Classical Syllogisms

Author: Timothy Eshing
Category: Logic and Reasoning, Historical Philosophy
Word Count: 999

Consider this argument:

All humans are mammals.
All mammals are animals.
Therefore, all humans are animals.

It’s an example of a classical syllogism. The logic of syllogisms, which are special kinds of deductive arguments, was famously discussed by the ancient Greek philosopher Aristotle about 2,400 years ago. His approach was developed for nearly two millennia.[1] Aristotelian syllogistic logic is an important precursor to modern-day quantificational (or “predicate”) logic.[2]

This essay discusses the classical Aristotelian approach to syllogisms. A companion [editor’s note: forthcoming] essay, Contemporary Syllogisms, discusses the modern approach.

The traditional "square of opposition."
The traditional “square of opposition.”[3]

1. Terms

Aristotle’s syllogistic logic is also known as “term logic”, because one can use it to reason with special parts of speech called terms.[4] Aristotle specified four kinds of terms:

  1. General terms are nouns (or phrases) that refer to whole classes or categories of things.[5] Examples include: philosophers, architects, houses, horses, race-winners, galaxies, etc.
  2. Singular terms are proper nouns (names) that refer to individuals who populate one or more categories. Examples include: Socrates (human, philosopher), Frank Lloyd Wright (human, architect), Fallingwater (house, thing-built-by-Wright), Seabiscuit (horse, race-winner), the Milky Way (galaxy), etc.
  3. Indefinite terms are ambiguous regarding whether they are general or singular. Aristotle’s example was “pleasure”.[6] Other examples include: hatred, water, gold, space, time, etc.
  4. Empty terms don’t refer to anything at all that exists. Aristotle’s example was “goat-stag”.[7] Other examples include: unicorns, talking donkeys, round squares, the present king of France, etc.

The classical approach to syllogisms only permits the use of general terms.[8]

2. Categorical Sentences

Only certain kinds of sentences, called categorical sentences, are used in classical syllogisms. A categorical sentence has a subject and a predicate, each of which is a general term, and it affirms or denies the predicate of the subject. That’s it.

We classify the different kinds of categorical sentences based on their quantity and quality:

  • Quantity
    • Universal: the subject refers to all members of a category
    • Particular: the subject refers to some members (at least one) of a category
  • Quality
    • Positive: the predicate is affirmed of the subject
    • Negative: the predicate is denied of the subject.[9]

Let “S” stand for any general subject term, and “P” stand for any general predicate term; now these are the four kinds of categorical sentences we can use in classical syllogisms:[10]

Positive Negative
Universal All S is P. No S is P.
Particular Some S is P. Some S is not P.

3. Moods and Figures

A classical syllogism consists of exactly three categorical sentences: two premises and one conclusion.

One premise contains the subject term of the conclusion, known as the “minor”, and the other contains the predicate term of the conclusion, known as the “major”.[11] Both premises share a third term, known as the “middle”, which isn’t in the conclusion.

We classify the different kinds of classical syllogism based on “moods” and “figures”.

The mood of a given categorical sentence is just which kind of sentence it is, given its quantity and quality. In the medieval period, scholars labeled each mood with a vowel:

  • A: universal positive
  • E: universal negative
  • I: particular positive
  • O: particular negative.[12]

The mood of a syllogism is just the order of the moods of its sentences. For example: AAA, EAO, IAI, etc.

The figure of a syllogism is just the placement of the middle term (“M”) in the premises:

  First Figure Second Figure Third Figure Fourth Figure[13]
First Premise MP PM MP PM
Second Premise SM SM MS MS
Conclusion SP SP SP SP

There are 64 possible syllogistic moods and 4 possible syllogistic figures, which means that there are 256 logically distinct syllogistic forms.

4. Classically Valid Syllogistic Forms

A syllogism is said to be “valid” if and only if, were its premises all true, then its conclusion would have to be true, too.

