regarding aristotelian logic as a sheffer stroke basic algebra · 2018-08-07 · ibrahim senturk lc...

ContentIntroduction and Background

PreliminariesCarroll’s Diagrams and The Elimination Method

The Calculus System SLCD and Its CompletenessRegarding Aristotelian Logic as a Sheffer Stroke Basic Algebra

References

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Regarding Aristotelian Logic as a Sheffer StrokeBasic Algebra

Ibrahim SENTURK and Tahsin ONER

ibrahim.senturk@ege.edu.tr

Faculty of Sciences – Department of MathematicsEge University

LOGIC COLLOQUIUM 2018Udine–ITALY

July 23–28, 2018

Ibrahim Senturk LC 2018 Regarding Aristotelian Logic as a Sheffer Stroke Basic Algebra1 / 76

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1 Introduction and Background

2 Preliminaries

3 Carroll’s Diagrams and The Elimination Method

4 The Calculus System SLCD and Its Completeness

5 Regarding Aristotelian Logic as a Sheffer Stroke Basic Algebra

6 References

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2 Preliminaries

6 References

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2 Preliminaries

6 References

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2 Preliminaries

6 References

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2 Preliminaries

6 References

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2 Preliminaries

6 References

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What we call today Aristotelian logic, it could be especially seen asthe theory of the syllogisms.

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The first systematic approach to the syllogisms dates back to thephilosopher Aristotle who searched them in the scope of reasoningand inference as a logical system in the Prior Analytics[J. Barnes, 1984].

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At the end of 1800s, Lewis Caroll used diagrammatic methods toanalyze the Aristotelian syllogisms in his book [L. Caroll, 1896].

Inaddition, Lukasiewicz interested with this topic comprehensivelyand he looked at this topic from the point of view of mathematicalfoundations in the middle of 1900s [J. Lukasiewicz, 1957]. Theseconstitute the bases of modern mathematical works on categoricalsyllogisms.

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At the end of 1800s, Lewis Caroll used diagrammatic methods toanalyze the Aristotelian syllogisms in his book [L. Caroll, 1896]. Inaddition, Lukasiewicz interested with this topic comprehensivelyand he looked at this topic from the point of view of mathematicalfoundations in the middle of 1900s [J. Lukasiewicz, 1957]. Theseconstitute the bases of modern mathematical works on categoricalsyllogisms.

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Nowadays, the topic is studied extensively and investigated withdifferent approaches.

For example, Stanley Burris examinedsyllogistic logic by using Boolean Algebras [S. Burris, 2013],Senturk and Oner examined by using Heyting Algebras[Senturk and Oner, 2016] and Esko Turunen used MV-Algebras forPeterson Intermediate Syllogisms [E. Turunen, 2014]. And also,syllogisms are used recently in different areas like as in computerscience[I. Pratt-Hartmann and L. S. Moss, 2009], in artificialintelligence [B. Kumova and H. Cakir, 2010], in engineering[B. A. Kulik, 2001], in traffic control systems[J. Niittymaki and E. Turunen, 2003] etc.

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Nowadays, the topic is studied extensively and investigated withdifferent approaches. For example, Stanley Burris examinedsyllogistic logic by using Boolean Algebras [S. Burris, 2013],Senturk and Oner examined by using Heyting Algebras[Senturk and Oner, 2016] and Esko Turunen used MV-Algebras forPeterson Intermediate Syllogisms [E. Turunen, 2014].

And also,syllogisms are used recently in different areas like as in computerscience[I. Pratt-Hartmann and L. S. Moss, 2009], in artificialintelligence [B. Kumova and H. Cakir, 2010], in engineering[B. A. Kulik, 2001], in traffic control systems[J. Niittymaki and E. Turunen, 2003] etc.

