Logical connectives
AND ${\displaystyle A\land B}$, ${\displaystyle A\cdot B}$, ${\displaystyle AB}$, ${\displaystyle A\&B}$, ${\displaystyle A\&\&B}$ equivalent ${\displaystyle A\equiv B}$, ${\displaystyle A\Leftrightarrow B}$, ${\displaystyle A\leftrightharpoons B}$ implies ${\displaystyle A\Rightarrow B}$, ${\displaystyle A\supset B}$, ${\displaystyle A\rightarrow B}$ NAND ${\displaystyle A{\overline {\land }}B}$, ${\displaystyle A\uparrow B}$, ${\displaystyle A\mid B}$, ${\displaystyle {\overline {A\cdot B}}}$ nonequivalent ${\displaystyle A\not \equiv B}$, ${\displaystyle A\not \Leftrightarrow B}$, ${\displaystyle A\nleftrightarrow B}$ NOR ${\displaystyle A{\overline {\lor }}B}$, ${\displaystyle A\downarrow B}$, ${\displaystyle {\overline {A+B}}}$ NOT ${\displaystyle \neg A}$, ${\displaystyle -A}$, ${\displaystyle {\overline {A}}}$, ${\displaystyle \sim A}$ OR ${\displaystyle A\lor B}$, ${\displaystyle A+B}$, ${\displaystyle A\mid B}$, ${\displaystyle A\parallel B}$ XNOR ${\displaystyle A\ {\text{XNOR}}\ B}$ XOR ${\displaystyle A{\underline {\lor }}B}$, ${\displaystyle A\oplus B}$ converse ${\displaystyle A\Leftarrow B}$, ${\displaystyle A\subset B}$, ${\displaystyle A\leftarrow B}$
Related concepts
Propositional calculus Predicate logic Boolean algebra Truth table Truth function Boolean function Functional completeness
Applications
Digital logic Programming languages Mathematical logic Philosophy of logic
Category
v t e

In logic, a truth function^[1] is a function that accepts truth values as input and produces a unique truth value as output. In other words: the input and output of a truth function are all truth values; a truth function will always output exactly one truth value, and inputting the same truth value(s) will always output the same truth value. The typical example is in propositional logic, wherein a compound statement is constructed using individual statements connected by logical connectives; if the truth value of the compound statement is entirely determined by the truth value(s) of the constituent statement(s), the compound statement is called a truth function, and any logical connectives used are said to be truth functional.^[2]

Classical propositional logic is a truth-functional logic,^[3] in that every statement has exactly one truth value which is either true or false, and every logical connective is truth functional (with a correspondent truth table), thus every compound statement is a truth function.^[4] On the other hand, modal logic is non-truth-functional.

Overview

A logical connective is truth-functional if the truth-value of a compound sentence is a function of the truth-value of its sub-sentences. A class of connectives is truth-functional if each of its members is. For example, the connective "and" is truth-functional since a sentence like "Apples are fruits and carrots are vegetables" is true if, and only if, each of its sub-sentences "apples are fruits" and "carrots are vegetables" is true, and it is false otherwise. Some connectives of a natural language, such as English, are not truth-functional.

Connectives of the form "x believes that ..." are typical examples of connectives that are not truth-functional. If e.g. Mary mistakenly believes that Al Gore was President of the USA on April 20, 2000, but she does not believe that the moon is made of green cheese, then the sentence

"Mary believes that Al Gore was President of the USA on April 20, 2000"

is true while

"Mary believes that the moon is made of green cheese"

is false. In both cases, each component sentence (i.e. "Al Gore was president of the USA on April 20, 2000" and "the moon is made of green cheese") is false, but each compound sentence formed by prefixing the phrase "Mary believes that" differs in truth-value. That is, the truth-value of a sentence of the form "Mary believes that..." is not determined solely by the truth-value of its component sentence, and hence the (unary) connective (or simply operator since it is unary) is non-truth-functional.

The class of classical logic connectives (e.g. &, →) used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by truth tables. Truth-functional propositional calculus is a formal system whose formulae may be interpreted as either true or false.

Table of binary truth functions

In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs P and Q. Any of these functions corresponds to a truth table of a certain logical connective in classical logic, including several degenerate cases such as a function not depending on one or both of its arguments. Truth and falsehood are denoted as 1 and 0, respectively, in the following truth tables for sake of brevity.

Zdroj:https://en.wikipedia.org?pojem=Truth-function
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Contradiction/False Notation Equivalent formulas Truth table Venn diagram ${\displaystyle \bot }$ "bottom" P ∧ ¬P Opq   Q 0 1 P 0    0   0  1    0   0	Tautology/True Notation Equivalent formulas Truth table Venn diagram ${\displaystyle \top }$ "top" P ∨ ¬P Vpq   Q 0 1 P 0    1   1  1    1   1
Proposition P Notation Equivalent formulas Truth table Venn diagram P p Ipq   Q 0 1 P 0    0   0  1    1   1	Negation of P Notation Equivalent formulas Truth table Venn diagram ¬P ~P Np Fpq   Q 0 1 P 0    1   1  1    0   0
Proposition Q Notation Equivalent formulas Truth table Venn diagram Q q Hpq   Q 0 1 P 0    0   1  1    0   1	Negation of Q Notation Equivalent formulas Truth table Venn diagram ¬Q ~Q Nq Gpq   Q 0 1 P 0    1   0  1    1   0