Types of Syllogisms:

The categorical syllogism
A categorical syllogism, as the name implies, involves an entire category. It is an unqualified proposition, and is characterized by words like all, every, or any. It may be directly expressed or clearly implied. There are some inherent problems with categoricals. The most obvious is that for existent objects, it is quite difficult to make such statements with certainty unless one has personally observed every existent member of the category, or has deductive proof that makes the conclusion certain. In making a statement about "all people" or "every grain of sand on the planet" we must make some sweeping generalizations unless we are talking about self-evident qualities.

The following tests should be applied to categoricals to determine their validity.
1. The categorical must have three terms. No more, no less. These are known as the major term, minor term, and middle term. It will follow this form:
Major Premise: All A's are B's
Minor Premise: C is an A
Conclusion: Therefore, C is a B.

In this form, A is the middle term. B is the Major term, and C is the minor term. (I know it's kind of a quirky order. Just memorize it.)

2. Every term must be used exactly twice.
See the example above.

3. Each term is only used once in any premise.

4. The middle term must be distributed in at least one premise. This means that it must be in at least one premise in an unqualified and/or universal sense.
MP: Some dogs (A) are poodles. (B)
mP: Sparky (C) is a dog (A)
C: Therefore, Sparky (C) is a poodle (B)
This example demonstrates an incorrect usage of the middle term. Note the word "some" violates the rule of the unqualified middle.

5. A term may be distributed in the conclusion ONLY if it has been distributed in the major or minor premise.
MP: All atheists (A) disbelieve in Allah (B).
mP: Bob (C) is not an atheist.(A)
C: Therefore, Bob (C) believes in Allah (B).
While this is intuitively incorrect, it takes some work to discover the error. Though (A) looks like a universal, it is not. There are people besides atheists who disbelieve in Allah. Therefore, the statement should read, "All atheists are among those (A) who disbelieve in Allah (B). Now, we can see that this is not actually distributed, and so may not be distributed in the conclusion.

6. At least one of the premises must be affirmative. In other words, two negatives cannot produce a positive.
MP: No Christians (A) believe in Allah (B).
mP: Bob (C) is not a Christian (A)
C: Therefore, Bob (C) ??????
As you can see, there is no way to form a conclusion about Bob's belief because nothing positive has been said.

7. If one premise is negative, the conclusion must be negative.
MP: No Christians (A) believe in Allah (B).
mP: Bob (C) is a Christian (A).
C: Therefore, Bob (C) does not believe in Allah (B).

The Disjunctive Syllogism
A disjunctive syllogism is one which offers us a choice between alternatives. These are marked by words like "either, or, neither, but, although." The disjunctive may be expressly stated or clearly implied.
MP: Either Congress will agree to continue funding the war or the president will veto the new budget.
mP: Congress will not agree to continue funding the war.
C: Therefore, the president will veto the new budget.

Tests for Disjunctive Syllogisms:
1. The major premise must include all possible alternatives.
MP: Either the universe exists as a steady state or it was created.
mP: Steady state has been refuted.
C: Therefore, the universe was created.
A good critical thinker will immediately recognize that there are other alternatives to explain the existence of the universe and dismiss this syllogism as invalid.

2. The alternatives must be mutually exclusive.
In a debate over immigration, it might be suggested that a wall on the Mexico-U.S. border is the alternative to illegal immigrants unlawfully collecting social security benefits. A good critical thinker would recognize that a wall would not eliminate all illegal immigration, nor would it prevent unlawful collecting of social security benefits. These are not mutually exclusive options.

3. The minor premise must affirm or contradict one of the alternatives.
MP: Congress must either raise taxes or lower spending.
mP: Congress will not lower their own salaries.
C: Therefore, Congress ???????
As you know, Congressional salaries are but a small portion of government spending. The knowledge that they will not lower their own salaries tells absolutely nothing about what they will do with the rest of the budget. Therefore, we have no meaningful information.

The Conditional Syllogism
This is also sometimes called a hypothetical syllogism. It contains a major premise dealing with an uncertain or hypothetical event or events. Words like "if, assuming," and "supposing" are expressly stated or clearly implied. Contained within the syllogism will be an antecedent which expresses the conditional, and a consequent, which necessarily follows the antecedent. This is simply expressed as an If-Then statement, with "If" being the antecedent and "then" being the consequent.

Tests for Conditional Syllogisms:
1. The minor premise must affirm the antecedent or deny the consequent.
MP: If the prime rate goes down, then more houses will be purchased.
mP: The prime rate will go down.
C: Therefore, more houses will be purchased.
Note that the minor premise affirms the antecedent and the conclusion affirms the consequent. The following example will demonstrate the minor premise denying the consequent.
MP: If the U.S. destroys all of its nuclear weapons, other nations will not attempt to acquire nuclear weapons
mP: Other nations will attempt to acquire nuclear weapons.
C: Therefore, the U.S. will not destroy all of its nuclear weapons.

