### Implication

Definition:

- Two propositions connected by 'if...then...'.

Example:

- "If I give you a penny then I will give you a quarter."

Following the logical convention of letting a single letter represent a simple, atomic proposition:

- P = "I give you a penny."
- Q = "I will give you a quarter."

And replacing 'if...then...' with a symbol for implication: '>'

- The above implication can be symbolized: (P > Q)

The component to the left of '>' is called the antecedent, and the component to the right of '>' is called the consequent. Note that an implication can be looked on as a promise. The above implication can be thought of as saying: I promise to give you a quarter if I give you a penny. And if I do not give you a penny nothing is promised: I'm free to give you the quarter or not give you the quarter. So an implication is only false when you break the promise: I give you the penny(true that I give you the penny) but I do not give you the quarter (false that I give you the quarter. In every other case the promise is not broken and the implication is true. The truth value of an implication is a function of the truth values of the antecedent and the consequent. In logic this is characterized by a truth table:

IMPLICATION

INPUT | OUTPUT
--- | ---
P......Q | P > Q
T......T | T
T......F | F
F......T | T
F......F | T

Examples of implication:
(A > B)
[A > (Y = ~S)]
[l . (N > M)] > J