Material nonimplication or abjunction () is a term referring to a logic operation used in generic circuits and

Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...

. It is the

negation In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...

of material implication. That is to say that for any two

proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...

P

and

Q

, the material nonimplication from

P

Q

is true

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

the negation of the material implication from

P

Q

is true. This is more naturally stated as that the material nonimplication from

P

Q

is true only if

P

is true and

Q

is false. It may be written using logical notation as

P \nrightarrow Q

P \not \supset Q

, or "L''pq''" (in Bocheński notation), and is logically equivalent to

\neg (P \rightarrow Q)

, and

P \land \neg Q

Definition

Truth table

Logical equivalences

Material nonimplication may be defined as the negation of material implication. In

classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...

, it is also equivalent to the negation of the

disjunction In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is ...

\neg P

and

Q

, and also the conjunction of

P

and

\neg Q

Properties

falsehood-preserving: The interpretation under which all variables are assigned a

truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...

of "false" produces a truth value of "false" as a result of material nonimplication.

Symbol

The symbol for material nonimplication is simply a crossed-out material implication symbol. Its Unicode symbol is 219B₁₆ (8603 decimal): ↛.

Natural language

Grammatical

"p minus q." "p without q."

Rhetorical

"p but not q." "q is false, in spite of p."

Computer science

Bitwise operation: A & ~B. This is usually called "bit clear" (BIC) or "and not" (ANDN). Logical operation: A && !B.

References

External links

* Logical connectives {{logic-stub