Propositional logic studies statements that can be true or false and the ways those statements can be combined. It provides a basic framework for analyzing reasoning.
A proposition is a statement with a truth value: it is either true or false. For example, London is the capital of England is a proposition. A question such as What is the capital of England? does not assert anything that can be true or false, so it is not a proposition. Commands, such as Close the door, are also not propositions.
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What is a proposition?
A declarative statement that can be assigned a truth value: true or false.
What is an atomic proposition?
A simple statement with no logical connective, treated as an indivisible unit in propositional logic.
What is a compound proposition?
A proposition formed by applying one or more logical connectives to propositions, as in P∧Q.
What does a biconditional express?
P↔Q means “P if and only if Q”; it is true when both sides have the same truth value.
What does propositional logic leave unanalyzed?
The internal structure of objects, predicates, and quantifiers within atomic statements.
Quiz questions
When is a conditional P→Q false in classical propositional logic?Answer: When P is true and Q is false.
How many truth-value assignments are possible for n distinct propositional variables?Answer: 2n assignments.
What makes a formula a tautology?Answer: It is true under every interpretation.
When is an argument valid in propositional logic?Answer: There is no interpretation with all premises true and the conclusion false.
In a semantic tableau testing an argument, what does an open branch represent?Answer: A possible counterexample with true premises and a false conclusion.