Material Nonimplication

Material nonimplication or abjunction ( Latin ''ab'' = "from", ''junctio'' =–"joining") is the negation of material implication. That is to say that for any two propositions $P$ and $Q$, the material nonimplication from $P$ to $Q$ is true if and only if the negation of the material implication from $P$ to $Q$ is true. This is more naturally stated as that the material nonimplication from $P$ to $Q$ 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 of $\neg P$ 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 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

References

External links

*

Logical connectives

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