We are going to present general rules that a syllogism has to follow in order to be logically valid. Some of them are formulated by Aristotle and some are of later origin. Each rule will be a necessary condition for the validity of any syllogism; it will not be a sufficient condition. This means that if a syllogism violates a certain rule it will be definitely invalid, but if it complies with a rule, it may or may not be valid because it may be violating another rule. However, we can select a set of such rules that together act as a sufficient condition for the validity of any syllogism. If a syllogism satisfies all of them, it will be logically valid, and if it violates at least one of them, it will be logically invalid.
Two of the rules are formulated in terms of the concept of distribution. Each term in a categorical sentence (the subject or the predicate) is either distributed or undistributed in it. Categorical sentences can be viewed as asserting a certain relation between the extensions of their subject and predicate, which are classes. Depending on whether the sentence is affirmative or negative, it is a relation of inclusion or exclusion. The subject or the predicate is distributed if it participates in that relation with its entire extension; otherwise, it is undistributed. As we will see, whether a term is distributed in a sentence depends entirely on its type (if it is an A, E, I, or O-sentence) and on whether the term is a subject or a predicate in it.
Uttering a universal affirmative (A) sentence “All S are P”, we commit ourselves to the fact that the whole extension of the subject S is included in the extension of the predicate P, but we are not committed to the fact that the whole extension of P is included in the extension of S. If all S are P, some part of the class of P’s will be included in the class of S’s but not necessarily the whole class. Therefore, the subject of an A-sentence participates in the relation of inclusion expressed by it with its entire extension and the predicate participates only with part of its extension, i.e. in an A-sentence, the subject is distributed and the predicate is undistributed.
Through a universal negative (E) sentence “No S are P”, we claim that the extensions of S and P are mutually exclusive, i.e. the whole extension of S excludes the whole extension of P, and vice versa. Therefore, S and P participate with their entire extensions in the relation of exclusion expressed by the sentence, i.e. in an E-sentence, both the subject and the predicate are distributed.
Uttering a particular affirmative (I) sentence “Some S are P”, one claims that a part of the extension of the subject S is included in the extension of the predicate P and (as a consequence) that part of the extension of P is included in the extension of S. The speaker does not commits oneself neither to the whole extension of S being included in the extension of P, nor to the whole extension of P being included in the extension of S. While these states of affairs are possible, they may well not be true. Therefore, neither S nor P participates with its entire extension in the relation of inclusion expressed by the sentence, i.e. in an I-sentence, both the subject and the predicate are undistributed.
Through a particular negative (O) sentence “Some S are not P”, one claims that part of the extension of the subject S excludes the entire extension of the predicate P. While it is possible that the entire extension of S is excluded from the extension of P, this is of course not necessary. So, the exclusion relation to which the speaker commits oneself is between part of the extension of S and the whole extension of P. Therefore, in an O-sentence, the subject is undistributed and the predicate is distributed.
The table below summarizes the facts about distribution of terms in categorical sentences.
|Subject (S)||Predicate (P)|
|Universal affirmative (A)||distributed||undistributed|
|Universal negative (E)||distributed||distributed|
|Particular affirmative (I)||undistributed||undistributed|
|Particular negative (O)||undistributed||distributed|
As the table shows, the subject is distributed in the universal sentences (A and E) and not distributed in the particular (I and O), and the predicate is distributed in the negative sentences (E and O) and not distributed in the affirmative (A and I). Therefore, the facts about distribution of terms in categorical sentences can be summarized (for easier memorization) as follows:
|The subject is distributed in the universal sentences, and the predicate is distributed in the negative sentences.|
Now we are ready to formulate the rules. As mentioned in the beginning, each of them is such that a syllogism must obey it in order to be valid, i.e. each of the rules is a necessary condition for the validity of any syllogism. On the other hand, each invalid syllogism violates one or more of the rules. This means that if a syllogism satisfies all of them, it will be valid, i.e. taken together, the rules are a sufficient condition for the validity of a syllogism. Taken separately, however, they are not such conditions. A syllogism may be invalid and satisfy one or more of the rules; what it cannot, however, is to satisfy all.
