Tautology and Contradiction

Context
Natural deduction

Theorems
A contradiction implies and is implied by the negation of a tautology:

That is, a falsehood can not be true, and a non-truth is a falsehood.

A tautology implies and is implied by the negation of a contradiction:

That is, a truth can not be false, and a non-falsehood must be a truth.

A conjunction with a contradiction:

$$p \land \bot \vdash \bot$$

A disjunction with a contradiction:

$$p \lor \bot \vdash p$$

Proofs
These are proved by the Tableau method.

$$\bot \vdash \lnot \top$$:

$$\lnot \top \vdash \bot$$:

$$\top \vdash \lnot \bot $$:

$$\lnot \bot \vdash \top$$:

$$p \land \bot \vdash \bot$$:

$$p \lor \bot \vdash p$$:

Comment
Note that the proofs of:


 * $$\lnot \bot \vdash \top$$
 * $$\lnot \top \vdash \bot$$

rely (directly or indirectly) upon the Law of the 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" and "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 intuitionist perspective, these results do not hold.