Syllogism

The syllogism (in Latin: syllogismus) is a type of deductive reasoning that is part of logic, of Greek origin. It consists of two propositions as premises and another as a conclusion, the latter being a necessarily deductive inference from the other two. It was first formulated by Aristotle. The syllogism is the central notion of Aristotelian logic, a fundamental pillar of scientific and philosophical thought since its invention more than two millennia ago.

Aristotle considered syllogisms in his collected logical work Organon, in the books known as First Analytics (Greek Proto Analytika, Latin Analytica Priora — language in which the work was known in Western Europe).

Aristotle. Louvre Museum.

Syllogisms according to Aristotle

Aristotle considered logic as a method of relating terms. Aristotelian syllogisms seek to establish the relationship between two terms: a subject and a predicate, which are joined or separated in judgments. The appearance of possible conclusions about the relationship between these two terms arises from their comparison, by means of judgments, with a third term that acts as the "middle term" 3. 4; (tertium comparationis). Thus, the syllogism consists of two judgments, a major premise and a minor premise, in which three terms are compared (subject, predicate and "middle term"), from whose comparison a new trial is obtained as a conclusion.

Syllogistic logic tries to establish the laws that guarantee that, from the truth of the comparative judgments, or premises, a new true judgment, can be obtained with a guarantee of truth. i> or conclusion.

Elements

According to what was explained in the previous paragraph, the elements of a syllogism are:

• A term subject S.

• A term preached P.

• A medium term M,.

• A backgroundwhich consists of two trials called premises.

• A Consequently, the trial resulting in conclusion.

Structure

A syllogism has the following structure:

• Higher priority, trial in which the term is greater or preached of the conclusion, P, compared to the medium term M.

• Minor premises, judgement in which the term minor or subject of the conclusion, S, compared to the medium term M.

• Consequential or Conclusion, judgment to which one arrives, which claims (une) or denies (separa) the relationship between S and P.

The judgments, that give rise to the major and minor premises, relate the terms to each other to constitute the argument. In this way, the syllogism argues establishing the conclusion as a relationship between two terms, derived from the comparison of both terms with a third term.

Quantity, or length of terms

The extension of the terms refers to a criterion of quantity. The terms S, P and M can be taken in their universal extension, encompassing all possible individuals - the domain of discourse - to which the concept may refer, or in its particular extension, when it refers only to some. For example, the relationship between S and P according to their extension can be:

• Universal: where Everything. S That's it. P. The names themselves have universal extension; for the one, as one, is equivalent to an individual who being unique is, therefore, all possible.
• Particular: where Some S They are. P

Quality, or relationship between terms

Specifically, the quality or relationship between terms can be:

• Affirmative or union: S That's it. P.
• Negative or separation: S It's not. P.

The predicate of an affirmation always has a particular extension, and the predicate of a negation is taken in its universal extension. When a concept, subject or predicate, is taken in all its extension, it is said to be distributed; when not, it is said to be undistributed.

Classification of lawsuits

According to the criteria of quantity and quality, judgments or premises can be grouped into the following classes:

The names of the classes A and I derive from the verb adfirmo (from Latin: I affirm) and that of the E and O of ne go (Latin: I deny).

Syllogistic figures and modes

Taking into account the arrangement of the terms in the premises and in the conclusion, the following syllogistic figures:

The syllogistic modes are the different combinations that can be made with the judgments that are part of the premises and the conclusion. Since judgments have four different classes (A,E,I,O), and to form figures they are taken three by three —two premises and a conclusion— there are 64 possible combinations. These 64 possible combinations are reduced to 19 valid modes, by applying the rules of the syllogism.

Rules of syllogism

When an error is made in the syllogism, the result is a fallacy.

Rules for terms

A syllogism cannot have more than three terms

This principle is limited to fulfilling the very structure of the syllogism: the comparison of two terms with a third. Although the rule is clear, its application is not always. It is what some call a four-legged syllogism, since a fourth term or quaternio terminorum is mistakenly introduced.

Example: if the following erroneous syllogism is analyzed:

The terms that appear as obvious are the words man, free, woman. But, as a non sequitur (a type of logical error) in the supposed major premise the word man is used in its meaning of species (Homo sapiens) while in the supposed minor premise of the quaternio terminorum the meaning of the word man has been changed using the meaning of gender (man as a synonym for male). That is to say, a fourth term has been surreptitiously included, hence the conclusion of the quaternio terminorum is wrong, a fallacy.

Terms should not be longer in the conclusion than in the premises

By the same structure of the syllogism; we can only draw conclusions about what we have compared in the premises.

