The Propositional Calculus
The Propositional Calculus (PC) is an astonishingly simple language, yet much can be learned (as we shall discover) from its study.
The Syntax of PC
The basic set of symbols we use in PC:
Sentence Letters or Propositional Variables
P, Q, R, S, T, U, V, ... (subscripted as necessary for indefinite supply.)
Connectives
∼, ∧, ∨, →, ↔ (read "squiggle", "caret", "vel", "arrow", and "double-arrow".)
Parentheses
(, )
Next we define a well-formed formula (or WFF) of PC:
1. Every sentence letter is a WFF of PC
2. If A and B are WFF's of PC, then so are
a. '∼A', and,
b. '(A ∧ B)', and,
c. '(A ∨ B)', and,
d. '(A → B)', and,
e. '(A ↔ B)'.
3. Nothing else is a WFF of PC
The Semantics of PC
With a few exceptions as noted, the semantics is likewise straightforward:
1. '∼A' is true iff A is false. (This is nothing more than negation.)
2. '(A ∧ B)' is true iff both A is true and B is true. (This is simple conjunction. We call A and B the left and right conjuncts, respectively.)
3. '(A ∨ B)' is true iff either A is true, or B is true, or both A is true and B is true. (This is the inclusive, as opposed to exclusive, disjunction. We call A and B the left and right disjuncts, respectively.)
4. '(A → B)' is true iff either A is false or B is true. (This known as the material conditional. A is called the antecedent, B the consequent.)
5. '(A ↔ B)' is true iff either both A is true and B is true or both A is false and B is false. (This is known as the bi-conditional.
Using these semantics, we can construct truth tables which show how the truth value of complex (or molecular) sentences in PC depend on the truth values of the individual sentence letters they contain. Once we've mastered that, we can use truth tables to show how one set of propositions entails a proposition, such that it is impossible for the entailed proposition to be false when all the entailing propositions are true. Indeed, we take this as one way of defining validity for PC, making it not a language simply, but a full-blown logic.