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A set C respects the implication A→B when ¬(C⊆A) or C⊆B. A formal context is a triple (G,M,I), where G and M are sets (of objects and attributes, respectively), and where I⊆G×M is a relation expressing which objects have which attributes. An implication that holds in such a formal context is called a valid implication
Logical consequence (also entailment or logical implication), the relationship between statements that holds true when one logically "follows from" one or more others; Material conditional (also material implication), a logical connective and binary truth function typically interpreted as "If p, then q"
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements.
The first implication suggests that S is a sufficient condition for N, while the second implication suggests that S is a necessary condition for N. This is expressed as " S is necessary and sufficient for N ", " S if and only if N ", or S ⇔ N {\displaystyle S\Leftrightarrow N} .
If natural language conditionals were understood in the same way, that would mean that the sentence "If the Nazis had won World War Two, everybody would be happy" is vacuously true. Given that such problematic consequences follow from a seemingly correct assumption about logic, they are called paradoxes .
Definition 1: Atomic propositional variables are formulas. ... Some of these connectives may be defined in terms of others: for instance, implication, ...
Implications for the US and its relations with Mexico Some legal experts said Trump's decision to rename the Gulf of Mexico says a lot about how he wants himself, and the US, to be perceived by ...
Now assume that Peter and the hearer both have access to the contextual information that Susan is ambitious. If she loses at something, she's pretty downcast. and that Peter intended the hearer to activate this knowledge. Then this is an implicated premise. The hearer can now draw the contextual implications that +> Susan needs to be cheered up.