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Syllogism

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Syllogisms are the cornerstone of the logical tradition, known as term logic or Aristotelian logic, that originated with Aristotle (see Prior Analytics, Book I, c. 1.) and survived broadly unchanged until the advent of modern predicate logic in the late nineteenth century. A syllogism (Greek sullogismos, meaning deduction) consists of three interrelated propositions, in which the last proposition is shown to be necessary if the first two propositions are true. Occasionally these are called categorical syllogisms because the propositions they contain involve relatively simple declarations about categories, objects, and the properties that apply to these categories and objects. Syllogisms are usually considered to be a form of inference, but the inferences drawn in syllogisms are generally viewed by nominalists as trivial and tautological. Instead, syllogisms are used to examine the validity of reasoning involved in a given statement by exposing questionable assumptions and incorrect category relations. This article provides an introduction to syllogisms (categorical syllogisms, term logic), giving some historical background, a description of the basic use and structure of syllogisms, an analysis of common errors in reasoning, and suggestions for further reading.

Non-categorical syllogisms deal with the truth values of proposition in relation to other propositions, rather than looking at their internal structure of propositions as is done here. Propositions are taken as single, non-decomposable elements in non-categorical syllogisms, each with a given truth value, without considering internal validity or derivation. See disjunctive and hypothetical syllogisms.

==Background==

Main article: Organon

Propositions and terms

Aristotle's logic is structured around propositions and terms. A proposition (Aristotle uses protasis, which is closer in meaning to the word premise) is a particular kind of sentence in which subject and predicate are combined to make an assertion that can be judged as true or false. Thus the statement "the sun rises" contains a subject (i.e., the sun) and a predicate that makes a claim about that subject (i.e., that it rises), a claim which may or may not be true. Propositional logic, which is not addressed here, is the branch of traditional logic that deals with the relationships between propositions that have known (or at least knowable) truth values.

The subjects and predicates noted above are terms. A term is the basic unit of a proposition, specifying a particular thing, group, category, action, or property. The meaning of the word term (both the original Greek word horos and the Latin terminus from which the modern word is derived) is "extreme" or "boundary", the implication being that a proposition is made of two terms which lie on the boundaries of the sentence, joined by an affirmation or denial of their connection. Propositions, in fact, are said to have the quality of being affirmative ("every sun rises") or negative ("no sun splutters"), depending on whether the terms are joined by an affirmation or a denial; and to have the quantity of being universal if the predicate applies to everything in the subject ("all suns rise") or of being particular if the predicate only applies to certain elements of the subject ("some suns are reddish"). Qualitity and quantity determine the mood of a proposition.

Terms are never in themselves true or false, and they do not need to relate to anything real, possible, or even imaginable.

There have been attempts to analyze the nature of terms in philosophical thinking. For early modern logicians like Arnauld (whose Port Royal Logic is the premier textbook of his period) a term is a psychological entity like an "idea" or "concept"; by contrast, Mill considers it merely a word. To date, however, no completely satisfactory understanding of the nature of terms has been arrived at. For instance, an assertion like "unicorns are invisible" is a perfectly acceptable proposition, but it does not appear to be an assertion about any actual thing. Nor are apparently non-problematical statement such as "all Greeks are men" entirely unambiguous. "All Greeks are men" does not, for instance, say that all ideas of Greeks are ideas of men, or that the word "Greeks" is the same as the word "men". A term in a proposition is an element of language without any necessary ontology, but it is seemingly more than mere sounds or letters. This is a problem about the meaning of language that is still not entirely resolved. (See the book by Prior below for an excellent discussion of the problem).

In modern philosophical understandings, a term is the assertion that results from the uttering of a sentence, and is regarded as something peculiarly mental or intentional. Writers before Frege-Russell, such as Bradley, sometimes spoke of the "judgment" as something distinct from a sentence, but this is not quite the same. As a further confusion the word "sentence" derives from the Latin, meaning an opinion or judgment, and so is equivalent to "proposition".

Decline of term logic

Term logic was the dominant form of logical thought throughout most of western history, until the advent of modern or predicate logic in the late nineteenth and early twentieth century. After that it fell into decline, due to the superiority predicate logic shows in mathematical reasoning. While the focus on categories and subjects in syllogisms reflects our natural way of thinking, the formal system precludes identity statements, multiple relations and multiple inferences, and any analysis of singular propositions, all of which are elementary in predicate logic. Term logic cannot, for example, make the inferential leap from "every car is a vehicle", to "every owner of a car is an owner of an vehicle", because it requires the identity statement "every owner of a car has a car".

