# Tautology and Contradiction

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## Contents

## 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 $(2)$