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Nowadays, the topic is studied extensively and investigated withdifferent approaches. For example, Stanley Burris examinedsyllogistic logic by using Boolean Algebras [S. Burris, 2013],Senturk and Oner examined by using Heyting Algebras[Senturk and Oner, 2016] and Esko Turunen used MV-Algebras forPeterson Intermediate Syllogisms [E. Turunen, 2014]. And also,syllogisms are used recently in different areas like as in computerscience[I. Pratt-Hartmann and L. S. Moss, 2009], in artificialintelligence [B. Kumova and H. Cakir, 2010], in engineering[B. A. Kulik, 2001], in traffic control systems[J. Niittymaki and E. Turunen, 2003] etc.

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But one of the main problems of all these areas is to find amathematical model for producing mechanically conclusions fromgiven premises.

More precisely, a system based on mathematicalfoundations that deduces conclusions from given premises. If it issucceeded, we can solve a lot of problems about systematicallythinking via a mathematical model.

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But one of the main problems of all these areas is to find amathematical model for producing mechanically conclusions fromgiven premises. More precisely, a system based on mathematicalfoundations that deduces conclusions from given premises.

If it issucceeded, we can solve a lot of problems about systematicallythinking via a mathematical model.

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But one of the main problems of all these areas is to find amathematical model for producing mechanically conclusions fromgiven premises. More precisely, a system based on mathematicalfoundations that deduces conclusions from given premises. If it issucceeded, we can solve a lot of problems about systematicallythinking via a mathematical model.

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In this work, our first aim is to construct a bridge between Shefferstroke basic algebra and categorical syllogisms together with arepresentation of syllogistic arguments by using sets in SLCD(Syllogistic Logic with Caroll Diagrams)[Senturk and Oner, 2018].

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A categorical syllogistic system consists of 256 syllogistic moods,15 of which are unconditionally and 9 are conditionally; in total 24of them are valid. Those syllogisms in the conditional group arealso said to be strengthened, or valid under existential import,which is an explicit assumption of existence of some S, M or P.So, we add a rule, which is “Some X is X when X exists”, toSLCD. Therefore, we obtain the formal system SLCD† from SLCD.

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Our conclusions in this work• Syllogism is valid if and only if it is provable in SLCD.

• Strengthened syllogism is valid if and only if it is provable inSLCD†.This means that SLCD is sound and complete. And also,

• We define a Sheffer stroke algebra by using sets which isobtained from syllogistic arguments.

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• Strengthened syllogism is valid if and only if it is provable inSLCD†.

This means that SLCD is sound and complete. And also,

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• Strengthened syllogism is valid if and only if it is provable inSLCD†.This means that SLCD is sound and complete. And also,

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Preliminaries

A categorical syllogism can be thought as a logical argument:

It consists of two logical propositions called premises and a logicalconclusion, where the premises and the conclusion have aquantified relationship between two objects which are given inTable 1. A syllogistic proposition or Aristotelian categoricalproposition indicates a quantified relationship between two objects.There are four different types of propositions presented as follows:

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Preliminaries

A categorical syllogism can be thought as a logical argument:It consists of two logical propositions called premises and a logicalconclusion, where the premises and the conclusion have aquantified relationship between two objects which are given inTable 1.

A syllogistic proposition or Aristotelian categoricalproposition indicates a quantified relationship between two objects.There are four different types of propositions presented as follows:

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Preliminaries

A categorical syllogism can be thought as a logical argument:It consists of two logical propositions called premises and a logicalconclusion, where the premises and the conclusion have aquantified relationship between two objects which are given inTable 1. A syllogistic proposition or Aristotelian categoricalproposition indicates a quantified relationship between two objects.There are four different types of propositions presented as follows:

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Table 1: Aristotle’s Syllogistic Propositions

Table: Aristotle’s Syllogistic Propositions

Symbol Statements Generic Term

A All X are Y Universal AffirmativeE No X are Y Universal NegativeI Some X are Y Particular AffirmativeO Some X are not Y Particular Negative

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We use S (for Subject term), M (for Middle term) and P (forPredicate term). That is, if there is a quantified relation betweenM and P (is said Major Premise), and a quantified relationbetween M and S (is said Minor Premise), then we deduce anyresult about a quantified relation between S and P (is saidConclusion).

