Conditional IF … THEN
In logic, a conditional is a compound statement formed by combining two sentences (or facts) using the words “if … then.” A conditional can also be called an implication.
The truth values for a conditional (implication) are hard to remember.
You will want to study this section.
The example below may help you remember the truth values for the conditional:
The statement: Your teacher tells you that “if you participate in class, then you will get extra points.”
fact 1: “you participate in class.”
fact 2: “you get participation points.”
When is the teacher’s statement true?
- If you participate in class (fact 1 true) and you get extra points (fact 2 true) then the teacher’s statement is true.
- If you participate in class (fact 1 true) and you do not get extra points (fact 2 false), then the teacher did not tell the truth and the statement is false.
- If you do not participate in class (fact 1 false), we cannot judge the truth of the teacher’s statement. The teacher did not tell you what would happen if you did NOT participate in class. Since we cannot accuse the teacher of making a false statement, we assign “true” to the statement.
“If you participate in class, then you will get extra points.” will be true in all cases except one: when you participate in class and you do NOT get the extra points.
Conditionals are FALSE only when the first condition (if) is true and the second condition (then) is false. All other cases are TRUE.
Mathematicians often use symbols and tables to represent concepts in logic. The use of these variables, symbols and tables creates a shorthand method for discussing logical sentences.
A truth table is a pictorial representation of all of the possible outcomes of the truth value of a compound sentence. Letters such as p and q are used to represent the facts (or sentences) within the compound sentence.
Truth table for conditional (if…then):
(notice the symbol used for “if…then” in the table below)
| p | q | p→q |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
REMEMBER:
IF…THEN is only FALSE when T implies F.
All other cases are TRUE.