Testing Validity Using Venn's Diagrams

TESTING VALIDITY USING VENN'S DIAGRAMS

To test the validity of a categorical syllogism, one can use the method of Venn diagrams. Since a categorical syllogism has three terms, we need a Venn diagram using three intersecting circles, one representing each of the three terms in a categorical syllogism. A three term diagram has eight regions (the number of regions being 2n where n is the number of terms).

The following chart gives the extension of the predicates in the various regions of the diagram.

Region S (Minor Term) P (Major Term) M (Middle Term) 1 yes no no 2 yes yes no 3 no yes no 4 yes no yes 5 yes yes yes 6 no yes yes 7 no no yes 8 no no no

In order to use a Venn diagram to test a syllogism, the diagram must be ﬁlled in to reflect the contents of the premises. Remember, shading an area means that that area is empty, the term represented has no extension in that area. What one is looking for in a Venn diagram test for validity is an accurate diagram of the conclusion of the argument that logically follows from a diagram of the premises. Since each of the premises of a categorical syllogism is a categorical proposition, diagram the premise sentences independently and then see whether the conclusion has already been diagramed. If so, the argument is valid. If not, then it is not.

Remember, one deﬁnition of validity is that the propositional (informational) content of the conclusion is already expressed in the premises. Venn diagram validity tests provide a graphic tool for using this approach to testing for validity. A categorical syllogism is valid if, but only if, a diagram of its premises produces a diagram that expresses the propositional content of its conclusion.

Begin the process by preparing a three term Venn diagram. As a convention, organize the diagram as above, 2 circles at the top of the diagram, one centered at the bottom. The upper left hand circle should represent the minor term (designated S as this term is the subject of the conclusion). The upper right hand circle should represent the major term (designated P as this term is the predicate of the conclusion). The lower circle should represent the middle term (designated M). Your diagram should look like this:

Consider the following argument:

All Greeks are mortal. (All M are P) All Athenians are Greek. (All S are M) So, all Athenians are mortal. (All S are P)

Next, diagram each of the premises. When doing this, act as if there are only 2 relevant circles. Begin with the ﬁrst premise (frequently the premise involving the major term, sometimes called the major premise). In our example you need to diagram the proposition "All M are P". Ignoring for a moment the circle representing the minor term, your diagram sho8uld look like this: Following the standard conventions we get:

Next, diagram the second premise--"All S are M"-- to get: Now, if we overlap the diagrams of the premises we get a diagram of the argument, and we are ready to determine whether the argument is valid or not:

Does this diagram express the informational content of the conclusion of the argument? Yes, all of the S's that remain are in region 5, and everything in region 5 is an S, an M, and a P. Since all the S's are in region 5, all the S's are P's and the argument is VALID.

Consider another argument:

All mathematicians are rational. (All P are M) All philosophers are rational. (All S are M) SO, all philosophers are mathematicians. (All S are P)

Beginning with the ﬁrst premise we get:

Adding the second premise we get:

Does this diagram express the informational content of the conclusion "All S are P"? NO. Region 4 of the diagram is not shaded (not empty) so it is possible that there is an S that is not a P. Accordingly, the argument is NOT VALID.

NOTE: Whenever a diagram of the premises of an argument produces exactly three shaded regions, the argument is not valid. Exactly three shaded regions indicates that there is a fallacy of distribution.

Similarly, if a diagram of the premises indicates 2 or more populated regions, then the argument is not valid.

Consider the argument:

All philosophers are logical. Some physicists are logical. So, some philosophers are physicists.

In the following diagram, the bar that crosses from region 5 into region 6 indicates that the argument is NOT VALID. All that we can be certain of is that there is either an SPM (region 5) or a PM non-S (region 6), but we don't know which. Since we don't know which, the conclusion does not follow logically from the premises.

If we modify the previous argument just a little, we see an important distinction. Consider the argument:

Some physicists are logical. (Some S are M) No philosophers are logical. (No P are M) So, some physicists are not philosophers. (Some S are not P)

In our new argument, a diagram of the ﬁrst premise produces:

The second premise, however, rules out the possibility that the S that is M is also a P, because the second premise guarantees that no P's are M's. Our diagram looks like this:

Does this diagram express the informational content of the conclusion "Some S are not P"? Yes, the S that is an M in region 4 is a non-P. So the argument is valid.

As useful as Venn's methods are, there are some severe limitations to their usefulness. First, while it is possible to construct a 16 region Venn-type diagram for a 4 term argument, and even a 32 region diagram for a 5 term argument, those diagrams are almost impossible to read or use. What is more, it is impossible to construct a 64 region diagram for a 6 term argument--there is no way to get exactly the right 64 regions in a 2 dimensional diagram, More signiﬁcantly, though, is that Venn's diagrams cannot capture the logic of quantiﬁed sentences that are more complex than simple categorical propositions. Fortunately, we have at our disposal an even more powerful analytical tool--modern predicate logic.

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