Syllogisms

SYLOGISMS

"One of the most beautiful discoveries of the human spirit" (Leibniz).

Proposiciones
Types of propositions

Aristotle distinguished between universal proposition and particular proposition, each of which in turn can be, in turn, affirmative or negative. This gives rise to four possible forms of propositions, which he identified by a vowel:

Vowel	Type	Form	Example
A	Universal affirmative	All A is B	Every man is mortal
E	Universal negative	No Ais B	No man is mortal
I	Particular affirmative	Some Ais B	Some man is mortal
O	Particular negative	Some A is not B	Some man is not mortal

The universal and the particular refer to quantity. Affirmation and negation refer to quality.

Relations between propositions

In the above table there are different relationships between the four propositions:

Contradictorias (A-O; E-I). One is the negation of the other. If one is true, the other is false, and vice versa.
Contrarias (A-E). They cannot both be true, but they can both be false.
Subcontrarias (I-O). They cannot both be false, but they can both be true.
Subalternas (A-I; E-O). If the universal is true, so is the particular; but not vice versa. And if the particular is false, so is the universal, but not vice versa.

Sylogisms
Structure of a syllogism

An Aristotelian syllogism has the following structure:

It consists of three propositions: two premises (major premise and minor premise) and a conclusion.
There are three terms: S (subject of the conclusion), P (predicate of the conclusion) and M (middle term).
Each proposition consists of two terms.
The two premises have a term in common M (middle term).
The conclusion consists of two terms, which are the terms of the premises that are not the common term (S and P).

The four figures

The figures are the forms of the syllogism, according to the position that the middle term occupies in the premises. There are four figures:

1st Figure	2nd Figure	3rd Figure	4th Figure
M en P S en M S en P	P is M P is M S is P	M is P M is S S is P	P is M M is S S is P

In the first figure the middle term acts as subject in the first premise and predicate in the second, etc.

Syllogistic modes

The syllogistic modes are the possible configurations that can be formed with these figures, depending on whether the propositions are A, E, I or O. In this way, each syllogism can be identified by 3 vowels, corresponding to the two premises and the conclusion. There are VR(4,3) = 43 = 64 possibilities: variations with repetition of 4 elements (the 4 types of propositions) taken 3 by 3 (the 3 propositions of the syllogism). Moreover, as there are 4 figures obtained by varying the position of the middle term, the total number of possible combinations or modes is 64 × 4 = 256.

The rules for the validity of syllogisms are:

At least one premise must be affirmative.
If a premise is negative, the conclusion must be negative.
If a premise is particular, the conclusion will also be particular.
The middle term must be universal at least once.
If a term is universal in the conclusion, it must also be universal in its corresponding premise.

Of the 256 possible modes only 25 meet these criteria, and of these 6 are useless. Therefore, there are only 19 valid modes.

Mnemonic of the valid modes

Medieval logicians named valid modes by mnemonic rules. Each valid mode receives a name whose 3 vowels indicate the type of the propositions used in the major premise, minor premise and conclusion, respectively. The valid modes are:

Figure 1	Figure 2	Figure 3	Figure 4
Barbara Celarent Darii Ferio	Cesare Camestres Festino Baroco	Darapti Felapton Datisi Feriso Disamis Bocardo	Bamalip Camenes Dimatis Fesapo Fresison

For example, the first figure:

Barbara	Celarent	Darii	Ferio
All M is P All S is M All S is P	No M is P All S is M No S is P	All M is P Some S is M Some S is P	No M is P Some S is M Some S is not P

Similarly for all other figures.

Perfect and imperfect modes. Reduction

The consonants of the names of the valid modes of syllogism also have meaning. It is related to the theory of reduction of imperfect modes to perfect modes. Aristotle called the modes of the first figure "perfect", understanding that their order was "more natural" than in the others, which made the passage to the conclusion more intuitive.

Kant wrote in 1762 the article "The False Subtlety of the Four Syllogistic Figures" in which he claimed that syllogistics should be restricted to only the first figure, which he considered the only perfect one, since all the others could be reduced to it.

Any imperfect mode can be reduced to a perfect one with an equivalent conclusion.The consonants of the mnemonic names are the key to the operations of reduction.

The initial of the imperfective mood indicates that it can be reduced to the mood of the figure with the same initial (e.g., a Fesapo can be reduced to a Ferio).
The presence of the letter "m" means that the order of the premises in the imperfective mode must be changed.
The letter "s" indicates that the proposition denoted by the vowel preceding it must be simply converted (we call simple conversion the permutation of the two terms of the proposition without change of quantity or quality (valid for categorical assertions E, I): "no Buddhist is a Catholic" => "no Catholic is a Buddhist"; "some intellectual is a politician" => "some politician is an intellectual").
The letter "p" means the accidental conversion in analogous conditions (the accidental conversion allows to pass, permuting terms, from any universal to the particular of the same quality, but not vice versa. Thus from "every fly is an insect" is inferred "some insect is a fly", but not vice versa; and the same with respect to E).
The letter "c" after one of the first two vowels will indicate that the premise in question is to be replaced by its negation in order to facilitate the reduction per impossibile of the mode (Baroco, Bocardo).

Here is an example of reduction from Disamis to Darii taken from Manuel Garrido's book [2007]:

Di	Some snakes are venomous animals	Da	All snakes are reptiles
sa	All snakes are reptiles	ri	Some venomous animals are snakes.
mis	Some reptiles are venomous animals	i	Some venomous animals are reptiles.

Specification in MENTAL
Aristotelian propositions

In the Aristotelian propositions appears the quantifying triad "All - Some - None" which has its correspondence in the triad of metaexpressions of MENTAL: Ω - α - θ.

