Questions: Introduction to Propositional Logic

Question 1

Which of the following is a proposition in the logical sense?

• A. Close the door!
• B. Is it raining?
• C. Either the light is on or the light is off.
• D. This sentence is false.

Answer: (C)

A proposition must be a declarative statement with a definite truth value. Commands (option A) and questions (option B) are not truth-apt. 'Either the light is on or the light is off' is a tautology — true by the law of excluded middle — making it a valid proposition. The paradoxical sentence in option D is self-referential in a way that produces no stable truth value, disqualifying it as a well-formed proposition in classical propositional logic.

Question 2

In propositional logic, the truth value of a compound statement like 'P AND Q' depends on understanding what P and Q actually mean in the real world.

• T. True
• F. False

Answer: False

This is the central feature — and the power — of propositional logic: truth values are determined entirely by the truth values of the component propositions, not by their content or meaning. If P is true and Q is false, then 'P AND Q' is false regardless of whether P means 'It is raining' or 'The Pythagorean theorem holds'. Logic is formal, not semantic.

Question 3

What distinguishes a proposition from a sentence, and why does this distinction matter for building a formal logical system?

Model answer: A sentence is a grammatical unit of language; a proposition is the abstract content of a declarative sentence that has a definite truth value. The distinction matters because the same proposition can be expressed by different sentences (in different languages or phrasings), and the same sentence can fail to express a proposition (if it is a question, command, or paradox). Formal logic operates on propositions — truth-bearing objects — not on sentences, which allows the system to be language-independent and unambiguous.

This abstraction is what makes propositional logic a 'formal' system: by stripping away linguistic content and working only with truth values, we can prove things about the structure of valid arguments that hold universally, regardless of subject matter. The sentence/proposition distinction is the first step in that formalization.