**1. Enthymeme**

An **enthymeme**, in its modern sense, is an informally stated syllogism (a three-part deductive argument) with an unstated assumption that must be true for the premises to lead to the conclusion. In an enthymeme, part of the argument is missing because it is assumed. In a broader usage, the term “enthymeme” is sometimes used to describe an incomplete argument of forms other than the syllogism.

## Enthymeme’s three parts

The following quotation is an example of an enthymeme (used for humorous effect).

“There is no law against composing music when one has no ideas whatsoever. The music of Wagner, therefore, is perfectly legal.” —Mark Twain.

The three parts:

There is no law against composing music when one has no ideas whatsoever. (Premise)

The music of Wagner, therefore, is perfectly legal. (Conclusion)

Wagner has no ideas. (Implicit premise)

## Further examples:

Socrates is mortal because he’s human.

The complete syllogism would be the classic:

All humans are mortal. (Major premise – assumed)

Socrates is human. (Minor premise – stated)

Therefore, Socrates is mortal. (Conclusion – stated)

Hidden premises are often an effective way to obscure a questionable or fallacious premise in reasoning. Typically fallacies of presumption (fallacies based on mistaken assumptions, such as ad hominem or two wrongs make a right) are attracted to enthymeme.

**2. Sorites**

An argument in which a conclusion is inferred from any number of premises through a chain of syllogistic inferences.

**Example:**

All babies are illogical persons.

All illogical persons are despised person..

No persons who can imagine crocodiles are despised persons.

Therefore, No babies are persons who can imgine crocodiles.

**Solution:**

** **This Sorites consists of two syllogisms, as follows:

All I is D No M is D

All B is I All B is D

Therefore, All B is D Therefore, No B is M.

**3. The Disjunctive Syllogism**

This syllogism presents two alternatives in an “either . . . or” form; one of the alternatives is for formal reasons assumed to be necessarily true, so that to deny one leaves the other as the only possibility. The two possibilities, called disjuncts, are stated in the major premise; one is and must be *denied* in the minor premise; and the other is affirmed in the conclusion. This is the valid form, which can be shown as follows:

Either A or B

Not A; therefore B

(Deny first disjunct; affirm the second)

Either A or B

Not B; therefore A

(Deny second disjunct; affirm the first)

The opposite procedure of first affirming and then denying is, however, incorrect. Except where the members are *explicitly* contradictory so that both could not possibly be true, the affirmation of one disjunct (in the minor premise) does *not* deny the other. For example, to say, “Either the power is off or the bulb is burned out; the power is off so the bulb is not burned out,” would be a fallacy, because, while we assume that one of the disjuncts is definitely true, both might be true–we did not check the bulb and so cannot be sure of its condition. Since the second disjunct has not been investigated, it cannot be denied by default. (Where the members of the disjunct are contradictory, as in “The plant is either alive or dead,” the argument should, to avoid confusion, be changed into the conjunctive form of syllogism and worked from there–see below, section #3.)

The fallacy, then, of first affirming one disjunct and then denying the other looks like this:

Either A or B

And A; therefore not B

Either A or B

And B; therefore not A

*Fallacy of Affirming a Disjunct* (AD)

**4. Hypothetical Syllogisms**

Hypothetical syllogisms are different from standard syllogisms and thus have their own rules. In a hypothetical syllogism the first premise (or major proposition) presents an uncertain condition (“if A, then B”) or a problem (“either A or B”; “S and T cannot both be true”) which must then be properly resolved by the second premise so that a valid conclusion can follow. The resolution of the problem is always in the form of affirmation or denial. In this article, the three types of hypothetical syllogism we will cover are the conditional syllogism, the disjunctive syllogism, and the conjunctive syllogism.

If P is true then Q is true.

If Q is true then R is true.

Therefore, If P is true then R is true.

If P is true then Q is true.

P is true.

Therefore, Q is true.

**5. The Dilemma**

A **dilemma** means double proposition, it is a problem offering at least two solutions or possibilities, of which none are practically acceptable; one in this position has been traditionally described as being **impaled on the horns of a dilemma**, neither horn being comfortable.

The dilemma is sometimes used as a rhetorical device, in the form “you must accept either A, or B”; here A and B would be propositions each leading to some further conclusion. Applied in this way, it may be a fallacy, a false dichotomy.

Horned dilemmas can present more than two choices. The number of choices of Horned dilemmas can be used in their alternative names, such as two-pronged (two-horned) or dilemma proper , or three-pronged (three-horned) or trilemma, and so on.

Constructive dilemmas–

1. (If X, then Y) and (If W, then Z).

2. X or W.

3. Therefore, Y or Z.

Destructive dilemmas–

1. (If X, then Y) and (If W, then Z).

2. Not Y or not Z.

3. Therefore, not X or not W.