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How Venn Diagrams are used to determine the validity of categorical syllogisms. It discusses the representation of categorical statements using Venn Diagrams and the steps to validate an argument using these diagrams. Two examples of valid and invalid arguments are provided for illustration.
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Recall the simple 2-circle representations of the meanings of our four categorical statements that we provided in §5.1:* The Venn Diagram method makes clever use of these representations to determine whether or not any given syllogism is valid.
All A are B
No A are B A B Some A are B
Some A are not B
Since every categorical syllogism consists of three categorical statements and contains a total of three terms — the minor term (S), the major term (P), and the middle term (M) — we can combine our 2 - circle representations of all three statements in a single diagram of the following form: The P and M circles together will be used to represent the content of the major premise: Minor term S P M Major term Middle term Minor term S P M Major term Middle term
Major premise
Recall that an argument is valid if it is not possible for the premises of the argument to be true and the conclusion false. The reason for this is that, in a valid argument, the content of the conclusion is already implicit in the premises; the argument simply draws this content out and makes it explicit. The Venn Diagram method enables us vividly to see when this connection between premises and conclusion holds. Specifically, the method consists of three steps:
Since both our premises are universal, we can diagram either premise first. So let’s just start with the major premise: Now let’s add the minor premise:
Thus, diagramming the minor premise first, we have: Diagramming the major premise in turn yields: E B L E B L X
And again we see that there is no work to be done to represent the content of the conclusion; we have an X in the overlap of Exceptional people and Beer lovers. So the argument is valid.
In contrast to what happens in the case of a valid argument, after diagramming the premises of an invalid argument there will be more work to do to diagram the conclusion. That is just what you’d expect, because in an invalid argument, the information expressed by the conclusion is not implicit in the premises; the conclusion says something more than the premises do. Let us see how this plays out with a couple more examples.
We diagram the minor premise first, since it is universal and the major premise is particular: But what do we do with the major premise? Where does the X go? It has to be placed inside the O circle but the outside the H circle, but where do we put it relative to M? We can’t put it inside M , since that would indicate that our arbitrary unhealthy, obsessive person is a marathon runner, and we don’t know that. But, similarly, we can’t put it outside M , since that would indicate that he or she is not a marathon runner, and we don’t know that either. Consequently, we must put the X in the only place that doesn’t indicate one way or the other, namely, right on the line:
And now we see that the information in the conclusion is not represented in the diagram. To capture that information the X would have to be fully inside the M circle. But it’s not, so the diagram shows that the argument is invalid; the information in the conclusion is not implicit in the premises.
