Chapter 18: Categorical syllogisms Ben Bayer ... - BenBayer.com

§6: DEDUCTIVE LOGIC
Chapter 18:
Categorical syllogisms
Ben Bayer
Drafted May 2, 2010
Revised August 24, 2010
Categorical syllogisms defined
In chapter 17, we observed that if an argument has all true premises but a
false conclusion, we don’t need any special method for judging its validity:
we know it is invalid. A valid argument is one that is such that if its premises
are true, its conclusion must be true, i.e., its conclusion cannot be false. So if
we have an argument whose premises are true while its conclusion is false,
we know that it is not one for which the truth of the premises forces the truth
of the conclusion. The conclusion is not true, so its premises did not have
that power.
But we don’t always know if the premises of an argument are true and
its conclusion false. Sometimes we don’t know enough about the subject
matter to know the truth or falsehood of any of the statements in the
argument. When this happens, we need a systematic method for determining
whether an argument is invalid. We examined a fairly systematic method in
the last chapter: the counterexample method. To use this method, we took an
argument of unknown validity and identified its abstract form. For example,
this argument
All philosophers are mortal.
All dogs are mortal.
Therefore, all philosophers are dogs
has the following abstract form:
All A is B
All C is B
Therefore, All A is C.
We took this abstract form and looked to see if we could identify examples
to substitute into the abstract variables, which would yield an argument with
all true premises and a false conclusion. For instance, the following
substitution yields the desired result, proving that the argument is invalid:
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All men are living.
All insects are living.
Therefore, all men are insects.
(T)
(T)
(F)
This method of finding counterexamples through substitution into the
abstract form is very effective, but it has at least two limitations. First, it
depends in part on the power of your imagination, or at least on your
patience for the repeated use of trial and error, in order to find substitution
instances that finally yield the desired T, T, F combination. Second, it is only
a method of demonstrating that the argument is invalid. We do not have a
comparable method for demonstrating that an argument is valid. You might
point out that if we try and try, and simply can’t find a counterexample
demonstrating the invalidity of a method, this is good enough to show that
an argument is valid. And it is often good enough for us to assume that
provisionally. But it is not a definitive proof. Perhaps our imagination or
patience are simply in short supply, and we haven’t tried hard enough.
The question of how we prove the validity of an argument—proving
whether or not it is a proof!—is a bigger question than we can answer in
what remains of this book. There are many forms of deductive reasoning and
proof that we will not examine here, and questions about logical “metatheory”
(the theory of how we prove that something is proven or provable),
are especially complicated. From the beginning, it has only been our goal to
examine the most basic principles of the most basic forms of deductive
reasoning, and we will continue to confine ourselves to this goal in this
chapter. We’ll be interested in just one method of showing that just one kind
of deductive argument is valid: the Venn diagramming method of
demonstrating the validity of categorical
syllogisms.
What is a categorical syllogism It is
the type of deductive argument that is
implicit in much of the flesh of everyday
thinking that concerns relationships among
categories. Consider these two separate
arguments:
Picture credit 1:
http://www.flickr.com/photos/mr_t_in_dc/480081967
4/
Saints cultivate some excellence. After all, saints are virtuous.
Yes, even the divine Socrates must be mortal. Alas, he is a man.
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If these arguments make sense to you, it’s because they rely implicitly on a
slightly more elaborate structure, which looks as follows:
All saints are virtuous people.
All virtuous people are cultivators of excellence.
Therefore, all saints are cultivators of excellence.
All men are mortal beings.
Socrates is a man.
Therefore, Socrates is a mortal being.
By now you are all too familiar with arguments of this type (especially the
second). They are examples of categorical syllogisms. Not every deductively
valid argument is a categorical syllogism. Here are some other arguments
widely given as exemplary deductive arguments, but which are not
categorical syllogisms:
If Edison was a hero, he overcame great obstacles in the pursuit of a
goal.
Edison was a hero.
Therefore, he overcame great obstacles in the pursuit of a goal.
Any inventor is either inspired or hard working.
Some inventor is not hard working.
Therefore, some inventor is inspired.
These arguments draw on forms of deductive logic that we will not examine
in this chapter, only because they are more advanced and better suited for a
longer discussion of their own. Notice that the first involves the use of a
premise involving an “if-then” statement. This makes it a form of
hypothetical syllogism (in particular, it is a “mixed hypothetical syllogism”).
The second also makes use of an “either-or” premise, as well as the
relationship between a premise using the quantifier “any” and another with
the quantifier “some.” It is a form of disjunctive syllogism, and one that is
said to involve a special logic of logic of quantified predicates.