Note, however, that the decline was a protracted affair. It is simply not true that there was a brief "Frege Russell" period 1890-1910 in which the old logic vanished overnight. The process took roughly 70 years. Even Quine's Methods of Logic devotes considerable space to the syllogistic, and Joyce's manual, whose final edition was in 1949, does not mention Frege or Russell at all.

The innovation of predicate logic led to an almost complete abandonment of the traditional system. It is customary, in fact, to revile or disparage term logic in standard textbook introductions. However, it is not entirely in disuse. Term logic was still part of the curriculum in many Catholic schools until the late part of the twentieth century, and is still taught in places even today. More recently, some philosophers have begun work on a revisionist programme to reinstate some of the fundamental ideas of term logic. Their main complaints about modern logic are:

  • that Predicate Logic is in a sense unnatural, in that its syntax does not follow the syntax of the sentences that figure in our everyday reasoning. It is, as Quine acknowledges, "Procrustean" employing an artificial language of function and argument, quantifier and bound variable.
  • that there are still embarrassing theoretical problems faced by Predicate Logic. Possibly the most serious are of empty names, and of identity statements.

Even orthodox and entirely mainstream philosophers such as Gareth Evans have voiced discontent:

"I come to semantic investigations with a preference for homophonic theories; theories which try to take serious account of the syntactic and semantic devices which actually exist in the language ...I would prefer [such] a theory ... over a theory which is only able to deal with [sentences of the form "all A's are B's"] by "discovering" hidden logical constants ... The objection would not be that such [Fregean] truth conditions are not correct, but that, in a sense which we would all dearly love to have more exactly explained, the syntactic shape of the sentence is treated as so much misleading surface structure." (Evans 1977)

Heeding the Paideia proposal from philosopher Mortimer J. Adler, advocates of the homeschooling movement in recent years have been trying to revive the traditional curriculum of the Trivium – grammar, logic, and rhetoric – and have argued that logic properly belongs in the language arts of a classical education, not in mathematics. The problem, as they see it, is the rampant nominalism in modern formal logic, which concerns itself with the manipulation of symbols and not with the whys and essences of things. Predicate logic is too difficult to be taught generally in school and is merely introduced in college. School children a hundred years ago were taught a usable form of formal logic, today – in the information age – they are taught nothing.

Structure of a Syllogism

Basic structure

Syllogisms consist of three things: a major premise, a minor premise, and a conclusion which follows logically from the first two. The major premise states a relationship between a category of objects and a general principle (or in some cases, a secondary, overarching category). The minor premise is category inclusion: it states that some specific object(s) belong in the category mentioned in the major premise. The conclusion, then, applies the general principle of the major premise to the specific object(s) of the minor premise. It follows from this (logically) that there can only be three terms in a syllogism: each term in the conclusion is taken from one of the premises, and of the four terms in the premises one is common to both. This leads to the following definitions (the letter designation shown is that commonly used in formal logic):

  • Major term (P): the term shared by the major premise and the conclusion
    • the property being assigned
  • Minor term (S): the term shared by the conclusion and the minor premise
    • the specific object in question
  • Middle term (M): the term shared by major premise and the minor premise
    • the category the object belongs to

The terms must have this cross-proposition relationship for the syllogism to be valid, but within each proposition (major premise, minor premise, and conclusion) the subject/predicate relationship may differ. Different forms are called moods and figures.

Mood

The mood of a syllogism is distinguished by the quality and quantity of the two premises; it signifies various forms of category inclusion and exclusion. There are two possible qualities (Affirmative and Negative) and two possible quantities (Universal and Particular), leading to the following four types:

A type
Affirmative and Universal
  • All horses have four legs
E type
Negative and Universal.
  • No horses have fangs
I type
Affirmative and Partial.
  • Some horses are brown
O type
Negative and Partial.
  • Some horses are not brown

The letters A, I, E, and O are taken - somewhat serendipitously - from the vowels of the Latin words Affirmo and Nego:


Asserit A, negat E, sed universaliter ambae
Asserit I, negat O, sed particulariter ambo
A asserts and E denies some universal proposition;
I asserts and O denies, but with particular precision.


