A truth table is a table used to establish the meaning of a logical connective as well as determining the validity of an argument. It is written as a two-dimensional array with n + 1 columns where n corresponds to the number of possible inputs. The last column, n + 1, is the column associated with the operation being performed. The number of rows in a truth table is dependent upon the type of statements given, either simple statements or compound statements. Truth tables are used in everything from binary logic to logic circuit tables in conjunction with Boolean operators.
During the late 1800s formal logic attempted to devise a complete, consistent formulation of mathematics. This formulation would be such that propositions could be formally stated and proved using a small number of symbols. Alfred North Whitehead's and Bertrand Russell's Principia Mathematica, published in 1925, showed that the problems with formal logic were too great to be overcome and that such a formulation was impossible. In lieu of formal logic a very simple form of logic was developed that relied on the study of truth tables and digital logic circuits. In the early 1900s Emil Leon Post published a paper on truth-table methods that introduced the concepts of completeness and consistency. Although Post, in this paper, attributed these methods to C. Keyser, it was previously accepted that C. Peirce and E. Schröder had formulated them. The studies of truth tables involve one or more outputs that depend on a combination of logical symbols and the input values. In such a formulation values at each step can take on values of only true or false. A useful principle for the analysis of truth tables is de Morgan's duality law. This law states that for every proposition involving logical addition and multiplication (logical operators for "or" and "and"), there are comparable statements that include the words addition and multiplication.
To construct a truth table, the required components need to be fully understood. A statement is a declaration whose validity can be determined to be true or false. In two-valued logic, the truth-value of a statement is either T if it is true or F if it is false. So for example, the statement "10 - 8 = 2" has a truth value of T. Compound statements are simple statements whose truth value is altered by one or more logical operators. The truth-value of a compound statement is dependent in a well-defined manner upon the truth-values of its simple components. Below is an example of a truth table that contains two simple statements, P and Q, and a conjunction ^ (the logical operator for "and") of those two statements:
As the truth table illustrates, for the conjunction of P and Q to be true, both simple statements have to have a truth-value of true. It is clear to see that truth tables can become very complicated when dealing with multiple statements. One of the most difficult parts in constructing a truth table is remembering the truth tables for different logical operators or connectives. Practically, truth tables can be used as proof tools only when there are no more than four different statements and the type of logic used is a two-valued logic, that is, that there are only two possible values: true or false.
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