Our focus will be categorical syllogisms. Since we’ve called all three
of these types of reasoning “syllogisms,” it is worth briefly defining that
term. A syllogism is a deductive argument from two premises. There are
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other deductive arguments that are not syllogisms. Consider, for instance,
this three premise argument:
If someone is an inventor, then everyone is helped by his creation.
If someone is an artist, then someone or other is inspired by her work.
Edison was an inventor, and Austen was an artist.
Therefore, everyone is helped by his creation, and someone is inspired
by her work.
So we know that a categorical syllogism is one kind of two-premise
deductive argument. But what sets it apart from the other kinds of syllogisms
briefly mentioned above The difference stems from the type of judgment
used in the premises and the conclusion. Here are examples of the type of
judgment used:
All saints are virtuous people.
Some saints are heroes.
No heroes are villains.
These are the type of judgments that describe relationships among
categories. Categories are just the classes or kinds into which we divide up
the world. They’re anything we mean when we use a general concept.
The first statement represents a relationship between categories also
shown using this circle diagram:
You can see that this shows the same relationship between saints and
virtuous people as “All saints are virtuous people,” because the circle of
“saints” is entirely contained within the circle of “virtuous people”; none of
the “saints” circle is outside of the circle of “virtuous people.” The second
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statement states a different kind of category relationship, this time as
represented by the following circle diagram:
We see that this diagram represents “Some saints are heroes” because only
part of the circle of “saints” overlaps the circle of “heroes.” The last
statement, “No heroes are villains,” separates these categories even further:
In fact these categories are entirely separate: hence, “No heroes are villains.”
These types of judgments are called categorical judgments, which are
judgments that relate a subject and predicate concept, each of which is
taken to stand for a class or category of objects. That means that a
categorical syllogism is just a syllogism whose premises and conclusions are
composed of categorical judgments. (In grammar as well as in categorical
logic, the subject term of a sentence is the one that refers to what one is
talking about; the predicate term is the one describing what one says of the
subject.)
Because a categorical syllogism has two categorical premises, and
each premise has two terms, we can expand its description as follows. A
categorical syllogism consists of
a) Three categorical propositions (two premises and one conclusion)
b) A total of three and only three terms, each of which appears twice
in distinct propositions.
With only this many terms in this many propositions, one of the three terms
must appear in a way that links the two premises but drops out in the
conclusion. This term is known as the middle term. In the following
argument, “saint” is the middle term:
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Some saints are not heroes.
All perfectionists are saints.
Therefore, some perfectionists are not heroes.
The other two terms also appear twice: once in a premise, and once in the
conclusion.
Categorical judgment types
Each categorical judgment has three important components that we’ll
need to be able to identify when evaluating categorical syllogisms in the
future. First, each such judgment contains quantifiers. Here are two (but not
the only) examples of quantifiers, “all” and “some”:
All heroes are saints.
Some saints are not heroes.
Using the same examples, we can also see the second important component
of categorical judgments, the subject and predicate terms:
All heroes are saints.
Some saints are not heroes.
“Heroes” is the subject term in the first example, and “saints” is in the
second. The other highlighted terms are predicate terms. The subject is the
thing or set of things a judgment is judging about, whereas the predicate is
what is judged about those thing or things. In grammar classes, you may
have learned to classify everything that isn’t the noun or noun phrase of a
sentence as the predicate. In logic, we are considering as the predicate only
the noun or noun phrase that comes after the third major component of a
categorical judgment, the copula:
All heroes are saints.
Some saints are not heroes.
The copula is just the form of the verb “to be” (or its negative counterpart)
which links the subject and predicate concepts.
Recall that the valid categorical syllogisms we’ve considered so far
have been valid because of their form, because of the abstract pattern among
the terms involved in categorical judgments, not because of anything special
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about the meaning of the terms themselves. As a result, it is instructive to
hold fixed the “matter” of a categorical judgment—the subject and predicate
concepts—and vary the form—the types of quantifier and copula. If we
know all of the ways in which these aspects of the form can differ, we will
be able to know how a difference in form can make a difference in whether
an argument is valid or not. Holding our subject and predicate concepts
fixed, here are the four types of categorical judgments—the four types of
categorical form—that are possible:
The convention here is to name these four categorical judgment types with
the letters A, E, I and O (which you may remember as the first four vowels).
