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The Venn Diagrams
 

The Venn Diagrams

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    The Venn Diagrams The Venn Diagrams Presentation Transcript

    • Venn Diagrams
    • Venn Diagrams • Venn diagrams represent the relationships between classes of objects by way of the relationships among circles. • Venn diagrams assume the Boolean interpretation of categorical syllogisms. • Shading an area of a circle shows that it is empty. • Placing an X in an area of a circle shows that there is at least one thing that is contained in the class represented by that area. • For universal propositions, shade (draw lines through) the areas that are empty. All S are P. All P are S. No S are P No P are S
    • • For particular propositions, place an X in the area that is inhabited. - Some S are P. - Some P are S. - Some S are not P. - Some P are not S.
    • • To test a syllogism by Venn diagrams, you diagram the premises to see whether the conclusion is also diagrammed. • This requires three interlocking circles, one for each term: • This divides the diagram into eight distinct regions (a line over a term means “not”):
    • Venn Diagrams: Examples • Consider the following syllogism: No arachnids are cows. All spiders are arachnids. No spiders are cows. • Let S represent the minor term (spiders), C represent the major term (cows), and A represent the middle term (arachnids). Since both premises are universals, let us begin by diagramming the major premise. We shade the area were S and C overlap: Now diagram the minor premise on the same diagram: Compare the diagram for the conclusion alone, if you wish: By diagramming the premises we have diagrammed the conclusion. The argument is valid.
    • Venn Diagrams: (pp. 162-166) • Most syllogistic forms are invalid. Consider the following: All P are M. All M are S. All S are P. • Diagram the major premise, then diagram the minor premise on the same diagram: We have diagrammed “All P are M,” which is not the conclusion. So the argument form is invalid.
    • • Consider an argument of the following form: All M are P. No M are S. No S are P. • An area has been shaded twice. So, we haven’t diagrammed the conclusion. The argument form is invalid Examples
    • • In summary: – Make sure you have exactly three terms. – If there is a universal premise and a particular premise, diagram the universal premise first. – If neither of the areas where the X could go is shaded, the X goes on the line. – No syllogism whose diagram places an X on the line or results in double-shading is valid. – It is valid if and only if shading the premises results in shading the conclusion.
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