Venn diagram

Venn diagram: Technique for Testing Syllogism.

Venn diagram is one of the most interesting topic in logic.  It’s quite obvious, most university in Nigeria offer the course irrespective of its course code like Phil 110, phil 101, Phil 112, Phil 102 this like.

Departments like linguistics, mass-com, law, History and international studies, History, philosophy, some educational course. Even some universities included these topic in their general studies (GST/GNS).

Also note, the mode of an examination types are being decided by the departments either to be theory or objectives. Let’s consider this wonderful topics ‘Venn diagram’ of which students in 100level, faculty Of Arts are meant to offer.

Venn diagram: Technique for Testing syllogism.

In Venn diagrams, they are three overlapping circles for which two premises of standard syllogism contain.

Three different terms — major term, minor term and middle term for which weawe abbrivaited as  S=subject, P=predicate and M=middle respectively.

Below are representing the overlapping circles

Before interpretation, let brings out the exact explanation of categorical syllogism displaying the major premises, minor premises and conclusion and also the major, minor and the middle terms.

From the picture the major premise are;

A. All great scientist are college graduates.

B. All artist are egotist

Minor Premise 

A. Some professional athletes are college graduates.

B. Some artist are Pauper


A. Some professional athletes are great scientists.

B. Some pauper are egotists.


Major terms: the major terms is the predicate of major premise and  (great scientist and egotists)

Minor term: the minor terms is the subject of the minor Premise (professional athletes and pauper).

Middle terms: the middle terms appears both in minor and major premises and its conclusion.

Next is the Venn diagram.  from the first picture above here are the given interpretations.

For example, in terms of the various different classes, determined by the class of all Swedes(S), the class of all Peasants(P) and the class of all Musicians(M).  SMP is the product of these three classes, which is the class of Swedish Peasants Musicians. 

Noted: for better understanding of the Discription of the Venn diagram above, Let small letters display (None included)

While as the capital letters means or shows this  (included). Or rather these below shows they aren’t included

∼S= same as the tilde on top of the S

∼P= same as the tilde on top of the P

∼m= same as the tilde on top of the M.

SP∼M is the product of these two classes and the complement of the third, which is the class of swedish peasants who are not musicians.

S∼PM is the product of the first and the third and complement of the second; the class of Swedish Musicians who are not Peasants.

SPM is the product of the second and third and complement of the first; the class of peasants musicians who are not swedish.

S∼P∼M is product of first and the complement of others; the class of swedish who are neither Peasants nor musicians.

SP∼M is the product of the second class with the complement of the two; the class of peasants who are neither Swedes nor musicians.

∼S∼PM is the product of the third class with the complement of the two; the class of musicians who are neither Swedes nor peasants.

Finally S∼P∼M is the product of the complement of three original classes: the class of all things that are neither Swedes nor peasants nor Muscians.

If we focus our attention on just two circles labelled P and M, it is clear that by shading out, or inserting an x, we can diagram any standard form categorical proposition whose two terms are P and M, regardless of which is the subject and predicate. This to diagram a proposition “All M are P” (M∼P=0), we shade out all of M that is not contain in(or overlapped by) P.  This area is seen includes both the portions label SPM and ∼S∼PM. The diagram becomes.

Example 2.

All S are M .  (S∼M=0)

Now we move to syllogism parts of it.

Example 3.

All dogs are mammals    → (P∼M=0)

­All cats are mammals    → (S∼M=0)

so, All cats are dogs.

Example 5.

All artists are egotists  (M∼P=0)

Some artists are pauper 

so, Some pauper are egotists 

Example 6.

All M are P  (M∼P=0)

Some S are not M 

Therefore Some S are not P 

Example 7.

All successful people are people who keenly interested in work.

No people who are keenly interested in work are people whose attention easily distracted when they are working

Therefore, No person whose attention, easily distracted when working are successful people. 

The interpretation for this above..

All  p are M (M∼P=0)

No M  are S

Therefore, No S are P.

Take a look at the Venn diagram.

Some questions may runs towards your mind on  how to identify x,  well, it’s quite easy if you already knew how to identify the terms in categorical syllogism.

Where do I Place the X in a Venn diagrams??

In Venn diagram representing a categorical syllogism, the three terms of the syllogism (minor, major and middle) are represented by three interlocking circles label S, P and M.( The choice of S and P reflects the fact that the minor and major term of syllogism correspond to the subject and predicate terms of its conclusion).

When one of the premises of syllogism calls for an to be replaced on a line in such a been diagram, we may ask.: Which line ? and Why ?

Where the answer to the questions, is that X is always placed on the line of the circle designating the class not mentioned in that premise.

Example: Suppose you were given as premise, “Some S are M. You may not be able to determine whether the x representing that “SOME” is a P or not p — so the x goes on the line of the P circle.


Another Example:

Suppose you are given a premises, “Some M is not P”. You may not be able to determine whether the M that is not P is  S or is not an S — so the  x goes on the line of the S circles, thus 

Tips: make sure you understand and can able to identify the major, minor and middle terms for easy comprehending.

Do comment if any questions.

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