Truth table
A truth table or truth table, and truth -value table or confusion matrix called, is a tabulation of the truth value of a logical progression statement. The truth table shows all possible mappings of a finite number (often two) truth values to the propositionally not separable part of statements that make up the overall presentation is composed, what truth value assumes the overall presentation of the respective assignment. The truth table is used to represent truth values of functions or Boolean functions or define and perform simple propositional evidence. For example, truth tables are used to define the meaning of connectives.
Representation of Boolean functions
For the divalent case, the truth value is 'true' as and hereinafter referred to as "wrong."
Generally, there is an m- valued logic, ie a logic with finitely many truth values whose number is m, n-ary truth-functional connectives or Boolean functions. For the bivalent propositional logic connectives so there's digit and two-digit connectives. Even for the trivalent propositional logic, there are one-digit and two-digit connectives.
The following table gives for each truth value of the statements a and b are the result of several divalent links to:
A special position (see Functional completeness and Sheffer operator ) have the following named after Henry Maurice Sheffer and Charles Sanders Peirce divalent functions:
The following table indicates two of 19,683 two -digit combinations of the three-valued logic; is the conjunction of the three-valued logic L3 by Jan Łukasiewicz (1920 ) and to the conjunction of the three-valued logic B3 by Dmitri Anatoljevich Bochvar ( Dmitry Anatol'evič Bočvar ) ( 1938).
The following table is a two digit of possible operators for the tetravalent logic again, the truth table for the conditional ( material implication ) in the logical system G4 of Kurt Gödel ( 1932).
Evidence and decision-making process
Truth tables are used to perform simple propositional evidence on the semantic model level, especially for the validity of the fundamental laws on which to build logical reasoning process. For example demonstrates the logical equivalence of the 3rd and 4th column in the following truth tables the validity of De Morgan 's laws:
In practice, this kind of argument is, however, only for statements with a small number of propositional variables, since the size grows exponentially in the number of variables.
For propositional logic with a finite number of truth values and classic inferential concept (see Classical logic) Truth tables are a decision procedure for many important issues, that is a process by which a particular issue can decide mechanically for each statement in a finite time. Thus, with the help of truth tables to decide the question whether a given statement is satisfiable, unsatisfiable or tautological (see satisfiability of propositional logic ); also enables you to decide whether an argument is valid or invalid.
Transformation into different forms of representation
The content of a truth table can be transferred for further processing or simplification in other equivalent representations, for example, in a Karnaugh Veitch diagram.
The History
If one understands the homomorphic mapping of truth values to the occurring in a statement atomic propositions under a truth table, then the truth table on Philo of Megara goes back, the defined this way in the 4th century BC the truth function for the material implication. Also in the of Soli Chrysippus influenced by Stoic logic truth tables have been used extensively in this sense.
In modern logic, George Boole used truth tables in 1847 under the name " modules of a function" for the semantic decidability of logical terms ( functions). Later, Gottlob Frege and Charles Sanders Peirce benützten this decision process in which Peirce the purpose of the determination of tautologies emphasized clearly. Truth tables in the literal sense as tables, however, were only introduced in 1921 by Emil Leon Post and Ludwig Wittgenstein; through their influence truth tables were as a method of decision for tautologies common knowledge.