Vowel	Form	MENTAL
A	All A is B	`⟨( x/A → x/B )⟩`
E	No A en B	`( {⟨( < b>x ← x/A ← x/B )⟩}# = 0 )`
I	Some A is B	`( {⟨( x ← x/A ← x/B )⟩}# > 0 )`
O	Some A is not B	`( {⟨( < b>x ← x/A ←' x/B )⟩}# > 0 )`

Here we can appreciate that the proposition A is the simplest.

These propositions can be specified in parameterized form as follows:

⟨( pA(A B) = ⟨( x/A → x/B )⟩ )⟩

⟨( pE(A B) = ( {⟨( x ← x/A ← x/B )⟩

⟨( pI(A B) = ( {⟨( x ← x/A ← x/B )⟩

⟨( pO(A B) = ( {⟨( x ← x/A ←' x/B )⟩}# > 0 )⟩

Another way to specify propositional forms is by means of sets.

Naming

⟨( {Z} = {⟨ x ← x/Z ⟩} )⟩

to the set of elements having property Z, and

⟨( {Z'} ={⟨ x ←' x/Z ⟩} )⟩

to the set of elements that do not have the property Z then the MENTAL coding is greatly simplified:

Vocal	Forma	MENTAL
A	All A is B	`( {A}∩{B} = {A} )`
E	No A is B	`( {A}∩{B} = {} )`
I	Some A is B	`( {A}∩{B} ≠ {} )`
O	Some A is not B	`( {A}∩{B'} ≠ {} )`

"Some" is expressed as nonzero intersection of sets.

"All" can be expressed as an intersection of sets that produces one of the sets, or it can be specified as an inclusion relation of one set in another. For example, "All A is B" can be specified as A∩B.

Relationships between sets are first order and syllogistic propositions (relating individual elements) are second order relationships.

Relations between propositions

Since "Any A is B" implies "Some A is B", we can express this property as follows:

⟨( pA(A B) → pI(A B) )⟩

Sylogisms

A valid syllogistic mode, e.g., "Barbara" (from the first figure) can be expressed as follows:


( Barbara(S M P) =: ⟨( pA(M P) → pA(S M M) → pA(S P) )⟩ )

Analogously for the rest of the valid syllogistic modes. All these expressions are logical properties, i.e., theorems and, therefore, of universal validity.

Another way of specifying syllogisms is by means of sets. For example, the syllogism,

Sylogisme	MENTAL
Some A are B All B are C Some A are C	`( {A}∩{B}) ≠ {} ) ( {B}∩{C}) = {B} ) ( {A}∩{C}) ≠ {} )`

Properties of quantifiers

An Aristotelian quantifier Q can be considered a relation between two arguments. For example, "All men are mortal" could be represented as "All(men, mortal)". In this case, "All" is a dyadic (two-argument) quantifier: "men" and "mortals". In general, if Q is a quantifier, and A and B are arguments, we can express it as Q(A, B).

Dyadic quantifiers can be symmetric or non-symmetric. The symmetric ones are those that satisfy the property: Q(A, B) → Q(B, A). In non-symmetrics this implication does not hold. For example:

The quantifier "Some" is symmetrical: E.g.: "Some men smoke" → "Some smokers are men".
The quantifier "None" is also symmetrical. For example, "No man smokes" → "No smoker is a man."
The quantifier "All" is asymmetrical. For example, "All men are living beings" does not imply "All living beings are men."

Another way to look at quantifiers is to consider them as functions. For example, the expression "All(men, mortals)" can be considered as a function of two arguments (men and mortals) and whose result is a truth value: V (true) or F (false). In this example, the result is V.

Advantages of MENTAL notation

It is descriptive and operational.
Propositions and syllogisms can be parameterized, resulting in more compact expressions.
In propositions, one can go beyond simple quantification (all-any-some) to specify a numerical expression: a particular number, a range, a number greater than a given number, etc.
Allows to automatically obtain conclusions.

Adenda
Aristotle not only invented logic, but he made three major findings:

The syllogism, a small formal axiomatic system consisting of two premises and a conclusion.
The propositional variables.
Quantification. Aristotelian syllogisms can be considered as the formal study of 4 basic quantifiers, their properties and relations: All, Some and their opposites: None and Not-All.

The fourth figure, strictly speaking, is not Aristotelian, since it was added by Theophrastus, a disciple of the Stagirite and director of the Lyceum after his death. The doubt is whether Theophrastus limited himself to systematize and order the work of his master, thus leaving the fourth figure as Aristotelian (posthumously), or on the contrary the appearance of the fourth figure is original to Theophrastus.

The mnemonic rules of the valid syllogistic modes were mainly contributed by Pedro Hispano (1205-1277) with his Summulae Logicales, which were the guideline for manuals of logic throughout the Middle Ages until the end of the Enlightenment. He introduced the nomenclature that became canonical, based on the vowels A, E, I and O taken from "AdfIrmo" and "nEgO".

Jan Lukasiewicz, in "Elementy logiki matematycznej" (1929), reworked the theory of syllogism with the techniques of the new logic. He emphasizes the discovery that all syllogistic modes are reduced to two: Barbara and Datisi.

Bibliografía

Bochénski, Joseph M. Historia de la Lógica formal. Gredos, 1985.
Capelle, Wilhelm. Historia de la Filosofía griega. Gredos, 2011.
Ferrater Mora, José; Leblanc, Hugues. Lógica Matemática. Fondo de Cultura Económica, 1962.
Garrido, Manuel. Lógica simbólica. Tecnos, 2007.
Kneale, William y Martha. El desarrollo de la lógica. Tecnos, 1980.
Vega, Luis. Una guía de historia de la lógica. UNED, 1997.