# Truth table

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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 ∨ Q) → R' (read as follows: "If either P or Q is true, then R is true") 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, and contains few of the sorts of things one needs to say to make these concepts clear to beginning logic students.

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 variables 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     |    F    |   T
T  |  F  |    F    |   T     |    T    |   F
F  |  T  |    F    |   T     |    T    |   T
F  |  F  |    F    |   F     |    F    |   T
```

Key:

T = true, F = false
& = and
∨ = 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
```

Contents of old truth tables page--to integrate with the above:

A truth table is the explicit depiction of the relationship of logical operators such as not, and, or, conditional, and biconditional. Generally limited to bivalent logic systems (where only two truth values are possible, true or false), the possible values of the terms involved are enumerated, as well as the result of performing the logical operation on the terms.

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 depicted in a truth table 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 also, then, 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
T   T      F     F         F
T   F      F     T         T
F   T      T     F         T
F   F      T     T         T
```

Because the enumeration of possible truth values for A and B yeilds the same truth value under both ¬ ( A ∧ B ) and ¬ A ∨ ¬ B, the two are logically equivalent, and may be substituted for each other (this particular equivalence is one of DeMorgan's Laws).