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We use S (for Subject term), M (for Middle term) and P (forPredicate term). That is, if there is a quantified relation betweenM and P (is said Major Premise), and a quantified relationbetween M and S (is said Minor Premise), then we deduce anyresult about a quantified relation between S and P (is saidConclusion).

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We assume that the relations between M and P, and between Mand S hold. If we cannot contradict with certain relation betweenS and P does not hold, then the syllogism is valid. Otherwise, thesyllogism is invalid.

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Syllogisms are grouped into distinct four subgroups which aretraditionally called Figures [E. Turunen, 2014]:

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Figures

Figure I

A quantity Q1 of M are P (Major Premise)

A quantity Q2 of S are M (Minor Premise)

A quantity Q3 of S are P (Conclusion)

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Figures

Figure II

A quantity Q1 of P are M (Major Premise)

A quantity Q2 of S are M (Minor Premise)

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Figures

Figure III

A quantity Q1 of M are P (Major Premise)

A quantity Q2 of M are S (Minor Premise)

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Figures

Figure IV

A quantity Q1 of P are M (Major Premise)

A quantity Q2 of M are S (Minor Premise)

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The mood of a syllogism is the sequence of the kinds of thecategorical propositions by which it is formed.

A categoricalsyllogistic system consisting of 64 syllogistic moods are possible foreach figure. Therefore, it has 256 moods for all figures.

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The mood of a syllogism is the sequence of the kinds of thecategorical propositions by which it is formed. A categoricalsyllogistic system consisting of 64 syllogistic moods are possible foreach figure. Therefore, it has 256 moods for all figures.

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A syllogism is determined by using not only its mood but also itsfigure. And they are examined in terms of whether it is valid ornot. So, we have some common properties which are called rulesof deduction for getting valid syllogisims.

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The rules of deduction of categorical syllogisms are thefollowing:Step 1: Relating to premises irrespective of conclusion or figure:

(a) No inference can be made from two particular premises.

(b) No inference can be made from two negative premises.

Step 2: Relating to propositions irrespective of figure:

(a) If one premise is particular, the conclusion must be particular.

(b) If one premise is negative, the conclusion must be negative.

Step 3: Relating to distribution of terms:

(a) The middle term must be distributed at least once.

(b) A predicate distributed in the conclusion must be distributed in the majorpremise.

(c) A subject distributed in the conclusion must be distributed in the minorpremise.

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We use ` symbol for valid syllogisms. For example, the syllogism

AMP ,ASM ` ASP

consists of from left to right major premise, minor premise andconclusion, respectively. Its mood is AAA, and it has first figure.

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A categorical syllogistic system has 256 moods for all figures. 15 ofthem are unconditionally and 9 of them are conditionally, totally 24of them are valid forms. We have unconditional valid forms ofsyllogism in Table 2. It means that these forms are valid withoutany condition in Syllogism.

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Table 2: Unconditionally Valid Forms

Table: Unconditionally Valid Forms

Figure I Figure II Figure III Figure IV

AMP ,ASM ` ASP EPM ,ASM ` ESP IMP ,AMS ` I SP APM ,EMS ` ESP

EMP ,ASM ` ESP APM ,ESM ` ESP AMP , IMS ` I SP IPM ,AMS ` I SP

AMP , I SM ` I SP EPM , I SM ` OSP OMP ,AMS ` OSP EPM , IMS ` OSP

EMP , I SM ` OSP APM ,OSM ` OSP EMP , IMS ` OSP

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Syllogistic forms in Table 3 are valid syllogistic forms depending onsome conditions. If these conditions hold, then these syllogisticforms are valid.

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Table 3: Conditionally Valid Forms

Table: Conditionally Valid Forms

Figure I Figure II Figure III Figure IV NecessaryCondition

AMP ,ASM ` ISP APM , ESM ` OSP APM , EMS ` OSP S existsEMP ,ASM ` OSP EPM ,ASM ` OSP S exists

AMP ,AMS ` ISP EPM ,AMS ` OSP M existsEMP ,AMS ` OSP M exists

APM ,AMS ` ISP P exists

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Remark

The syllogisms in the Table 2 are referred to simply as syllogisms,those in Table 3 are referred as strengthened syllogisms.

