# Junction

A junction (from the Latin iungere “to link, connect”) is a logical connection between statements within the propositional logic , ie a logical operator . Junctions are also called connective, connectors, sentence operators, sentence links, sentence links, propositional links, logical connective words, linking symbols or functors and classified as logical particles .

Linguistically, there is often no distinction between the respective link itself (for example the conjunction ) and the word or linguistic sign that characterizes it (for example the word “and” or the sign “∧”).

In programming languages , propositional logics are also used, but they differ in essential points from the usual propositional logics. There they are mainly referred to as logical operators .

In (formal) logic , a statement that is composed of other statements with the help of particles such as “and”, “or”, “if – then” and “it is not the case that” is called complex or compound Statement or as a link to statements . A statement that is not composed of other statements is called an atomic statement .

Example: When Anna vacation has, then she goes to the sea.

The question of which of the theoretically possible connectives should be used for a logical system is - of course beyond the requirement of functional completeness - of a purely pragmatic nature. In classical propositional logic (see classical logic ) the following junctions are most common (related to two statements and ): ${\ displaystyle P}$${\ displaystyle Q}$

• the negation corresponds to a negation${\ displaystyle \ neg P}$
• the material implication , also called subjunction or conditional , corresponds to the sufficient condition "(already) if P, then Q"${\ displaystyle P \ rightarrow Q}$
• the biconditional , also called bisubjunction or equivalence , corresponds to a sufficient and necessary condition, "Q if and only if P"${\ displaystyle P \ leftrightarrow Q}$
• the conjunction , the logical and: "Both P and Q"${\ displaystyle P \ land Q}$
• the disjunction , the inclusive or: "Either P or Q or both"${\ displaystyle P \ vee Q}$

## Extensionality

An operator is called truth-functional or extensional if the truth value of a compound sentence formed by it is uniquely determined by the truth values ​​of its sub-sentences. The connectives of classical propositional logic are extensional in this sense. For a more precise definition of extensionality see the extensionality principle .

### Truth tables

Scheme: Truth table for a two-digit connector of a two-valued logic
${\ displaystyle P}$ ${\ displaystyle Q}$ ${\ displaystyle P \ circ Q}$
w w ${\ displaystyle WHW (P \ circ Q)}$,
for w and w ${\ displaystyle WHW (P) =}$${\ displaystyle WHW (Q) =}$
w f ${\ displaystyle WHW (P \ circ Q)}$,
for w and f ${\ displaystyle WHW (P) =}$${\ displaystyle WHW (Q) =}$
f w ${\ displaystyle WHW (P \ circ Q)}$,
for f and w ${\ displaystyle WHW (P) =}$${\ displaystyle WHW (Q) =}$
f f ${\ displaystyle WHW (P \ circ Q)}$,
for f and f ${\ displaystyle WHW (P) =}$${\ displaystyle WHW (Q) =}$
" " And " " are any two statements, " " stands for the connection as a logical operation, " " for truth value, "w" for the truth value "the true", "f" for the truth value "the false".${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle \ circ}$${\ displaystyle WHW}$

The so-called truth tables are a method to clearly display the truth value curve of extensional connectives in a logic with a finite number of truth values . In these, the truth value of the overall statement is specified in each line for an overall statement made up of individual statements using the junction for every possible assignment of truth values ​​to the individual statements . For a two-digit join of a two-digit logic, a truth table could look like the table on the right:

### Possible junctions

The number of statements that (or with which) an operator connects to a new statement is called its arity : A one-digit operator connects to a single statement to make a new statement, and two-digit joins to two statements to make a new statement and so on. In general, an n-digit juncture combines with n statements to form a new one.

The arity is not to be confused with the value, i. H. with the question of how many truth values ​​are permitted (see principle of bivalence ).

In classical logic , the most important single-digit juncture is the negation . Important two-digit joiners are the conjunction and the disjunction (often only these two are used). Classic three- and multi-digit junctions can also be traced back to combinations of one- and two-digit junctions.

