Conjunction (logic)

Venn diagram of The intersection of sets is defined by the conjunction${\ displaystyle A \ land B}$
Technical realization of the conjunction in the AND gate : When the buttons E1 and E2 are pressed, the lamp lights up.

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 )${\ displaystyle {\ land}}$
• 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 : ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$

 ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle A \ land B}$ true true true 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". ${\ displaystyle {A \ land B}}$${\ displaystyle A \ cap B}$${\ displaystyle AB}$

A conjunction itself is a Boolean expression . In digital technology , conjunctively linked variables are also called product term .

The following important laws apply to conjunction:

• Idempotence :${\ displaystyle A \ land A = A}$
• Associative law :${\ displaystyle A \ land (B \ land C) = (A \ land B) \ land C}$
• Commutative law :${\ displaystyle A \ land B = B \ land A}$
• De Morgan's rules :
${\ displaystyle \ neg {(A \ land B)} = \ neg {A} \ lor \ neg {B}}$
${\ displaystyle \ neg {(A \ lor B)} = \ neg {A} \ land \ neg {B}}$

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 . ${\ displaystyle {A \ land B}}$${\ displaystyle {A \ land B}}$${\ displaystyle A}$${\ displaystyle B}$

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:

${\ displaystyle T (A, B)}$ is a conjunction if:

• Commutativity :${\ displaystyle T (A, B) = T (B, A)}$
• Associativity :${\ displaystyle T (A, T (B, C)) = T (T (A, B), C)}$
• Monotony :${\ displaystyle A> B \ Rightarrow T (A, C) \ geq T (B, C)}$
• Single element :${\ displaystyle T (1, A) = A}$

Other useful but not necessary properties are continuity and idempotence .

For example, the following conjunctions were set up in three-valued logics :

Conjunction
in the three-valued logic Ł3

by Jan Łukasiewicz (1920)

${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle A \ land B}$
1 1 1
1 0.5 0.5
1 0 0
0.5 1 0.5
0.5 0.5 0.5
0.5 0 0
0 1 0
0 0.5 0
0 0 0

Conjunction
in the three-valued logic B3

by Dimitri Analtoljewitsch Bočvar (1938)

${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle A \ land B}$
1 1 1
1 0.5 0.5
1 0 0
0.5 1 0.5
0.5 0.5 0.5
0.5 0 0.5
0 1 0
0 0.5 0.5
0 0 0

The logical conjunction and the word "and"

The natural language word “and” is not identical with the conjunction in the sense of logic. On the one hand, the word “and” is not always used in the sense of the logical conjunction. Examples:

• “And then”
“I ate and (then) went home.” Here the word “and” is used to represent a sequence in time.
• “And therefore”
“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:

• "but"
"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.