We will introduce five rules that, in their entirety, are sufficient to distinguish valid from invalid syllogisms. Then, for convenience, we will introduce two more, formulated by Aristotle. When a syllogism satisfies the five rules in question, it is valid; when it breaks one (or more) of them, it is invalid. Two of the rules relate to the distribution of terms and the other three to negative sentences; the additional two refer to the particular (I and O) sentences.
What underlies all of the rules is the mediating role of the middle term. A syllogism is valid, when the relation between the extensions of the minor and the major term stated in the conclusion is a consequence of their relations to the extension of the middle term stated in the premises. If the middle term is undistributed in both premises, it relates to the extensions of both the minor and the major term only with a part of its extension. Then it will be possible that the two parts in question have no common elements, because of which the middle term will not be able to carry out its mediating role.
An example of a syllogism that violates this rule and therefore is invalid is AAI-2:
|PaM||All dolphins are mammals.|
|SaM||All seals are mammals.|
|SiP||Some seals are dolphins.|
Because the middle term (“mammal”) is undistributed in both premises, it allows only part of the mammals to be dolphins and only part of the mammals to be seals. In this case, the two parts have no common elements, which makes the conclusion false.
If any of the terms in the conclusion (minor or major) is undistributed in the premise, it relates to the extension of the middle term only with a part of its extension. If the middle term succeeds in doing its mediating job, then the term of the conclusion in question can participate in the relation stated in the conclusion only with the part of its extension with which it relates to the middle term, i.e. it cannot be distributed in the conclusion. In essence, the rule tells us that the conclusion cannot contain more than the premises.
There are two ways to violate this rule. It can be broken either by the minor or by the major term (or by the both). Here is an example of a violation by the major term, AEE-1:
|МаР||All dolphins are mammals.|
|SеM||No seal is a dolphin.|
|SеP||No seal is a mammal.|
The problem is in the major term (“mammal”), which is distributed in the conclusion (being the predicate of a negative sentence there) but undistributed in the premise (being the predicate of an affirmative sentence).
Here is an example of a violation by the minor term, AAA-3:
|МаР||All humans can laugh.|
|МaS||All humans are animals.|
|SaP||All animals can laugh.|
The minor term “animal” is distributed in the conclusion (as it is the subject of a general sentence there) but undistributed in the premise (as it is the predicate of an affirmative sentence there).
A negative sentence (including particular in quantity) allows the extensions of the subject and the predicate to be mutually exclusive (no common elements). So, the two premises being negative allows for the classes of S and P to be entirely excluded from the class of M (the middle term). However, it is obvious that then all possible relations between the classes of S and P can be a fact – they can be wholly or partially excluded from each other and wholly or partly included in each other. Therefore, in this case it is not possible to draw a definite conclusion. An example of a syllogism violating that rule (EEE-1):
|МeР||No fish are mammals.|
|SeM||No dolphins are fish.|
|SeP||No dolphins are mammals.|
This rule is equivalent (by transposition) to the rule “If the conclusion is affirmative, both premises must be affirmative”. Assume that one of the premises is negative. The other premise has to be affirmative (because of the previous rule). An affirmative sentence (including particular in quantity) allows the subject and predicate to have the same extension – this is always a possibility. Therefore, the extensions of one of the extremes (the minor or the major terms) and the middle term may be the same. At the same time, the negative premise always allows the extensions of the middle term and the other extreme to be mutually exclusive. Thus, if one of the premises is negative, there is always the possibility for the minor and the major term to be mutually exclusive. This possible case excludes an affirmative conclusion (in it, the conclusion will be false while the premises are true). An example of a syllogism violating that rule (EII-2):
|PeM||No snakes can fly.|
|SiM||Some fishes can fly.|
|SiP||Some fishes are snakes.|
The rule is similar to the previous one, but it is in the other direction – from the conclusion to the premises. It is equivalent (by transposition) to the rule that if both premises are affirmative, then the conclusion must also be affirmative. We will justify it in that form. Assume that both premises are affirmative. As an affirmative sentence allows the subject and the predicate to have the same extension, there is always the possibility that the extensions of the minor, major and middle term are the same class. This case excludes a negative conclusion (in it, the conclusion will be false while the premises are true). An example of a syllogism violating that rule (АAО-2):
|РаМ||All mammals are animals.|
|SaM||All cats animals.|
|SоP||Some cats are not mammals.|
The above syllogism also violates the rule that the middle term should be distributed in at least one of the premises. This can happen – if a syllogism is invalid, it violates at least one of the rules, but it may violate more rules.
