Truth table

Truth tables are a tool developed by [Charles Pierce]? in the 1880s and used in logic to determine whether an expression? is [necessarily true]?, [contingently true]?, or [necessarily false]?, or whether an argument is valid or not. For example, the [truth value]? of an expression such as '(P or Q) → R' depends on the truth values of the variables? 'P', 'Q', and 'R'. The truth table exhaustively lists, in separate rows, the possible combinations of truth values of all the variables in the expression, and then outputs the truth value of the complete expression for each row of the table. If an expression is true in every row of the table, it is necessarily true; if false in every row, it is necessarily false; if it is true in some and false in others, it is neither necessarily true nor necessarily false, but logically contingent.

We need a lot more here--the above is just a rough first attempt.

Truth tables are also used in Finite Mathematics and its applications to describe the boolean? outputs of an expression, circuit, or other computational entity for each possible value of its inputs. The input variable?s and output expressions are listed as column headings. The rows of the table are filled by listing each possible combination of inputs, one combination per row, and filling in the outputs that result from each combination of inputs.

Example of a truth table in logic:

    P  |  Q  |  P & Q  | P ∨ Q  | P xor Q | P → Q
  ----------------------------------------------------
    T  |  T  |    T    |   T     |    T    |   T
    T  |  F  |    F    |   T     |    F    |   F
    F  |  T  |    F    |   T     |    F    |   T
    F  |  F  |    F    |   F     |    T    |   T

Key:

T = true, F = false

& = and

&or = [inclusive or]?

xor = [exclusive or]?

→ = [logically implies]?)

Example of a truth table in finite mathematics:

    x  |  y  |  x and y  | x or y  | x xor y
  -------------------------------------------
    0  |  0  |     0     |   0     |    0
    0  |  1  |     0     |   1     |    1
    1  |  0  |     0     |   1     |    1
    1  |  1  |     1     |   1     |    0