Within syllogisms, only eight of the sixteen possible combinations of moods for major and minor premises are valid: AA, AI, AE, AO, IA, EA, EI, OA. Note that a major premise which is partial (type I or O) can only be followed by a minor premise of type A, and a major premise which is negative (type E or type O) can only be followed by a minor premise that is affirmative.

Figure

The figure of a syllogism is determined by the relative positions of the middle term within the major and minor premises. There are four permutations of the middle term M, as follows:

figure 1: M is subject in the major, predicate in the minor.
  • All men are mortal
  • Socrates is a man
figure 2: M is predicate in both premises.
  • Mortality is the lot of man
  • Socrates is a man
figure 3: M is subject in both premises.
  • All men are mortal
  • A man is Socrates
figure 4: M is predicate in the major, subject in the minor.
  • Mortality is the lot of man
  • A man is Socrates

Symbolically, this looks as follows (noting that the conclusion does not in this case vary, for simplicity).

Figure 1 Figure 2 Figure 3 Figure 4
Major Premise M-P P-M M-P P-M
Minor Premise S-M S-M M-S M-S
Conclusions S-P S-P S-P S-P

Aristotle thought that only the first or perfect figure displayed the process of reasoning with complete transparency, and never discussed the fourth figure (assumedly it is as ungainly in Greek as it is in English). The other figures he designated as imperfect syllogisms, because the structure of their reasoning could only be made evident by converting the mood of the individual premises to create a syllogism of the first figure.

Conversion

Conversion is the process of changing one proposition into another by re-arranging the terms, without changing the meaning of the proposition. Thus:

Some S is a P converts to Some P is an S
  • Some dogs are pets Some pets are dogs
No S are P converts to no P are S
  • No cows are pets No pets are cows

But caution must be exercised. For example, the proper conversion of "some dogs are not pets" is not "some pets are not dogs", but rather "some (things that are) not pets are dogs"

Some S is not P converts to Some not P is S

Conversion per accidens involves changing a proposition into another which is implied, but does not carry exactly the same meaning. Thus:

All S are P converts to Some S are P

Notice that for conversion per accidens to be valid, there is an existential assumption involved in "all S are P"; if no S exist, there is no distinction between All S and Some S.

Reduction

As explained, Aristotle thought that only in the first or perfect figure was the process of reasoning completely transparent. The validity of an imperfect syllogism is only evident, when by conversion of its premises, it can be turned into some mood of the first figure. This was called reduction by the scholastic philosophers. Thus, given a syllogism of the third figure as follows:

All Greeks are athletes
Greeks include all Athenians

Reduction involves converting the minor premise to read:

All Greeks are athletes
All Athenians are Greeks

which is in the first figure.

The rules of reduction, historically, were captured in mnemonic lines, first introduced by William of Shyreswood (1190 - 1249), and usually memorized as verse. This system is mostly archaic, though the terms barbara and celarent still arise in logic discussions (as they represent the most conventional affirmative and negative first figure syllogisms). Two common versions are:

  • Barbara, Celarent, Darii, Ferioque, prioris
  • Cesare, Camestres, Festino, Baroco, secundae
  • Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet
  • Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison

and

  • Barbara, Celarent, Darii, Ferio, Baralipton
  • Celantes, Dabitis, Fapesmo, Frisesomorum
  • Cesare, Campestres, Festino, Baroco, Darapti
  • Felapton, Disamis, Datisi, Bocardo, Ferison

Each word represents the formula of a valid mood and is interpreted according to the following rules:

  • The first three vowels indicate the quantity and quality of the three propositions, thus Barbara: AAA, Celarent, EAE and so on.
  • The initial consonant of each formula after the first four indicates that the mood is to be reduced to that mood among the first four which has the same initial
  • "s" immediately after a vowel signifies that the corresponding proposition is to be converted simply during reduction,
  • "p" in the same position indicates that the proposition is to be converted partially or per accidens,
  • "m" between the first two vowels of a formula signifies that the premises are to be transposed,
  • "c" appearing after one of the first two vowels signifies that the premise is to be replaced by the negative of the conclusion for reduction per impossibile (a syllogism that produces a conclusion known to be impossible, in order to show - because the syllogism is itself valid - that one of the premises must be impossible).

