Tautology and Contradiction
- $\bot \dashv \vdash \neg \top$
That is, a falsehood can not be true, and a non-truth is a falsehood.
- $\top \dashv \vdash \neg \bot$
That is, a truth can not be false, and a non-falsehood must be a truth.
- $p \land \top \dashv \vdash p$
- $p \lor \top \dashv \vdash \top$
- $p \land \bot \dashv \vdash \bot$
- $p \lor \bot \dashv \vdash p$
Note that the proofs of:
- $\neg \bot \vdash \top$
- $\neg \top \vdash \bot$
- $p \vdash p \land \top$
- $p \lor \top \vdash \top$
rely (directly or indirectly) upon Law of Excluded Middle - and it can be seen that they are just another way of stating that truth.
- If it's not false, it must be true
- If it's not true, it must be false
are indeed valid only in the context where there are only two truth values.
From the intuitionistic perspective, these results do not hold.