All corporations that overcharge their customers are unethical
businesses.
Some unethical businesses are investor-owned utilities
Some investor owned utilities are corporations that overcharge their
customers.
LESSON #16
Venn Diagrams
Reading Assignment: 5.2 (pp. 243-256)
Click here to skip the following discussion and go straight to the assignments.
Venn diagrams are another method which can be used to determine whether a categorical syllogism is valid or invalid. As always, an argument is valid if the premises force the conclusion. In other words, if the premises are assumed to be true (even if they are not actually true we can pretend that they are momentarily) . . . would the conclusion then necessarily be true as well? (Recognize this?) If the answer is yes, then the argument is valid, and if the answer is no, then the argument is invalid. In the language of Venn diagrams, the argument is valid if all of the information on the conclusion diagram is in the premise diagram.
Venn diagrams for syllogisms are made similarly to Venn diagrams for propositions.
1. make the usual two circles. These represent the conclusion, and also one term from each of the premises. These two circles are your S (left) and P (right) terms.
2. Add a third overlapping circle on top. This is your middle term--the one that is in the premises but not the conclusion.
3. Enter the information from the premises into the diagram.
4. Then read it and see whether the conclusion can be read back out of it. In other words, see whether the conclusion is necessarily true based on what is already there from the premises.
Hint: Sometimes, if they are very confused, students do well to draw a separate proposition (two circles) diagram for the conclusion so they are clear what to look for when looking at the syllogism diagram. If the information on the proposition diagram is in the main one, the syllogism is valid.
It is very important to remember that when making a Venn diagram you enter only the two premises and then try to read the conclusion back out. Obviously, if you were to enter the conclusion into the diagram, you would be able to read it back out, and then all your arguments would seem to be valid.
When entering the premises, keep in mind the following points:
- When you are entering one premise you are dealing with the relation between two terms only and are therefore dealing with only two circles. Try to ignore the third circle as much as possible.
- If you have a universal premise and a particular premise, enter the universal premise first.
- An E statement, i.e., a "No S are P" statement, will always make a football shape and an A statement, i.e., an "All S are P" statement, will always look like a cookie with a bite out of it; just as you have been doing, in the last chapter. But in this case you will be shading over a line that segments the section you are used to shading. Ignore the line.
- I and O statements are still symbolized with an X. Perhaps the most difficult part of doing Venn diagrams is figuring out where the X goes. Because now, since each area is bisected by the circle you are not momentarily working with, there will sometimes be two places the X could go. Sometimes, if you had a universal premise and entered it first (as you are supposed to) one of the possible segments is shaded out, so it is clear where you put your X--in the unshaded segment. But if there is no shading, put the X on the line between the possible segments. Do not just randomly choose one segment and put your X in it. (Do not do eenie-meenie-miney-mo, either. ) To do so would be to add information to the diagram that isn’t contained in the premises. You don't want to add new information that wasn't in the premises or an invalid argument might look valid.
Reading the diagram
You have entered both premises into the diagram, and must see whether your conclusion is contained therein. For example, if your conclusion is "Some S are not P" we must look at our completed diagram to see whether there is an X that is in the S area but that is not in the P area. If there is such an X, then the argument is valid and if there is not such an X then the argument is invalid.
If the conclusion is a universal statement, for example, "All S are P" then you must look at your completed diagram to see whether indeed the only S area that is still open is entirely contained within the P circle. If so, the argument is valid, and if not, it is invalid.
A sticky point in trying to figure out if an argument is valid is when you have an X on a line between two areas. Remember that an argument is valid only if you are forced into the conclusion with no other way around it. So, when you have a diagram with an X on a line, the argument is valid only if both sides of the line would satisfy the conclusion. Otherwise, if one area satisfies the conclusion and the other area does not, then you are not forced into your conclusion, because it could always be the other area which could be used for the premise and your conclusion would not necessarily be contained in the diagram. In this situation, the argument is invalid.
Demonstration: If you would like to see a step by step demonstration, click on the diagram below. (It's somewhat crude, but helpful anyway, I hope)
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This is the first problem on p. 271
All corporations that overcharge their customers are unethical
businesses. |
Aristotelian Standpoint:
The other sticky point is syllogisms that have two universal premises and a particular conclusion. By now, you should know what is coming. . . yes, conditionally valid syllogisms. If (when) you come across one of these syllogisms it will be invalid by the steps you have followed so far. But if you can certify that the pertinent S term exists you can add add your "stamp of approval" as you did in lesson 12--a circled X in the appropriate unshaded area(s). The thing to look for is a circle that has three shaded segments and only one unshaded segment. If you have such a circle, check whether the term it denotes exists. If it does, add your stamp of approval. Maybe after you have done this there will be an X (a circled one) in the correct area for the conclusion, and the argument will be valid.
Logic Coach Assignment: 5.2 I 1-15
Assignment: (20 points each)
Use Venn diagrams to determine whether the following standard form categorical syllogisms are Valid or Invalid. Use P for the major term, S for the minor term and M for the middle term. State what P, S and M represent. Draw the diagrams as in the Hurley text, with one circle at the top (middle term) and two circles at the bottom (minor and major terms). e.g., #1: M = Students anxious to learn
1. No students anxious to learn are failures
Some students anxious to learn are romantics.
Some romantics are not failures.
2. No individuals truly concerned with the plight of suffering
humanity are persons motivated purely by self interest.
All television evangelists are persons motivated purely by
self interest.
Some television evangelists are not individuals truly
concerned with the plight of suffering humanity.
3. No bald eagles are lizards.
Some parakeets are not lizards.
Some parakeets are not bald eagles.
4. All greyhounds are fast runners.
All dogs owned by Mary Lou are greyhounds.
All dogs owned by Mary Lou are fast runners.
5. All fire-breathing dragons are fearsome creatures.
Some fearsome creatures are things to be remembered.
Some things to be remembered are fire-breathing dragons.