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In the case that the hypothesis is true, the result is the same as conclusion. Otherwise, the whole statement is true regardless the value of conclusion.
The same [[binary relation]] is called [[inclusion]] (for [[set]]s), [[subsumption]] (for [[concept]]s), or [[implication]] (for [[proposition]]s). This relation might also be represented by the sign <math>\<</math>; because it has formal properties analogous to those of the mathematical relation <math>\lt</math> ([[less than]]) or more exactly <math>\leq</math>, especially the relation of not being symmetrical.
In the conceptual interpretation, when <math>a</math> and <math>b</math> denote concepts, the relation <math>a \ltin b</math> signifies that the concept <math>a</math> is subsumed under the concept <math>b</math>; that is, it is a species with respect to the genus <math>b</math>. From the extensive point of view, it denotes that the class of <math>a</math>'s is contained in the class of <math>b</math>'s or makes a part of it; or, more concisely, that ``"All <math>a</math>'s are <math>b</math>'s''". From the comprehensive point of view it means that the concept <math>b</math> is contained in the concept <math>a</math> or makes a part of it, so that consequently the character <math>a</math> implies or involves the character <math>b</math>. Example: "All men are mortal"; "Man implies mortal"; "Who says man says mortal"; or, simply, "Man, therefore mortal".
In the propositional interpretation, when <math>a</math> and <math>b</math> denote propositions, the relation <math>a \ltRightarrow b</math> signifies that the proposition <math>a</math> implies or involves the proposition <math>b</math>, which is often expressed by the hypothetical judgement, "If <math>a</math> is true, <math>b</math> is true"; or by "<math>a</math> implies <math>b</math>''; or more simply by "<math>a</math>, therefore <math>b</math>". We see that in both interpretations the relation may be translated approximately by "therefore".
''Remark''. -- Such a relation is a proposition, whatever may be the interpretation of the terms <math>a</math> and <math>b</math>.
Consequently, whenever a <math>\ltRightarrow</math> relation has two like relations (or even only one) for its members, it can receive only the propositional interpretation, that is to say, it can only denote an implication.
A relation whose members are simple terms (letters) is called a ''primary'' proposition; a relation whose members are primary propositions is called a ''secondary'' proposition, and so on.
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