In logical aristotelician, the syllogism is a logical reasoning with two proposal S (also called premises ) leading to a conclusion that Aristote was the first to be formalized. For example, All the men are mortals, but the Greeks are men, therefore the Greeks are mortals is a syllogism; both Prémisse S (known as “major” and “minor”) are proposals given and presumedly true making it possible to check the formal veracity of the conclusion. The science of the syllogisms is the syllogistic , in which, inter alia, the thinkers were interested of the medieval Scolastique, Gottfried Leibniz and Emmanuel Kant.

Etymology

Of the Greek συν (syn) , which means " avec" or " plus" , and of the Greek (Logos) , which can mean:
  • Word
  • Speech
  • Study
  • Reasoning
logos is used here in the direction of " raisonnement". Let us note that logos is a concept, more than one simple mot.
Syllogism thus means literally " with raisonnement".

Introduction

The syllogism makes it possible to put in report/ratio in a conclusion two terms , the major one and the minor, by means of medium term. The major one and the minor should appear only once each one in the premises, medium term is present in each premise (since it allows the setting in report/ratio of the two other terms) while the conclusion presents the relationship between the major one and the minor, so that the syllogism is a “report/ratio of reports/ratios” (expression of Renouvier, Traité ):

Paradox 1

The application without understanding of the rule which precedes can lead to some strangenesses:

This conclusion can disturb and will encourage to handle the syllogism with precautions (see Paradoxe). Here, it leads to an aberration. The paradox of the emmental (and not Gruyere) is another example. See also Apagogie/Reasoning by the absurdity.

It is acted here in fact of a sophism being appeared as syllogism. It is not a valid syllogism owing to the fact that the second premise contains in fact two proposals. “This cheap horse is rare” means in fact: “this horse is cheap and it is rare”.

It should be noted that this type of paradox, unlike that of Emmental, is mainly due to the proposals themselves. For example, " All the rare things are chères" is false, since a counterexample is found. This not being very complicated, the proposal is not true, and does not have to thus be used in the establishment of the syllogism because preventing Boolean transitivity necessary to its checking. (logically, if C = (HAS AND B), C is false if HAS OR B is false)

In short, there is contradiction of direction. The major one is the anthology of the minor one. Something expensive cannot be at the same time cheap. The syllogism being used much in philosophy, one says in this field that something cannot equalize its opposite. However this erroneous syllogism tends to prove that yes. One could give like example symbolically mathematical:

A value of direction is identified, and similarities there are found

means (T) erme = rare Things, (M) ajeure = low Cost, (- M) ineure = high Cost

  • M

  • - M = T thus
  • If T = T Then

  • - M

What this example states is that the major one and the minor one must have different directions, and nonopposite. This example is not official but shows well the resemblance between a syllogism and a report/ratio of equations. That would thus state that our paradox Ci-high is truly invalid and that a cheap horse is not expensive.

Paradox 2 also explains the problems of induction of the proposals.

Paradox 2

How can I affirm that all the Greeks are mortals? Strictly speaking , I can be about it certain only if I saw well dying all the Greeks, including Socrate. It will thus be conceived that in practice a deductive syllogism is seldom applicable without a more or less retracted share of induction. One formerly could believe that a syllogism explained something on the real-world at one time when one believed in the gasolines , i.e. where it was thought that the word defined the thing, and not the reverse (see Induction (logical) , Réalisme vs. Nominalisme ).

Case of premises without relationship with reality

For seeking to include/understand the operation of the syllogisms, it is necessary to take guard at a point of most important: to say of a syllogism which it is concluding is to affirm that its form is valid. Its material Truth, however, does not import. Thus, the syllogism
Toutes the creatures with Dent S are kleptomaniac,
But the hens have Dent S,
Donc the hens are kleptomaniac
is formally valid. It has, on the other hand, no value of material truth.

Proposals

Subject and predicate of the proposals

The syllogisms consist of proposals, or made assertions of a prone (indicated by S ) connected by a copula to a Prédicat (indicated by P ), of type
S {prone} is {copula} P {predicate}.

These proposals must be built in a precise order: the subject of the conclusion, indeed, must be present in one of the premises (normally the minor one), its predicate in the other (most of the time the major one), so that the syllogism is valid. Medium term (M) draws up the report/ratio: * {M is P } gold { S is M} thus {S is P}.
Note: the order in which the premises appear does not import. The use is to quote in first that which contains the major one, i.e. the predicate of the conclusion.

