As a conjunction ( Latin coniungere, connect ') is called in logic a particular combination of two statements or propositional functions. Read the conjunction of two propositions A and B usually as " A and B". In classical logic, the conjunction of two statements is exactly true if both linked statements are true.
Can be meant by the word Conjunction
- The associated statement as a whole ( the phrase " A and B")
- The concatenation character ( connective )
- The linking word "and"
- In the case of truth-functional conjunction lets the truth function "et" with which to determine the truth value of the linked statement " A and B" of the truth values of its subsets (A, B)
The conjunction in the classical two-valued logic
In classical logic, the conjunction of two propositions A and B is only true if both A and B are true, and false if and only if at least one of the two statements A, B is false. This relationship is clearly shown in the truth table of the corresponding truth value function, the co - feature:
Common notations for the conjunction are, " A & B", " A ▪ B", " A ∩ B" ( Peano ) and "AB". In Polish notation, the conjunction is written as " Kab ".
A conjunction is itself a Boolean expression. In digital technology conjunctively linked variables are also called product term.
For the conjunction among other things, the following important laws are:
- Associative law:
- De Morgan's rules:
In calculi of natural deduction can be used as inference rules for the conjunction of the Konjunktionseinführung and Konjunktionsbeseitigung. With the Konjunktionseinführung can be of two propositions A, B close to their conjunction; with the Konjunktionsbeseitigung can be inferred from the conjunction to each of the conjuncts A and B respectively.
The conjunction in many-valued logics
When setting up a multivalued conjunction efforts are generally to maintain as many properties of the classical conjunction, in particular the associativity and commutativity. Thus, a multivalent conjunctions are defined axiomatically as follows:
Is a conjunction if:
- One element:
Other useful, but not necessary properties are continuity and idempotence.
In trivalent logics example, the following conjunctions have been identified:
The logical conjunction and the word "and "
The natural language word "and" is not identical with the conjunction in the sense of logic. On one hand, the word "and " is not always used in the sense of logical conjunction. Examples:
- "And then"
- " And therefore"
On the other hand, the conjunction may also be expressed by other linguistic tools. example: