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

[Universal Statements] [Deduction] [Universal Syllogisms]

Introduction to Non-Universal Syllogisms

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.

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.)