Syllogism

A syllogism is a kind of logical argument that arrives at a conclusion based on two "premises" that are asserted to be true. A syllogism can be either valid or invalid, depending on whether it follows the rules of syllogistic logic. A valid syllogism "preserves" the truth of its premises. In other words, if a syllogism is valid and the premises are true, the conclusion will also be true. However, if either the syllogism is invalid or either of the premises are untrue (i.e., not sound), the truth of the conclusion is not guaranteed.

Syllogisms are the basic tools of "term logic," also known as "traditional logic," "classical logic," or "Aristotelian logic." Term logic was created by Aristotle and was widely used until the creation of predicate logic in the late 19th century. Although it has fallen out of favor, it is still useful for understanding basic formal fallacies, and for understanding texts written before the rise of predicate logic. Another advantage is that it mostly uses regular syntax and language, allowing it to be understood more readily by the layperson, while predicate logic uses a more formal artificial language.

Basic structure
The basic building blocks of classical logic are "terms," "propositions," and "syllogisms."

Terms
Terms are words and phrases that mean something but are not necessarily true or false. Examples of terms include "men", "mortals", and "Socrates". These words represent an object or idea but cannot be said to be true or false in and of themselves.

Propositions
Propositions combine two terms (a subject and predicate) to create an assertion that is either true or false. Specifically, the proposition asserts whether or not members of the predicate group are members of the subject group. In the example "All men are mortals," all members of the group "men" are asserted to be members of the group "mortals."

Propositions can be affirmative or negative. This is easy enough to understand. The proposition "All men are mortals." is affirmative, while the proposition "No dogs are cats." is negative. Propositions can also be universal or particular. A universal proposition deals with all members of the subject. A particular proposition deals with only some members of the subject. The proposition "All men are mortals." is universal, while the proposition "Some men are liars." is particular.

We also need to understand whether each term within a proposition is distributed or undistributed (this becomes important when we analyze the validity of syllogisms). A distributed term is a term that claims to know something about all the things referred to by that term; it is universal. An undistributed term is a term that claims to know something about only some of the things referred to by that term; it is particular. We already looked at whether the subjects of propositions were universal or particular. It is easy to determine the distribution of these terms because they have the words "all" and "some" in front of them. This tells us that the universal subjects ("all") are distributed and that the particular subjects ("some") are undistributed. But the predicates of propositions also have a distribution, which is indicated by the copula of the proposition, i.e. the affirmative or negative quality of the proposition that determines the distribution of the predicate. Thus the predicate of an affirmative proposition is always undistributed and the predicate of a negative proposition is always distributed. This is a universal rule.

While it is important to understand why terms are distributed and undistributed, it is usually easier to simply memorize the distribution of terms in the four types of propositions. The distribution is always the same in each type of proposition.

Syllogisms


Syllogisms are simple deductive argument that consists of three propositions.
 * The major premise is the proposition that contains the predicate of the conclusion (the major term) in addition to the middle term.
 * The minor premise is the proposition that contains the subject of the conclusion (the minor term) in addition to the middle term that it shares with the major premise.
 * The conclusion contains the major term (the predicate) and minor term (the subject). While major and minor premises can occur in any order, the conclusion appears at the end of the syllogism.

The most simple kind of syllogism consists of three "A" propositions (an AAA syllogism), taking the following form (with the symbol "∴" representing the word "therefore"):


 * All humans are mortals, (major premise)
 * All Socrates are human, (minor premise)
 * ∴ All Socrates are mortal. (conclusion)

The validity of this argument can be affirmed by thinking in terms of Euler Diagrams. If the circle "Socrates" is inside the circle "humans", which is also inside the circle "mortals", then it obviously follows that the circle "Socrates" is also inside the circle "mortals".

Modes
Modes are the letter types of the three propositions in the syllogism. If a syllogism has three A statements, it's mode is AAA. In the Middle Ages, people memorised a poem to help remember all the modes, but it isn't necessary to. It goes like this:
 * Barbara, Celarent, Darii, Ferio ← direct first figure
 * Baralipton, Celantes, Dabitis, Fapesmo, Frisesomorum ← indirect first figure
 * Cesare, Camestres, Festino, Baroco ← second figure
 * Darapti, Felapton, Disamis, Datisi, Bocardo, Ferison ← third figure

Every vowel in the word tells you what the mode of the syllogism is. Words in italics are valid but weren't written into the poem at its time of conception because medieval logicians considered them weaker than the ones in the poem. For instance, AAI is valid, but making an A conclusion over an I conclusion (AAA vs AAI) is stronger; therefore, AAI was left out of the original poem.

Figure
Figure refers to where the middle term in a syllogism is placed. Figure one is the middle term is in the S position of the first premise and in the P position of the second premise. Figure two means the middle term is in the P position of both premises, figure three means the middle term is in the S position of the premises, and figure four means the middle term is in the P position of the first premise and the S position of the second premise. Figure and mode work together to make sure a syllogism has a valid structure. There are technically 256 types of syllogisms, but only the ones in the mode chart are valid syllogisms.

Rules of validity
Hundreds of different kinds of syllogisms can be arranged, but the vast majority of these are invalid. To be considered valid, a syllogism must follow seven basic rules.