The middle ground cannot enter the conclusion

Due to the very structure of the syllogism, the function of the middle term is to serve as an intermediary, as a term of comparison.

Example: We can represent the cited mathematical axiom as follows:

A=B

B=C

∴ A = C

We see that the role of the middle term (B) is that of the third quantity, that is, to equalize the two extremes. In other words, his job is to show the relationship that exists between the major term (C) and the minor term (A). Therefore, it has nothing to do in the conclusion; its true place is in the premises as antecedent.

The middle term must be taken in its universal extension in at least one of the premises

For the comparison to be such, it is necessary that the middle term be compared in its entirety. Otherwise, one term could be compared with one part and the other with the other, thus actually constituting a four-term syllogism.

Example: Consider the following erroneous syllogism:

Which is obviously not a valid mode, since "Spanish" in the major premise the being predicated of an affirmative is taken in its particular extension.

Rules of propositions

No conclusion can be drawn from two negative premises

Two negative premises do not adapt to the structure of the syllogism, since if we deny S of M, and P of M, we do not know what relationship there can be between S and P. To establish the relationship, at least one of the terms has to be identified with M. Therefore one of the two premises has to be affirmative.

No negative conclusion can be drawn from two affirmative premises

In effect, if S identifies with M, and P also identifies with M, it does not make sense to establish a negative relationship with between S and P. The conclusion will be affirmative.

The conclusion always follows the weaker part. The weak part is understood to be the negative versus the affirmative, and the particular versus the universal

Let's look at the two cases separately:

Negative conclusion of an affirmative premise and the other negative.

If a relationship is affirmed between two terms (X, M), but one of them is denied with another (Y, M), being M the middle term, there can be no more conclusion than denying the relationship between the first (X) and the last (Y) being one subject and the other predicated of the conclusion.

Particular conclusion of a universal and other particular premise (taking into account that two particular premises cannot be, as we shall see in the following rule).

There may be two cases: one being affirmative and the other negative, or the two being affirmative.

1. Two statements. (The preaching of an affirmative is taken in its particular extension, and the preaching of a refusal in its universal extension).
   As both assertives their preaching are particular. The term of the universal must necessarily be the medium term, the conclusion must have a particular subject.
2. One affirmative and one negative: there must be two universal terms. One of them has to be the middle term, the other has to be the preaching of the conclusion, because the conclusion will have to be negative (almost a) of this same rule. Therefore the term remaining will be the subject of the conclusion with particular extension.

No valid conclusion can be obtained from two particular premises

It also has two possible cases: that one is affirmative and the other is negative, or that both are affirmative.

Affirmative and negative

Some A is B - Some A is not C.

There is only one universal term that is the predicate of negative, which therefore has to be the middle term. The conclusion will have to be negative (almost a) of the previous rule), and therefore the preacher will have to be universal, and it cannot be the middle term therefore there can be no conclusion.

Two statements

Some A is B - Some A is C.

The three terms are particular, and therefore there can be no medium term with universal extension, and therefore there is no possible conclusion.

Valid modes

The mode of the syllogism is the form it takes according to the quantity and quality of the premises and the conclusion. From the application of the laws of syllogisms to the 64 possible modes, only 19 are valid and they are the ones that are traditionally memorized according to the valid modes of each figure with its premises and conclusion.

Nota bene: The subaltern moods BARBARI, CELARONT are also valid for the first figure; for the second: CESARO, CAMESTROP; and for the fourth: CAMENOP.

Graphical representation of modes as class logic using Venn diagrams

Convention for the Graphical Representation of Type A Trial

These modes can be represented by Venn diagrams with the following conventions:

• Each term of syllogism is represented by S, P, M, by a colourless circle that represents all possible members of a class.
• The conclusion appears as a result of the relationship of the terms S and P in their relationship with M.
• The absence is shown as a colored area.
• Individual existence is affirmed by an X: At least one, or some.
• The relation of the terms is constituted as belonging or not belonging to the class.
• The inclusion relationship, All S is P, is represented as “There is no S other than P” as it shows the image shown on the margin.

Graphical representation of valid modes in Venn diagrams

Taking into account the problematic of Aristotelian logic, discussed below, the problem of "existential commitment" it affects the modes Darapti, Felapton, Bramalip, and Fesapo that are not shown in the graphs, since they are not accepted as valid by some and, above all, the graphical representation does not make the conclusion plausible, due to the lack of "existential commitment", as discussed below.

Problems of syllogistic logic

The previous exposition is the simplest and most schematic form traditionally presented as Aristotelian logic.

However, the problem that Aristotle deals with is much more complex. Aristotle defines:

Silogism is an argument in which, established certain things, necessarily results from them, as they are, another different thing.

