Syllogisms & Venn Diagrams - Amateur Logician

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Categorical Syllogisms & Venn Diagrams

In a categorical syllogism, there are three and only three terms:
(1) Major Term (appears as the predicate in conclusion)
(2) Minor Term (appears as the subject in conclusion)
(3) Middle Term (appears in both premises but never the conclusion)

And there are three and only three propositions in a categorical syllogism:
(1) Major Premise (major term appears in this)
(2) Minor Premise (minor term appears in this)
(3) Conclusion (both minor and major terms appear in this)

Validity can be tested with Venn diagrams!

Our diagram shows three overlapping circles. Each circle represents one of a syllogism’s three terms: minor term (S), major term (P), and middle term (M).

Area 1 = non-S, non-P, and non-M
Area 2 = S, non-P, and non-M
Area 3 = non-S, P, and non-M
Area 4 = non-S, non-P, and M
Area 5 = S, non-P, and M
Area 6 = non-S, P, and M
Area 7 = S, P, and non-M
Area 8 = S, P, and M

Since there are a total of three terms, we need three circles. Notice that, in their overlap, there are multiple areas.

The circle S represents all members of S.
The circle P represents all members of P.
The circle M represents all members of M.

Testing Validity with Venn Diagrams

Any valid categorical syllogism has it that its conclusion can be “read off” its representative Venn diagram. First, we only diagram the syllogism’s premises. Second, we determine if the conclusion can be “read off” this diagram.

Remember, in a Venn diagram, a “filled in” or “shaded in” area indicates that there is nothing there. An “X” indicates that at least one thing exists there. (It gives “existential import.”) An area not “shaded in” signifies possibility: members may or may not be there.

Warning! Your intuition might tell you to “shade in” areas that indicate something is there. But you would be mistaken! I cannot stress enough how wrong that intuition is.

(I have that wrong intuition, too. So, I have to be careful.)

Also, be sure to review how to do Venn diagrams for a single proposition.
See “Euler’s Circles & Venn Diagrams.”

Don’t get careless.
Make these steps a habit.

First: Look at your universal premise(s).
– “Shade out” any area that’s known not to have any members.
– Always do this first!
– Don’t do particular propositions before universal propositions.
– That’s because you don’t want to put an “X” in a shaded out area.

Second: Look at your particular premise(s).
– Only focus on the unshaded areas.
– Never place an “X” in a shaded area.
– If an area is known to have at least one member, place an “X” in that area.
– If there’s uncertainty over which of two areas members belong, place an “X” on the border between those two areas.

Third: Is your conclusion particular with universal premises?
– If so, follow this simple step.
– This third simple step only works from a so-called “Aristotelian” point of view.
– Place a circled “X” in an unshaded area to indicate that at least one member exists there according to a universal premise (if such a member really exists).

Fourth: Does the finished Venn diagram show the conclusion?
– If it does, the argument is valid.
– If it does not, the argument is invalid.

With examples below, I’m going to (somewhat) ignore the advice above.

I’m going to treat the two premises in isolation from each other and then combine them. The first premise will be diagrammed. The second premise will be diagrammed separately without consideration to the first. And then the two diagrams combined.

First Example (“the classic example”) . . .
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Premise: All men are mortal.
Premise: Socrates is a man.
Conclusion: Therefore, Socrates is mortal.*

First, what are the minor, major, and middle terms?
Minor: Socrates (because he is the subject of the conclusion).
Major: Mortal (because it is the predicate of the conclusion).
Middle: Man (because it appears only in the premises).

Second, draw Venn diagram with its three terms.
By default, have the top circle the middle term.

Third, diagram the first premise.
“All men are mortal.”
This only concerns the circles “man” and “mortal.”

Since “all men are mortal,” we must “shade in” areas 4 and 5.
Don’t worry about those areas in terms of the circle “Socrates.”
That’s not our concern at present.

That “shaded in” area represents something that is empty. Since “all men are mortal,” there is no “man” outside the circle of “mortal.”

Fourth, diagram the second premise.
“Socrates is a man.”
This only concerns the circles “Socrates” and “man.”

Since “Socrates is a man,” all of Socrates is a man.
Thus, we must “shade in” areas 2 and 7.

Fifth, put everything together and determine validity.
Both premises are now in our Venn diagram.
Is it the case that “Socrates is mortal” ?

Look at the circle “Socrates.” Most of it is empty. But the part that is not is contained within the circle “mortal.” Therefore, “Socrates is mortal.” The argument is valid.

*[Note: I’m treating all propositions as universal in this argument. A contemporary textbook might treat the second premise and conclusion as “existential.” We can place a circled x in area 8.]

Second Example. . .
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Premise: No logician is funny.
Premise: Some teachers are logicians.
Conclusion: Therefore, some teachers are not funny.

First, what are the minor, major, and middle terms?
Minor: Teachers (because it is the subject of the conclusion).
Major: Funny (because it is the predicate of the conclusion).
Middle: Logicians (because it appears only in the premises).

Second, draw Venn diagram with its three terms.
By default, have the top circle the middle term.

Third, diagram the first premise.
“No logician is funny.”
This only concerns the circles “logicians” and “funny.”