In general, for -value logic, i. H. for a logic with finitely many truth values, the number of which is m, -digit truth-functional connectives. For the two-valued propositional logic there are therefore single-digit connectors and two-digit connectors. Even for the three-value propositional logic there are single-digit and two-digit connectives. ${\ displaystyle m}$${\ displaystyle m ^ {m ^ {n}}}$ ${\ displaystyle n}$${\ displaystyle 2 ^ {2 ^ {1}} = 4}$${\ displaystyle 2 ^ {2 ^ {2}} = 16}$${\ displaystyle 3 ^ {3 ^ {1}} = 27}$${\ displaystyle 3 ^ {3 ^ {2}} = 19 \, 683}$

The sixteen two-digit junctions of the two-valued logic are shown in the table below.

Table of the two-digit joiners of a two-valued logic
Names Truth values Symbols formula
 ${\ displaystyle P}$ ${\ displaystyle Q}$ w w
 ${\ displaystyle P}$ ${\ displaystyle Q}$ w f
 ${\ displaystyle P}$ ${\ displaystyle Q}$ f w
 ${\ displaystyle P}$ ${\ displaystyle Q}$ f f
Contradiction f f f f ${\ displaystyle \ bot}$ ${\ displaystyle P ~ \ land ~ \ neg P}$
conjunction w f f f ${\ displaystyle \ wedge}$ ${\ displaystyle P ~ \ land ~ Q}$
Post section, P only f w f f ${\ displaystyle \ not \ rightarrow}$, ${\ displaystyle \ not \ supset}$ ${\ displaystyle P ~ \ land ~ \ neg Q}$
Prependency, identity of P w w f f ${\ displaystyle \ rfloor}$ ${\ displaystyle P}$
Pre-section, Q only f f w f ${\ displaystyle \ not \ leftarrow}$, ${\ displaystyle \ not \ subset}$ ${\ displaystyle ~ \ neg P ~ \ land ~ Q}$
Post pendenz, identity of Q w f w f ${\ displaystyle \ lfloor}$ ${\ displaystyle Q}$
Contravalence , exclusive disjunction, XOR f w w f ${\ displaystyle \ not \ leftrightarrow}$, , , ,${\ displaystyle \ not \ equiv}$${\ displaystyle \ veebar}$${\ displaystyle {\ dot {\ vee}}}$${\ displaystyle \ oplus}$ ${\ displaystyle \ neg (P ~ \ leftrightarrow ~ Q)}$
Disjunction , adjunction w w w f ${\ displaystyle \ vee}$ ${\ displaystyle P ~ \ lor ~ Q}$
Peirce function , NOR, nihilition, rejection f f f w ${\ displaystyle \ downarrow}$, ${\ displaystyle {\ overline {\ vee}}}$ ${\ displaystyle ~ \ neg P ~ \ land ~ \ neg Q}$
Biconditional , Bijunktion , equivalence w f f w ${\ displaystyle \ leftrightarrow}$, ${\ displaystyle \ equiv}$ ${\ displaystyle P ~ \ leftrightarrow ~ Q}$
Postnon pendency, negation of Q f w f w ${\ displaystyle \ lceil}$ ${\ displaystyle \ neg Q}$
Replication w w f w ${\ displaystyle \ leftarrow}$, ${\ displaystyle \ subset}$ ${\ displaystyle P ~ \ leftarrow ~ Q}$
Prenon pendency, negation of P f f w w ${\ displaystyle \ rceil}$ ${\ displaystyle \ neg P}$
Subjunction , implication , conditional w f w w ${\ displaystyle \ rightarrow}$, ${\ displaystyle \ supset}$ ${\ displaystyle P ~ \ rightarrow ~ Q}$
Sheffer function , NAND , exclusion f w w w ${\ displaystyle \ mid}$, ,${\ displaystyle \ uparrow}$${\ displaystyle \ barwedge}$ ${\ displaystyle ~ \ neg P ~ \ lor ~ \ neg Q}$
tautology w w w w ${\ displaystyle \ top}$ ${\ displaystyle P ~ \ lor ~ \ neg P}$

