Syllogistic theories

Syllogistic ( ancient-Greek. Συλλογιστικός inference ) - the theory of logical inference , examining the conclusions, consisting of the so-called. categorical statements (judgments). In syllogistics, for example, conclusions from a single premise (so-called direct conclusions), “complex syllogisms”, or polysillogisms with at least three premises are considered. However, the syllogistic focus focuses on the theory of a categorical syllogism, which has exactly two premises and one conclusion of this type. The classification of various forms (modes) of syllogisms and their justification was given by the founder of logic as a science, Aristotle . Subsequently, the syllogistic was perfected by various schools of antiquity (peripatetics, Stoics) and medieval logicians. Despite the limited nature of the application, noted by F. Bacon , R. Descartes , J. S. Mill and other scientists, syllogistic has long been an integral traditional element of “classical” humanitarian education, which is why it is often called traditional logic . With the creation of calculi of mathematical logic, the role of syllogistic became very modest. It turned out, in particular, that almost all of its content (namely, all conclusions that do not depend on the assumption that the subject domain is not empty) that is characteristic of syllogistic, can be obtained by means of a fragment of the predicate calculus - the so-called. single calculus predicates. A series of axiomatic statements of syllogistics in terms of modern mathematical logic has also been obtained (starting with J. Lukasevich , 1939 ).

Types of judgment

The statement in which it is affirmed that all objects of a class possess or do not possess a certain property is called general (respectively, generally affirmative or generally negative). The statement in which it is affirmed that some objects of a class possess or do not possess a certain property is called private (respectively, affirmative or particular negative). According to Aristotle, all simple sentences are divided into the following six types: single-affirmative, single-negative, general-affirmative, general-negative, private-affirmative, private-negative.

Types of simple utterances belonging to classes of objects are denoted by vowel letters of the Latin alphabet: A - generally positive, E - generally negative, I - privately affirmative, O - privately negative. Further, the class of objects is denoted by the letter S , the property by the letter P. In this case, S is called the subject, and P is called the predicate. These four types of simple sentences have the following general logical form:

A (general statement): "All items of class S have the property P ". ("All S is P ".) Symbolically: SaP ;

E (General Negative Judgment): "Not a single object of class S possesses the property P ". ("No S is P ".) Symbolically: SeP ;

I (private affirmative judgment): "Some items of class S have the property P ". ("Some S are P 's.") Symbolically: SiP ;

O (partial negative judgment): "Some items of class S do not have the property P ". ("Some S are not P ".) Symbolically: SoP .

All these judgments in the language of predicate logic are:

$A\colon (\forall x){\bigl (}S(x)\to P(x){\bigr )}.$ ${\ displaystyle A \ colon (\ forall x) {\ bigl (} S (x) \ to P (x) {\ bigr)}.}$

$E\colon (\forall x){\bigl (}S(x)\to \lnot P(x){\bigr )}.$ ${\ displaystyle E \ colon (\ forall x) {\ bigl (} S (x) \ to \ lnot P (x) {\ bigr)}.}$

$I\colon (\exists x){\bigl (}S(x)\land P(x){\bigr )}.$ ${\ displaystyle I \ colon (\ exists x) {\ bigl (} S (x) \ land P (x) {\ bigr)}.}$

$O\colon (\exists x){\bigl (}S(x)\land \lnot P(x){\bigr )}.$ ${\ displaystyle O \ colon (\ exists x) {\ bigl (} S (x) \ land \ lnot P (x) {\ bigr)}.}$

The same formulas can be equivalently transformed as follows:

$A\colon \lnot (\exists x){\bigl (}S(x)\land \lnot P(x){\bigr )}.$ ${\ displaystyle A \ colon \ lnot (\ exists x) {\ bigl (} S (x) \ land \ lnot P (x) {\ bigr)}.}$

$E\colon \lnot (\exists x){\bigl (}S(x)\land P(x){\bigr )}.$ ${\ displaystyle E \ colon \ lnot (\ exists x) {\ bigl (} S (x) \ land P (x) {\ bigr)}.}$

$I\colon \lnot (\forall x){\bigl (}S(x)\to \lnot P(x){\bigr )}.$ ${\ displaystyle I \ colon \ lnot (\ forall x) {\ bigl (} S (x) \ to \ lnot P (x) {\ bigr)}}.}$

