Conjunction with Tautology

Theorem

A conjunction with a tautology:

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

Proof

Proof by Natural deduction

This is proved by the Tableau method:

$p \land \top \vdash p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p \land \top$	P	(None)
2		1	$p$	$\land \mathcal E_1$	1

$\blacksquare$

$p \vdash p \land \top$

Line		Pool	Formula	Rule	Depends upon
1		1	$p$	P	(None)
2		-	$q \lor \neg q$	LEM	(None)
3		-	$\top$	LEM	2
4		1	$p \land \top$	$\land \mathcal I$	1, 3

$\blacksquare$

Comment

Note that this relies directly 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.

Proof by Truth Table

We apply the Method of Truth Tables to the proposition.

As can be seen by inspection, in each case, the truth values in the appropriate columns match for all models.

$\begin{array}{|c|ccc||c|} \hline p & p & \land & \top & \top \\ \hline F & F & F & T & T \\ T & T & T & T & T \\ \hline \end{array}$

$\blacksquare$