It is thus excluded that medium term appears in the conclusion or that one of the premises connects the two latest dates (terms minor and major).

Relationship between the subject and the predicate

In fact, the copula is introduced a relationship between the two concepts S and P. This report/ratio must be apprehended under the angle of the comprehension (indicates in logic the whole of qualities and the characteristics suitable for a unit, or class , objects) and extension (the whole of the objects which have these joint qualities and properties). S is P must thus be included/understood at the same time like:
  • comprehension: “the unit S has the attribute of P”;
  • extension: “the unit S belongs to the unit P”.

Thus, all the men are mortals includes themselves/understands doubly:

  • comprehension: “the whole of the men has the characteristics of the whole of the mortals”;
  • extension: “the whole of the men belongs to the whole of the mortals”.

One thus sees, in addition to the distribution of the terms within the premises, one second constraint to take shape: a proposal must be made up of proposals in which the predicate is a superset of the subject. A syllogism can thus be summarized as follows:

⊂ P)   ∧ (S ⊂ M) ⊃ (S ⊂ P) .

However, a Truth table makes it possible to check that this expression is a Tautologie (with the logical direction):

This truth table must be read as follows: “the conjunction of “M is implied by P” and of “S is implied per M” implies well that S is implied by P”. Indeed, the implication (carrying number 4 in the table) is true whatever the values of M, P and S.

Classes of proposals

The table considering higher makes it possible to include/understand why, for little that it is correctly built, a syllogism is valid formally. It however makes it possible to consider only the syllogisms of which all the proposals would be affirmative and universal. They are not the only possibilities.

There exist indeed four classes of proposals, distinguished by their quality and their quantity:

  • quality: affirmative or negative proposals;
  • quantity: universal proposals (the subject relates to all the extension) or particular (part of the extension).

These four classes are indicated traditionally by letters mnemotechnics as used by the medieval Scolastique:

  • has = affirmative universal: “all the men are mortals”;
  • E = universal negation: “no man is immortal”;
  • I = particular assertion: “some men are painters”;
  • O = particular negation: “some men are not painters”.
Pour to retain these letters: has ff' i' rmo (Latin “I affirm”), e' g' O (“I deny”).

Two proposals having same the subject and predicate can be opposed by their quality and/or their quantity. Thus the oppositions which can be created are as follows:

  • Two proposals contradictoires' are proposals which are opposed by the quality and the quantity  
  • Two proposals contrary are universal proposals which are opposed by the quality  
  • Two proposals subcontraires are particular proposals which are opposed by the quality  
  • Two proposals subordinates are proposals which are opposed by the quantity.

One establishes the logical Carré thus of the opposition of the proposals.

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However, a syllogism must consider the class of its proposals and the order in which they appear to remain valid: the diagram ⊂ P) ∧ (S ⊂ M) ⊃ (S ⊂ P) is not enough, would not be this because one has sometimes to make with exclusions of units, and not of only inclusions.

Modes

The position of medium term: concept of figure

It was said, the order in whom appear the premises is not relevant. What is it, on the other hand, it is the distribution of the subject and the predicate of the conclusion within the premises, indicated by that of medium term.

The canonical form of a syllogism is ⊂ P)   ∧ (S ⊂ M) ⊃ (S ⊂ P). In this case, medium term is prone the major one and predicate of the minor one. That draws what one names first figure the , in which the major term is predicate of the major premise and the prone minor term of the minor premise. Three other figures are however possible:

  • 1st figure: ⊂ P)   ∧ (S ⊂ ''' M ''');
  • 2nd figure: ⊂ ''' M ''')   ∧ (S ⊂ ''' M ''');
  • 3rd figure: ⊂ P)   ∧ (''' M ''' ⊂ S);
  • 4th figure: ⊂ ''' M ''')   ∧ (''' M ''' ⊂ S). Cette fourth figure was not analyzed by Aristote (considering that it returns to the first figure whose premises would be reversed) but by Galien in IIe century of the Christian era. It is named also galenic figure .

