The Categorical Syllogism - Amateur Logician

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Categorical Syllogisms

Logic is able to extend our knowledge.

In the categorical syllogism, we start from two categorical propositions (which serve as premises) to derive a third categorical proposition (which is the conclusion). Insofar as we start with true premises, and provided we follow the formal rules of logic, the derived conclusion is then guaranteed to be true.

It’s important to realize that the conclusion “follows from” the premises by the “structure” of the syllogistic argument. Hence a valid argument — that is, an argument that follows the formal rules of logic — can bring us a false conclusion.

For example, the following “structure” is valid. . .
Premise 1: All A’s are B’s.
Premise 2: All C’s are A’s.
Conclusion: Thus, all C’s are B’s.

The following argument is thus valid. . .
Premise 1: All giraffes are fish.
Premise 2: All monkeys are giraffes.
Conclusion: Thus, all monkeys are fish.

It’s a valid argument but has a false conclusion.
Validity doesn’t guarantee a true conclusion.

The following argument is also valid. . .
Premise 1: All animals are living things.
Premise 2: All humans are animals.
Conclusion: Thus, all humans are living things.

This argument is valid and has a true conclusion.
The particular content — also called “material” — itself did not make it valid.
The conclusion “followed from” the premises by the “structure” of the argument.

Remember. . .
— Propositions are true or false.
— Arguments are valid or invalid.

An argument that is both valid and has true premises is sound.
An argument that is not sound is unsound.

A sound argument always will get us a true conclusion.

If we have all true premises and a valid argument, then the conclusion is true.
It’s impossible for any valid argument to have all true premises but a false conclusion.

Notice that an argument is guaranteed to be invalid insofar as it contains all true premises but a false conclusion. That’s logical!

Since insofar as we have all true premises and a valid argument, then the conclusion must be true. Therefore, if the conclusion is false, it must be the case that either we didn’t have all true premises or a valid argument; but if, we also know that the premises are all true, it must be that the argument is invalid.

Introducing the Categorical Syllogism. . .

Constitutive Rules of the Categorical Syllogism:
— It contains only three categorical propositions: two of which are premises and the third of which is the conclusion.
— It contains only three terms.

The Middle Terms and the Extremes:
— It’s through the “middle term” that we can arrive at a new proposition.
— The middle term relates the other two terms, called the “extremes.”
— The middle term will always appear in both premises.
— The middle term never appears in the conclusion.

The Classic Example. . .

Premise 1: All men are mortal.
Premise 2: Socrates is a man.
Conclusion: Therefore, Socrates is mortal.

“Men” is the middle term.
— It appears in the premises, but not in the conclusion.
“Mortal” and “Socrates” are the extremes.
— Together both extremes appear in the conclusion.
— They appear separately in the premises.

The middle term, “men,” connects the extremes, “mortal” and “Socrates.”
“Men” is a logical part of one of the terms and a logical whole for the other term.
Thus we can conclude that “Socrates is mortal.” He must be, given the premises.

It’s fair to claim we “proved“ that conclusion.
That’s because the argument has true premises and is a valid argument.

The “Invitation to Logic” page provided a “big picture” preview of this example. This example — or any other — implicitly involves the “three acts” of the mind. Syllogistic reasoning depends upon judgement, and judgement depends upon conceptualization. A deep appreciation of logic requires an understanding of each.

Another Example. . .

Premise 1: No logician is funny.
Premise 2: Some teachers are logicians.
Conclusion: Therefore, some teachers are not funny.

“Logician” is the middle term.
“Funny” and “teachers” are the extremes.

This example has a disconnect between the extremes. Logicians and being funny are apart. Since some teacher is a logician, and no logician is funny, it must be that some teachers are not funny (that is, at least one teacher must be not funny).

Of course, while this argument is valid, it is unsound. The first premise is false, and that can be shown by one example of a funny logician. The truth of “some logician is funny” (an I-type proposition) cancels out the truth of “no logician is funny” (an E-type proposition).