2. If the minor premise denies the antecedent or affirms the consequent, no valid conclusion can be drawn.
MP: If the rate of abortion increases, the population will decrease.
mP: The rate of abortion will not increase.
C: Therefore, ???????
On the surface, it appears that we may safely conclude that the population will increase, but this is not a valid conclusion. Nothing has been stated about whether the rate of abortion will remain constant, or whether it will decrease. Furthermore, nothing has been stated about the effect of either of these conditions, so there is nothing that can be concluded.

THE ENTHYMEME
An enthymeme is a syllogism in which one of the premises is not stated. Despite the fact that very few people have ever heard of this word, it is a description of the way we generally speak. For example, I might give the following argument:
mP: The bill up for consideration would lead to restrictions on free speech.
C: Therefore, this bill is a bad idea.
Enthymemes are trickier to examine because it is possible to either intentionally or unintentionally obscure the unspoken premise. In this case, if the major premise is "All bills that restrict free speech are bad ideas" then the discovered syllogism would be valid. However, the real major premise might be "Almost all bills that restrict free speech are bad ideas." Indeed, most people agree that some restrictions, such as preventing people from yelling "Fire!" in crowded theaters, are good ideas. In this case, we can see that the middle term is not distributed and therefore, the argument is invalid.

A second form of enthymeme is one that deals with probability rather than certainty. This kind of syllogism may or may not omit one of the premises. It is extremely important to note that this kind of enthymeme carries ABSOLUTELY NO FORMAL VALIDITY. In other words, it is not useful in establishing certainty in a deductive argument. This is not to say that it is not useful in a practical sense. Indeed, we use this kind of thinking daily.
MP: All food additives that promote cancer should be removed from food.
mP: Saccharin might promote cancer.
C: Therefore, Saccharin should be removed from food.
We can intuitively understand that this argument proves nothing, and that the conclusion is not necessarily true. If this were the only argument presented for removing saccharin from food, we would properly dismiss the argument. The person trying to prove this would need to demonstrate exactly how much certainty there is in the belief that saccharin can cause cancer. Is it probable, possible, or plausible? Are there empirical studies showing a possible correlation? How strong is the correlation? These are questions that we could ask to determine the relative strength of this argument, but it must be remembered that this argument can NEVER, under any circumstances, be used to definitively prove anything!

One tactic that is often used to try to fool an audience in a debate is chaining enthymemes. A speaker may state only the conclusion of an enthymeme, and then use that conclusion as a premise for a second enthymeme, omit the second portion, and state the next conclusion, then continue the chain in a similar manner until a desired conclusion is reached. It is crucially important that a good critical thinker learns to spot this potentially deceptive practice, and insist that the speaker fully articulate each and every premise of his argument. Debates can often be won simply by cornering the speaker on this tactic, for it is commonly used when the speaker knows that he cannot reveal each premise for fear of destroying his own argument. Put another way, if a speaker refuses to articulate every premise, it is highly probable that his argument is invalid, and in general, the conclusion should be regarded with extreme distrust.

FORMAL VALIDITY and MATERIAL TRUTH
In each of the examples I have given thus far, it has been assumed that each premise of a syllogism is absolutely true. With enthymemes, the assumption is that the premises are probably true. Within syllogisms, if the premises are absolutely true, then the conclusion is absolutely certain. Within enthymemes, if the probability of the premises are absolutely true, then the probability of the conclusion is equally true. If, however, any one of the premises is false, then its conclusion is necessarily invalid, and completely worthless with regard to formal validity.
MP: All Caucasians are bad at dancing.
mP: John is a Caucasian.
C: Therefore, John is bad at dancing.
There are four important things to notice in this syllogism. First, the conclusion is obviously invalid because there are clearly Caucasians who dance well. Second, if we could only think of one Caucasian who was a good dancer, the syllogism would be equally invalid! Please note that in any categorical syllogism, even one counter example is sufficient to render the entire argument invalid. Because of the existence of one Caucasian who can dance, we have absolutely no basis on which to come to a conclusion about whether or not John can dance. Third, note that if we were presented with evidence of the probability that Caucasians are bad at dancing, we could form an enthymeme that would reflect the probability of John's dancing prowess, but without such evidence, we can say absolutely nothing about John's dancing ability.

Lastly, if you reduce this syllogism to letters and check it against all the tests, you will see that it is a valid syllogism. It is crucially important to good critical thinking skills that we recognize that the validity of an argument is only relevant if the facts within the premises are verifiably correct.

Conversely, we should recognize that the material truth of a conclusion does not have conclusive bearing on the material truth of the premises. Furthermore, the truth of both the premises and the conclusion is irrelevant to the validity of the syllogism.
MP: All fish can breathe underwater.
mP: The lungfish is a fish.
C: Therefore, the lungfish can breathe outside of the water.
Note that all three of the statements are true, but that the conclusion cannot be drawn from within the premises. This is an example of an invalid syllogism with completely true premises and conclusion. Recall the essay on tests of reasoning and evidence, and you will see that this example violates quite a few.