While there are 256 logically distinct syllogistic forms, only 24 of these forms are classically valid.[14] Grouped by figure, they are:[15]

Barbara. All M is P. All S is M. ∴[16] All S is P.

Celarent. No M is P. All S is M. ∴ No S is P.

Darii. All M is P. Some S is M. ∴ Some S is P.

Ferio. No M is P. Some S is M. ∴ Some S is not P.

Barbari.S All M is P. All S is M. ∴ Some S is P.

Celaront.S No M is P. All S is M. ∴ Some S is not P.

Baroco. All P is M. Some S is not M. ∴ Some S is not P.

Camestres. All P is M. No S is M. ∴ No S is P.

Cesare. No P is M. All S is M. ∴ No S is P.

Festino. No P is M. Some S is M. ∴ Some S is not P.

Camestros.S All P is M. No S is M. ∴ Some S is not P.

Cesaro.S No P is M. All S is M. ∴ Some S is not P.

Bocardo. Some M is not P. All M is S. ∴ Some S is not P.

Datisi. All M is P. Some M is S. ∴ Some S is P.

Disamis. Some M is P. All M is S. ∴ Some S is P.

Ferison. No M is P. Some M is S. ∴ Some S is not P.

Darapti.M All M is P. All M is S. ∴ Some S is P.

Felapton.M No M is P. All M is S. ∴ Some S is not P.

Camenes. All P are M. No M is S. ∴ No S is P.

Dimaris. Some P is M. All M is S. ∴ Some S is P.

Fresison. No P is M. Some M is S. ∴ Some S is not P.

Calemos.S All P is M. No M is S. ∴ Some S is not P.

Fesapo.M No P is M. All M is S. ∴ Some S is not P.

Bamalip.P All P is M. All M is S. ∴ Some S is P.[17]

The names of the forms were chosen to help students remember how they work.[18]

4. Beyond Classical Syllogisms

Aristotle’s approach to syllogisms was enormously influential, but nowadays we approach syllogisms a bit differently. For more information, see the companion essay on Contemporary Syllogisms.

Notes

[1] Aristotle originally defined a “syllogism” [Greek: συλλογισμός, syllogismos = inference] as “a logos [= account, discourse, speech, text] in which, certain things being posited, something other than what is posited follows of necessity from their being so” (Prior Analytics 1.2.24b8). 

Aristotle laid out his theory of non-modal (also known as “assertoric”) syllogistic in Prior Analytics 1.1–2&4–7. That is basically the theory that is discussed in this essay. He laid out his theory of modal (“apodeictic”) syllogistic in Prior Analytics 1.3&8–22. In contrast to his assertoric syllogistic, Aristotle’s modal syllogistic is widely regarded as problematic; scholars continue to debate how best to make sense of it.

For a broad discussion of Aristotelian logic, including his syllogistic, see Smith 2017. For an earlier influential treatment of Aristotelian (esp. non-modal) syllogistic, see Łukasiewicz 1957. For compelling alternative treatments, see Corcoran 1973 or Read 2017. For recent treatments of Aristotle’s modal syllogistic specifically, see Rini 2011 or Malink 2013; the latter is arguably the best reconstruction to date. For introductory treatments of classical and contemporary syllogistic, see respectively Rescher 1966 part II (chs. 7–11) and Gensler 2017 ch. 2. For a discussion of the relevance of Aristotelian logic to contemporary philosophy of logic, see Lear 1980.

In this essay, I treat “classical” and “Aristotelian” as synonymous. But there was an ancient debate between the followers of Aristotle, called the Peripatetics, and another philosophical school called the Stoics about the proper nature and function of syllogistic. Stoic logic differs in important ways from Aristotelian logic — whereas Aristotelian syllogistic is a logic of terms and thus a progenitor of modern quantificational or predicate logic, Stoic syllogistic is a logic of propositions and a progenitor of modern sentential or propositional logic. For more information on the difference between Aristotelian/Peripatetic and Stoic syllogistic, see Frede 1987 [1956].