$O\colon \lnot (\forall x){\bigl (}S(x)\to P(x){\bigr )}.$ ${\ displaystyle O \ colon \ lnot (\ forall x) {\ bigl (} S (x) \ to P (x) {\ bigr)}.}$

Syllogistic Inference

Aristotle distinguishes the most important type of deductive reasoning - the so-called syllogistic reasoning, or syllogism. Aristotelian syllogism is a scheme of logical inference (inference), consisting of three simple statements S, M, P of one of the four specified types A, E, I, O : the first two statements S, M are the premises, the third P is the conclusion. As a result, perhaps only 4 types of syllogisms:

${\begin{array}{cccc}{\dfrac {\begin{matrix}{\text{Figure I}}\\[2pt]MxP\\SyM\end{matrix}}{SzP}}&\quad {\dfrac {\begin{matrix}{\text{Figure II}}\\[2pt]PxM\\SyM\end{matrix}}{SzP}}&\quad {\dfrac {\begin{matrix}{\text{Figure III}}\\[2pt]MxP\\MyS\end{matrix}}{SzP}}&\quad {\dfrac {\begin{matrix}{\text{Figure IV}}\\[2pt]PxM\\MyS\end{matrix}}{SzP}}\end{array}}$ ${\ displaystyle {\ begin {array} {cccc} {\ dfrac {\ begin {matrix} {\ text {Figure I}} \\ [2pt] MxP \\ SyM \ end {matrix}} {SzP}} & \ quad {\ dfrac {\ begin {matrix} {\ text {Figure II}} \\ [2pt] PxM \\ SyM \ end {matrix}} {SzP}} & \ quad {\ dfrac {\ begin {matrix} { \ text {Figure III}} \\ [2pt] MxP \\ MyS \ end {matrix}} {SzP}} & \ quad {\ dfrac {\ begin {matrix} {\ text {Figure IV}} \\ [2pt ] PxM \\ MyS \ end {matrix}} {SzP}} \ end {array}}}$

Here $x,y,z\in \{a,e,i,o\}$ {\ displaystyle x, y, z \ in \ {a, e, i, o \}} and the SzP entry (like MxP and SyM , etc.) denotes, depending on the value of z, one of the four judgments of the types A, E, I, O. Each figure delivers the following number of syllogisms (diagrams): $4\cdot 4\cdot 4=64$ ${\ displaystyle 4 \ cdot 4 \ cdot 4 = 64}$ . Since figures 4, we get $4\cdot 64=256$ ${\ displaystyle 4 \ cdot 64 = 256}$ syllogisms.

The task of the Aristotelian syllogistic, brilliantly solved by Aristotle himself, is to discover all those syllogisms (inference schemes) that are valid, that is, represent logical following. There are exactly 19 syllogisms, as Aristotle established, the rest are incorrect. In this case, 4 of the 19 correct syllogisms are conditionally correct.

To memorize the correct syllogisms by medieval scholastics, the following mnemotechnical Latin poem was invented:

BARBARA, CELARENT, DARII, FERIO que prioris;

CESARE, CAMESTRES, FESTINO, BAROCO sedundae;

Tertia DARAPTI *, DISAMIS, DATISI, FELAPTON *, BOCARDO, FERISON habet; quarta insuper addit

BAMALIP *, CAMENES, DIMATIS, FESAPO *, FRESISON.

Here, words in large letters, or rather, vowels in these words, mean A, E, I, O judgments, substituted into the x, y, z place in each syllogism figure (the words in the first line of the verse correspond to the first figure, the second line - the second, etc.) That is, for the first figure, the variants of syllogisms (so-called moduses) of the first line will be correct: BARBARA (AAA), CELARENT (EAE), DARII (AII), FERIO (EIO):

