﻿ 2.4 Syllogisms

2.4 Syllogisms

What is a syllogism?

Traditional logic distinguishes between immediate inference (which have only one premise) and “mediate” inferences (which have more than one). In the previous section, we discussed the immediate inferences – subalternation, conversion, obversion and contraposition. We now turn to the most basic and important of the mediate inferences – the syllogisms. Here are two examples:

 (1) No metals are insulators. Some metals are liquids. Some liquids are not insulators.
 (2) All humans are mortal. No mortals are gods. No gods are humans.

Every syllogism has two premises (and one conclusion) and contains exactly three terms. In (1) the terms are “liquid”, “metal” and “insulator”; in (2) they are “god”, “mortal” and “human”. As always in traditional logic, the premises and the conclusion are categorical sentences and the three terms are their subjects or predicates. Each term occurs twice – each time in a different sentence. One of the terms occurs in both premises and does not occur in the conclusion (“metal” in the first example and “mortal” in the second). The other two terms occur in the conclusion and in one of the premises (“liquid” and “insulator” in (1) and “god” and “human” in (2)).

The terms are called minor, middle and major term. To identify them by name we start from the conclusion. The conclusion’s subject is the minor term (“liquid” in (1) and “god” in (2)). The conclusion’s predicate is the major term (“insulator” in (1) and “human” in (2)). Except in the conclusion, the minor and major term occur in one of the premises – each one in a different premise. In both examples, except in the conclusion, the major term occurs in the first premise and the minor term occurs in the second premise. The minor and the major terms are collectively called extremes (minor extreme and major extreme). The term that does not occur in the conclusion but occurs in both premises is the middle term (“metal” in (1) and “mortal” in (2)).

The two premises also have names. They are referred to by the minor or the major term that occurs in them. The premise containing the major term (the predicate of the conclusion) is called major premise (in both examples it is the first sentence), and the premise containing the minor term (the subject of the conclusion) is called minor premise (the second sentence in the examples).

It is important to keep in mind that the major premise is not the premise that appears first and the minor premise is not the premise that appears second – both premises may come first, second, or even after the conclusion. Here is an example of an argument in the form of a syllogism in which the conclusion is stated first, the minor premise second, and the major premise last:

 God is just, because He is perfect, and everything that is perfect is just.

What makes a premise minor is the fact that it contains the minor term – the subject of the conclusion. Similarly for the major premise and the major term. This is the reason way the logical analysis of a syllogism begins with the conclusion – the subject and predicate of the conclusion determine which the minor and the major terms are (the remaining term is the middle term), and the minor and the major terms determine which the minor and the major premises are. To analyze the above syllogism, we begin by determining which sentences is the conclusion. Obviously, “God is just”. Its subject (“God”) is the minor term and its predicate (“just”) is the major term. Having determined the minor and the major terms, we are in a position to determine the minor and the major premises. The sentence containing the minor term (“God”) is the minor premise and the one containing the major term (“just”) is the major premise.

For the purpose of logical analysis, we will order the premises and the conclusion of a syllogism in the way they are ordered in the examples (1) and (2) above – first the major premise, then the minor premise, then the conclusion. When the three sentences are ordered in that way we will say that the syllogism is in standard form. To be in standard form is important for the correct analysis of a syllogism. We will shortly see why.

Every syllogism has a mood. It is determined by the types (A, E, I, O) of the sentences it consists of and is represented by a sequence of three letters corresponding to those types. The order of the letters is the order of the sentences of the syllogism in standard form. That is, the first letter indicates the type of the major premise, the second letter – the type of the minor premise, and the third letter – the type of the conclusion. The mood of the syllogism (1) above is EIO since the major premise is a universal negative (E), the minor premise – a particular affirmative (I), and the conclusion – a particular negative (O) sentence. Accordingly, the mood of the syllogism (2) is AEE. Being able to unambiguously specify the mood is one of the reasons for specifying a standard form for a syllogism – the order of the sentences determines the mood only if the syllogism is in standard form.

The mood of a syllogism is not enough to characterize its logical form completely. Consider the following syllogisms:

 (3) All cats have tails. All trees are plants. All cats are animals. All flowers are plants. Some animals have tails. Some flowers are trees.

