Syntax of Propositional Calculus

(20th-July-2020)

• A proposition is a sentence, written in a language, that has a truth value (i.e., it is true or false) in a world. A proposition is built from atomic propositions using logical connectives.

• An atomic proposition, or just an atom, is a symbol that starts with a lower-case letter. Intuitively, an atom is something that is true or false.

• For example, ai_is_fun, lit_l1, live_outside, and sunny can all be atoms.

• In terms of the algebraic variables of the preceding chapter, an atom can be seen as a statement that a variable has a particular value or that the value is in a set of values. For example, the proposition classtimeAfter3 may mean ClassTime>3, which is true when the variable ClassTime has value greater than 3 and is false otherwise. It is traditional in propositional calculus not to make the variable explicit and we follow that tradition. A direct connection exists to Boolean variables, which are variables with domain {true, false}. An assignment X=true is written as the proposition x, using the variable name, but in lower case. So the proposition happy can mean there exists a Boolean variable Happy, where happy means Happy=true.