A new statement or proposition combing two statements are called compound statement or compound proposition.
The basic operators are "not" (¬), "and" (∧, or ), "or" (∨), "conditional" (→), and "biconditional" (↔). "Not" is a unary operator--it takes a single term ( ¬ P ). The rest are binary operators, taking two terms to make a compound statement ( P ∧ Q, P ∨ Q, P → Q, P ↔ Q ).
Note the similarity between the symbols for "and" () and "set theoretic intersection" ( ∩ ); likewise for "or" ( ∨ ) and "union ( ∪ ). This is not a coincidence: the definition of the intersection uses "and" and the definition of union uses "or".
Truth tables for these connectives:
| P | Q | ¬P | P ∧ Q | P ∨ Q | P → Q | P ↔ Q |
|---|---|---|---|---|---|---|
| T | T | F | T | T | T | T |
| T | F | F | F | T | F | F |
| F | T | T | F | T | T | F |
| F | F | T | F | F | T | T |
In order to reduce the number of necessary parentheses, one introduces precendence rules: ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So for example, P ∨ Q ∧ ¬ R → S is short for (P ∨ (Q ∧ (¬ R)) → S.
Note that the logical equivalence of certain compound statements entails that not all of these operators are necessary for a full-blooded logical calculus. For example, ¬ P ∨ Q is logically equivalent to P → Q; since logical equivalence means that equivalent terms may be subsituted for each other in an expression, it's not necessary to have a conditional operator. The five operators listed above are the basic set for the sake of convenience (and brevity).
One can also consider other connectives, such as NAND, XOR and NOR. It can be shown that all connectives can be expressed with NAND alone, and they can also all be expressed with NOR alone.