Venn Diagrams

An alternate method for determining the validity of categorical syllogisms is the Venn diagram method. The conventions of this method are 1.) to represent categorical claims with interlocking circles; 2.) each circle represents a term; 3.) an asterisk indicates that at least one thing exists in the area where it is placed; 4.) stroking out an area indicates that there is nothing in that area. The 4 kinds of categorical propositions can then be diagrammed as follows:
 
All S are P 
No S are P 
 
Some S are P 
Some S are not P 
 
The only diagram likely to cause some difficulty in that for the A proposition. The idea is that if no S's exist outside the P circle (diagrammed by stroking that area out), then all the S's that can exist are also P's (i.e. All 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.

Example

We will diagram this argument:
No M are P
All S are M
No S are P
 
First Diagram
No M are P 
Second Diagram
No M are P
All S are M 
Third Diagram
No S are P 
 
The first diagram shows the information given in the major premise. The second diagram adds to the first diagram the information from the minor premise. The third diagram here is unnecessary, but is included merely to show which areas must be shaded for the conclusion to be diagrammed. Since these areas are indeed shaded in the second diagram, the argument is valid. The fact that the second diagram contains more information than needed to diagram the conclusion does not matter.

 

Unfortunately complications may arise. Consider this argument:

Some M are P
No M are S
Some S are not P
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.
 
First Step 
 
No M are S		 Second Step 
 
Some M are P 
No M are S
 
Notice now that the conclusion requires that an asterisk be placed in the part of the S circle outside the P circle (the leftmost area in the S circle). Since no asterisk is in this area, the argument is invalid.

 

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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