Aristotle An. Pr. I 24 b 18-23

Two aspects to highlight in its definition:

• The need, which considers syllogism as categorical, considering that the judgments that make it are equally categorical.

• The basis of such need, for "being what they are."

Talking about the categorical syllogism means talking about what is necessary and unconditioned. And precisely unconditioned because it is based on the “being of things”.

Aristotle is thinking of a predicate apprehended from experience and attributed by understanding to a subject. In apophantic language, the syllogism manifests the truth, because human understanding (agent understanding, according to Aristotle) is capable of reaching a direct intuition of the real, even through a process of abstraction.

It starts from the assumption that P is a “true” predicate of S (in the sense that P manifests the "identity" of the being of S), which raises a metalogical question. See true.

Aristotle thinks that judgment manifests “what is” as true. The problem then is, how is a subject predicated what "is not"? (V.: aporetic).

Aristotelian logic encounters the problem of negative judgments, which it does not solve quite well.

In fact, in the opposition box of judgments, Aristotle studied in great detail problems that have not been taken into account later; he actually considered three figures and not all 19 valid modes. Aristotle considers perfect modes those whose validity appears evident, while the others are imperfect since they must be proven by means of the perfect modes, which are those corresponding to the first figure: BÁRBARA, CELARENT, DARII, FERIO.

He even came to consider such modes as the axioms of the entire logical system.

The judgment as “attribution” of a true predicate to a subject, (in the sense that P manifests the "identity" as "being of the subject", while P known reality), raises the problem of a false predicate, that is, a non-predicate. How do we know a non-predicate?...

Linguistically, the problem is disguised by negating the verb instead of the predicate as an attribute (grammar). In this way instead of saying "Antonio is a non-horse", (what is a non-horse?), we say "Antonio is not a horse". But this second is only intelligible from the extensional point of view of the concepts, that is, from the point of view of being an element of a set defined by a property, or what is the same by its membership or non-membership of a certain class; which brings us to the logic of classes.

Modern symbolic logic, merely formal logic, has no connection to any truth content and clearly overcomes these difficulties; especially with the advantage of being able to treat polyadic propositions, so called because they have more than two terms (for example: "Jupiter is greater than the Earth and less than the Sun"), and greatly facilitate logical calculation, so that, in fact, Aristotelian logic, as such, is clearly out of use.

Hans Reichenbach studies the opposition table of the judgments considering the judgments A, E, I, O, as a class relationship and considers that the negative judgments E, O, which are the problematic ones, can be eliminated by noting the denial of the complementary class.

The notation is made by establishing between the subject S and the predicate P, the lowercase letter corresponding to the type of judgment. Thus we have that:

SeP▪ ▪ SaP! ! {displaystyle SePleftrightarrow Sa{bar {P}}}

SorP▪ ▪ SiP! ! {displaystyle SoPleftrightarrow Si{bar {P}}}

In this way, not only is the notation simplified, but in ways that have traditionally been considered invalid, a valid conclusion can be obtained, which the classical notation made impossible.

For all these reasons, the current interpretation of Aristotelian logic as a syllogism is its interpretation as class logic. Such is the merit of Lukasiewicz's work.

But considering universal concepts as classes raises the problem of the individual's existence as instantiation or existential commitment. For the class as an independent property can be considered as a universal abstract. But predicates, as attributes, have no meaning without a grammatical subject of which they are predicated because it possesses said property.

Traditional logic did not consider the problem of the existence or non-existence of the individual with respect to universal concepts, since it is assumed that these have arisen from abstraction from the knowledge of singulars or existing individuals.

The syllogism in formal logic

Current formal logic considers the relationship S and P as a merely syntactic relationship without any material content, either in a class relationship or a propositional function of predicates. Aristotle considers this formality, of course, from the point of view of the relationship between two terms S (subject) and P (predicate) that at the same time have a linguistic-grammatical function, since for Aristotle the terms represent aspects of being and therefore so much of reality

But the formality of current logic converts the deduction into an inference, as a logical consequence, instead of an implication with transmission of content in an apophantic language that transmits truth as Aristotle intended for the language of science.

In the new form of syntactic relation, all relation of the terms with the grammar of the language and possible "meaning" are lost. The syllogism thus loses its formality of being categorical, transmitter of the necessary truth, "because things are as they are", to acquire a hypothetical formality.

Being S the subject, P the predicate and M the middle term, the syllogism is now interpreted as class logic, and its logical scheme would be of the following type:

If the class S represents the class as a property of being Greek; class M represents class as a property of being a man; and the class P represents the property of being mortal, then the syllogism in Barbara would be:

If all men are mortal and all Greeks are men then all Greeks are mortal.