### Reducibility and functional completeness

It is possible to express individual links through others; For example, the conjunction can be expressed by disjunction and negation as or conditional by the disjunction . In general, a set of connectives related to a logical system is called functionally complete or semantically complete if all other connective of the logical system can be expressed with the help of the connective concerned. For example, for classical propositional logic, the sets of junctions , and are functionally complete. This means that all connectives in classical propositional logic can be traced back to negation and conjunction, negation and disjunction, or negation and conditional. Junktorenmengen commonly used are , , . ${\ displaystyle A \ land B}$${\ displaystyle \ neg (\ neg A \ lor \ neg B)}$ ${\ displaystyle P \ rightarrow Q}$ ${\ displaystyle \ neg P \ vee Q}$${\ displaystyle \ {{\ neg}, {\ land} \}}$${\ displaystyle \ {{\ neg}, {\ lor} \}}$${\ displaystyle \ {{\ neg}, {\ rightarrow} \}}$${\ displaystyle \ {{\ neg}, {\ land}, {\ lor} \}}$${\ displaystyle \ {{\ neg}, {\ land} \}}$${\ displaystyle \ {{\ neg}, {\ lor} \}}$

In fact, it is possible to represent all links with the help of a single link, namely with the Sheffer function (NAND), but also with the Peirce function (NOR).

### Sheffer operators

When dealing with a joint alone, i.e. H. express all other junctures without adding any further junctions, then this juncture is called the Sheffer operator or Sheffer function (after Henry Maurice Sheffer ). There are exactly two Sheffer operators for classical propositional logic: the Sheffer's dash , also called NAND ( or ) and the Peirce operator , also called NOR . ${\ displaystyle \ uparrow}$${\ displaystyle \ vert}$${\ displaystyle \ downarrow}$

## Intensional operators

Logical operators in which the truth value of a sentence formed from them is not uniquely determined by the truth values ​​of their sub-clauses are called intentional joiners . Intensional are e.g. B. the one-digit modal operators “it is necessary that” and “it is possible that” (see modal logic ): The fact that a statement is true does not mean that this statement is also necessary. That a statement is wrong does not mean that it is impossible. The modalities therefore cannot be dealt with truth-functionally.

For the interpretation of intensional connectives one needs more complex models than the extensional truth tables. The first significant formal semantics of intensional joiners is probably the Kripke semantics originally developed by Saul Aaron Kripke for the interpretation of modal logic (see modal logic ). Kripke semantics are also suitable for interpreting intuitionist logic.

## Examples

Truth table for conjunction
in two-valued classical logic
Truth table for disjunction
in two-valued classical logic
Truth table for the material implication
in two-valued classical logic
Truth table for the conjunctor
in the three-valued logic Ł3
by Jan Łukasiewicz (1920)
${\ displaystyle P}$ ${\ displaystyle Q}$ ${\ displaystyle P \ land Q}$
true true true
true not correct not correct
not correct true not correct
not correct not correct not correct
${\ displaystyle P}$ ${\ displaystyle Q}$ ${\ displaystyle P \ lor Q}$
true true true
true not correct true
not correct true true
not correct not correct not correct
${\ displaystyle P}$ ${\ displaystyle Q}$ ${\ displaystyle P \ rightarrow Q}$
true true true
true not correct not correct
not correct true true
not correct not correct true
${\ displaystyle P}$ ${\ displaystyle Q}$ ${\ displaystyle P \ land Q}$
1 1 1
1 ½ ½
1 0 0
½ 1 ½
½ ½ ½
½ 0 0
0 1 0
0 ½ 0
0 0 0
Truth table for the conjunctor
in the three-valued logic B3
by Dimitri Analtoljewitsch Bočvar (1938)
In dialogic logic
${\ displaystyle P}$ ${\ displaystyle Q}$ ${\ displaystyle P \ land Q}$
1 1 1
1 ½ ½
1 0 0
½ 1 ½
½ ½ ½
½ 0 ½
0 1 0
0 ½ ½
0 0 0
opponent Proponent
${\ displaystyle P \ rightarrow Q}$
${\ displaystyle P?}$ The subjunctive assertion is attacked according to the subjunctive rule : The preceding one is asserted. ${\ displaystyle P}$
${\ displaystyle Q}$ The following is mentioned as a defense , this can be defended by adopting the previous line. Depending on the rule set, the statement can be attacked first. ${\ displaystyle Q}$${\ displaystyle P}$${\ displaystyle P}$