There are four types because there are four possible combinations of
the two kinds of quantity (as represented by choice of quantifier) and two
kinds of quality (as represented by choice of copula). A universally
quantified judgment is one that says something about every member of a
category. Notice that even “No heroes are saints” does this: it says of all
heroes that they are not saints. A judgment with a particular quantity is one
that does not say something about every member of the category, but does
say something about at least one such member. The representative of an
affirmative quality indicates that a predicate is affirmed of a subject, i.e., the
predicate is said to be true of the subject. A judgment possesses a negative
quality when it claims that a predicate is not true of a subject.
Notice, of course, that since we are keeping the same “S” and “P”
terms through these four examples, it is impossible for all four of these
statements to be true at the same time. (In particular, if the first statement,
named “A,” is true, then at the very least the statement named “O” cannot be
true. Ordinarily, you would think that if “A” is true, “E” could not be true,
either.)
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If you understand the ways in which these four judgment types differ,
in terms of both quantity and quality, you should be able to take any
particular judgment and change just one of the dimensions without the other.
Consider the following examples:
1. Some orcs are not urukai.
2. No dwarves are men.
3. All urukai are orcs.
4. Some hobbits are reliable hobbitses.
What do you get when you change the quality of each, but not the quantity
Or when you change the quantity, but not the quality
Diagramming categorical judgments
Earlier, we used intersecting circle diagrams to represent various kinds of
categorical judgments. These are called Euler diagrams, after the
mathematician who popularized their use. Although these Euler diagrams
are straightforward when it comes to representing single judgments, it turns
out that it is not always easy to use them to represent the relationship
between two separate judgments in the way we need to represent syllogisms.
For this reason, we will adopt a slightly more complicated form of circle
diagram which, while it is slightly less intuitive, will turn out to be easier to
use to represent syllogisms. Here are the simpler Euler diagrams:
We will now learn about the slightly more counterintuitive kind of
diagram, called a Venn diagrams (again, named after their originator). (Note:
among teachers and business professionals, the phrase “Venn diagram” is
sometimes used to refer to any intersecting circle diagram. This is
inaccurate; only the kind of intersecting circle diagram we’re about to learn
is properly a Venn.) Every Venn diagram we’ll use begins with a simple
template, which otherwise looks like the diagram of “Some S is P” above.
But it does not mean “Some S is P”: as a mere template, it doesn’t actually
mean anything yet:
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Because we’re treating this as a template, it doesn’t acquire any meaning
until we do something to fill it in.
Here is how we will represent the “All S is P” categorical judgment:
When you see the left portion of the “S” circle filled in, you tend to think of
it as a form of marking territory, as a way of highlighting something that is
present. But you should really think of this filling in as a blackening out.
When this portion is blackened out, the diagram is, in effect, lopping off this
portion of the original “S” circle. All that is now left of “S” is what is still
white, what is inside “P.” Do you now see how this is equivalent to the
original Euler “All S is P” diagram In the Euler diagram, the “S” circle is
entirely contained within the “P” circle. Here, all that remains of the “S”
circle is entirely contained within the “P” circle. The idea and the topology
is the same.
You can see the same logic at work in the Venn diagram for “No S is
P”:
Again, don’t think of the middle portion as filled in or as an overlap between
the two classes; instead, think of it as a blackened-out deletion. This diagram
is now saying that there is no overlap between the “S” circle and “P” circle.
(Think of the diagram as two kissing Pac-Men.) In the same way, the
original Euler diagram simply shows the “S” and “P” circles as entirely
separate, i.e., as not overlapping.
For “Some S is P” and “Some S is not P,” we introduce a new
symbolic element, the asterisk. Here is “Some S is P”:
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This diagram says that there is at least one S that is also P, i.e., “Some S is
P.” By the same token, the following says there is at least one S that is not
also P (“Some S is not P”):
We have now encountered three distinct symbolic elements of Venn
diagrams: black space, which means nothing is in the area designated, the
asterisk, which means something is present, and white space. What is the
white space, if not something or nothing It is ignorance. It indicates that
there may or may not be something present in the area designated, that the
premise in question simply does not tell us.
