Introduction to Syllogisms

This is the place where critical thinking teachers and textbooks normally introduce the idea of a syllogism by presenting two paradigms of a universal syllogism that exactly parallel the two valid forms of a conditional. We are not going to follow the pattern, however, for three good reasons:

  1. because you can deal already with all forms of a valid universal syllogism, simply by restating them as conditionals;
  2. because the rule-based approach to all syllogisms that will be presented in the next section covers universal and non-universal syllogisms, so there is no reason to study two different approaches.
  3. and because students are often confused when asked to learn and apply two different solutions to the same problem.
However, should you desire to learn the paradigm system for analyzing syllogism, please go to the section on universal syllogisms

Before we begin to explain the relationship between a conditional and a syllogism, we should first define the term. A syllogism is any argument that has two premises and a conclusion. If you think that a conditional fits that definition--well, you are right! In fact, to be precise, what we have been calling "conditionals" might more accurately be termed "conditional syllogisms," and what we are going to call "syllogisms" might more accurately be termed "categorical syllogisms." But we will use the terms "conditionals" and "syllogisms" because they are shorter and widely accepted.

Even though all conditionals can be expressed as a syllogism, and all syllogisms can be expressed as a conditional, there are important differences between the two models. One difference is that a conditional involves only two terms (P and Q), while a syllogism involves three. But that turns out to be a minor difference, because the third term is actually hidden within the conditional form. Take, for example, the argument:

If a player is injured, then you may activate another one.
Armida is injured.

Therefore, you may activate another player.
In this example, "player is injured" is P, "activate another" is Q--but what about "Armida"? She would be the third terms in a syllogism expressing the same argument:
All injured players may be replaced.
Armida is injured.

Therefore, Armida may be replaced.
Not all third terms in a conditional are as obvious as this, but with a little practice you will realize that all syllogisms can be expressed as conditionals, and vice versa.

The drawback for conditionals is their difficulty in dealing with qualified terms. The claim "All dogs are retrievers" can be expressed "If P then Q," where P is "dogs" and Q is "retrievers." But what of the claim "Some dogs are retrievers"? We could still use "If P then Q," where P is "some dogs," but if the antecedent is to be confirmed, the P of the second premise must also be "some dogs"--we are unable to deal with "no dogs" or "all dogs," because they simply would not be "P." But syllogisms, by putting the terms into categorical instead of conditional relationships, avoid this problem. Thus, the following argument can be expressed as a syllogism:

Some dogs are retrievers.

No retriever is white.


Therefore, some dogs are not white.
--or--

Some X are Y.

No Y is Z.


Therefore, some X are not Z.
The point here is not whether this syllogism is valid--though, as you will learn in the next section, it is--but simply that a syllogism can deal with the qualified terms that the argument presents here, and a conditional cannot.

There is no reason to ignore conditionals completely, however. Having learned the two valid forms, you will find that analyzing some arguments in a conditional form is both efficient and accurate--enough reason to practice restating claims as conditionals. Using the three rules to check the validity of a syllogism, which you will learn to do in the next section, may seem complicated by comparison. While you can use the rule-based approach in every case, most people use a combined analysis of argumentation, employing both conditionals and syllogisms


The Three Rules

Before beginning, you may want to review some of these sections:

[Universal Statements] [Deduction]


In other sections, the uses of paradigms to check the validity of deductive conditionals and syllogisms are covered. These are common techniques, but they suffer from two major shortcomings:

  1. They can only be applied to universal claims. If the second premise of a conditional or either premise of syllogism includes non-universal terms (for example, "Some P" or "Some X are Y"), the paradigms cannot be applied, and the validity of the argument is unknown.
  2. They are dependent on specific sequences, of terms within a claim and of premises within an argument, and sometimes require the extra step of re-ordering the claims. Consider this argument: "All kayaks are boats, and all boats are watercraft, so all kayaks are watercraft." Though this is obviously a valid argument, it doesn't seem to fit the paradigm, since it has the form "All X are Y, all Y are Z, therefore all X are Z." To make it valid, we must either re-order the two premises, or treat this as a chain argument with an unstated premise; either way, we have made a simple argument much more complicated.

These problems can be avoided by applying three simple rules to any syllogism. The first rule concerns negatives, like "no" and "not."

1. If the conclusion is a negative claim, one (but not both) of the premises must be a negative claim; if the conclusion is not a negative claim, neither of the premises can be negative.
The next two rules concern what is usually called a distributed term, but which is exactly what we have been calling a universal term elsewhere: that is, a term which is qualified by the equivalent of "all" or "none."
2. If a term is distributed in the conclusion, it must be distributed in one of the premises.
3. The "middle term," the term repeated in each of the premises, must be distributed in at least one of the premises.

The table below identifies which terms are distributed, or universal, in the four possible claims, and which of the claims are negative.

All A are B.

No A is B.

Some A are B.

Some A are not B.

A, but not B.

Both A and B.

Neither A nor B.

B, but not A.

Positive

Negative

Positive

Negative

In general, only terms with "all," "no," or "not" in front of them are distributed. The exception is the claim "No A is B," where both A and B are distributed. One way to understand this is to remember that "No A is B" is equivalent to "No B is A"; since both terms can have the "no" in front of them, they both must be distributed. (See Statements and Conversions.)