These figures have an importance in the research of the conclusive modes because they determine, in addition to the place of the predicate, that of the major and minor terms; however, according to whether a term is prone or predicate, and according to the quality of the proposal (affirmative or negative), the extension of this term varies. If one remembers that the syllogism functions on the inclusion of classes within other classes, one understands that the extension of the terms is fundamental: to say that all the men are mortals, but the Greeks are men thus the Greeks are mortals requires that the units men , Greek mortals and are taken in the same extension of an end to the other of the syllogism or at least in a less extension in the conclusion. If, for example, Greek corresponded in the premises to only the Greeks of Béotie and in the conclusion to all the Greeks , the syllogism would not have any direction: the class all the Greeks is not included in the class Greek of Béotie . Knowing that the extension of the terms changes according to the quality of the proposal and their place into his center, it is advisable, if one wants to respect their identity from one end to another of the syllogism, to know the following rules:

  • with affirmative proposal, particular predicate;
  • with negative proposal, universal predicate;
  • with universal proposal, prone universal;
  • with particular proposal, prone private individual.

Indeed, in:

  • all the Greeks are mortals , the class Greek is included in that of the mortals ; one cannot however say that the class mortals is limited to that of Greek ( all the Greeks are mortals ≠ all the mortals are Greek ). One thus considers part of the extension of mortals ;
  • no Greek is immortal , the class immortal is seized in its entirety: the entirety of the class immortal does not have any common point with that of Greek . One can thus say that no Greek is immortal is equivalent to no immortal is not a Greek ;
  • as for the subjects, they are quantified directly according to the quantity of the proposal where they appear: in any man is mortal , the class man is taken in integrality, in some men wear a beard in a particular way.
One can also summarize the questions of extension by considering the classes of proposals:

The extension of the subjects and the predicates, one will see it low, plays in the determination of the conclusive modes.

Conclusive modes

Knowing that there exist four classes of proposals (has, E, I and O), that a syllogism is composed of three proposals and that medium term draws four figures, it thus exists 4 ³ × 4 = 256 modes. From these two hundred and fifty six, only nineteen are valid, or conclusive .

Indeed, several rules (which one deduces from other logical rules concerning the extension of the terms; to see low) are to be considered:

  • the extension of the terms of the conclusion can be more important only in the premises;
  • medium term must be universal at least once in the premises;
  • one cannot draw from conclusion starting from two particular premises;
  • one cannot draw from conclusion starting from two negative premises;
  • two affirmative premises cannot give a negative conclusion;
  • the conclusion must be as weak as the weakest premise.

Of kind, it is possible to count the conclusive modes. Those are since the Middle Ages indicated by names without significance whose vowels indicate the classes of the proposals. Thus, the syllogism B' a' rb' a' r' has must be included/understood as being composed of two premises and a conclusion affirmatives and universal (A) .

Note: the names of these modes can vary; the logicians of Port-Royal say them “Barbari”, “Calentes”, “Dibatis”, “Fespamo” and “Fresisom”.

First figure (“perfect modes”)

Diagram: ⊂ P)   ∧ (S ⊂ M) ⊃ (S ⊂ P); these modes are known as “perfect” because they were used in Aristote to show the conclusive character of the modes of the other figures (or “imperfect modes”). Indeed, any syllogism can be reduced to the one of the four perfect modes. Each one of these modes gives a conclusion of one of the classes:
  • Barbara: very M is P, but any S is M, therefore any S is P ;
  • Celarent: no M is P, but any S is M, therefore no S is P ;
  • Darii: very M is P, but some S is M, therefore some S is P ;
  • Ferio: no M is P, but some S is M, therefore some S is not P .
See also : Examples of syllogisms of the first figure

Second figure

Diagram: ⊂ M)   ∧ (S ⊂ M) ⊃ (S ⊂ P); all these modes have a negative conclusion:
  • Baroco: any P is M, but some S is not M, therefore some S is not P ;
  • Camestres: any P is M, but no S is M, therefore no S is P ;
  • Cesare: no P is M, but any S is M, therefore no S is P ;
  • Festino: no P is M, but some S is M, therefore some S is not P .
See also : Examples of syllogisms of the second figure

Third figure

Diagram: ⊂ P)   ∧ (M ⊂ S) ⊃ (S ⊂ P); each mode of this figure implies a particular conclusion:
  • Bocardo: some M is not P, but very M is S, therefore some S is not P ;
  • Darapti: very M is P, but very M is S, therefore some S is P by supposing M nonempty ;
  • Datisi: very M is P, but some M is S, therefore some S is P ;
  • Disamis: some M is P, but very M is S, therefore some S is P ;
  • Felapton: no M is P, but very M is S, therefore some S is not P by supposing M nonempty ;
  • Ferison: no M is P, but some M is S, therefore some S is not P .
See also : Examples of syllogisms of the third figure

Fourth figure, known as “galenic”

