Venn diagram, graphical method of representing categorical propositions and also testing the validity that categorical syllogisms, devised by the English logician and also philosopher john Venn (1834–1923). Long recognized for your pedagogical value, Venn diagrams have actually been a standard part of the curriculum of introduce logic since the mid-20th century.

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Venn introduced the diagrams that bear his name together a way of representing relationships of inclusion and also exclusion in between classes, or sets. Venn diagrams consists of two or three intersecting circles, each representing a class and also each labeled v an uppercase letter. Lowercase x’s and also shading are used to suggest the existence and also nonexistence, respectively, of some (at the very least one) member the a given class.

Two-circle Venn diagrams are offered to stand for categorical propositions, who logical relations were an initial studied systematically by Aristotle. Such propositions consists of two terms, or class nouns, referred to as the topic (S) and the property (P); the quantifier all, no, or some; and also the copula are or are not. The proposition “All S space P,” referred to as the universal affirmative, is stood for by shading the component of the circle labeling S that does not intersect the circle labeled P, indicating that there is nothing that is an S the is not additionally a P. “No S are P,” the universal negative, is stood for by shading the intersection the S and also P; “Some S space P,” the certain affirmative, is represented by place an x in the intersection that S and P; and also “Some S space not P,” the particular negative, is represented by placing an x in the part of S that does not intersect P.

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Three-circle diagrams, in which each circle intersects the other two, are supplied to stand for categorical syllogisms, a type of deductive debate consisting of two categorical premises and also a categorical conclusion. A usual practice is to brand the circles with funding (and, if necessary, additionally lowercase) letters matching to the topic term that the conclusion, the predicate term of the conclusion, and the middle term, which appears once in each premise. If, after ~ both premises are diagrammed (the global premise first, if both room not universal), the conclusion is additionally represented, the syllogism is valid; i.e., the conclusion complies with necessarily indigenous its premises. If not, it is invalid.


Three examples of categorical syllogisms are the following.

All Greeks space human. No people are immortal. Therefore, no Greeks are immortal.

Some mammals space carnivores. Every mammals are animals. Therefore, some pets are carnivores.

Some sages room not seers. No seers room soothsayers. Therefore, some sages are not soothsayers.

To chart the premises of the an initial syllogism, one shades the component of G (“Greeks”) the does not intersect H (“humans”) and the part of H that intersects i (“immortal”). Due to the fact that the conclusion is represented by the shading in the intersection that G and I, the syllogism is valid.

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To diagram the 2nd premise that the second example—which, since it is universal, need to be diagrammed first—one shades the component of M (“mammals”) that does not intersect A (“animals”). To diagram the first premise, one places an x in the intersection that M and also C. Importantly, the component of M the intersects C yet does not intersect A is unavailable, since it to be shaded in the diagramming of the an initial premise; thus, the x should be put in the component of M the intersects both A and also C. In the resulting diagram the conclusion is stood for by the illustration of one x in the intersection the A and also C, so the syllogism is valid.

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To diagram the universal premise in the 3rd syllogism, one shades the part of Se (“seers”) that intersects so (“soothsayers”). To diagram the specific premise, one areas an x in Sa (“sages”) top top that part of the border of So the does no adjoin a shaded area, i beg your pardon by an interpretation is empty. In this way one suggests that the Sa that is no an Se might or might not it is in an so (the sage that is not a seer may or may not it is in a soothsayer). Because there is no x that shows up in Sa and also not in So, the conclusion is not represented, and also the syllogism is invalid.

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Venn’s Symbolic Logic (1866) contains his fullest advancement of the technique of Venn diagrams. The mass of that work, however, was specialized to defending the algebraic interpretation of propositional logic presented by the English mathematician George Boole.