Law of Detachment

This law can be proven by making a truth table to see that the compound statement is always true.

[(p \Rightarrow q) \land p] \Rightarrow q

The proof consists of three steps.

Determine the Number of Columns

To determine the number of columns in the truth table, the compound statement can be broken down into simpler statements until the simplest statements $p$ and $q$ are obtained.

The compound statement is broken down into simpler statements

The columns are formed by ordering each unique step of simplification from the simplest to the most complex. The last column of a truth table is always the original statement.

$p$	$q$	$p \Rightarrow q$	$(p \Rightarrow q) \land p$	$[(p \Rightarrow q) \land p] \Rightarrow q$
$?$	$?$	$?$	$?$	$?$

In this case, the truth table has six columns.

Determine the Number of Rows

A truth table has as many rows as there are combinations of truth values of the variables. Those combinations are the simplest statements that make up the complex statement. In this case, the complex statement was broken down into two simplest statements.

\underline{Simplest Statements} p and q

Each of these statements can be either true (T) or false (F). Therefore, since there are four possible combinations for these truth values, the table will have four rows.

$p$	$q$	$p \Rightarrow q$	$(p \Rightarrow q) \land p$	$[(p \Rightarrow q) \land p] \Rightarrow q$
T	T	$?$	$?$	$?$
T	F	$?$	$?$	$?$
F	T	$?$	$?$	$?$
F	F	$?$	$?$	$?$

Fill in the Cells

The last step in making a truth table is to fill in the remaining empty cells, column by column. The first column that will be filled in is the conditional statement $(p \Rightarrow q) .$ A conditional statement is false only when the hypothesis is $true$ and the conclusion $false .$

$p$	$q$	$p \Rightarrow q$	$(p \Rightarrow q) \land p$	$[(p \Rightarrow q) \land p] \Rightarrow q$
$T$	$T$	$T$	$?$	$?$
$T$	$F$	$F$	$?$	$?$
$F$	$T$	$T$	$?$	$?$
$F$	$F$	$T$	$?$	$?$

The next column is the conjunction $(p \Rightarrow q) \land p .$ A conjunction is $false$ unless both statements are $true .$

$p$	$q$	$p \Rightarrow q$	$(p \Rightarrow q) \land p$	$[(p \Rightarrow q) \land p] \Rightarrow q$
$T$	T	$T$	$T$	$?$
$T$	F	$F$	$F$	$?$
$F$	T	$T$	$F$	$?$
$F$	F	$T$	$F$	$?$

Similarly to the third column, the last column is a conditional statement.

$p$	$q$	$p \Rightarrow q$	$(p \Rightarrow q) \land p$	$[(p \Rightarrow q) \land p] \Rightarrow q$
T	$T$	T	$T$	$T$
T	$F$	F	$F$	$T$
F	$T$	T	$F$	$T$
F	$F$	T	$F$	$T$

As the truth table describes, the compound statement is always

true .

Therefore, the law is proven.

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Law of Detachment

Proof

Extra

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$p$	$q$	$p \Rightarrow q$	$(p \Rightarrow q) \land p$	$[(p \Rightarrow q) \land p] \Rightarrow q$
$T$	$T$	$T$	$?$	$?$
$T$	$F$	$F$	$?$	$?$
$F$	$T$	$T$	$?$	$?$
$F$	$F$	$T$	$?$	$?$

$p$	$q$	$p \Rightarrow q$	$(p \Rightarrow q) \land p$	$[(p \Rightarrow q) \land p] \Rightarrow q$
$T$	$T$	$T$	$?$	$?$
$T$	$F$	$F$	$?$	$?$
$F$	$T$	$T$	$?$	$?$
$F$	$F$	$T$	$?$	$?$

$p$	$q$	$p \Rightarrow q$	$(p \Rightarrow q) \land p$	$[(p \Rightarrow q) \land p] \Rightarrow q$
$T$	$T$	$T$	$?$	$?$
$T$	$F$	$F$	$?$	$?$
$F$	$T$	$T$	$?$	$?$
$F$	$F$	$T$	$?$	$?$