The rules in summary:
|Distribute the middle term in at least one premise.|
|Any term distributed in the conclusion must be distributed in the premises.|
|Avoid two negative premises.|
|If either premise is negative, the conclusion must be negative.|
|If the conclusion is negative, one of the premises must be negative.|
Taken together, the five rules are enough to exclude all invalid syllogisms without excluding valid ones. Every invalid syllogism violates one or more rules and every valid one satisfies all of them. Thus, in their entirety, the rules are a sufficient condition for the validity of any syllogism; and each rule is a necessary condition for the validity of any syllogism. This gives us the following algorithm to check the validity of an arbitrary syllogism. We start checking whether it satisfies the rules one by one. It is better to start with the last three rules (about negative sentences), as they are easier to use. If a rule is violated, we stop the check – the syllogism is invalid. If all rules are satisfied, the syllogism is valid.
While the above rules are sufficient to distinguish valid syllogisms from invalid ones, we will add to them the following two rules, which are due to Aristotle and which are useful for rejecting invalid syllogisms:
These two rules can be deduced from the above five. Each is a necessary condition for the validity of a syllogism. The first rule is similar to the rule that both premises cannot be negative, and the second rule is similar to the rule that if a premise is negative, the conclusion must also be negative. The analogue of rule that if the conclusion is negative at least one of the premises must also be negative is not valid. It is not true that if the conclusion is particular (I or O), at least one of the premises must be particular. For example, AAI-3 and AAI-4 are valid syllogisms.1
Although we can go without the two additional rules, they are convenient for rejecting invalid syllogisms, as they are easier to apply than the rules about distribution of terms. For example, for IOO-1 it is very easy to see that it is invalid because it has two particular premises, while from the above five rules it breaks only one of the rules about distribution, which is not so easy to see:
The violated rule is that if a term is distributed in the conclusion it has to be distributed in the premise – the major term P is distributed in the conclusion (as it is the predicate of a negative sentence) and undistributed in the major premise (as it is the predicate of an affirmative sentence).
|(1) Determine which rules are violated by the following syllogisms:|
|(2) The following arguments are syllogisms. Represent them symbolically and determine through the syllogistic rules whether they are logically valid.|
|1)||Some good physicists do not a have high spatial intelligence, because all good physicists are good mathematicians and some good mathematicians have low spatial intelligence.|
|2)||If a person has an IQ of less than 50, he (or she) is an imbecile, and no one who can graduate at a university has an IQ of less than 50. Therefore, no one who can graduate at a university is an imbecile.|
|3)||Some central bank governors have extreme left-wing beliefs and no one with such beliefs is a liberal. Therefore, some liberals are not central bank governors.|
|4)||All managers of large companies are actively opposed to the raising of corporate tax because all members of the Chamber of Commerce are managers of large companies and all active opponents of the corporate tax raise are members of the Chamber of Commerce.|
|5)||Some narcotics are medicines that can be bought without a prescription. No medicines that can be bought without a prescription cause dependence. Therefore, some narcotics do not cause dependence.|