This lecture explains our fourth logical operation: the conditional.
- A conditional tells us that if the first part is true, the second part must be true as well.
- We use => as the conditional operator. So P => Q is read “if P, then Q.”
- If-then statements are the most common version of conditional statements. However, “only if,” “required,” “necessary,” and “sufficient” are indicators of alternative ways of making if-then statements. To translate these into the correct conditional statement, think about how to rewrite them in English as if-then statements while keeping the intended meaning. From there, it is easy to translate them into the correct logical conditional.
- In P => Q, the P part is sometimes called the antecedent or sufficient condition; the Q part is sometimes called the consequent or necessary condition.
- If the antecedent is false, we define the conditional as being vacuously true. The reason for this convention will become clear when we begin discussing truth tables.
- P => Q is logically equivalent to ~P v Q. (We will see why when we cover replacement rules.) This means that the conditional is technically unnecessary notation.