Conditionals

A proposition of type $A \rightarrow B$ is called a conditional proposition which could be understood as follows: “If $A$, then $B$.” A conditional proposition has two equivalents:

• $\neg B\rightarrow (\neg A)$

• $\neg A \vee B$

In the conditional, proposition $A$ is called a hypothesis or antecedent, and proposition $B$ is a conclusion or consequent. If the antecedent is true but its consequent is false, as the result we get a false conditional. All other combinations of propositions generate a true conditional. The following table clarifies it:

$\text{Truth table for conditional } p\rightarrow q$

Remember, the negation of a conditional type $A \rightarrow B$ will be

$$\neg (A\rightarrow B) = A \wedge (\neg B).$$

If Rafael goes to the beach, then his girlfriend will go with him.

Negation: Rafael goes to the beach, but his girlfriend doesn't go with him.