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Mnemonic Names of All Valid Forms

Here are the traditional mnemonic names of 24 of the forms,arranged by figures:

1 2 3 4Barbara Cesare Darapti ∗ Bramantip ∗Celarent Camestres Felapton ∗ CamenesDarii Festino Disamis DimarisFerio Baroco Datisi Fesapo ∗Barbari † Camestrop † Bocardo FresisonCelaront † Cesaro † Ferison Camenop †

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Carroll’s Diagrams and The Elimination Method

Carroll’s diagrams, thought up in 1884, are Venn-type diagramswhere the universes are represented by a square [L. Caroll, 1896].Nevertheless, it is not clear whether Carroll studied his diagramsindependently or as a modification of John Venn’s. Still, Carroll’sscheme looks like a sophisticated method summing up severaldevelopments that have been introduced by researchers stuying inthis area.

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Let X and Y be two terms and let X ′ and Y ′ be the complementsof X and Y , respectively. For two-terms, Carroll divides the squareinto four cells, and he gets the so-called bilateral diagram, asshown in below:

X ′ X

Y ′ X ′Y ′ XY ′

Y X ′Y XY

Each of these four cells can have three possibilities, when weexplain the relations between two terms. They can be 0 or 1 orblank. In this method, 0 means that there is no elementintersection cell of two elements, 1 means that it is not empty andblank cell means that we don’t have any information about thecontent of the cell, therefore it could be 0 or 1.

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X ′ X

Y ′ X ′Y ′ XY ′

Y X ′Y XY

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X ′ X

Y ′ X ′Y ′ XY ′

Y X ′Y XY

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As above method, let X , Y , and M be three terms and X ′, Y ′,and M ′ be their respective complements. To examen all relationsbetween three terms, he added one more square in the middle ofbilateral diagram which is called the trilateral diagram, as thefollowing:

Each cell in a trilateral diagram is marked with a 0, if there is noelement and is marked with a I if it is not empty and another usingof I, it could be on the line where the two cell is intersection, thismeans that at least one of these cells is not empty. So, I isdifferent from 1. In addition to these,if any cell is blank, it has twopossibilities, 0 or I.

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In order to get the conclusion of a syllogism, the data of twopremises are written on a trilateral diagram.

This presentation ismore effective than Venn Diagram method. So, one can extractthe conclusion truer and quicker from trilateral diagram. Underfavour of this method, we transfer the data shown by the trilateraldiagram into a bilateral diagram, involving only two terms thatshould occur in the conclusion and consequently eliminating themiddle term.

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In order to get the conclusion of a syllogism, the data of twopremises are written on a trilateral diagram. This presentation ismore effective than Venn Diagram method. So, one can extractthe conclusion truer and quicker from trilateral diagram.

Underfavour of this method, we transfer the data shown by the trilateraldiagram into a bilateral diagram, involving only two terms thatshould occur in the conclusion and consequently eliminating themiddle term.

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In order to get the conclusion of a syllogism, the data of twopremises are written on a trilateral diagram. This presentation ismore effective than Venn Diagram method. So, one can extractthe conclusion truer and quicker from trilateral diagram. Underfavour of this method, we transfer the data shown by the trilateraldiagram into a bilateral diagram, involving only two terms thatshould occur in the conclusion and consequently eliminating themiddle term.

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This method can be used in accordance with the rules below[L. Carroll, 1896]:

First Rule: 0 and I are fixed up on trilateral diagrams.Second Rule: If the quarter of trilateral diagram contains a ”I” ineither cell, then it is certainly occuppied, and one may mark thecorresponding quarter of the bilateral diagram with a ”1” toindicate that it is occupied.Third Rule: If the quarter of trilateral diagram contains two ”0”s,one in each cell, then it is certainly empty, and one may mark thecorresponding quarter of the bilateral diagram with a ”0” toindicate that it is empty.

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This method can be used in accordance with the rules below[L. Carroll, 1896]:First Rule: 0 and I are fixed up on trilateral diagrams.