Why Categorical Syllogisms Work: Dictum de Omni et Nullo

Syllogistic reasoning is based on the Maxim of All and None. Whatever is affirmed (or denied) universally of a subject is thereby affirmed (or denied) of every logical part of that subject. This is self-evidently true. We can understand it immediately.

To deny this maxim is to violate the Law of Contradiction.
For if “all men are mortal” is true, it is impossible that “some men are not mortal” is true.

Six Basic Inferential Rules of the Categorical Syllogism

Two Rules of Quantity. . .
(1) At least one premise must be universal.
(2) If there is a particular premise, then the conclusion must be particular.

Two Rules of Quality. . .
(1) At least one premise must be affirmative.
(2) If there’s a negative premise, then the conclusion must be negative. And if there’s a negative conclusion, then there must be a negative premise.

Two Rules of Distribution. . .
(1) The middle term must be distributed at least once.
(2) If a term is distributed in the conclusion, then it must be distributed in the premise from which it came.

Distribution Review. . .
(For more details, please see the page on Distribution.)
A-type propositions distribute their subject.
I-type propositions distribute no term.
E-type propositions distribute their subject and predicate.
O-type propositions distribute their predicate.

There are slightly different ways to formulate — and verify — these rules.
For now, let’s algorithmically use them to check for validity.

Back to “The Classic Example”. . .

Premise 1: All men are mortal. (HAM.)
Premise 2: Socrates is a man. (SAH.)
Conclusion: Therefore, Socrates is mortal. (SAM.)

When it comes to quantity, notice that there is at least one premise that is universal. Both Premise 1 and Premise 2 are universal. There is no particular premise; thus, that a particular premise always demands a particular conclusion has been “vacuously satisfied.”

When it comes to quality, both rules have been satisfied as well. There is at least one affirmative proposition. Any negative premise will always demand a negative conclusion (and vice versa). Here there is no negative premise nor a negative conclusion; thus, the rule is “vacuously satisfied.”

And when it comes to distribution, we are also good to go. The middle term is H (“men”). Premise 1, HAM, is a universal affirmative proposition. H is a distributed term in it. Also, the Conclusion, SAM, has the distributed term S. It happens to be distributed in Premise 2, SAH.

Back to “Another Example”. . .

Premise 1: No logician is funny. (LEF.)
Premise 2: Some teachers are logicians. (TIL.)
Conclusion: Therefore, some teachers are not funny. (TOF.)

When it comes to quantity, everything passes the checklist. Premise 1 is universal. Premise 2 is particular and we have a particular Conclusion.

When it comes to quality, things look great. Premise 2 is affirmative. Premise 1 is negative and there is a negative Conclusion.

And when it comes to distribution, all is good. The middle term is L (“logician”). It’s distributed in Premise 1, LEF. The Conclusion, TOF, has F (“funny”) as distributed. This term appears in Premise 1 and is distributed there.

A New Example. . .

Premise 1: All cats have four legs. (CAL.)
Premise 2: My dog has four legs. (DAL.)
Conclusion: Therefore, my dog is a cat. (DAC.)
Referred to as “Politician’s Logic” in old TV show Yes, Prime Minister.

This example is technically known as the “Fallacy of the Undistributed Middle.” The middle term, “four legs,” is not distributed. Thus, the middle term doesn’t relate the extremes, “cats” and “my dog.”

The diagram on the right shows that both extremes are members of the middle term. That’s all the premises tell us, however. There’s no way to compare one extreme to another to reach a conclusion.

That example has the same “structure” as this:
Premise 1: We must do something. (CAL.)
Premise 2: This is something. (DAL.)
Conclusion: Therefore, we must do it. (DAC.)
It’s the “Politician’s Syllogism.”

Find the Missing Premise Example. . .

The “Rule Checklist” is one way to find a missing premise in an argument.
For example: Premise BAC and Conclusion BED. Find the Second Premise, if possible.

Notice that the middle term must be C (as it doesn’t appear in the conclusion).
There are eight possibilities: CAD, DAC, CED, DEC, CID, DIC, COD, and DOC.

(Remember there are four categorical propositions, and for each one there are two possibilities with the terms C and D in it as subject or predicate.)

