LESSON #12
Venn Diagrams and the Modern Square of Opposition
Reading Assignment: 4.3 (pp. 207-214) 4.6 (pp. 234-240)
Click here to skip the following discussion and go straight to the assignments.
I'm going to be frank with you here: I think Hurley has had a hard time deciding on the order of presentation for the rest of the chapter to keep from being confusing. If you look at the other six editions of this book, shuffling the modern and Aristotelian squares of opposition, Venn diagrams, and conversion, obversion and contraposition into different sections is one of the main changes he makes. And I fear there will be yet another edition soon, because I don't think he has yet achieved his goal. (!) Part of the reason for the confusion is that each concept is easier to grasp if the others are already in place.
I have wrestled with the same issue. Should I strike out on my own and present the material in a different order from the one Hurley has settled on? Or would that make things yet more confusing for you? I have decided to mix things up a bit. In this lesson you will focus on Venn diagrams, later you will do both types of squares of opposition. It would be a good idea, however, to read the information on the modern and traditional squares of opposition before you start working on this exercise, just so you get a sense of how things fit together.
Venn Diagrams in General
These are standardized drawings that help us visualize and represent categorical propositions, and later, with modifications, categorical syllogisms. There is no real reason for them to be the way they are, rather than the little doodles you are probably already making to visualize what's going on in categorical propositions, except to standardize them so they are recognizable to anybody. (at least anybody who knows the conventions.)
The text should be pretty self-explanatory here. Just remember:
Þ The basic diagram for a proposition is two interlocked circles, side by side.
Þ The left circle represents the S class. Label it with a letter that suits the particular term you are diagramming.
Þ The right circle represents the P class. Label it with a different letter.
Venn Diagrams for Boolean Propositions:
The difference between the Boolean diagrams and Aristotelian or traditional diagrams, which you'll learn soon, lies in the way you draw universal statements. Boole figured that universal statements made no assumptions about the existence of the objects being talked about. In the modern interpretation, "All unicorns are animals with one horn" means "if you do encounter a unicorn it will have one horn." or "There are no unicorns that are not also one horned animals." So the diagrams for these simply indicate where individuals named would be found if there are any, and where they aren't in any case.
Þ Universal statements (A, E) will be represented with shading.
I. Shading indicates empty space (where nothing exists).
II. There are only two shapes you can shade:
1. For an E statement you shade a football shape.
2. For an A statement (All S are P) you shade a cookie with a bite out of it.
Think of it this way: Shading means nothing is in an area. (There would be no room to draw an X)
· To represent All S are (in) P, you must shade out all the S that is not in P.
· To represent No S are (in) P, you must shade out all the S that is in P.
Þ Particular statements (I, O) will be represented with an X
Both Aristotle and Boole interpreted particular statements the same way. They mean at least one S is a P, or is not a P. There is a claim about existence implicit in this interpretation. An X indicates that thing which is claimed to exist. We only draw one X because the statement is ambiguous as to how many more individuals exist, if any. "Some" only means "at least one."
· To represent some S are (in) P you must put an X in the S area that is also a P area.
· To represent some S are not (in) P, you must put an X in the S area that is not also a P area (which is outside of P).
Negating Venn Diagrams
This can be confusing the first couple of times you do it, but it isn't hard at all. (when you learn about the square of opposition refer back to this and see whether it looks familiar--you are basically doing contradictory)
There are two steps:
1. Decide what you would do if the proposition were true.
2. Do the opposite. If you would have put an 'x' in an area shade it instead, and visa-versa.
EXAMPLE: Make a Venn diagram for "It is false that All S are P"
· First decide what you would do for "All S are P"--shade the area of S that is outside the P circle.
· So, instead put an 'x' there
Testing Immediate Inferences
An immediate inference is an argument with only one premise, and the conclusion is expected to also be true, just because of the laws of logic. To test these using Venn diagrams, draw one diagram for the premise and one for the conclusion, then compare them. If there is nothing new on the conclusion that wasn't on the premise, then the argument is valid. If the conclusion has something that isn't on the premise, then obviously it doesn't follow necessarily from the premise and is invalid. It doesn't matter if the premise has information that isn't on the conclusion.
Venn Diagrams and the Traditional Standpoint (4.6)
As mentioned earlier, the Boolean standpoint differs from the Aristotelian according to how it interprets the existence of the individuals in the S class of universal statements. Because it is restricted in this way, many immediate inferences that intuition would tell you are obviously valid, come out to be invalid when tested with Boolean diagrams. If you can certify that the individuals being discussed do exist--and that is what most of us talk about most of the time, isn't it?--you can use modified Venn Diagrams that are more useful than the Boolean ones.
The steps of the process I'd like for you to use are a bit different from Hurley's.
1) Test the inference according to the Boolean interpretation.
A) If the diagram shows a valid result, stop. It is valid under any interpretation.
B) If the diagram shows an invalid result and the conclusion has no X on it, stop. It would be invalid under any interpretation.
C) If the diagram shows an invalid result, and has an X in the conclusion, check whether there is shading in part of the S circle. If not, stop. It would be invalid under any interpretation.
2) If the premise has shading on part of the S circle, check whether it represents something that exists. If it does not, stop. The argument is invalid because it commits the existential fallacy. DO NOT draw a circled X. (Hurley tells you to--I say no, this indicates that at least one S exists--don't indicate that if it doesn't)
If the S circle does denote something that exists, certify this existence with a "stamp of approval"--a circled X in the unshaded part of the S circle.
A) IF the argument is now valid (that is if the premise has the information on it that the conclusion has) Stop. It's valid.
B) If your stamp of approval didn't make your argument valid, stop. It is invalid under any interpretation.
Logic Coach Assignment: 4.3 I all; III all; 4.6 all
Assignment 1:
Draw Venn diagrams for each of the following statements:
1. All electric motors are machines that depend on magnetism.
2. No tax audits are pleasant experiences for cheaters.
3. Some people who attend Grateful Dead concerts are people who don't use drugs.
4. Some people who listen to Henry Rawlins' CDs are not violent people.
5. It's false that some birds are fish.
Assignment 2:
Test the following immediate inferences using the Venn Diagrams and the Traditional standpoint (Modified Venn diagrams) to test the following immediate inferences.
1) All
Vulcans are good logicians.
Therefore, some Vulcans are good logicians.
2) All
radio talk shows are celebrations of ignorance.
Therefore it is false that some radio talk shows are not celebrations of
ignorance.
3) Some
advertisements are works of art.
Therefore, some advertisements are not works of art.
4)
It is false that no cellular phones are digital devices.
Therefore, all cellular phones are digital devices.
5) No fast foods are low fat meals.
Therefore, it is false that all fast foods are low fat meals.
