Syllogism (logic)

A logical argument that allows to deduct a new proposition by combining two propositions as premises.

Originally, the term denoted (almost) any form of deductive inference, but today it is used exclusively for forms that are based on exactly two premises with exactly three terms.

For example:

All Athenians are Greeks.
Sokrates is an Athenian.
Therefore: Sokrates is a Greek.

The first premise is called the major (Lat.: praemissa major), the second the minor (Lat.: praemissa minor). Both together are referred to as the “premises”. Accordingly, these two propositions can also be called the “first” or “second” premise.

The new proposition is called the “conclusion” (from Lat.: conclusio). We can also speak about it as the “result”.

The conclusion contains two terms, one from the major and one from the minor premise (in the example: “Greeks” and “Socrates”). These are named accordingly the “major” and the “minor” terms.

Finally, there is a third term that connects the two premises (in the example: “Athenians”). This is called the middle term.

A syllogism cannot be true or false, but only valid or invalid. A valid form preserves the truth value of the premises in its conclusion, i.e. if the premises are true, the final sentence is also true; if the premises are false, so is the conclusion.

In the case of invalid syllogisms, the truth value of the conclusion is indeterminate, as is the case with mixed premises (i.e. a true and a false premise).

In syllogisms, the following four types of basic statements (so-called “categorical propositions”) can be used, which have different properties:

These statements can be combined in the following ways (in this table: F = major term [first], S = minor term [second] and M = middle term).

The combination of four types of statements with four figures gives a total of 256 (theoretically) possible syllogisms. Of these, however, the vast majority (exactly 232) are invalid, therefore 24 valid forms remain (see below for an overview).

In classical logic, artificial names were invented that make it easier to remember the valid syllogism forms. These are formed according to a number of rules, which are reproduced here in a simplified manner:

• The first letter indicates from which “syllogism family” they come; E.g. all syllogisms that begin with B are related to Modus Barbara and can be transformed back to this form.
• Each name contains exactly three vowels, which reflect the forms of major, minor and conclusion (in exactly this order) through the vowels A, E, I or O. See the table above for typing.
• Certain consonants (“c”, “s” and “m”) provide information on how a conversion to the basic form is to be carried out.

On this basis, the following 24 forms are obtained, which can be grouped according to their four basic forms:

• Modus Barbara, with variants: Baroco, Bocardo, Barbari and Bamalip.
• Modus Celarent, with variants: Calemes, Calemos, Camestres, Camestros, Celaront, Cesare and Cesaro.
• Modus Darii, with variants: Dimatis, Datisi, Disamis and Darapti.
• Modus Ferio, with variants: Festino, Ferison, Felapton, Fesapo and Fresison.

Every (valid) syllogism consists of exactly two premises and one conclusion, all of the above types. These contain exactly three terms in total. If there are more different terms (typically four), it forms what is called a “four-term fallacy”.

This usually occurs when one of the terms is used in two different meanings (equivocation) or referring to different extensions of the term. This is most often seen in relation to the middle term, which, when it refers to two different meanings, leads to the fallacy of the ambiguous middle, a variant of the aforementioned “four-term fallacy”.

The term “distributivity” describes whether an expression refers to the whole or to a part of the extension set. Obviously, one cannot draw a conclusion that refers (directly or indirectly) to the total from a statement that refers to a subset. More on this in fallacies of distribution.

The middle term must appear in at least one premise in a distributed position (e.g. in an 'A' form as the subject). A syllogism in which this does not happen is called a “fallacy of the undistributed middle”.

Similarly, the principle applies that a term that is distributed in the conclusion must also be in a distributed position in its premise (either the major or minor). Traditionally, a distinction is made here between undistributed majors and minors, but for simplicity these are both described here under “illicit process”.

An important (and often only implicit) precondition for the validity of some syllogisms is that the terms occurring in them do not describe empty sets, i.e. that the extensions of the terms are not empty, but that they refer to actually existing objects. If this is not the case, the empty extension set fallacy may occur.

• Syllogism on Wikipedia
• Categorical proposition on Wikipedia