Notice that the conclusion is universal; ergo, all premises must be universal:
CAD, DAC, CED, DEC, ~~CID~~, ~~DIC~~, ~~COD~~, and ~~DOC~~.
Notice that the conclusion is negative; ergo, a premise must be negative:
~~CAD~~, ~~DAC~~, CED, DEC, ~~CID~~, ~~DIC~~, ~~COD~~, and ~~DOC~~.

We now have two possibilities: CED or DEC.
But by simple conversion, CED and DEC are equivalent.
Therefore, either one is a perfectly fine answer.

Reflecting on the Rules (once again). . .

Among the “constitutive rules,” it simply follows from the definition of a categorical syllogism that it can only contain three terms. The entire point is to derive something new (namely, a conclusion) from two premises. The middle term serves as the link between the conclusion’s subject and predicate.

That link must be air tight or the conclusion is invalid. This requires that the middle term’s meaning must be exactly the same in both premises. Otherwise, we have an ambiguous middle. An ambiguous middle, essentially, implies there are four terms, not three terms. It is thereby fallacious, i.e., invalid.

To have an air tight link between the conclusion’s subject and predicate requires that the middle term be distributed at least once. This means that its entire extension is being referred to. Otherwise, we have an undistributed middle. This is invalid. There is then no way to precisely relate the two extremes. Each extreme is related to a middle term, but nothing is telling us if both extremes reference the same part of the middle term’s extension.

Similarly, insofar as a term is distributed in the conclusion, it must be distributed in a premise. Otherwise, the conclusion would invalidly go beyond what the premises reveal to us. It cannot be valid to reference a greater extension than what is provided by those premises.

An illicit process of the major is when the conclusion’s predicate is invalidly distributed. An illicit process of the minor is when the conclusion’s subject is invalidly distributed.

Given the paramount importance of being able to relate terms, we can also see that both premises cannot be negative in quality. Any new conclusion inferred from them is invalid — which is also known as the fallacy of exclusive premises. So at least one premise must be affirmative. Otherwise, both extremes are declared to have no relation to the middle term and, therefore, there is no way to derive something new through that middle term.

Although when there is one premise negative in quality, it must follow that the conclusion must be negative. This goes both ways: when the conclusion is negative, it follows that one premise must be negative. Because there’s a denial of the extremes in the conclusion, this must mean that one extreme is not part of the middle term’s extension (while the other extreme is a part).

Finally, we can consider why at least one premise must be universal and why a particular premise will demand a particular conclusion as it narrows things down so that only a particular conclusion will do.

Let’s assume that particular premises can get us a new conclusion.

Basically, we can do a kind of “indirect proof.”

There are three possibilities (as premises): II, IO, and OO. We already know that two negative premises won’t work. Hence, we now have two possibilities: II and IO. We can rule out II because the middle will be undistributed. Hence, we now have one possibility: IO. This won’t work either. The conclusion would have to be negative, yet this demands its predicate be distributed. Since two terms must be distributed in the premises (i.e., the middle term and one extreme), and only one term is distributed in the premises, nothing validly can be derived. Therefore, the assumption is false; and, therefore, one premise must be universal.

And let’s assume that a syllogism with a particular premise can get us a universal conclusion.

We can consider these possibilities (as premises): AI, II, EI, OI, AO, IO, EO, and OO. No valid syllogism will have only negative premises. So, we now should consider this refined list of possibilities: AI, II, EI, AO, and IO. No valid syllogism can only have particulars. So, now we have: AI, EI, and AO. Note that a universal conclusion will have a distributed subject. Because of that AI won’t work. It will only have one distributed term (and either that will be the middle term or an extreme), but for it to work it would need two distributed terms (both the middle term and an extreme). Therefore, it’s impossible that the conclusion could be a universal. Neither will EI or AO work. Both have two distributed terms respectfully. Yet both EI and AO both demand a negative proposition. This requires three distributed terms in the premises. Therefore, AI and AO cannot produce a universal proposition. Our assumption leads to logical impossibilities; and therefore, one particular premise will demand a particular conclusion.

Therefore, all rules are justified. ■