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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"
There is no sharp cutoff between implicatures, which are part of the intentional meaning of an utterance, and unintended implications the addressee may draw. For example, there may be no consensus whether ?+> Peter wants me to buy Susan some chocolate to cheer her up. is an implicature of the above utterance.
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 material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol → {\displaystyle \rightarrow } is interpreted as material implication, a formula P → Q {\displaystyle P\rightarrow Q} is true unless P {\displaystyle P} is true and Q {\displaystyle Q} is false.
An implication A→B is simply a pair of sets A⊆M, B⊆M, where M is the set of attributes under consideration. A is the premise and B is the conclusion of the implication A→B . A set C respects the implication A→B when ¬(C⊆A) or C⊆B.
In ordinary English (also natural language) "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. For example, being a man is a necessary condition for being a brother, but it is not sufficient—while being a man sibling is a necessary and sufficient condition for being a brother.
The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning.
Such a logical connective as converse implication "" is actually the same as material conditional with swapped arguments; thus, the symbol for converse implication is redundant. In some logical calculi (notably, in classical logic ), certain essentially different compound statements are logically equivalent .