Second Rule: If the quarter of trilateral diagram contains a ”I” ineither cell, then it is certainly occuppied, and one may mark thecorresponding quarter of the bilateral diagram with a ”1” toindicate that it is occupied.Third Rule: If the quarter of trilateral diagram contains two ”0”s,one in each cell, then it is certainly empty, and one may mark thecorresponding quarter of the bilateral diagram with a ”0” toindicate that it is empty.

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This method can be used in accordance with the rules below[L. Carroll, 1896]:First Rule: 0 and I are fixed up on trilateral diagrams.Second Rule: If the quarter of trilateral diagram contains a ”I” ineither cell, then it is certainly occuppied, and one may mark thecorresponding quarter of the bilateral diagram with a ”1” toindicate that it is occupied.

Third Rule: If the quarter of trilateral diagram contains two ”0”s,one in each cell, then it is certainly empty, and one may mark thecorresponding quarter of the bilateral diagram with a ”0” toindicate that it is empty.

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This method can be used in accordance with the rules below[L. Carroll, 1896]:First Rule: 0 and I are fixed up on trilateral diagrams.Second Rule: If the quarter of trilateral diagram contains a ”I” ineither cell, then it is certainly occuppied, and one may mark thecorresponding quarter of the bilateral diagram with a ”1” toindicate that it is occupied.Third Rule: If the quarter of trilateral diagram contains two ”0”s,one in each cell, then it is certainly empty, and one may mark thecorresponding quarter of the bilateral diagram with a ”0” toindicate that it is empty.

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The Calculus System SLCD and Its Completeness

In this section, we correspond a set to each possible form of anysyllogistic bilateral diagrams and also define universes of major andminor premises and conclusions in the categorical syllogisms.Moreover, we give a definition of a map which obtains a conclusionfrom two possible forms of premises. Then, we generalize it forconclusion of any two premises and also valid forms in syllogisms.

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Our aim is to construct a complete bridge between Sets andAristotelian Logic :

Table: The Paradigm for the Representation of Syllogistic Arguments byusing Sets

LOGIC DIAGRAMS SETS

PREMISES PropositionsTranslate−−−−−→ Sets

↓CONCLUSION Propositions

Translate←−−−−− Sets

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Our aim is to construct a complete bridge between Sets andAristotelian Logic :

Table: The Paradigm for the Representation of Syllogistic Arguments byusing Sets

LOGIC DIAGRAMS SETS

PREMISES PropositionsTranslate−−−−−→ Sets

↓CONCLUSION Propositions

Translate←−−−−− Sets

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Let X and Y be two terms and their complements are denoted byX ′ and Y ′, respectively. Assume that pi shows a possible form ofany bilateral diagram, such that 1 ≤ i ≤ k , where k is the numberof possible forms of bilateral diagram, as follows:

Table: Bilateral diagram for a quantity relation between X and Y

pi X ′ X

Y ′ n1 n2

Y n3 n4

where n1, n2, n3, n4 ∈ {0, 1}. Given throughout this paper thesymbols R(A), R(E), R(I ) and R(O) represent “All”, ”No”, “Some”and “Some − not” statements, respectively.

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pi X ′ X

Y ′ n1 n2

Y n3 n4

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pi X ′ X

Y ′ n1 n2

Y n3 n4

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”All S are P”

We examine All S are P, it means that there is no element in theintersection of S and P ′ cell. This is shown in the followingbilateral diagram:

Table: Bilateral diagram for ”All S are P”

R(A) =

P ′ P

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”All S are P”

We examine All S are P, it means that there is no element in theintersection of S and P ′ cell. This is shown in the followingbilateral diagram:

Table: Bilateral diagram for ”All S are P”

R(A) =

P ′ P

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From the above Table, we obtain all possible bilateral diagramshaving 0 in the intersection of S and P ′ cell:

Table: Possible forms of ”All S are P”

p1 P ′ P

S ′ 0 0

p2 P ′ P

S ′ 0 0

p3 P ′ P

S ′ 0 1

p4 P ′ P

S ′ 1 0

p5 P ′ P

S ′ 0 1

p6 P ′ P

S ′ 1 0

p7 P ′ P

s ′ 1 1

p8 P ′ P

S ′ 1 1

These tables show all possible forms of “All S are P”.