${\begin{array}{cccc}{\dfrac {\begin{matrix}{\text{BARBARA}}\\[2pt]MAP\\SAM\end{matrix}}{SAP}}&\quad {\dfrac {\begin{matrix}{\text{CELARENT}}\\[2pt]MEP\\SAM\end{matrix}}{SEP}}&\quad {\dfrac {\begin{matrix}{\text{DARII}}\\[2pt]MAP\\SIM\end{matrix}}{SIP}}&\quad {\dfrac {\begin{matrix}{\text{FERIO}}\\[2pt]MEP\\SIM\end{matrix}}{SOP}}\end{array}}$ ${\ displaystyle {\ begin {array} {cccc} {\ dfrac {\ begin {matrix} {\ text {BARBARA}} \\ [2pt] MAP \\ SAM \ end {matrix}} {SAP}} & \ quad {\ dfrac {\ begin {matrix} {\ text {CELARENT}} \\ [2pt] MEP \\ SAM \ end {matrix}} {SEP}} & \ quad {\ dfrac {\ begin {matrix} {\ text {DARII}} \\ [2pt] MAP \\ SIM \ end {matrix}} {SIP}} & \ quad {\ dfrac {\ begin {matrix} {\ text {FERIO}} \\ [2pt] MEP \\ SIM \ end {matrix}} {SOP}} \ end {array}}}$

similarly, for other figures of the syllogism, the modes from the verse line corresponding to the figure number are applied.

It should be noted that in the Aristotelian logic all classes M, P, S are considered non-empty, that is, having at least one element. If this is not taken into account, then there are obvious errors. Example of Russell : Let M mean the class of (empty) "golden mountains", P - the class of "golden objects", and S - the class of "mountains". Then we have the third figure in DARAPTI mode:

All the golden mountains are golden.

All the golden mountains are mountains. -

Hence some mountains are golden.

Thus, from two true (tautological) statements, we will get by no means a tautological, but obviously incorrect statement.

Since in modern mathematics, physics and even structural linguistics often work with empty sets, in this case it is impossible to apply the modes, marked with asterisks (DARAPTI, FELAPTON, BAMALIP, FESAPO)

Set-theoretic interpretation of Aristotelian syllogistic

Formalization of the theory of the Aristotelian syllogisms

The described formalization was invented in the 1950s by the Polish logician Lukasevich.

Let the lowercase Latin letters a, b, c, ... denote the variable terms of syllogistic, two uppercase letters A and I - two syllogical binary relations: Aab : “Every a is b ”, Iab : “Some a is b ”.

The concept of a formula is given by the following inductive definition:

1) Aab and Iab are simple (or atomic) syllogistic formulas;

2) if $F, G$ ${\ displaystyle F, G}$ - syllogistic formulas, then syllogistic formulas will also be $(F\wedge G),(F\vee G),(F\to G),(\neg F)$ ${\ displaystyle (F \ wedge G), (F \ vee G), (F \ to G), (\ neg F)}$ ;

3) there are no other formulas other than those obtained according to the rules of clauses 1 and 2.

Formulation of axioms. First, we believe that there is some formalized calculus of statements , so that its axioms open the list of axioms of formal syllogistic. The following axioms are accepted as special axioms:

$(FS1)\colon Aaa;$ ${\ displaystyle (FS1) \ colon Aaa;}$

$(FS2)\colon Iaa;$ ${\ displaystyle (FS2) \ colon Iaa;}$

$(FS3)\colon (Abc\wedge Aab)\to Aac$ ${\ displaystyle (FS3) \ colon (Abc \ wedge Aab) \ to Aac}$ (Barbara syllogism);

$(FS4)\colon (Abc\wedge Iba)\to Iac$ ${\ displaystyle (FS4) \ colon (Abc \ wedge Iba) \ to Iac}$ (syllogism Datisi).

Using the following definitions, we introduce two more syllogical binary relations E ' and O : Eab means $\neg Iab$ ${\ displaystyle \ neg Iab}$ , Oab means $\neg Aab$ ${\ displaystyle \ neg Aab}$ .

As rules of inference in the system of formalized syllogistic FS , two rules of substitution are accepted and the rule of inference modus ponens

Literature

Aristotle. Works: In 4 t. - M., 1976-1981.
Akhmanov A.S. The logical teachings of Aristotle .. - M., 1960.
Igoshin V.I. Mathematical logic and theory of algorithms. - Academia, 2008.
Lukasevich J. Aristotelian syllogistic from the point of view of formal logic: Trans. from English .. - M., 1959.

Links

Radlov, EL. Syllogism // Encyclopedic Dictionary of Brockhaus and Efron : in 86 tons (82 tons and 4 extras). - SPb. , 1890-1907.