Propositions and Propositional Variables

Math for CS · Axiom Academy

True/false statements and the building blocks of logic In logic, a proposition is a declarative statement that is either true or false, but not both. "Paris is the capital of France" (TRUE) "The moon is made of cheese" (FALSE) "What time is it?" (question, not a statement) "Close the door!" (command, not a statement) "x + 5 = 10" (open sentence, truth depends on x) "This sentence is false" (paradoxical, can't have a definite truth value) Propositions can be combined to form more complex statements. Examples of atomic propositions: Examples of compound propositions: "It is raining AND the sun is shining" (p ∧ q) "I will go to the park OR stay at home" (r ∨ s) Every proposition has a truth value: either TRUE or FALSE. In logic, we use variables (letters) to represent propositions, making it easier to work with complex statements. Convention: Use lowercase letters (p, q, r, s, ...) for propositions. Statement: "If it is raining, then the streets are wet" → p → q This notation allows us to reason about the structure of arguments without getting distracted by the specific content.

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