10.2 The Language of Logic

Just as with any other language, logic sentences are written in a special syntax. Every logical sentence is code for a proposition about a world that may or may not be true. For example, the sentence “the floor is lava” may be true in our agent’s world, but probably not true in ours. We can construct complex sentences by stringing together simpler ones with logical connectives to create sentences like “you can see all of campus from the Big C and hiking is a healthy break from studying”. There are five logical connectives in the language:

• \(\neg\), not: \(\neg P\) is true if and only if (iff) \(P\) is false. The atomic sentences \(P\) and \(\neg P\) are referred to as literals.
• \(\wedge\), and: \(A \wedge B\) is true iff both \(A\) is true and \(B\) is true. An ‘and’ sentence is known as a conjunction and its component propositions the conjuncts.
• \(\vee\), or: \(A \vee B\) is true iff either \(A\) is true or \(B\) is true. An ‘or’ sentence is known as a disjunction and its component propositions the disjuncts.
• \(\Rightarrow\), implication: \(A \Rightarrow B\) is true unless \(A\) is true and \(\) is false.
• \(\Leftrightarrow\), biconditional: \(A \Leftrightarrow B\) is true iff either both \(A\) and \(B\) are true or both are false.