Both syllogisms are of mood AAI but the first is logically valid while the second is not. Whether an inference is logically valid or invalid depends on its logical form. The difference in validity shows that the logical form of these syllogisms is different. To see what that difference consists in, let us represent them symbolically. Usually, the minor term (the subject of the conclusion) is represented by “S”, the major term (the predicate of the conclusion) by “P”, and the middle term by “M”. Substituting these symbols for the terms in (3), we get:

 All M are P. All P are M. All M are S. All S are M. Some S are P. Some S are P.

The entirely symbolic representations are:

 MaP PaM MaS SaM SiP SiP

Now it becomes clear that the difference in the logical forms of the two syllogisms is in the position of the middle term M (“cat” and “plant”, respectively). In the first syllogism, the middle term is a subject in both premises while in the second syllogism it is a predicate in both premises. Thus, in addition to the mood, the position of the middle term is also important. It determines the figure of the syllogism. There are four figures as there are four possible positions of the middle term in the premises:

 1-st: M – P 2-nd: P – M 3-rd: M – P 4-th: P – M S – M S – M M – S M – S S – P S – P S – P S – P

In the first figure, the middle term is а subject in the major premise and а predicate in the minor premise. In the second figure, it is a predicate in both premises. In the third figure, it is a subject in both premises. In the fourth figure, it is a predicate in the major premise and a subject in the minor premise.

The correct determination of the figure of a syllogism is another reason for specifying a standard form for it. Obviously, the order of the premises is important for properly distinguishing between the first and the forth figure.

The logical form of a syllogism is completely specified by its mood and its figure. The first syllogism in (3) is mood AAI in 3-rd figure, which we will denote by “AAI-3”, and the second is mood AAI in 2-nd figure (denoted “AAI-2”). Every syllogism that has the first form (AAI-3) will be logically valid and every syllogism that has the second form (AAI-2) will be logically invalid. The reader should verify that the logical form of the syllogism (1) above is EIO-3 and that of (2) is AEE-4. Both are logically valid.

As a rule of inference, a syllogism is completely determined by its mood and its figure. So, to find out the total number of all possible kinds of syllogisms (valid or invalid), we should find the number of all possible combinations of moods and figures. The possible moods are 64 for the following reason. There are 4 possibilities for the major premise (A, E, I or O), which are combined with another 4 (A, E, I or O) for the minor premise (4x4=16), which are combined with another 4 (A, E, I or O) for the conclusion (16x4=64). If we list exhaustively and without repetitions all possible combinations of three letters (“AAA”, “AAE”, “AAI”, …, “OOO”), this list will have 64 elements. A syllogism in a certain mood may be in one of four possible figures. Therefore, the combinations between all moods and all figures are 64x4=256, which means that the total number of different kinds of syllogisms is 256.

Among all 256 syllogisms, only 24 are logically valid (according to traditional logic) – the rest are logically invalid. Let us take two examples – of a valid and invalid syllogism and see what makes them so.

We started this section with two valid syllogisms, (1) and (2), which have the forms EIO-3 and АЕЕ-4, respectively. Take АЕЕ-4. It has the following symbolic representation:

 PaM MeS SeP

We may use the following diagram to verify its validity:

The major premise states that all P are M. Therefore, the class of all P (the extension of P) is a subset of the class of all M (the extension of M). This is represented in the diagram by the circle of P being included in the circle of M (drawing so, we do not exclude the possibility that the extensions of P and M coincide). The minor premise states that no M are S, which means that the extension of M has no common elements with the extension of S. This is represented by the circle of S being outside the circle of M. The diagram shows that the extensions of S and P are mutually exclusive. As the class of P is completely included in the class of M and class of M is completely excluded from the class of S, the class of S must be completely excluded from the class of P, i.e. no S are P, as stated in the conclusion.

As an example of an invalid syllogism, take ААА-4:

The major premise states that all P are M, which is represented in the diagram by the circle of P being included in the circle of M. The minor premise states that all M are S, which is represented by the circle of M being included in the circle of S. The conclusion is that all S are P, but the diagram shows that this can be fulfilled only when the extensions of P, M and S coincide. When this is not the case (which is, of course, completely possible and which is in fact the state of affairs shown in the diagram), there will be things that are S but not P, in which case the conclusion will be false, although the premises are true. Therefore, the syllogism is invalid.

