Form of argument that, in its most commonly discussed instances, has two categorical propositions as premises and one categorical proposition as conclusion. An example of a syllogism is the following argument: Every human is mortal (every M is P); every philosopher is human (every S is M); therefore, every philosopher is mortal (every S is P). Such arguments have exactly three terms (human, philosopher, mortal). Here, the argument is composed of three categorical (as opposed to hypothetical) propositions, it is therefore a categorical syllogism. In a categorical syllogism, the term that occurs in both premises but not in the conclusion (human) is the middle term; the predicate term in the conclusion is called the major term, the subject the minor term. The pattern in which the terms S, M, and P (minor, middle, major) are arranged is called the figure of the syllogism. In this example, the syllogism is in the first figure, since the major term appears as predicate in the first premise and the minor term as subject of the second.
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Each of the three distinct terms represents a category, in this example, "human," "mortal," and "Socrates." "Mortal" is the major term; "Socrates," the minor term. The premises also have one term in common with each other, which is known as the middle term — in this example, "human." Here the major premise is universal and the minor particular, but this need not be so. For example:
Here, the major term is "die", the minor term is "men," and the middle term is "[being] mortal things." Both of the premises are universal.
A sorites is a form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion. For example, if one argues that a given number of grains of sand does not make a heap and that an additional grain does not either, then to conclude that no additional amount of sand will make a heap is to construct a sorites argument.
The premises and conclusion of a syllogism can be any of four types, which are labelled by letters as follows.
|A||All||S||are||P||universal affirmatives||All humans are mortal.|
|E||No||S||are||P||universal negatives||No humans are perfect.|
|I||Some||S||are||P||particular affirmatives||Some humans are healthy.|
|O||Some||S||are not||P||particular negatives||Some humans are not clever.|
(See Square of opposition for a discussion of the logical relationships between these types of propositions.)
By definition, S is the subject of the conclusion, P is the predicate of the conclusion, M is the middle term, the major premise links M with P and the minor premise links M with S. However, the middle term can be either the subject or the predicate of each premise that it appears in. This gives rise to another classification of syllogisms known as the figure. The four figures are:
|Figure 1||Figure 2||Figure 3||Figure 4|
Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, although this makes no difference logically). Each premise and the conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the number for the figure. For example, the syllogisms above are AAA-1.
Of course, the vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from the premises). The table below shows the valid forms of syllogism. Even some of these are sometimes considered to commit the existential fallacy, thus invalid. These controversial patterns are marked in italics.
|Figure 1||Figure 2||Figure 3||Figure 4|
A sample syllogism of each type follows.
Forms can be converted to other forms, following certain rules, and all forms can be converted into one of the first-figure forms.
Syllogism dominated Western philosophical thought until The Age of Enlightenment in the 17th Century. At that time, Sir Francis Bacon rejected the idea of syllogism and deductive reasoning by asserting that it was fallible and illogical. Bacon offered a more inductive approach to logic in which experiments were conducted and axioms were drawn from the observations discovered in them.
In the 19th Century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Kant famously claimed that logic was the one completed science, and that Aristotelian logic more or less included everything about logic there was to know. Though there were alternative systems of logic such as Avicennian logic or Indian logic elsewhere, Kant's opinion stood unchallenged in the West until Frege invented first-order logic.
Still, it was cumbersome and very limited in its ability to reveal the logical structure of complex sentences. For example, it was unable to express the claim that the real line is a dense order. In the late 19th century, Charles Peirce's discovery of second-order logic revolutionized the field and the Aristotelian system has since been left to introductory material and historical study.
For instance, given the following parameters: some A are B, some B are C, people tend to come to a definitive conclusion that therefore some A are C. However, this does not follow. For instance, while some cats (A) are black (B), and some black things (B) are televisions (C), it does not follow from the parameters that some cats (A) are televisions (C). This is because first, the mood of the syllogism invoked is illicit (III), and second, the supposition of the middle term is variable between that of the middle term in the major premise, and that of the middle term in the minor premise (not all "some" cats are by necessity of logic the same "some black things").
Determining the validity of a syllogism involves determining the distribution of each term in each statement, meaning whether all members of that term are accounted for.
In simple syllogistic patterns, the fallacies of invalid patterns are: