Tautology is Negation of Contradiction

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Theorem

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.


Proof by Natural Deduction

By the tableau method of natural deduction:

$\top \vdash \neg \bot$
Line Pool Formula Rule Depends upon Notes
1 1 $\top$ Premise (None)
2 2 $\bot$ Assumption (None) If a contradiction were assumed ...
3 2 $\neg \top$ Rule of Explosion: $\bot \mathcal E$ 2
4 1, 2 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 1, 3
5 1 $\neg \bot$ Proof by Contradiction: $\neg \mathcal I$ 2 – 4 Assumption 2 has been discharged

$\Box$


By the tableau method of natural deduction:

$\neg \bot \vdash \top$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \bot$ Premise (None)
2 2 $\neg \top$ Assumption (None) To assume a non-truth ...
3 2 $\bot$ Sequent Introduction 2 from above result
4 1 $\top$ Reductio ad Absurdum 2 – 3 Assumption 2 has been discharged

$\blacksquare$


Proof by Truth Table

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, the truth values in the appropriate columns match.

$\begin{array}{|c||cc|} \hline \bot & \neg & \top \\ \hline F & F & T \\ \hline \end{array}$

$\blacksquare$


Proof by Boolean Interpretation

Let $p$ be a propositional formula.

Let $v$ be an arbitrary boolean interpretation of $p$.


Then:

$\map v p = T \iff \map v {\neg p} = F$

by the definition of the logical not.

Since $v$ is arbitrary, $p$ is true in all interpretations if and only if $\neg p$ is false in all interpretations.


Hence:

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

$\blacksquare$


Law of the Excluded Middle

This theorem depends on the Law of the Excluded Middle, by way of Reductio ad Absurdum.

This is one of the axioms of logic that was determined by Aristotle, and forms part of the backbone of classical (Aristotelian) logic.

However, the intuitionist school rejects the Law of the Excluded Middle as a valid logical axiom. This in turn invalidates this theorem from an intuitionistic perspective.


That is, the proposition:

If it's not false, it must be true

is valid only in the context where there are only two truth values.


Also see


Sources