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.