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VENN DIAGRAMS FOR SYLLOGISMS. In a previous section, we used Venn di- agrams to test the validity of immediate inferences. Immediate inferences contain only two terms or classes, so the corresponding Venn diagrams need only two overlapping circles. Categorical syllogisms contain three terms or classes. To reflect this, we will use diagrams with three overlapping circles. If we use a bar over a letter to indicate that things in the area are not in the class (so that Sindicates what is not in S), then our diagram looks like this: $ SPM SPM SPY SPM SPM SPM SPM SPM This diagram has eight different areas, which can be listed in an order that resembles a truth table: M S S S S s 5 5 5 P M P P M P M P M P M P M р м Notice that, if something is neither an Snor a P nor an M, then it falls com pletely outside the system of overlapping circles. In every other case, a thing is assigned to one of the seven compartments within the system of overlapping circles. Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May nor be copied, scarined, or duplicated, in whole or in part. VALIDITY FOR CATEGORICAL ARGUMENTS 197 TESTING SYLLOGISMS FOR VALIDITY. To test the validity of a syllogism us- ing a Venn diagram, we first fill in the diagram to indicate the information contained in the premises. Remember that the only information contained in a Venn diagram is indicated either by shading out an area or by putting an asterisk in it. The argument is valid if the information expressed by the conclusion is already contained in the diagram for the premises. To see this, consider the diagrams for examples that we have already given: All rectangles have four sides. All squares are rectangles. All squares have four sides Here's the diagram for the premises: Squares Things having four sides Rectangles Here's the diagram for the conclusion: Squares Things having four sides This diagram for the conclusion contains only the information that nothing is in the circle for squares that is not also in the circle for things having four sides. In the diagram for the premises, all the things that are squares are cor- ralled into the region of things that have four sides. Thus, the diagram for the premises contains all of the information in the diagram for the conclusion. That shows that this syllogism is valid. Next, let's try a syllogism with a negative premise: No ellipses have sides. All circles are ellipses. No circles have sides. Here's the diagram for the premises: Gedes Things having sides Elipses Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. CHAPTER 7 CATEGORICAL LOGIC We diagram the conclusion "No circles have sides" as follows: Cirdes @ Things having sides That information is clearly already contained in the Venn diagram for the premises, so this syllogism is also valid. Let's try a syllogism with a particular premise All squares have equal sides. Some squares are rectangles. Some rectangles have equal sides. It is a good strategy to diagram a universal premise before diagramming a particular premise. The diagram for the above argument then looks like this: Rectangles Things having qual sides Squares Here's the diagram for the conclusion that there is something that is a rectangle that has equal sides: < Rectangles Things having equal sides The asterisk in the middle area of this diagram says that something is in both circles, and that information already appears in the diagram for the premises, so this argument is valid. So far we have looked only at valid syllogisms. Let's see how this method applies to invalid syllogisms. Here is one: All pediatricians are doctors. All pediatricians like children. All doctors like children We can diagram the premises at the left and the conclusion at the right: Copyright 2010 Cengage Leaming, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. VALIDITY FOR CATEGORICAL ARGUMENTS 199 Premises: All pediatricians are doctors. All pediatricions like children Conclusion: All doctors like children. Doctors People who like children Doctors People who like children Pediatricians It is evident that the information in the diagram for the conclusion is nof already contained in the diagram for the premises. The arrow shows dif- ferences in informational content. Thus, this form of syllogism is not valid. Notice that the difference between these diagrams not only tells us that this form of syllogism is invalid; it also tells us why it is invalid. In the diagram for the premises, there is no shading in the upper left area, which includes people who are doctors but are not pediatricians and do not like children. This shows that the premises do not rule out the possibility that some people are doctors without being pediatricians or liking children But if anyone is a doctor and not a person who likes children, then it is not true that all doctors like chil- dren. Because this is the conclusion of the syllogism, the premises do not rule out all of the ways in which the conclusion might be false. As a result, this con- clusion does not follow by virtue of categorical form. Here is an example of an invalid syllogism with particular premises. Some doctors are golfers. Some fathers are doctors. Some fathers are golfers. Premises: Conclusion: Fathers Golfers Fathers Golfers Doctors Examine this diagram closely. Notice that in diagramming "Some doctors are golfers," we had to put an asterisk on the boundary of the circle for Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned or duplicated, in whole or in part. CHAPTER 7 CATEGORICAL LOGIC 200 fathers, because we were not given information saying whether anything falls into the category of fathers or not. For the same reason, we had to put an asterisk on the boundary of the circle for golfers when diagramming "Some fathers are doctors." The upshot was that we did not indicate that anything exists in the region of overlap between fathers and golfers. But this is what the conclusion demands, so the form of this syllogism is not valid. Here is an invalid syllogism with negative premises: No babies are golfers. No fathers are babies. No fathers are golfers. Premises: Conclusion: Fothers Golfers Fathers Golfers Babies Again, we see that the form of this syllogism is not valid, because the entire area of overlap between the circles is shaded in the diagram for the conclu- sion, but part of that area is not shaded in the diagram for the premises. The method of Venn diagrams is adequate for deciding the validity or in- validity of all possible forms of categorical syllogism. To master this method, all you need is a little practice. < EXERCISE VII Using Venn diagrams, test the following syllogistic forms for validity: 1. All Mis P 4. All Pis M All Miss Some Miss All Sis P Some Sis P. 2. All Pis M 5. All Pis M All Miss Some Sis M All Sis P Some Sis P 3. All Mis P 6. All Pis M Some Miss Some S is not M. Some Sis P. Some S is not Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
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