Tautology and Contradiction

Theorems

Contradiction Negation of Tautology

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

$\bot \dashv \vdash \neg \top$

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

Tautology Negation of Contradiction

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

$\top \dashv \vdash \neg \bot$

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

Conjunction with Tautology

A conjunction with a tautology:

$p \land \top \dashv \vdash p$

Disjunction with Tautology

A disjunction with a tautology:

$p \lor \top \dashv \vdash \top$

Conjunction with Contradiction

A conjunction with a contradiction:

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

Disjunction with Contradiction

A disjunction with a contradiction:

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

Comment

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 the Law of the Excluded Middle - and it can be seen that they are just another way of stating that truth.

The propositions:

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.

Sources

• D.J. O'Connor and Betty Powell: Elementary Logic (1980): $\S 1.3$