[chuckles](S M)∧ ∧ (M P)→ → (S P)]{displaystyle [(Ssubset M)wedge (Msubset P)rightarrow (Ssubset P){big ]}

When the reference of existential instantiation is with respect to individuals, Aristotelian judgments can be formalized in the following way as predicate logic:

Aristotelian Judgment	Logic of preaching
All S is P	Русский Русский x(S(x)→ → P(x)){displaystyle forall x(S(x)rightarrow P(x))}
No S is P	¬ ¬ consuming consuming x(S(x)∧ ∧ P(x)){displaystyle neg exists x(S(x)land P(x))}
Some S is P	consuming consuming x(S(x)∧ ∧ P(x)){displaystyle exists x(S(x)land P(x)}
Some S is not P	consuming consuming x(S(x)∧ ∧ ¬ ¬ P(x)){displaystyle exists x(S(x)land neg P(x))}

The syllogism is thus interpreted as:

If all (or some) of the individuals who belong (or do not belong) to the S (Subject Set) belong (or do not belong) to the M (Second Medium Term) set, and all (or some) of the individuals who belong (or do not belong) to the M set (or do not belong) belong to the P set, then all (or some) of the individuals who belong (or do not belong) to the P set).

And the syllogism in Barbara is formalized as follows:

Русский Русский x(M(x)→ → P(x))∧ ∧ Русский Русский x(S(x)→ → M(x))→ → Русский Русский x(S(x)→ → P(x)){displaystyle forall x(M(x)rightarrow P(x)))land forall x(S(x)rightarrow M(x))rightarrow forall x(S(x)rightarrow P(x)} Being M the middle, S the subject and P the preached of syllogism.

If M(x) symbolizes "Be man", being M=be man what can be preached about a variable x whose commitment to existence would be given by the existential quantification of the reference of that function, whether it be a universal quantifier, all x: Русский Русский x{displaystyle forall x}a particular quantifier, some x: x{displaystyle lor x}; or a determined individual constant: a, b, c...; and P(x) "being mortal" and M(x) "being Greek", then the formula Русский Русский x(M(x)→ → P(x))∧ ∧ Русский Русский x(S(x)→ → M(x))→ → Русский Русский x(S(x)→ → P(x)){displaystyle forall x(M(x)rightarrow P(x)))land forall x(S(x)rightarrow M(x))rightarrow forall x(S(x)rightarrow P(x)} represents a material syllogism in beard. The logic of preaching thus solves the problem of existential instance.

In both cases (as logic of classes or as logic of predicates) the syllogism is expressed in formulas of hypothetical relation; and since there is no truth statement in the premises, the conclusion is conditioned and not implied.

In both cases, as class relation or as predicate logic, the classic categorical syllogism:

All men are mortal. All Greeks are men. Therefore all Greeks are mortal.

It becomes a hypothetical syllogism:

If all men are mortal and all Greeks are men, then all Greeks are mortal.

Which, without a doubt, is a transformation of Aristotelian logic.

Trial of terms

The judgment of terms is the comparison of two concepts, either logically or extracted from experience, through which we believe or affirm the relationship of one with respect to the other as an objective truth. Thus the true belief in the Aristotelian judgments of classical logic was justified.

For example: in snow is white, the mind settles that whiteness is a truly predicable property of snow. Such has been the consideration of Aristotelian judgments in the syllogism of traditional logic.

Today, formal and symbolic logic does not accept such judgments that are interpreted as belief, since it does not require its linguistic or conceptual formulation, as the scholastics already considered. On the other hand, the possibility of a categorical, as Aristotle thought, is seriously questioned from Kant and his Critique of Pure Reason. Currently, in logic such a relationship is formally considered:

• As a result of the discourse domain of the relationship of two logical classes.
• As the attribution of a preacher to a quantified individual logic variable.

On judgment and proposition

Although in the Aristotelian syllogism one speaks of judgment, nowadays one would speak of proposition. The difference between judgment and proposal is important. The proposition affirms a fact as a whole, which is or is not, as the logical content of knowledge. Instead, the judgment attributes a predicate to a logical subject of knowledge, giving terms both a linguistic function of meaning (semantics) and a formal logical function (syntactic). This directly influences the very concept of the content of a judgment and of a proposition, especially in cases of negation, as considered later in syllogistic logic.

We keep here the denomination of judgment because it is more in line with the traditional. It should be noted that this type of logic, as such, is deprecated, replaced by symbolic logic, in which syllogistic logic is interpreted as class logic.^{[citation required ]}