

Some S are P 

In order to use these diagrams to test for validity we must link three circles together. A categorical syllogism has three terms and since each circle represents one term, three circles will be needed. Any of the three propositions in the syllogism can be diagrammed by using the two circles which represent that proposition's terms and (in some ways) ignoring the third circle.
To test for validity one diagrams only the two premises. Then one looks at the diagram to see whether anything would need to be added to diagram the conclusion. Since the conclusions of valid arguments do not claim more than the information given in their premises, if more would have to be added to diagram the conclusion, that conclusion must claim more than the information given in the premises and hence be invalid. In the event of an invalid argument one leaves the conclusion undiagrammed to demonstrate the invalidity of the argument.
No M are P
All S are M
No S are P



Unfortunately complications may arise. Consider this argument:
When one tries to diagram the major premise one finds a line passing through the area where one must place the asterisk. Which side of the line does one place the asterisk? Or does one place it on the line? Often times one has no choice but to place the asterisk right on the line. When one places an asterisk on the line it means that one does not know, on the basis of the information given in the premises, which side of the line it goes; one does not know which area has at least one member. But in this case the minor premise is a universal proposition. Whenever one has a universal premise and a particular premise, one should diagram the universal premise first, because it may give us information about where the asterisk cannot go, by eliminating one side of the line.Some M are P
No M are S
Some S are not P
First Step
No M are S 
Second Step
Some M are P No M are S 
A last example is needed to show how to handle asterisks when they are on a line.
Some M are P
Some S are M
Some S are P
First Step
Some M are P 
Second Step
Some M are P Some S are M 
In this argument the conclusion diagrammed would place an asterisk
in the area common to both the S circle and the P circle. Clearly the lower
portion of that common area is empty. But the top area has two asterisks
on its border lines. Is this enough for the argument to be valid? No. Remember
what an asterisk on the line means: one does not know to which side of
the line it belongs. But the conclusion claims that it is known to belong
in the very middle area, which is clearly more information than the diagram
gives us. So the argument is invalid.
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