Over the centuries, Aristotle’s approach to assertoric syllogistic was tidied and codified, especially by the medieval “schoolmen”. In this essay, such superficial improvements are taken for granted (including how we now express categorical sentences, discussed in §2 of this essay, and the names of the twenty-four classically valid syllogistic forms, discussed in §4 of this essay). To be clear: the presentation of classical syllogisms in this essay differs in notable ways from what you’ll find if you turn to Aristotle’s original texts (cf. note 15 below). For more information on medieval developments, see Lagerlund 2022.

For a brief history of logic, including discussions of classical and contemporary syllogistic, see Gensler 2017 ch. 16. For a slightly longer history of logic, including excellent sections on syllogistic, see Hintikka & Spade 2020. For a long and detailed history of logic, including thorough discussions of the same, see Kneale & Kneale 1962.

[2] For more information on quantificational logic (also known as predicate or first-order logic), see my Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic.

[3] The traditional square of opposition depicts the relationship between the four categorical sentences discussed in §§2&3 of this essay, from top left to bottom right: A E I O. It says that A and E are contraries (they can both be false, but they can’t both be true), that I and O are subcontraries (they can both be true, but they can’t both be false), that A and O as well as E and I are contradictories (one but not both must be true), and that I and O are subalternates to A and E respectively (the truth of latter entails the truth of the former by existential import; see notes 14 and esp. 17 below as well as §4 of the companion essay Contemporary Syllogisms for more information about that notion).

[4] The term “term” comes from the Latin terminus, meaning “boundary” or “limit”. The terms of a categorical sentence (discussed in §2 of this essay) are, as it were, its beginning and end.

[5] Aristotle lays out his theory of categories in a work now fittingly known as the Categories. According to Aristotle, there are ten mutually exclusive and jointly exhaustive categories: substance, quantity, quality, relation, place, time, attitude, condition, action, and affection. On the one hand, these may be thought of as ten distinct kinds of things that are said. On the other hand, they may be thought of as ten distinct ways for something to be. No matter how we ultimately understand this tenfold classification, Aristotle is clear that the terms so-classified are simple parts of speech, without composition or structure, and thus fit to be the subjects and predicates of categorical sentences. By contrast, categorical sentences are complex, possessing composition or structure, and thus fit to be true or false. For more information on categorical sentences, see §2 of this essay.

[6] Aristotle, Prior Analytics 1.1.24a17.

[7] Aristotle, De Interpretatione 1.16a16.

[8] However, the contemporary approach also permits singular and empty terms. For more information, see §1 of the companion essay Contemporary Syllogisms. Neither approach permits indefinite terms.

[9] Alternatively, we can use “affirmative” and “privative” (or “affirmation” and “denial”) in place of “positive” and “negative” respectively.

[10] This table reproduces the traditional square of opposition described in note 3 above.

[11] The premise sharing the conclusion’s subject term is known as the “minor premise” and the premise sharing the conclusion’s predicate term is known as the “major premise”. Traditionally, the major premise comes first, followed by the minor premise, and then the conclusion. Nowadays we often switch up the order of premises for ease of reading, but such reordering has no effect on a syllogism’s validity.

[12] See, e.g., Lagerlund 2022 §1 for more information.

[13] Though Aristotle was aware of the fourth figure syllogisms, he did not treat them separately, and it’s unclear why. Generations of scholars have pondered the issue and debated whether or not to include the fourth figure as a separate class. The fourth figure is equivalent to the first with the terms in the conclusion converted, i.e., place-swapped. For this reason, the first figure syllogisms came to be known as “direct” and the fourth figure syllogisms as “indirect”.