Since “no logician is funny,” we must “shade in” areas 6 and 8.
Those areas are empty. There exists no funny logician.
At present we’re not concerned with the “teachers” circle.

Fourth, diagram the second premise.
“Some teachers are logicians.”
Warning! There may be uncertainty.

We diagram particular propositions with an “X.”
All circles must be considered with particular propositions.

We’re given that “some teachers are logicians.”
Ignoring the first premise entirely, we would be uncertain if that entails they are funny or not funny.

Hence the “X” cannot be definitely placed in area 2 (for that would imply that the teacher must be not funny) nor can it be definitely placed in area 7 (for that would imply that the teacher must be funny). To account for this uncertainty, we place the “X” on the line of the “funny” circle between area 5 and area 8.

**“Thus, some teachers are not funny.“**

Fifth, put everything together and determine validity.
Both premises are now in our Venn diagram.
Is it the case that “some teachers are not funny” ?

Look at the “X” in the circle “teachers.” This tells us that there exists at least one “teacher.” The “X” touches the “funny” circle.

So, this “teacher” may or may not be “funny.”
However, area 8 is “filled in” (i.e., empty). Therefore, this “teacher” cannot be “funny.”
The argument is accordingly valid.

Third Example. . .
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Premise: No diamonds are inexpensive.
Premise: All diamonds are pretty.
Conclusion: Therefore, some pretty things are not inexpensive.

First, what are the minor, major, and middle terms?
Minor: Pretty (because it is the subject of the conclusion).
Major: Inexpensive (because it is the predicate of the conclusion).
Middle: Diamonds (because it appears only in the premises).

Second, draw Venn diagram with its three terms.
By default, have the top circle the middle term.

Third, diagram the first premise.
“No diamonds are inexpensive.”

Since “no diamonds are inexpensive,” we must “shade in” areas 6 and 8. Those areas are empty. There exists no “diamond” that is inexpensive. This only concerns the circles “diamonds” and “inexpensive.”

Fourth, diagram the second premise.
“All diamonds are pretty.”

Since “all diamonds are pretty,” we must “shade in” areas 4 and 6. Those areas are empty. All “diamonds” exist in the “pretty” circle. This only concerns the circles “diamonds” and “pretty.”

**“Therefore, some pretty things are not inexpensive” ?**

Fifth, put everything together and determine validity.
Both premises are now in our Venn diagram.

Can we “read off” the conclusion from that diagram?
Is it the case that “some pretty things are not inexpensive” ?

It is valid, though can we really “see” it in the diagram presently?
Strictly speaking, we should modify our diagram to make it clearer when dealing with premises that are all universal but a conclusion that’s particular.

Given “all diamonds are pretty,” we will assume that at least one diamond exists and place an “X” in the remaining portion of the “diamonds” circle that hasn’t beeen “shaded in.” This is in area 5.

We now can clearly “read off” the conclusion. There exists at least one “pretty” thing that is not “inexpensive.” The argument is valid.

Fourth Example. . . (with invalidity)
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Premise: All M are P.
Premise: No S are M.
Conclusion: Therefore, no S are P.

First, what are the minor, major, and middle terms?
Minor: S (because it is the subject of the conclusion).
Major: P (because it is the predicate of the conclusion).
Middle: M (because it appears only in the premises).

Second, draw Venn diagram with its three terms.
By default, have the top circle the middle term.

Third, diagram the first premise.
“All M are P.”

Since “all M are P,” we must “shade in” areas 4 and 5.

Fourth, diagram the second premise.
“No S are M.”

Since “no S are M,” we must “shade in” areas 5 and 8.

Fifth, put everything together and determine validity.
Both premises are now in our Venn diagram.

Can we “read off” the conclusion from that diagram?
Is it the case that “no S are P” ?

No! This is invalid. Area 7 is not shaded in. It is thereby left open if some “S” are “P.” So, we cannot infer that “no S are P.”

Fifth Example. . . (with invalidity)
– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Premise: Some S are M.
Premise: Some M are not P.
Conclusion: Therefore, some S are not P.

First, what are the minor, major, and middle terms?
Minor: S (because it is the subject of the conclusion).
Major: P (because it is the predicate of the conclusion).
Middle: M (because it appears only in the premises).

Second, draw Venn diagram with its three terms.
By default, have the top circle the middle term.

Third, diagram the first premise.
“Some S are M.”

Warning! There is an uncertainty here.
We’re told that “some S are M;” it is left open if some S are P.

Hence the “X” cannot be definitely placed in area 5 (for that would imply that no S are P) nor can it be definitely placed in area 8 (for that would imply that the S must be a P). To account for this uncertainty, we place the “X” on the line of the P circle between area 5 and area 8.

Fourth, diagram the second premise.
“Some M are not P.”

Warning! There is an uncertainty here.
We’re told that “some M are not P;” it is left open if some M are S.

Hence the “X” cannot be definitely placed in area 4 (for that would imply that no M are S) nor can it be definitely placed in area 5 (for that would imply that the M must be a S). To account for this uncertainty, we place the “X” on the line of the S circle between area 4 and area 5.

No! This is invalid. There is no “X” in areas 2 or 5. So, we cannot infer that “some S are not P.”