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From the above Table, we obtain all possible bilateral diagramshaving 0 in the intersection of S and P ′ cell:

Table: Possible forms of ”All S are P”

p1 P ′ P

S ′ 0 0

p2 P ′ P

S ′ 0 0

p3 P ′ P

S ′ 0 1

p4 P ′ P

S ′ 1 0

p5 P ′ P

S ′ 0 1

p6 P ′ P

S ′ 1 0

p7 P ′ P

s ′ 1 1

p8 P ′ P

S ′ 1 1

These tables show all possible forms of “All S are P”.Ibrahim Senturk LC 2018 Regarding Aristotelian Logic as a Sheffer Stroke Basic Algebra39 / 76

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Now in order to correspond bilateral diagrams and sets, let us forma set consisting of numbers which correspond to possible formsthat each bilateral diagram possesses. To do this, first we definethe value which corresponds to the bilateral diagram.

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Definition [A. E. Kulinkovich, 1979]

Let r valj denote the value corresponding to a possible bilateraldiagram pj and ni is the value that the i-th cell possesses, then thevalue of this possible bilateral diagram is calculated by using theformula

r valj =4∑

2(4−i)ni , 1 ≤ j ≤ k,

where k is the number of all possible forms.

Definition

Let Rset be the set of the values which correspond to all possibleforms of any bilateral diagram; that isRset = {r valj : 1 ≤ j ≤ k, k is the number of all possible forms}.The set of all these Rset ’s is denoted by RSet .

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Definition [A. E. Kulinkovich, 1979]

Let r valj denote the value corresponding to a possible bilateraldiagram pj and ni is the value that the i-th cell possesses, then thevalue of this possible bilateral diagram is calculated by using theformula

r valj =4∑

2(4−i)ni , 1 ≤ j ≤ k,

where k is the number of all possible forms.

Definition

Let Rset be the set of the values which correspond to all possibleforms of any bilateral diagram; that isRset = {r valj : 1 ≤ j ≤ k, k is the number of all possible forms}.The set of all these Rset ’s is denoted by RSet .

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The set representations of all categorical propositions as follows:

- All X are Y: It means that the intersection of X and Y ′ isempty set:

R(A) =

X ′ X

Y ′ 0

YThen the set representation of ”All X are Y ” is

Rset(A) = {0, 1, 2, 3, 8, 9, 10, 11}.

- No X are Y: No element in the intersection cell of X and Y

R(E) =

X ′ X

Rset(E) = {0, 2, 4, 6, 8, 10, 12, 14}.

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R(A) =

X ′ X

Y ′ 0

Rset(A) = {0, 1, 2, 3, 8, 9, 10, 11}.

R(E) =

X ′ X

Rset(E) = {0, 2, 4, 6, 8, 10, 12, 14}.

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R(A) =

X ′ X

Y ′ 0

Rset(A) = {0, 1, 2, 3, 8, 9, 10, 11}.

R(E) =

X ′ X

Rset(E) = {0, 2, 4, 6, 8, 10, 12, 14}.

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- Some X are Y: There is at least one element which belongs Xand Y

R(I ) =

X ′ X

Rset(I ) = {1, 3, 5, 7, 9, 11, 13, 15}.

- Some X are not Y: If some elements of X are not Y , thenthey have to be in Y ′.

R(O) =

X ′ X

Y ′ 1

Rset(O) = {4, 5, 6, 7, 12, 13, 14, 15}.

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- Some X are Y: There is at least one element which belongs Xand Y

R(I ) =

X ′ X

Rset(I ) = {1, 3, 5, 7, 9, 11, 13, 15}.

- Some X are not Y: If some elements of X are not Y , thenthey have to be in Y ′.

R(O) =

X ′ X

Y ′ 1

Rset(O) = {4, 5, 6, 7, 12, 13, 14, 15}.