We mentioned above that out of all 256 types of syllogisms, only 24 are valid. The table below shows which they are in each of the four figures:

 First figure Second figure Third figure Fourth figure AAA-1 “Barbara” EAE-2 AAI-3 AAI-4 ЕАЕ-1 “Celarent” AEE-2 IAI-3 AEE-4 AII-1 “Darii” EIO-2 AII-3 EAO-4 EIO-1 “Ferio” AOO-2 EAO-3 IAI-4 (AAI-1) (EAO-2) EIO-3 EIO-4 (EAO-1) (AEO-2) OAO-3 (AEO-4)

There are six valid syllogisms in each figure. Five of them are included in parentheses – two in the first and the second figure and one in the fourth figure. The conclusions of those syllogisms are particular in quantity – I or O sentences. For each of them, in the same figure another syllogism has the same premises but а universal conclusion. For example, AAI-1 in the first figure has the same premises (two A sentences) as AAA-1 but it has a particular (I) conclusion. The same holds for EAO-1 and EAE-1 in the same figure – they have the same premises but the conclusion of EAO-1 is an O rather than E-sentence. By subalternation, universal sentences entail the particular sentences of the same quality (and the same subject and predicate) – A entails I, and E entails O. Therefore, the syllogisms in parentheses are simply weakened versions of other valid syllogisms that have the same premises but a general conclusion. Take, for example, these two syllogisms:

 No mammals are fish. No mammals are fish. All dolphins are mammals. All dolphins are mammals. No dolphins are fish. Some dolphins are not fish.

The first is of type EAE-1 and the second of type EAO-1. They have the same premises and are logically valid, but from the conclusion of the first, “No dolphins are fish”, we can deduce the conclusion of the second, “Some dolphins are not fish”, which is a weaker sentence (it is entailed by the former but does not entail it itself). The fact that a stronger conclusion can be drawn from the same premises makes the five weakened versions put in parentheses pointless. Disregarding them, the valid syllogisms in traditional logic become nineteen – four in the first figure, four in the second figure, six in the third figure, and five in the fourth figure.

In first column of the table are included the names that the medieval logicians gave to the valid syllogisms in the first figure. The vowels in a name correspond to the mood (the three “a” in “Barbara” to AAA, etc.). The valid syllogisms in the other figures also have such names but they are not included as they are not important or very useful. I have given the names of the syllogisms of the first figure because of the importance Aristotle attaches to them. In Prior Analytics, he accepts the syllogisms of the first figure as axioms and deduces through them the valid syllogisms of the second and the third figure. For some reason he disregards the fourth figure.

The table shows some characteristic features of the different figures. All valid syllogisms in the second figure have negative conclusions. All valid syllogisms in the third figure have particular conclusions. Barbara (AAA-1) is the only valid syllogism with a universal affirmative (A) conclusion. That there is no such syllogisms in the other figures is one of the reasons for the importance attached by Aristotle to the first figure.

Before we see how Aristotle derives the logical validity of all valid syllogisms through the validity of the syllogisms in the first figure, let us convince ourselves of the logical validity of Barbara, Celarent, Darii, and Ferio (the four valid syllogisms of the first figure) by diagrams.

Barbara’s premises state that all M are P and that all S are M. Once the class of M’s is included in the class of P’s and the class of S’s is included in the class of M’s, there is no way that the class of S’s will not be included in the class of P’s, i.e. all S will be P.

Celarent’s premises state that all S are M and that no M are P. As the class of S’s is included in the class of M’s and the class of M’s is excluded from the class of P’s, the class of S will be excluded from the class of P’s, i.e. no S will be P.

Darii’s premises state that all M are P and that some S are M. If M is fully included in P and some part of S is included in M, then that same part of S will be included in P, i.e. some S will be P. (That only a part of the extension of S is included in the extension of M is depicted in the diagram by drawing only a part of the circle of S in the circle of M.)

The premises of Ferio state that no M are P and that some S are M. If the extension of M is excluded from the extension of P and some part of the extension of S is included in the extension of M, then the same part of the extension of S will be excluded from the extension of P, i.e. some S will not be P as stated in the conclusion.