Diagram: ⊂ M)   ∧ (M ⊂ S) ⊃ (S ⊂ P); the conclusion of the modes of this figure cannot be universal affirmative. The galenic modes were not recognized conclusive by Aristote.
  • Bamalip: any P is M, but very M is S, therefore some S is P ;
  • Camenes: any P is M, but no M is S, therefore no S is P ;
  • Dimaris: some P is M, but very M is S, therefore some S is P ;
  • Fesapo: no P is M, but very M is S, therefore some S is not P ;
  • Fresison: no P is M, but some M is S, therefore some S is not P .
See also : Examples of syllogisms of the fourth figure

Validation of the conclusive modes

One higher indicated of the common rules to all the figures allowing to locate the conclusive modes without explaining the major reasons of them, if is not to evoke the importance of the extension of the terms. Thus, how to explain that Bamalip galenic (any P is M, but very M is S, therefore some S is P) is conclusive but not possible “a Bamalap” galenic (any P is M, but very M is S, therefore any S is P) ?

It is necessary, with this intention, to study by the menu the rules of formation of the syllogisms.

The extension of the terms of the conclusion can be more important only in the premises

The extension of the terms of the conclusion (its subject and predicate) cannot exceed that which they have in the premises. Since the conclusion rises from the premises, it is necessary that the units which are indicated there are or the same ones or of smallest so that the play of inclusion of classes within other classes functions. That explains why the mode Bamalip (any P is M, but very M is S, therefore some S is P) fourth figure cannot have universal conclusion: in this figure, the minor term (prone of the conclusion) is always predicate, but, in this mode, it is taken in particular since the proposal is affirmative. It must thus be particular in the conclusion.

Medium term must be universal at least once in the premises

Medium term ensuring the relationship between the terms of the conclusion, this one must at least once be used under its universal extension. Indeed, this report/ratio functions only if medium term has a clear identity. However, if medium term were considered twice only partly, nothing would make it possible to affirm that these two parts are identical or that one is included in the other. This explains why the syllogisms of the second figure, in which medium term is always predicate, therefore taken particularly, cannot follow a diagram AAA: nothing indicates that in the two premises this medium term would be the same one: the cherries are spherical, but the eyes are spherical, therefore the eyes are cherries . In the premises, the two classes of the spherical objects evoked are not recut: the relationship between the minor term and the major one cannot be assured in the absence of nonambiguous medium term.

One cannot draw from conclusion starting from two particular premises

This case of figure is impossible. Indeed, if the two premises would be affirmative particular, all the terms would be particular (see higher table), of which the means. However, medium term must obligatorily be taken at least once universally (see higher). If one of the two premises would be negative particular (two negative being impossible; to see low), the conclusion should be negative and the syllogism should contain two universal terms. The predicate of negative would be universal, and only a universal premise would make it possible to obtain a universal subject.

One cannot draw from conclusion starting from two negative premises

The subject and the predicate of the conclusion being put in report/ratio by medium term, if this report/ratio is denied twice, one cannot naturally establish bond. Thus, it cannot exist of syllogism EEE or OOO (or an unspecified mixture of these two classes), which would resemble that: no animal is immortal, but no god is an animal, therefore no god is immortal .

Two affirmative premises cannot give a negative conclusion

Two affirmative premises link the terms of the conclusion by medium term. One cannot thus obtain a negative conclusion, i.e. an absence of bond between the terms. That excludes all modes AAE, AAO, IIE and IIO.

The conclusion must be as weak as the weakest premise

One understands by a “weak” hierarchy within qualities and of the quantities:
  • the particular one is weaker than the universal one;
  • the negative one that affirmative.

When one of the premises is negative (the case where two premises would be negative not being possible; to see higher), the relationship drawn up by medium term between the major term and the minor is double: one of the classes is included or identical to that of medium term, the other is excluded from medium term. There cannot thus be union between the major one and the minor.

In the same way, to suppose that a conclusion is universal affirmative, its premises will have also to be affirmative and contain each one a universal term, the extension of the terms of the conclusion not being able to exceed that of the terms of the premises. If the conclusion is universal negative, is needed that the premises contains three universal terms, are negative (universal predicate), and two universal subjects. ---- These rules make it possible to explain the conclusive character of all the syllogistic modes by excluding those which would not be convincing because of extension of the terms. The use of nonconclusive syllogisms however often meets within the framework of the Argumentation; one speaks in this case about Sophisme , most of the time by generalization, or sophism secundum quid .

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