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Chapter 16: Venn Diagrams. Venn Diagrams (pp. 159-160). 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.

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venn diagrams pp 159 160
Venn Diagrams (pp. 159-160)
  • 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.
venn diagrams pp 159 1603
Venn Diagrams (pp. 159-160)
  • 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.

venn diagrams pp 159 1604
Venn Diagrams (pp. 159-160)
  • 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.

venn diagrams for syllogisms pp 162 166
Venn Diagrams for Syllogisms (pp. 162-166)
  • 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:
venn diagrams for syllogisms pp 162 1666
Venn Diagrams for Syllogisms (pp. 162-166)
  • This divides the diagram into eight distinct regions (a line over a term means “not”):
venn diagrams for syllogisms pp 162 1667
Venn Diagrams for Syllogisms (pp. 162-166)
  • Diagram the premises to see whether you have diagrammed the conclusion.
    • You should always set up the diagram in the same way: upper left circle for the minor term; upper right circle for the major term; bottom circle for the middle term.
    • If by diagramming the premises you have diagrammed the conclusion, the argument is valid.
    • If by diagramming the premises you have not diagrammed the conclusion, the argument is invalid.
venn diagrams for syllogisms pp 162 1668
Venn Diagrams for Syllogisms (pp. 162-166)
  • If you have both a universal premise and a particular premise, you should diagram the universal premise first: this will sometimes “force” the X into a determinate region.
  • If you have a particular premise and the X is not forced into a determinate section of the diagram, it goes “on the line.” The line in question is always the line of the circle not mentioned in the premise.
  • It might be helpful to draw a separate, two-circle diagram of the conclusion; but never add anything to the three-circle diagram other than the diagrams of the premises.
venn diagrams examples pp 162 166
Venn Diagrams: Examples(pp. 162-166)
  • Consider the following syllogism:

All logicians are critical thinkers.

All philosophers are logicians.

All philosophers are critical thinkers.

  • Where L represents the middle term and C represents the major term, and P represents the minor term, the diagram for the major premise looks like this:
venn diagrams examples pp 162 16610
Venn Diagrams: Examples(pp. 162-166)

Now you diagram the minor premise on the same diagram:

venn diagrams examples pp 162 16611
Venn Diagrams: Examples(pp. 162-166)
  • If you’re so inclined, compare the diagram for the conclusion alone.
  • Since the premises require that all of P that is outside of C is shaded, we have diagrammed the conclusion in diagramming the premises. The argument is valid.
venn diagrams examples pp 162 16612
Venn Diagrams: Examples(pp. 162-166)
  • If you find the process a bit odd, consider an argument of the following form:

All P are M.

No M are S.

No S are P.

Draw a two circle diagram for each of the premises:

venn diagrams examples pp 162 16613
Venn Diagrams: Examples(pp. 162-166)
  • Roll them together to form a three-circle diagram:
  • You have diagrammed the conclusion by diagramming the premises. The argument form is valid.
venn diagrams examples pp 162 16614
Venn Diagrams: Examples(pp. 162-166)
  • 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:
venn diagrams examples pp 162 16615
Venn Diagrams: Examples(pp. 162-166)

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 examples pp 162 16616
Venn Diagrams: Examples(pp. 162-166)
  • Consider the following syllogism:

Some lizards are reptiles.All reptiles are beautiful beasts. Some beautiful beasts are lizards.

  • Here we have a particular premise and a universal premise. When you have both, you diagram the universal premise first. “Why?” you ask. The X for representing the particular should always go into a determinate area if possible. If you diagram the universal first, the X is forced into a determinate area of the diagram:
venn diagrams examples pp 162 16617
Venn Diagrams: Examples(pp. 162-166)

Then add the X.

The argument is valid. The diagram shows that there is at least one thing (X) that is a beautiful lizard, so the argument is valid.

venn diagrams examples pp 162 16618
Venn Diagrams: Examples(pp. 162-166)
  • If you’d diagrammed the particular premise first the X would have gone on the line, since the X goes on the line except when the area on one side of the line is shaded. So, if you’d diagrammed the particular premise first, the diagrams would look like this:
  • It is bad form to have an X on the line if the area on one side of the line is shaded. You would have to erase and place it in the unshaded area.
venn diagrams examples pp 162 16619
Venn Diagrams: Examples(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.

venn diagrams examples pp 162 16620
Venn Diagrams: Examples(pp. 162-166)
  • 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
venn diagrams examples pp 162 16621
Venn Diagrams: Examples(pp. 162-166)
  • Consider the following syllogism:

Some aardvarks are not sheep, and no sheep are trumpets, so all aardvarks are trumpets.

After making sure there are exactly three terms, you could represent the form as follows:

No S are T.

Some A are not S.

All A are T.

venn diagrams examples pp 162 16622
Venn Diagrams: Examples(pp. 162-166)
  • You diagram the major premise, since it’s universal:
  • Now you diagram the particular. The X has to be in A and outside of S. Since it could be in either of two areas, neither of which is shaded, you place the X on the T circle that divides A into two parts. It looks like this:
venn diagrams examples pp 162 16623
Venn Diagrams: Examples(pp. 162-166)

The X is on the line. That is sufficient to show that the argument form is invalid. If you prefer, you could compare the top two circles to the two-circle diagram for the conclusion. You’d notice that you have not diagrammed the conclusion. (The diagram for a universal is always a strictly shady affair.)

venn diagrams examples pp 162 16624
Venn Diagrams: Examples(pp. 162-166)
  • Consider the following:

All mice are rodents, so some mice are bothersome beasts, since some rodents are bothersome beasts.

  • There are three terms, so we may set out the form as follows:

Some R are B.

All M are R

Some M are B.

venn diagrams examples pp 162 16625
Venn Diagrams: Examples(pp. 162-166)
  • This time the major premise is a particular, and the minor premise is a universal. So, we diagram the minor premise first:
  • Now we diagram the major, placing an X in the area where B and R overlap. The X goes on the line:
venn diagrams examples pp 162 16626
Venn Diagrams: Examples(pp. 162-166)

The argument is invalid.

venn diagrams examples pp 162 16627
Venn Diagrams: Examples(pp. 162-166)
  • 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.