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Logical Thought
A Brief Guide to Venn Diagrams and Categorical Propositions
(with an emphasis on complements)
Part I: The Basics
1. Suppose S is a term (e.g., ‘dogs’). Then we can represent the extension of S (e.g., the set of
all dogs) with a circle:
S
2. The complement of a term (S) is another term (nonS) that refers to the set of all things not in
the extension of the term. (We add the box to make it easier to read the diagram) For
example, the complement of ‘dogs’ is ‘nondogs’, and ‘nondogs’ refers to the set of all things
that are not dogs.
nonS
S
Part II: The 4 Standard Categorical Propositions
3. The universal propositions are represented using Venn Diagrams as follows:
All S are P
S
No S are P
P
S
P
A shaded region is an empty region. For example, in All S are P, the region of S that does not
overlap with P is shaded in because the proposition asserts that there are no S’s that are not P’s –
that region is empty.
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4. The particular propositions are represented using Venn Diagrams as follows:
Some S are P
S
x
P
Some S are not P
S
P
x
Here, we use an ‘x’ to indicate that there is at least one thing in a certain region. For example, the
proposition Some S are not P says there is at least one thing that is an S and is not a P, so the
‘x’ goes in S but not in P. We do not shade in any region, because, even if there is at least one S
that is not a P, there may still be an S that is a P. For example, even if there is at least one dog
that is not vicious, there may still be dogs that are vicious.
Part III: Contraposition and Venn Diagrams
Two categorical propositions are equivalent if they cannot differ in truth value – when one is true,
so is the other, and when one is false, so is the other. If two propositions are equivalent, then it is
valid to reason from the truth of one to the truth of the other – that is, the truth of the first
guarantees the truth of the second.1 We used Venn Diagrams to check to see if propositions are
equivalent: If two propositions have the same diagram, then they are equivalent, and it is valid to
reason from the truth of one to the truth of the other.
Now, it can be tricky using Venn Diagrams and complements.
Example #1: The contraposition of No S are P is No nonP are nonS. We know that, if the
two propositions have different Venn Diagrams, then it is invalid to reason from the first to the
second (or vice versa). And we know how to diagram the first proposition (see above). But how
do we diagram No nonP are nonS?
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Note that, if two propositions have different diagrams, it may still be valid to infer from the truth of one to
the truth of the other. In particular, in subalternation we infer from the truth of a universal proposition to
the truth of the corresponding particular proposition (‘truth flows downwards’). These two propositions
have two different diagrams, but the inference is nonetheless valid (assuming the traditional interpretation
of the square of opposition).
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Try thinking it through stepwise. First, consider nonP. As noted above, the extension of nonP
is everything not in the extension of P. Suppose we mark the extension of nonP using vertical
blue lines:
nonP
P
Second, let’s do the same thing for nonS, using horizontal red lines to indicate where nonS is in
the diagram:
nonS
S
Third, combine the two diagrams. Since the red lines indicate the extension of nonS, and the
blue lines indicate the extension of nonP, wherever we find both red and blue lines is where we
find those things that are both nonS and nonP, i.e., that’s where nonS and nonP intersect.
S
P
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Finally, the proposition No nonP are nonS asserts that, if you pick anything that is nonP, it
will not be nonS. That is, nothing that is nonP is also nonS – the intersection of nonP and
nonS is empty.
S
P
Now we can see that the Venn Diagrams for No nonP are nonS and No S are P (given
above) are different, and so we cannot reason from the truth of one to the truth of the other.
Example #2: The contraposition of All S are P is All nonP are nonS. How do we diagram
the latter proposition?
First, follow the same initial steps as before: If we indicate nonP with blue vertical lines, and
nonS with red horizontal lines, and then combine the two diagrams, we get:
S
P
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Second, the proposition All nonP are nonS asserts that, if we pick anything that is nonP, it
will be nonS. That is, there is nothing that is nonP and S.
For example, suppose S = dogs and P = mammals. Then All nonP are nonS = All nonmammals are non-dogs, which means that there is nothing that is both a non-mammal and a
dog.
Remember, the blue vertical lines indicate the set of things that are nonP, and red horizontal
lines indicate the set of things that are nonS. So if there is nothing that is both nonP and S,
then there is nothing in the region of the diagram where there are blue vertical lines and no red
horizontal lines:
S
P
If you compare this diagram with the diagram for All S are P, you will see that they are
identical. So anytime All S are P is true, so is All nonP are nonS – it is a valid inference to
reason from the truth of All S are P to the truth of All nonP are nonS, or vice versa.