Concept
Material nonimplication
Overview
Material nonimplication or abjunction (from Latin ab 'away' and junctio 'to join') 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}
.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
The copula is the sign denoting that there is an affirmation or denial; and thereby enabling the hearer or reader to distinguish a proposition from any other kind of discourse. Thus, in the proposition, The earth is round, the Predicate is the word _round_, which denotes the quality affirmed, or (as the phrase is) predicated: _the earth_, words denoting the object which that quality is affirmed of, compose the…
John Stuart Mill, A System of Logic
States what Material nonimplication is.
This is a question which we can not be exempted from answering: but the consideration of it belongs to what is called the Theory of Probability, which will form the subject of a chapter hereafter. It is seen, however, at once, that the conclusion does amount to a practical certainty after a sufficient number of instances, and that the method, therefore, is not radically vitiated by the characteristic…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Material nonimplication is.
In accordance with the above peculiarity of our Understanding it cannot happen that the whole shall contain the ground of the possibility of the connexion of the parts (which would be a contradiction in discursive cognition), but only that the _representation_ of a whole may contain the ground of the possibility of its form and the connexion of the parts belonging to it. Now such a whole would be an effect…
Immanuel Kant, Kant's Critique of Judgement
States what Material nonimplication is.
Philosophers who discussed Material nonimplication
Related ideas
- If and only if
- Negation
- Truth value
- Logical conjunction
- Boolean algebra
- Classical logic
- Material conditional
- Logical disjunction
- Proposition
Place in the tradition
Part of Logic gate.