Material nonimplication or abjunction is a term referring to a logic operation used in generic circuits and Boolean algebra. It is the negation of material implication. That is to say that for any two propositions P {\displaystyle P} and Q {\displaystyle Q} , the material nonimplication from P {\displaystyle P} to Q {\displaystyle Q} is true if and only if the negation of the material implication from P {\displaystyle P} to Q {\displaystyle Q} is true. This is more naturally stated as that the material nonimplication from P {\displaystyle P} to Q {\displaystyle Q} is true only if P {\displaystyle P} is true and Q {\displaystyle Q} is false.
It may be written using logical notation as P ↛ Q {\displaystyle P\nrightarrow Q} , P ⊅ Q {\displaystyle P\not \supset Q} , or "Lpq" (in Bocheński notation), and is logically equivalent to ¬ ( P → Q ) {\displaystyle \neg (P\rightarrow Q)} , and P ∧ ¬ Q {\displaystyle P\land \neg Q} .
Definition
Truth table
| A {\displaystyle A} | B {\displaystyle B} | A ↛ B {\displaystyle A\nrightarrow B} |
|---|---|---|
| F | F | F |
| F | T | F |
| T | F | T |
| T | T | F |
Logical equivalences
Material nonimplication may be defined as the negation of material implication.
| P ↛ Q {\displaystyle P\nrightarrow Q} | ⇔ {\displaystyle \Leftrightarrow } | ¬ ( P → Q ) {\displaystyle \neg (P\rightarrow Q)} |
| ⇔ {\displaystyle \Leftrightarrow } | ¬ {\displaystyle \neg } |
In classical logic, it is also equivalent to the negation of the disjunction of ¬ P {\displaystyle \neg P} and Q {\displaystyle Q} , and also the conjunction of P {\displaystyle P} and ¬ Q {\displaystyle \neg Q}
| P ↛ Q {\displaystyle P\nrightarrow Q} | ⇔ {\displaystyle \Leftrightarrow } | ¬ ( {\displaystyle \neg (} | ¬ P {\displaystyle \neg P} | ∨ {\displaystyle \lor } | Q ) {\displaystyle Q)} | ⇔ {\displaystyle \Leftrightarrow } | P {\displaystyle P} | ∧ {\displaystyle \land } | ¬ Q {\displaystyle \neg Q} |
| ⇔ {\displaystyle \Leftrightarrow } | ¬ ( {\displaystyle \neg (} | ∨ {\displaystyle \lor } | ) {\displaystyle )} | ⇔ {\displaystyle \Leftrightarrow } | ∧ {\displaystyle \land } |
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."
"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.
See also
External links
- Media related to Material nonimplication at Wikimedia Commons
References
Berco, Dan; Ang, Diing Shenp; Kalaga, Pranav Sairam (2020). "Programmable Photoelectric Memristor Gates for In Situ Image Compression". Advanced Intelligent Systems. 2 (9): 5. doi:10.1002/aisy.202000079. https://doi.org/10.1002%2Faisy.202000079 ↩