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Law of Detachment
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Rule
Law of Detachment
For two statements p and q, the following compound statement is always true.
[ (p⇒ q) ∧ p ] ⇒ q
This can also be written as follows.
If p ⇒ q and p are both true, then q is true.
The Law of Detachment is used in deductive reasoning. Therefore, it can be applied to reach conclusions in mathematical proofs.
Proof
This law can be proven by making a truth table to see that the compound statement is always true.
[ (p⇒ q) ∧ p ] ⇒ q
The proof consists of three steps.
1
Determine the Number of Columns
expand_more To determine the number of columns in the truth table, the compound statement can be 
broken downinto simpler statements until the simplest statements p and q are obtained.
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⇒ q | (p⇒ q) ∧ p | [ (p⇒ q) ∧ p ] ⇒ q |
|---|---|---|---|---|
In this case, the truth table has six columns.
2
Determine the Number of Rows
expand_more 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.
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⇒ q | (p⇒ q) ∧ p | [ (p⇒ q) ∧ p ] ⇒ q |
|---|---|---|---|---|
| T | T | |||
| T | F | |||
| F | T | |||
| F | F |
3
Fill in the Cells
expand_more 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⇒ q). A conditional statement is false only when the hypothesis is true and the conclusion false.
| p | q | p⇒ q | (p⇒ q) ∧ p | [ (p⇒ q) ∧ p ] ⇒ q |
|---|---|---|---|---|
| T | T | T | ||
| T | F | F | ||
| F | T | T | ||
| F | F | T |
The next column is the conjunction (p⇒ q) ∧ p. A conjunction is false unless both statements are true.
| p | q | p⇒ q | (p⇒ q) ∧ p | [ (p⇒ q) ∧ p ] ⇒ 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⇒ q | (p⇒ q) ∧ p | [ (p⇒ q) ∧ p ] ⇒ 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.
Extra
ExampleThe law may sound more complicated than it actually is. For example, consider the following two statements that are both true.
- If a person lives in LA, then they live on the West Coast.
- Diego lives in LA.
According to the Law of Detachment, it can be concluded that Diego lives on the West Coast.
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