Validity of Syllogisms

Basics of validity

As noted previously, syllogisms consist of three interrelated statements: a major premise, which assigns a property to a category; a minor premise, which establishes category inclusion for some specific object(s); and a conclusion, which transfers the property of the major premise to the specific object(s) of the minor premise. The classic example of a 'barbara' (see reduction) mode first figure syllogism is:

Humans are mortal (major premise)
Socrates is human (minor premise)
Therefore, Socrates is mortal (conclusion)

or drawn out in detail:

Everything in the category 'Human Being' has the property 'Mortality'
The individual object named 'Socrates' belongs in the category 'Human Beings'
Therefore, the individual object named 'Socrates' has the property 'Mortality'

Syllogisms concerns themselves only with the formal structure of the statements, generally referred to as their validity. Content is irrelevant, as is the truth of the conclusion or either of the premises. See category mistake. In fact, the validity of a syllogism - though it may feel strange - is completely unrelated to the truth of its conclusion. For instance (using another 'barbara' mode syllogism):

All humans are goatherders
Socrates is human
Therefore, Socrates is a goatherder

is a perfectly valid syllogism that is not true. Or:

All giant squids are great philosophers
Socrates is a giant squid
Therefore, Socrates is a great philosopher

is a valid syllogism whose conclusion is true, even though its premises are absurd. The value of syllogisms does not lie in their ability to produce true statements per se, but in their ability to draw out and expose errors of reasoning. As you can see, without seeing the previous syllogism one would never know that the above author's reason for respecting Socrates is that he or she believes that Socrates is a giant squid.

A syllogism's ability to explore errors of reasoning, however, relies on its structural validity. Not every trio of propositions creates a valid syllogism, and often - particularly when variant moods and figures are involved - the validity of a syllogism becomes ambiguous and difficult to ascertain. e.g.:

Some giant squids are not great philosophers
Some Greeks are giant squids
Therefore, some Greeks are not great philosophers?

There are, however, some simple and common errors (the technical term is logical fallacies) in which an invalid construction is taken for a valid construction: the fallacy of four terms, the fallacy of the undistributed middle, an illicit processes of the major or minor term, and the fallacy of negative premises.

Common Errors

First, a valid syllogism in the barbara mode (for later comparison)

All giant squids are great philosophers
Socrates is a giant squid
Therefore, Socrates is a great philosopher
Category A has property Z
Object B belongs to A
Therefore, object B has property Z

Four terms fallacy

In this fallacy a fourth term is surreptitiously introduced into the conclusion, usually by association or an indirect relation. This is common in casual discourse and thought, lying at the root of certain of prejudices and stereotypes, as well as certain kinds of jokes and humor.

All monsters are scary
The creature Dr. Frankenstein created was a monster
Therefore, Dr. Frankenstein is scary
Category A has property Z
Object B belongs to A
Therefore, object C (often indirectly related to B) is (improperly) given property Z

This is not a fallacy if conversion per accidens applies—i.e., if the fourth term is a proper subset of the minor term.

Undistributed middle

This confuses the major term and the middle term. It appears to be a second figure syllogism where the middle term is "designer clothes", but in fact it is a first figure syllogism where the appropriate middle term is "beautiful people"; the minor premise and conclusion have been reversed. While this is an error in logic, it is closely related to analogy and metaphor, which have their virtues, and is heavily exploited in modern advertising.

All beautiful people wear designer clothes
Sally wears designer clothes
Therefore, Sally is beautiful
Category A has property Z

minor premise and conclusion reversed

Object B has property Z
Therefore, object B is (improperly) included in category A

This is not a fallacy if the speaker actually intends to say that wearing designer clothes makes one a beautiful person, which would make this a valid second figure syllogism. The ambiguity of intention in the first line (when compared with the second) creates the problem.

Illicit processes

This fallacy contains a double error. First, the minor premise has actually become a second major premise (in that it assigns a new property to the category). Second, the conclusion assigns the property of the major premise to this second property (as though it were, instead, a category equivallent to the major term).

Technically, the first example is called an illicit process of the major term and the second an illicit process of the minor term, though the structure of the error is the same.

