Material nonimplication or abjunction (

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...

''ab'' = "from", ''junctio'' =–"joining") is the negation of material implication. That is to say that for any two

proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

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" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...

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, it is also equivalent to the negation of the

disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...

\neg P

and

Q

, and also the

conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy), in which two astronomical bodies ...

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''). Computing In some pro ...

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).

Definition

Truth table

Logical Equivalences

Properties

Symbol

Natural language

Grammatical

Rhetorical

Computer science

See also

References

External links