[Math Lair] Categorical Syllogisms

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The syllogism is a type of logical argument in which several (usually two) premises that are assumed to be true are presented, and a conclusion based on those premises is evaluated. There are several types of syllogisms, including conditional syllogisms, categorical syllogisms, and disjunctive syllogisms. This article will primarily discuss categorical syllogisms.

Aristotle was the first person to investigate categorical syllogisms. A categorical syllogism has three propositions. The three propositions use three terms in total: a subject, a predicate, and a middle term that connects the subject and the predicate. The three propositions are:

  1. The major premise, which relates the predicate to the middle term
  2. The minor premise, which relates the subject to the middle term
  3. The conclusion, which relates the subject to the predicate.

Propositions can be universal, asserting something about all things of a certain type, or particular (or existential), asserting something about at least one thing of a certain type. Propositions can also be affirmative or negative. In categorical syllogisms, the propositions contain words called quantifiers, which provide information about how many members of the class are under consideration. There are four possible types of quantifiers. Each of these are given a single-letter label as given below:
Type of quantifierEnglish wordLabelExample
Universal affirmativeallAAll men are mortal.
Universal negativenoENo men are mortal.
Particular affirmativesomeISome men are mortal.
Particular negativenot all
or some... are not
OSome men are not mortal.

The letters for the affirmative quantifiers some from the Latin word affirmo, "I affirm," while the letters for the negative quantifiers come from the Latin word nego, "I deny."

Let's take a look at an example of a syllogism:

All men are mortal.
Socrates is a man.
  [Therefore] Socrates is mortal.

Classification of Syllogisms

Each proposition in a syllogism has the form:

[quantifier] [one term] [is/are] [other term]
(looking at the above example, we could look at the minor premise as being "All Socrates' are men" to fit the pattern better.)

For each of the three propositions, there are four different ways quantifiers that could be used, which results in 64 possibilities. As well, the middle term can be either the subject or predicate of the major and minor premises. The position of the middle term in the two premises is referred to as the figure of the syllogism. There are four figures. They are illustrated in this table, where S refers to the subject, P to the predicate, and M to the middle term:
 Major PremiseMinor Premise
Figure 1M-PS-M
Figure 2P-MS-M
Figure 3M-PM-S
Figure 4P-MM-S

So, since there are 64 possible combinations of quantifiers and four figures, there are 64 × 4 = 256 possible types of syllogisms. We can label these syllogisms by listing the three quantifiers followed by the figure, so the syllogism illustrated above is AAA-1.

Obviously, not all of these are going to be valid modes of reasoning, either because they don't connect the subject and the predicate in any meaningful manner, or because the conclusion doesn't otherwise follow from the relation between subject, middle term, and predicate. For example:

All men are mortal.
Socrates is a man.
   Socrates is not mortal. (AAE-1)

Some animals are birds.
Some animals are dogs.
   Some dogs are birds. (III-3)

No birds are mammals.
No ducks are mammals.
   Some ducks are not birds. (EEO-2)

Activity: Before reading on, try constructing some other syllogisms. See if you can determine some patterns as to which are valid.

There are three rules for determining whether a syllogism is valid:

  1. The middle term must be distributed at least once
  2. No term may be distributed in the conclusion that is not distributed in one of the premises
  3. The number of negative premises must be equal to the number of negative conclusions.

The word distributed needs some explanation. For a more in-depth explanation, see distribution of terms, but here's a brief summary. In theory, a term is said to be distributed with respect to a certain proposition when it refers in its context to all the members of the class of objects that it denotes. With a quantifier of "All", the subject is distributed and the predicate is not distributed. With a quantifier of "Some", neither term is distributed. With a quantifier of "No", both subject and predicate are distributed. With a quantifier of "Not all", the subject is not distributed and the predicate is distributed. This last one doesn't make a lot of sense if you think about it, but the theory requires that the predicate be distributed in this case.

Breaking one of these three rules results in a formal fallacy. The fallacy of the undistributed middle results when the first rule is broken. The illicit major or illicit minor fallacies occur when the major/minor term is undistributed in the major/minor premise but distributed in the conclusion. The fallacy of exclusive premises occurs when any conclusion is drawn from a negative premise. The fallacy of affirmative conclusion from a negative premise occurs when a positive conclusion is drawn from a syllogism with at least one negative premise.

It turns out that, out of the 256 possible syllogisms, 19 (plus another five that are weaker versions of the five that have universal conclusions) are valid syllogisms. Medieval logicians created mnemonics to help remember these, with the first three vowels showing the quality and quantity of the premises and conclusion, using A, E, I, and O as above. The consonants also have some meaning (how to convert the syllogism to a syllogism of the first figure).
Figure 1 Barbara, Celarent, Darii, Ferio (AAA, EAE, AII, EIO)
Figure 2 Cesare, Camestres, Festino, Baroco (EAE, AEE, EIO, AOO)
Figure 3 Darapti, Felapton, Disamis, Datisi, Bocardo, Ferison (AAI, EAO, IAI, AII, OAO, EIO)
Figure 4 Bramantip, Camenes, Dimaris, Fesapo, Fresison (AAI, AEE, IAI, EAO, EIO)

By the way, if you don't know Latin, and the words seem meaningless to you, don't worry; they aren't real words in Latin, either.

Activity: Make your own set of mnemonics that is more meaningful to you.

There are five additional valid forms where the particular affirmative or negative is found instead of the universal, making them weaker versions of some of the ones listed above. They are:
Figure 1 Barbari, Celaront (AAI, EAO)
Figure 2 Cesaro, Camestros (EAO, AEO)
Figure 4 Calemos (AEO)

Syllogistic or term logic represented an important type of logic for over 2,000 years after Aristotle's death. In the nineteenth century, it was replaced with first-order logic. First-order logic can represent many arguments that can't be represented using traditional syllogisms, such as:

All dogs are mammals.
   All owners of dogs are owners of mammals.

Robert is richer than John.
Steven is richer than Robert.
   Steven is richer than John.

Another potential pitfall in syllogisms that is avoided in first-order logic is the existential fallacy. See the formal fallacies page for more information about this fallacy.

Sources used (see bibliography page for titles corresponding to numbers): 19.