All Greeks are intelligent
All Greeks are athletes
Therefore, all athletes are intelligent

- or -

All Greeks are athletes
No Swedes are Greek (read that as All Greeks are not Swedes)
Therefore, no Swedes are athletes
Category A has property Z
Category A has property Y
Therefore, property Y (improperly taken as a category) is (improperly) given property Z

This first is not a fallacy if the minor premise is an identity statement, and the second is not a fallacy if the major premise is an identity statement, because identity statements allow the conversion from "All Greeks are athletes" to "All athletes are Greeks." As noted, however, syllogisms are not capable of dealing with identity statements effectively.

Negative premises

Double-negative reasoning: is saying "it doesn't not have this property" the same as saying "it has this property"? Here the given object is taken to have the property in question because it does not belong to a category that does not have the property. Examples of this fallacy abound in political discourse.

Politicians cannot be trusted
Our president is not a politician (e.g., he's a statesman)
Therefore, our president can be trusted
Category A does not have property Z

and now for the spin-doctoring...

Object B does not belong to A
Therefore, object B is (improperly) given property Z

Negative Premises is not a fallacy in binary systems: this is part of the reason it is such a common fallacy. If everything except category A has property Z - i.e., if politicians are the only people in the world who cannot be trusted - then, yes, everyone who is not a politician must be trustworthy. Syllogistically:

Only politicians cannot be trusted
  • - converts correctly to "all people who are not politicians can be trusted
Our president is not a politician
Therefore, our president can be trusted

A Note on Time in Syllogisms

Classical syllogisms are not designed with time in mind. Each statement in a syllogism is expected to be a timeless universal. Thus:

All humans are mortal

is taken to be true of all humans, everywhere and always. Time-bound events can be included in syllogisms, however, if they are first set up as a timeless universals. For instance, the following syllogism is technically incorrect:

All humans are mortal
Socrates is a human
Therefore, Socrates will die

because the phrase 'will die' implies a future state that hasn't been accounted for. The error can be corrected through a separate syllogism:

Everything mortal will die
All humans are mortal
Therefore, all humans will die

which establishes the property 'will die' as a timeless universal of 'being mortal'.

Note how the minor premise of the second syllogism is the major premise of the first syllogism. This is not a typical use of syllogisms - as noted, syllogisms are not an effective tool for dealing with multiple inferences. This example is merely intended to show that syllogisms are designed for exploring categorical relationships, not for working with time and causal relations.


End Material

See also

References to paper texts

  • I. M. Bocheński, I. M., 1951. Ancient Formal Logic. North-Holland, Amsterdam.
  • Louis Couturat, 1961. La Logique de Leibniz. Georg Olms Verlagsbuchhandlung, Hildesheim.
  • Gareth Evans, 1977. 'Pronouns, Quantifiers and Relative Clauses'. Canadian Journal of Philosophy.
  • Peter Geach, 1976. Reason and Argument. University of California Press.
  • Hammond and Scullard, 1992. The Oxford Classical Dictionary. Oxford University Press, ISBN 0198691173.
  • Joyce, G.H., 1949. [http://uk.geocities.com/frege@btinternet.com/joyce/principlesoflogic.htm Principles of Logic. London, 3rd edition. A manual written for Catholic schools, probably in the early 1910s. It is spendidly out of date, there being no hint even of the existence of modern logic, yet it is completely authoritative within its own subject area. There are also many useful references to medieval and ancient sources.
  • Jan Lukasiewicz, 1951. Aristotle's Syllogistic, from the Standpoint of Modern Formal Logic. Clarendon Press, Oxford.
  • John Stuart Mill, 1904. A System of Logic. London, 8th edition.
  • Parry and Hacker, 1991. Aristotelian Logic. State University of New York Press, Albany.
  • Terence Parsons, 1999. 'Traditional Square of Opposition'. Article at the Stanford Encyclopedia of Philosophy.
  • Arthur Prior, 1976. The Doctrine of Propositions & Terms. London.
  • Lynn E. Rose, 1968. Aristotle's Syllogistic. Clarence C. Thomas, Springfield.
  • Robin Smith, 2004. 'Aristotle's Logic'. Article at the Stanford Encyclopedia of Philosophy.

Abbreviatio Montana article by Prof. R. J. Kilcullen of Macquarie University on the medieval classification of syllogisms. The Figures of the Syllogism is a brief table listing the forms of the syllogism. Stanford Encyclopedia of Philosophy entry on Medieval Theories of Syllogisms.

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