 * 1) A syllogism must contain exactly three terms. The violation of this rule is called the Four-term Fallacy.
 * 2) A syllogism must have exactly two propositions and one conclusion.
 * 3) The middle term must be distributed at least one time. Violating this rule results in the fallacy of the undistributed middle. (When checking for this and the next rule, it is useful to mark the distribution of every term in the syllogism.)
 * 4) No term that is undistributed in the premise may be distributed in the conclusion. The violation of this rule is either the fallacy of the illicit major or the fallacy of the illicit minor depending on whether the minor or major term contains the fallacy (abbreviated to IP — see illicit process).
 * 5) A syllogism cannot have two negative premises. Doing this results in the fallacy of exclusive premises, which is abbreviated to EP. Any syllogism which contains the premises EE, EO, OE, or OO is invalid by default because of this fallacy.
 * 6) If a syllogism contains a negative premise, the conclusion must be negative; conversely, if it contains a negative conclusion, it must contain a negative premise (see — affirmative conclusion from a negative premise and negative conclusion from affirmative premises).
 * 7) 'If a syllogism contains a particular premise, the conclusion must be particular The violation of this rule is the only fallacy an IAA, IIA or OIE syllogism commits.

As an example, we can check the following syllogism for validity: (the distributed and undistributed terms are marked for convenience)


 * 1) The syllogism passes the first rule, as it contains exactly three terms: "war," "justified," and "genocide".
 * 2) The syllogism passes the second rule, as it consists of exactly two premises and a conclusion.
 * 3) The syllogism violates the third rule, as the middle term ("war," recognizable because it is not in the conclusion but is in both the major and minor premise) is not distributed at least once.
 * 4) The syllogism passes the fourth rule, as there is no distributed term in the conclusion that is undistributed in the premise.
 * 5) The syllogism passes the fifth rule, as it does not have two negative premises.
 * 6) The syllogism passes the sixth rule, as it does not even contain any negative premises.
 * 7) The syllogism passes the seventh rule, as it does not even contain any universal premises.

Because the syllogism violated one of the rules, it is invalid.

However, a syllogism can be valid but unsound because one or both of the premises are false in reality. In the "Destiny of the Daleks" episode of Doctor Who the following syllogism is presented:

This argument is unsound because the first premise, "All elephants are pink", is false, but the argument is perfectly valid. It's important not to confuse validity with soundness. Note that this argument also isn't in standard logical form.

Invalid syllogisms or syllogistic fallacies are logical fallacies in which categorical syllogisms are used incorrectly.

Polysyllogisms
A polysyllogism (or complex syllogism) is a longer argument composed of several categorical syllogisms or enthymemes. Most arguments used in rhetoric or conversation can be analyzed as polysyllogisms. Consider the following argument:
 * Drugs are addictive things.
 * Addictive things are things that should not be used.
 * Drugs are things that should not be used.
 * Things that should not be used are things that should be illegal.
 * Therefore, drugs are things that should be illegal.

This argument is actually composed of two categorical syllogisms, with the conclusion of the first making up the first premise of the second:
 * All addictive things are things that should not be used.
 * All drugs are addictive things.
 * ∴ All drugs are things that should not be used.


 * All things that should not be used are things that should be illegal.
 * All drugs are things that should not be used.
 * ∴ All drugs are things that should be illegal.

By breaking a polysyllogism into its composite parts, one can easily test the validity of each different syllogism in these complex arguments.

Hypothetical syllogism
A hypothetical syllogism takes the form:
 * If P is true then Q is also true.
 * If Q is true then R is also true.
 * Therefore if P is true then R is also true.

We can use shorthand, with arrows representing the logical relationship:
 * P → Q
 * Q → R
 * ∴ P → R

The hypothetical therefore takes essentially the same form as the AAA syllogism we analyzed with Venn Diagrams earlier: If A=B, and B=C, then A=C. The difference is that a hypothetical syllogism qualifies the statements with a conditional. It does not necessarily state that the premises are true, only that there is a logical relationship if they are true. Replacing the notation with actual terms gives us the following example:


 * If I get paid today, I will need to buy groceries.
 * If I need to buy groceries, I will need to go to the store.
 * Therefore, if I get paid today, I will need to go to the store.

Disjunctive syllogism
The disjunctive syllogism (such as modus ponens or modus tollens) is similar to the hypothetical syllogism, but uses a "disjunctive" premise (an "either-or"). The disjunctive syllogism takes the form:


 * Either P or Q.
 * Not P.
 * Therefore Q.

Using shorthand (with "∨" representing the "either-or" relationship and "¬" negating the truth of a premise) it becomes:


 * P ∨ Q
 * ¬P
 * ∴ Q

It is important to note that the second premise should negate a premise, and not affirm it. Because of the ambiguous nature of the disjunctive relationship, in some logical problems both premises can be considered true. Thus, affirming one of the premises does not necessarily exclude the other premise; however, since the disjunctive relationship does imply at least one of the premises must be true, negating one premise will guarantee the truth of the other premise. Using ordinary language, the disjunctive syllogism would look like this:


 * Either I will wear a coat or a sweater today.
 * I will not wear a coat today.
 * Therefore, I will wear a sweater today.