Truth table

Truth tables are a tool developed by Charles Peirce in the 1880s. A truth table is a kind of mathematical table used in logic to determine whether an expression is true or whether an argument is valid.

Truth tables show the values, relationships, and the results of performing logical operations on logical expressions. The logical operators are not, and, or, conditional, and biconditional. The logical expressions are necessarily true, contingently true, or necessarily false.

The column headings on a truth table show the input variables and output expressions. The rows show each possible combination of inputs, one combination per row, and the outputs that result from each combination of inputs.

Truth tables are usually limited to bivalent logic systems where only two truth values are possible, true or false.

For example, take two terms, A and B, and the logical operator "and" (∧), signifying the conjunction "A and B". In common English, if A is true and B is true, then the conjunction "A and B" is true; under all other possible assignments of truth values to A and B, the conjunction is false. This relationship is defined as follows:

A B A ∧ B T T T T F F F T F F F F

In a bivalent logic system, all the operators can be explicitly defined this way. For example, the not (~) relationship is defined as follows:

  A   ~A
  T    F
  F    T

The or (∨) relationship is defined as follows:

A B A ∨ B T T T T F T F T T F F F

Compound expressions can be constructed, using parenthesis to denote precedence. The negation of conjunction [ ~( A ∧ B ) ], is depicted as follows:

A B A ∧ B ~( A ∧ B ) T T T F T F F T F T F T F F F T

Truth tables can be used to prove logical equivalence. The truth table for the disjunction of not-A and not-B is:

  A   B    ~A    ~B     A ∧ B    ~(~A ∨ ~B)
  T   T     F     F       T            T
  T   F     F     T       F            F
  F   T     T     F       F            F
  F   F     T     T       F            F

(Because the enumeration of possible truth-values for A and B yields the same truth-value under both ( A ∧ B ) and ~(~A ∨ ~B), the two are logically equivalent, and may be substituted for each other. This equivalence is one of DeMorgan's Lawss.)

An example of a truth table in logic:

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

Key:

T = true, F = false

& (or ∧) = and

∨ = inclusive or

⊕ = exclusive or

→ = conditional or "if-then

↔ = biconditional or "if-and-only-if

Johnston diagrams, similar to Venn diagrams or Euler Diagrams, provide a way of visualizing truth tables. An interactive Johnston diagram illustrating truth tables is at LogicTutorial.com

Truth-tables and Truth-Functions can be found in Tractatus Logico-Philosophicus, Proposition 5.101.