Let us now see how Aristotle proves the the validity of the valid syllogisms in the second, third and fourth figure through the validity of Barbara, Celarent, Darii, and Ferio (the valid syllogisms in the first figure). In addition, the proofs use the immediate inference of conversion and the fact that A and O-sentences, on the one hand, and E and I-sentences, on the other, are exact negations of each other, i.e. the contradictory relation in the square of opposition.

As a first example, let us prove the validity of EAE-2:

 РeМ SаM SеP

We will show that if the two premises are true, the conclusion is also true. The idea is by converting one of the premises to be able to use one of the four syllogisms in the first figure, whose validity we have verified by diagrams above

 1. РeМ 2. SaM / SеP

From 1., by conversion, we may infer MeP:

 3. MeP 1, conversion

Now, from 3. and 2., by the valid syllogism of the first figure EAE-1 (Celarent), we can derive SeP (what we are aiming at). In that inference, we use 3. as a major premise and 2. as a minor premise:

 МеР SаM SеP
 4. SeP 3, 2, EAE-1

That completes the proof.

As a next example, we will prove the validity of AEE-4, which is the logical form of (2) above:

 PaM MeS SeP
 1. PaM 2. MeS / SeP 3. PeS 2, 1, EAE-1 4. SeP 3, conversion

3. is derived from 2. and 1. on the basis of EAE-1 (Celarent). In that step, 2. plays the part of the major premise and 1. the part of the minor premise:

 MeS РаМ РeS

Accordingly, in that inference P is the minor term and S is the major term. The syllogism we are proving has the same premises but their roles in terms of minor and major premise (as well as the roles of the terms) are switched, and so the figure is forth rather than first.

Not all valid syllogisms of the second, third and fourth figure can be derived directly from the valid syllogisms in the first figure using conversion. An example of a valid syllogism for which this is not possible is AOO-2:

 РаМ SoM SoP

As we know, particular negative sentences cannot be validly converted. Therefore, while proving that syllogism, we can only convert the major premise PaM, which will result in MiP. Then we will have two particular sentences – MiP and SoM. However, in the first figure there is no valid syllogism with two particular premises. In fact, there is no such valid syllogism at all. Therefore, we need another approach here. In such cases, Aristotle uses indirect proofs (reductio ad absurdum). Let us see how this works with AOO-2:

 1. PaM 2. SoM / SoP 3. SaP (¬SoP) assumption 4. SaM 1, 3, AAA-1 (S – minor, Р – middle, М – major term) – contradicts 2 5. SoP 3–4, reductio ad absurdum

In 3., we have assumed the negation of the conclusion SoP. This is the same as to affirm SaP since, as we know from the square of opposition, A and O-sentences are exact negations of each other. For the same reason, in 4., SaM contradicts 2. (SoM). As a whole, the proof goes like this: we assume the negation of what we want to prove (SoP). Using Barbara (AAA-1) we get to a contradiction (in case the premises are true). Therefore, SoP is a logical consequence of the premises.

Each valid syllogism of the second, third and fourth figure (which are given in the table above) can be deduced in one of the two ways shown – by a direct or indirect proof.

In fact, we do not need to accept as axioms all four valid moods in the first figure. Assuming only Barbara and any one of the rest, by reductio ad absurdum we can deduce the other two. For example, if we accept as axioms Barbara and Celarent, then Darii and Ferio are derived as follows:

 1. MaP 2. SiM / SiP 3. SeP (¬SiP) assumption 4. PeS 3, conversion 5. MeS 4, 1, EAE-1 (М – minor, Р – middle, S – major term) 6. SeM 5, conversion – contradicts 2 7. SiP 3–6, reductio as absurdum

 1. MeP 2. SiM / SoP 3. SaP (¬SoP) assumption 4. PeM 1, conversion 5. SeM 4, 3, EAE-1 (S – minor, Р – middle, M – major term) – contradicts 2 6. SoP 3–5, reductio as absurdum

As for the invalid types of syllogisms, Aristotle proves that they are such by finding counterexamples – inferences having the logical form in question that have true premises and a false conclusion. For example, the second syllogism in (3) above can serve as a proof of the invalidity of AAI-2:

 РаМ SaM SiP

Interpreting S as “flower”, P as “tree”, and M as “plant”, we obtain a syllogism with true premises and a false conclusion:

 All trees are plants. All flowers are plants. Some flowers are trees.

This shows that the truth of the premises does not guarantee the truth of the conclusion, so the inference scheme is invalid.