At this point, you might be wondering why we include an asterisk in
“Some S is P,” but not in “All S is P.” The absence of the asterisk in “All S
is P” suggests that we do not know if there is at least one S that is also P. But
doesn’t “All S is P” imply that “Some S is P” When we say that all heroes
are virtuous, aren’t we implying that there are some heroes
The answer is somewhat controversial among logicians. The logicians
who defend the Venn diagram of “All S is P” without the asterisk argue for
what is called the Boolean interpretation of “All S is P” (after the logician
who originated the idea). According to this interpretation, we should not
suppose that “All S is P” implies the existence of any S, because there are
seemingly straightforward examples of universal affirmative categorical
judgments which we accept as true without supposing that they refer to any
real S, such as “All unicorns have horns,” or “All urukai are orcs.” The
Boolean interpretation says we should just assume that all “All S is P”
statements are the same way; with this assumption, we will never draw any
mistaken implications of existence. Here, then, are the original Euler
diagrams, paired with their Boolean Venn counterparts:
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Other logicians—who defend what is called an Aristotelian
interpretation, after the great Greek philosopher and logician—will argue
that the common sense is right, and that “All S is P” does imply “Some S is
P.” In this case, “All S is P” implies the existence of at least one S. They will
point out that of course “All unicorns have horns” does not imply the
existence of any unicorns, but that the sense in which we take this statement
to be true is not the same in which we take “All heroes are virtuous” to be
true. When we use statements taken from fiction, we are speaking in a
different voice than when we speak literalities. Accordingly, provided that
we know that we are speaking in a literal voice, we presuppose that the
subject term in question really exists, and it is fine if we place an “asterisk”
in the diagrams of universal categorical statements:
Diagramming to test simple arguments for validity
To show how these diagrams can be used to test the validity of categorical
syllogisms, we should first illustrate how they can be used to test the validity
of arguments even simpler than syllogisms: one-premise arguments called
immediate inferences..
Consider the following examples of an immediate inference:
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All saints are producers
Therefore, some saints are producers.
You’ll notice that this is an example of an implication from an “All S is P”
judgment to a “Some S is P” judgment. We know this implication holds only
on Aristotelian assumptions, and we can show this if we use an Aristotelian
diagram of both the premise and the conclusion:
Recall that an argument is valid if it is such that its conclusion cannot
be true if its premises are assumed to be true. In the same way, you can see
that this pair of diagrams is such that the second diagram contains no more
information than that contained in the diagram of the premise. The first has a
blackened left portion, and an asterisk; the second is missing the blackened
portion, but it has the asterisk. Remember that missing black space, i.e.
white space, represents nothing but ignorance: i.e., the lack of information.
So there is no information in the conclusion that is not in the premise (the
white space on the left is not information; it’s the lack of information). This
tells us that the argument is valid.
Now, consider an example of an invalid argument, demonstrated
using the same kind of method:
All urukai are orcs.
Therefore, some urukai are orcs.
This time we are speaking about things we know not to exist—the mythical
race of the urukai, the “men crossed with orcs” from The Lord of the Rings.
Since we are speaking of fictional items, we have to be sure to use Boolean
Venn diagrams to make sure we don’t infer unwarranted conclusions:
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Given the Boolean presumption against inferring “Some S is P” from “All S
is P,” we can see how this conclusion does in fact contain more information
than the premise. Right there in the middle, we see an asterisk that does not
originally appear in the premise. This is new information, not implied by the
premise. So the argument is invalid.
We could have realized in advance that the first argument was valid,
and the second invalid. We didn’t need diagrams to understand it. But now
that we know how to construct diagrams to reveal validity or invalidity, we
can use this technique to test arguments whose validity or invalidity is not as
obvious.
Diagramming to test syllogisms for validity
We now have all of the pieces we need to use Venn diagrams to test validity
of full-fledged syllogisms, i.e. two premise deductive arguments composed
of categorical judgments. To illustrate, let’s test the validity of the following
argument:
No heroes are villains.
All perfectionists are heroes.
No perfectionists are villains.
There are three steps in the method:
1. Construct a diagram with three interlocking circles, each representing
one of the terms of the syllogism.
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2. Fill in the diagrams for each of the premises.
To perform this second step, it is important to be able to focus
selectively on one pair of circles at a time. To diagram the first premise, “No
heroes are villains,” we focus exclusively on the relationship between the
“H” and “V” circles. Just for the sake of simplicity, we’ll start by using only
Boolean diagrams. The Boolean Venn diagram for a “No S is P” judgment
blackens out the overlap between the two circles, so that is what we do here.
Then, to diagram the second premise, “All perfectionists are heroes,” we
focus just on the relationship between the “P” and “H” circles, and
reproduce the Boolean Venn diagram of an “All S is P” judgment only
between these two circles:
3. See if you can “read off” the desired conclusion from the diagram of
the premises.
Just for the sake of reference, it is useful to know what the diagram of
the conclusion would look like, were it true. Here, the conclusion is “No
perfectionists are villains,” so we should focus just on the relationship
between the “P” and “V” circles, and use the appropriate Boolean Venn:
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This is just the conclusion that we would like to find in the diagram of the
premises, if we like valid arguments. So do we find it In this case, we do.
