A conjunction ( Latin coniungere ' to connect'; common language: and connection ) is a specific connection between two statements or propositional functions in logic . The conjunction of two statements A and B is usually read as "A and B". In classical logic , the combination of two statements "A and B" is true if both statement components, "A" and "B", are each true.
Can be meant by the word conjunction
- the linked statement as a whole (the sentence "A and B")
- the link mark ( connective )
- the linking word "and"
- in the case of a truth-functional conjunction, the truth function “et”, with which the truth value of the linked statement “A and B” can be determined from the truth values of its sub-clauses (A, B)
The conjunction in classical, two-valued logic
In the classical logic is the conjunction of two statements and only true if both and are true, and if and wrong if at least one of the two statements , is incorrect. This relationship is clearly shown in the truth table of the corresponding truth value function , the et function :
|true||not correct||not correct|
|not correct||true||not correct|
|not correct||not correct||not correct|
Common spellings for the conjunction are , "A & B", "A ▪ B", " " (Peano) and " ". In Polish notation , the conjunction is written as "Kab".
The following important laws apply to conjunction:
In calculi of natural inference , the introduction and elimination of the conjunction are used as rules of inference for the conjunction. With the introduction of the conjunction, two statements A, B can be used to infer their conjunction ; With the elimination of the conjuncture , each of the conjuncts or can be inferred from the conjunction .
The conjunction in multi-valued logics
When setting up a multi-valued conjunction, one generally tries to retain as many of the properties of the classical conjunction as possible, especially associativity and commutativity. A multi-valued conjunction can thus be defined axiomatically as follows:
is a conjunction if:
For example, the following conjunctions were set up in three-valued logics :
in the three-valued logic Ł3
by Jan Łukasiewicz (1920)
in the three-valued logic B3
by Dimitri Analtoljewitsch Bočvar (1938)
The logical conjunction and the word "and"
“I ate and (then) went home.” Here the word “and” is used to represent a sequence in time.
“The patient took the drug and got (therefore) well.” Here a causal relationship is expressed.
On the other hand, the conjunction can also be expressed through other linguistic means . Example:
- "It's spring and it's raining."
- "It's spring, but it's raining."
- From a propositional point of view, these two sentences are equivalent.