{ p, Mp } => { LLp, Lp }

```

[Pledger, 1972, p270-271]