[14] More precisely, a classically valid syllogism must obey the following six rules (adapted from Rescher 1964 §10.2):

  1. It must (i) have exactly three terms, each of which occurs twice and in the same sense, or (ii) be equivalent to one that does.
  2. Its middle term must be distributed in at least one premise.
  3. If a term is distributed in the conclusion, then it must be distributed in the premises.
  4. It must not have two negative premises.
  5. If one of its premises is negative, then its conclusion must be negative.
  6. If both of its premises are universal, then its conclusion must not be particular, unless it would still be valid if a universal premise were replaced by the corresponding particular sentence.

Regarding rules 2 and 3: a term is said to be “distributed” if it occurs as part of a statement that applies to every member of the category to which the term refers; this notion is discussed in §2 of the companion essay Contemporary Syllogisms. Regarding rule 6: this is equivalent to the assumption of existential import; for more information about that notion, see §4 of the companion essay Contemporary Syllogisms, as well as note 17 below.

[15] What you’ll actually find in Aristotle’s texts is a little different, so it’s worth describing what the “true form” of an Aristotelian syllogism actually looks like. First, Aristotle didn’t express categorical sentences the way we do now. For example, in place of “All A is B” Aristotle put either “B is predicated of all A” or “B belongs to all A”. Second, Aristotle didn’t use three separate sentences, nor the entailment indicator “therefore” on conclusions. Instead, he expressed every syllogism as a single complex sentence known as a conditional (“If … then …”), where (i) the antecedent (“If …” clause) is a conjunction (“… and …”) of the two premises and (ii) the consequent (“then …” clause) is a necessary claim (“it must be the case that …”).

So, for example, in place of our:

  • All A is B. All B is C. Therefore, all A is C.

Aristotle would put something like:

  • If B is predicated of all A and C is predicated of all B, then C must be predicated of all A.

For more information on the “true form” of an Aristotelian syllogism, see esp. Łukasiewicz 1957 §1.

[16] Note: the “∴” symbol is read as “therefore”; it indicates which sentence is the conclusion. Cf. §1 of Thomas Metcalf’s Formal Logic: Symbolizing Arguments in Sentential Logic.

[17] Nine of these syllogistic forms — those marked with superscripts — are invalid on the contemporary approach, because they depend on a feature called existential import, which is discussed in §4 of the companion essay Contemporary Syllogisms. Basically, the validity of some of these syllogisms depends on one of the terms being non-empty. For those marked “S”, the minor term “S” must be non-empty. For those marked “M”, the middle term “M” must be non-empty. For those marked “P”, the major term “P” must be non-empty. (Note: the other fifteen syllogistic forms listed above are still valid on the contemporary approach, as are many other argument forms disallowed by the classical approach, namely those with singular or empty terms or more or fewer premises.)

[18] Aristotle regarded the first figure syllogisms as self-evident, or “perfect”. The other figures are said to be “imperfect”, because to make it plainly evident how their conclusions follow from their premises they require a reordering of premises or additional (“tacit”) premises that (i) are drawn from the first figure syllogisms, (ii) present a reductio per impossibile (an argument which demonstrates the truth of a claim by showing that its negation is impossible by deriving a contradiction or absurdity), or (iii) apply one of three presumably self-evident principles of conversion (discussed in Prior Analytics 1.2):

  • A-conversion: swap the terms of a universal positive and make it particular (i.e., “All A is B” becomes “Some B is A”)
  • E-conversion: swap the terms of a universal negative (i.e., “No A is B” becomes “No B is A”)
  • I-conversion: swap the terms of a particular positive (i.e., “Some A is B” becomes “Some B is A”).

