### Logical Connectives:Conditional

A conditional statement has the form, " if p, then q." In such a statement, p is called the antecedent and q the consequent. The connection between them, the if-then relationship, is represented by the horseshoe symbol
().

A statement involving the conditional connective says that if the antecedent is true, the consequent is true as well. In propositional logic, such a statement is defined by the following truth table:

 p q pq T T T Conditional F T T T F F F F T

Example:

Consider the statement, "If I study hard, I'll pass the exam." The antecedent is "I study hard," and the consequent is "I'll pass the exam."

Which combinations of truth values for the antecedent and consequent are consistent with the truth of the conditional statement as a whole?

Suppose first that I do study hard and I do pass the exam: p and q are both true, as in the first line of the truth table. That's certainly consistent with the truth of the conditional.

Likewise, if I don't study hard, I don't pass--the last line of the table.

What about the second line? Suppose I don't study hard, but I still pass. Would that prove the conditional statement false? No -- the statement doesn't say that studying hard is the only way to pass; perhaps the exam was easy, or perhaps I already knew the material well.

But if I do study hard and I don't pass--the situation represented by the third line--then we'd have to conclude that the conditional is false.

In propositional logic, therefore, pq is defined as false in line 3, where p is true and q false, and defined as true in the other lines.

A conditional statement in English need not take the form, "if p, then q." Each of the following statements is also a conditional:

A. I will go camping this weekend if I finish my work.

B. I will go camping this weekend only if I finish my work.

C. We will go camping this weekend unless it rains.

Each of these statements can be translated systematically into an if-then structure, and symbolized according to the following rules:

A. p if q = If q, then p = qp

B. p only if q = If p, then q = pq

C. p unless q = If ~q, then p = ~q p

Thus, A says that if I finish my work, then I will go camping. B makes the different statement that if I go camping, I have finished my work. And C says that if it does not rain, then I will go camping.

Conjunction | Negation | Disjunction | Conditional
Logical Connectives