Notice that the darker shading between the “H” and the “V,” together with
the (here) lighter shading between “P” and “H” together fully shade the
overlap between “P” and “V.” This means there is no overlap between “P”
and “V” possible, which, when translated, means that the diagram of our
premise implies “No perfectionists are villains.” This is a valid argument.
Now for a quick example of an invalid argument:
All moralists are ideologues.
No pragmatists are moralists.
No pragmatists are ideologues.
Here is our diagram of the premises:
Here is the diagram of the conclusion:
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The question is: do we see the conclusion already contained in the
premises As you see, the question concerns the overlap between the “P”
and the “I” circle. It is filled in entirely in the diagram of the conclusion, but
not in the diagram of the premises. The premises only fill in half of that
overlap. That means this conclusion contains more information—the ruling
out of any overlap between “P” and “I”—than the premises. This argument
is invalid.
Now for a third example, but this time a syllogism using a particular
categorical judgment—not just universal judgments:
Some people are thinkers.
All thinkers are focusers.
Some people are focusers.
The usual tendency when diagramming an argument like this is to
diagram the first premise first, the second, second. But what happens when
you try doing this If you focus on the relationship between the “P” and “T”
circles, you’ll notice that the intersection between these two circles divides
between an area that is in the “F” circle and an area that is not. Where are we
to put the asterisk Inside “F” or not We simply don’t know. The first
premise doesn’t tell us whether the people who are thinkers are focusers or
not. So we would have to make the asterisk “straddle” the line between “F”
and not “F”:
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These “straddling” asterisks can be a source of consternation, since they do
not reveal as much information as they would if the asterisk were on one
side of the line or the other. It would be far better if we could force a
decision. If we avoid the temptation to diagram the first premise first, we can
do that. The second premise above is a universal premise that eliminates the
possibility that the asterisk falls on one side. If we diagram the second
premise first, diagramming the first premise will be easier:
With this diagram of the premises in hand, let’s compare it to the
diagram of the conclusion, “Some people are focusers”:
Here you need to be careful. What does the diagram of the conclusion mean
Here again we see the straddling asterisk. But this time it is not a source of
consternation, but of liberation. The straddle says that these people who are
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focusers may be either thinkers, or not. Our diagram of the premises has an
asterisk in the “T” part of the overlap, so it affirms the existence of people
who are focusers who are thinkers. This is one of the two options allowed by
the conclusion. As a result, the conclusion contains no more information
than the premises. If it had definitively affirmed that the star was on the
south side of the “T”/non-“T” divide, then it would be saying something that
the premises do not say. But since it straddles, it makes no commitment and
claims nothing that the premises do not claim. Hence, its information is
already contained in the premises, and this argument is valid.
Let’s rehearse one last example, this time one involving particular
categorical judgments, but one which turns out to be invalid.
Some dreamers are angels.
All people are dreamers.
Therefore, some people are angels.
Here is our diagram of the premises:
And here is our diagram of the conclusion, “Some people are angels”:
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As before, our conclusion contains a “straddling” asterisk. But does
the uncertainty of this asterisk help us or hurt us when it comes to
determining validity This time, it hurts, because you’ll notice that the
asterisk in the diagram of the premises also straddles. Suppose that the
asterisk in the premises were right in the middle of the three circles, in the
intersection of “P,” “A,” and “D.” If it were there, the premises would be
giving us a definite statement about where to find existing dreamers who are
angels. But it does not. As a result, we cannot say that these premises imply
a claim that is consistent with the claim of the conclusion. If it turns out that
the asterisk in the premises straddles only because there are dreaming angels
who are not people, then the conclusion, that there are people who are
angels, does not follow. This argument is invalid.
We have now gone through most of the nuances of using Venn
diagrams to determine validity, but we have only used Boolean Venn
diagrams, not Aristotelian diagrams. Because Aristotelian diagrams for
universal judgments contain more information than Boolean diagrams for
universal judgments, the Aristotelian interpretation yields more valid
arguments than the Boolean. But because these diagrams are slightly more
complicated, we will not go into them here, and will leave their use as an
exercise for the student. Here are some syllogisms for which the Aristotelian
interpretation will sometimes deliver the same answers as the Boolean, and
sometimes not:
1. Some saints are not heroes.
All truth-tellers are saints.
Some truth-tellers are not heroes
2. All people are choosers.
All people are valuers .
All valuers are choosers.
3. All heroes are saints.
Some villains are evaders.
All evaders are heroes.
4. All good people are truth-tellers.
All truth-tellers are focusers.
Some focusers are good people.
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5. All Dryads are Tree spirits.
All Tree spirits are magical creatures.
Some magical creatures are Dryads
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