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In propositional logic, material implication [1] [2] is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not- P {\displaystyle P} or Q {\displaystyle Q} and that either form can replace the other in ...
For example, carrying on from the previous example, one can say that knowing that someone is called Socrates is sufficient to know that someone has a Name. A necessary and sufficient condition requires that both of the implications S ⇒ N {\displaystyle S\Rightarrow N} and N ⇒ S {\displaystyle N\Rightarrow S} (the latter of which can also be ...
An example of a conventional implicature is "Donovan is poor but happy", where the word "but" implicates a sense of contrast between being poor and being happy. [ 7 ] Later linguists introduced refined and different definitions of the term, leading to somewhat different ideas about which parts of the information conveyed by an utterance are ...
Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. [1] A sentence is said to be a logical consequence of a set of sentences, for a given language , if and only if , using only logic (i.e., without regard to any personal interpretations of the sentences) the sentence must ...
A set C of attributes is a concept intent if and only if C respects all valid implications. The system of all valid implications therefore suffices for constructing the closure system of all concept intents and thereby the concept hierarchy. The system of all valid implications of a formal context is closed under the natural inference.
A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.
Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it.. For example, the two-place truth function that always returns false is not definable from → and arbitrary propositional variables: any formula constructed from → and propositional variables must receive the value true when all of its variables are ...
In philosophy and artificial intelligence (especially, knowledge based systems), the ramification problem is concerned with the indirect consequences of an action. It might also be posed as how to represent what happens implicitly due to an action or how to control the secondary and tertiary effects of an action.