Questions: Variables in Logic

Question 1

In the statement 'If P, then Q,' what does the variable P represent?

• A. A number we do not know yet
• B. A statement that is either true or false
• C. A question that needs an answer
• D. The probability of an event

Answer: (B)

In logic, P is a propositional variable — it stands for a declarative statement with a definite truth value (true or false). This is different from algebraic variables, which represent numbers. P might stand for 'it is raining' or 'the number is prime' — any statement that is either true or false.

Question 2

The logical statement 'If P, then Q' can primarily be used when P means 'it is raining' and Q means 'the ground is wet.'

• T. True
• F. False

Answer: False

That is one specific substitution, but the whole point of using variables is generality. 'If P, then Q' captures the structure of ANY if-then argument. P could mean 'you practice daily' and Q could mean 'you improve.' The form works regardless of what specific statements P and Q represent.

Question 3

Translate this argument into variable form: 'If the battery is dead, then the car will not start. The battery is dead. Therefore, the car will not start.'

Model answer: Let P = 'the battery is dead' and Q = 'the car will not start.' The argument is: If P, then Q. P. Therefore Q.

The translation strips away the specific content and reveals the logical skeleton. This skeleton — 'If P, then Q; P; therefore Q' — is a valid argument form called modus ponens. Any argument with this shape is valid, regardless of what P and Q represent. That is the power of working with variables.