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 ...
s
and
, the material nonimplication from
to
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
to
is true. This is more naturally stated as that the material nonimplication from
to
is true only if
is true and
is false.
It may be written using logical notation as
,
, or "L''pq''" (in
Bocheński notation), and is logically equivalent to
, and
.
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 ...
of
and
, 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 ...
of
and
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).
Natural language
Grammatical
"p minus q."
"p without q."
Rhetorical
"p but not q."
Computer science
Bitwise operation: A&(~B)
Logical operation: A&&(!B)
See also
*
Implication
*
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 variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
References
External links
*
Logical connectives
{{logic-stub