Now here is how to interpret the medieval names of the classically valid syllogistic forms (adapted from Rini 2011 pp. 14–15, Read 2017 fn. 13, and Hintikka & Spade 2020):

  • vowels indicate the syllogism’s mood (e.g., the mood of Barbara is AAA, that is, all its categorical sentences are universal positives)
  • the first letters (“B”, “C”, “D”, or “F”) of the imperfect syllogisms indicate which perfect syllogism is needed to complete their proofs (e.g., the second figure Camestres requires the first figure Celerent)
  • “m” means swap the order of the premises (as in, e.g., Disamis)
  • “c” following a vowel means use a reductio per impossibile for that premise (only needed for Baroco and Bocardo)
  • “p” after an “a” in the middle means use A-conversion on that premise (as in, e.g., Darapti); a “p” at the end means use A-conversion on the conclusion of the perfect syllogism involved in the proof (as in, e.g., Bamalip) — this was known as accidental or partial conversion
  • “s” after an “e” or “i” in the middle means use E- or I-conversion on that premise (as in, e.g., Festino and Ferison); an “s” at the end means use E- or I-conversion on the conclusion of the perfect syllogism involved in the proof (as in, e.g., Camenes and Dimatis) — this was known as simple conversion
  • the other letters (“b” and “d” when not first, plus “l”, “n”, “t”, and “r”) are purely for ease of pronunciation, and don’t indicate anything special (which is why many alternative spellings abound, e.g., Bramantip).

In spite of all this detail, the medieval names lack any indicators for which figure a given form is in, so they don’t provide all the information needed to construct an actual syllogism; how they’re classed into the four figures must be memorized separately.

For more information on the technicalities discussed above, see Lear 1980 ch. 1, Łukasiewicz 1957 ch. 3, Malink 2013 part I, or Rini 2011 ch. 1. Smith 2017 §5.4 contains a helpful table that shows at a glance the deduction procedure, as described by Aristotle, for fourteen of the figures (excluded are the four forms in the first two figures that require simple conversion, plus all of the forms in the fourth figure).

Note: some authors (e.g., Rescher 1964 part II) just name the classically valid syllogistic forms by mood and figure as follows:

  • AAA-1, EAE-1, AII-1, EIO-1, AAI-1, EAO-1
  • AOO-2, AEE-2, EIO-2, AEO-2, EAO-2
  • OAO-3, AII-3, IAI-3, EIO-3, AAI-3, EAO-3
  • AEE-4, IAI-4, EIO-4, AEO-4, AAI-4.

References

Aristotle: Categories, De Interpretatione, and Prior Analytics.

Corcoran, John: “A Mathematical Model of Aristotle’s Syllogistic” in Archiv für Geschichte der Philosophie 55(2): 191–219 (1973).

Frede, Michael: “Stoic vs. Aristotelian Syllogistic” in Essays in Ancient Philosophy (University of Minnesota, 1987 [1956]).

Gensler, Harry: Introduction to Logic (3rd ed., Routledge, 2017).

Hintikka, Jaako and Paul Vincent Spade: “history of logic” in Encyclopedia Britannica (2020).

Kneale, William & Martha Kneale: The Development of Logic (Clarendon, 1962).

Lagerlund, Henrik: “Medieval Theories of the Syllogism” (Stanford Encyclopedia of Philosophy, 2022).

Lear, Jonathan: Aristotle and Logical Theory (Cambridge, 1980).

Łukasiewicz, Jan: Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic (2nd ed., Oxford, 1957).

Malink, Marko: Aristotle’s Modal Syllogistic (Harvard, 2013).

Read, Stephen: “Aristotle’s Theory of the Assertoric Syllogism” (2017).

Rescher, Nicholas: Introduction to Logic (St. Martins, 1964).

Rini, Adriane: Aristotle’s Modal Proofs: Prior Analytics A822 in Predicate Logic (Springer, 2011).

Smith, Robin: Aristotle’s Logic (Stanford Encyclopedia of Philosophy, 2017).

Related Entries

Contemporary Syllogisms [editor’s note: forthcoming] by Timothy Eshing

Critical Thinking: What is it to be a Critical Thinker? by Carolina Flores

Formal Logic: Symbolizing Arguments in Sentential Logic by Thomas Metcalf

Formal Logic: Symbolizing Arguments in Quantificational or Predicate Logic by Timothy Eshing

About the Author

Timothy Eshing is a philosophy graduate student at Saint Louis University.

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