Tautology and Contradiction

Theorems

Contradiction is 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 is 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

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

Disjunction with Tautology

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

Conjunction with Contradiction

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

Disjunction with 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 Law of 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 intuitionistic perspective, these results do not hold.

Sources

• 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{I}: 3$: Logical Constants $(2)$