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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 ...
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.
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 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 ...
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.
Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is ...
Ramification (mathematics), a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. Ramification (botany), the divergence of the stem and limbs of a plant into smaller ones; Ramification group, filtration of the Galois group of a local field ...