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Example-Validity of AAA

If All S are M and All M are P, then All S are P. This syllogism,called Barbara, is valid. We show this truth by using eliminationmethod from trilateral daigram to bilateral diagram.

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All S are M: it means that the intersection of cell S and M ′ is 0without any condition. It is shown as follows:

R(A) =

S ′ S

M ′ 0

All M are P: it means that the intersection cell of M and P ′ is 0without any condition. It is also shown as follows:

R(A) =

P ′ P

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R(A) =

S ′ S

M ′ 0

R(A) =

P ′ P

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R(A) =

S ′ S

M ′ 0

R(A) =

P ′ P

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R(A) =

S ′ S

M ′ 0

R(A) =

P ′ P

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Now, we input the data on the trilateral diagram:

By the elimination method, we obtain the relation between S andP on the bilateral diagram:

R(A) =

P ′ P

This means ”All S are P”. So, we can say that this syllogism isvalid.

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R(A) =

P ′ P

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R(A) =

P ′ P

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R(A) =

P ′ P

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Example

Let pi be one possible form of the bilateral diagram having arelation between P and M, and pj be one possible form of thebilateral diagram having a relation between S and M. Then, wecan obtain a relation between S and P. We take possible formsgiven as below:

P ′ P

M ′ 0 0

and pj =

S ′ S

M ′ 0 0

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We input the data on the trilateral diagram as follows:

By using the elimination method, we can obtain a relation betweenS and P as below:

P ′ P

S ′ 1 1

r vali = 2 corresponds to possible form pi , and r valj = 3 corresponds

to possible form pj , we obtain that r vall = 12 corresponds to plthat is a possible conclusion.

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P ′ P

S ′ 1 1

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P ′ P

S ′ 1 1

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P ′ P

S ′ 1 1

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P ′ P

S ′ 1 1

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Question

Can we generalize it for all possible bilateral diagrams?

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After these examples, we try to generalize them by formula. So, wedefine an operation and a theorem as follows:

Definition

The syllogistic possible conclusion mapping, denoted ∗, is amapping which gives us the deduction set of possible forms ofmajor and minor premises sets.

Theorem

Let r vali and r valj correspond to the numbers of possible forms of

major and minor premises, respectively. Then, r vali ∗ r valj equals thevalue given by the intersection of row and column numberscorresponding to r vali and r valj in Table 4.

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Definition

Theorem

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Definition

Theorem

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We deal with the operation table below given by Kulinkovich[A. E. Kulinkovich, 1979]. It is used for finding valid syllogisms bymeans of set theoretical representation of bilateral diagrams.

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Table 4: Operation Table

∗ 0 1 2 3 4 8 12 5 10 6 9 7 11 13 14 150 01 1 4 52 2 8 103 3 12 H4 1 4 58 2 8 10

12 3 12 H5 1 4 5 5 5 5 5 5 5

10 2 8 10 10 10 10 10 10 106 3 12 9 6 11 14 7 13 159 3 12 6 9 7 13 11 14 15

7 3 12 13 7 H4 H′3 7 13 H′

111 3 12 14 11 H3 H′

4 11 14 H′2

13 3 12 7 13 7 13 H4 H′3 H′

114 3 12 11 14 11 14 H3 H′

4 H′2

15 3 12 15 15 H1 H2 H1 H2 H

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In the above Table, considering possible conclusion mapping. Butsome possible forms of premises have more than one possibleconclusions, given as below:

H = {6, 7, 9, 11, 13, 14, 15}, H1 = {7, 11, 15}, H ′1 = {6, 7, 9, 11, 13, 15},H2 = {13, 14, 15}, H ′2 = {11, 14, 15}, H3 = {6, 7, 11, 14, 15},

H ′3 = {6, 7, 13, 14, 15}, H4 = {7, 9, 11, 13, 15}, H ′4 = {9, 11, 13, 14, 15}

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Indeed, the possible conclusion is an image of possible premisesunder a mapping.

Definition

Universes of values sets of major premises, minor primises, andconclusions are denoted by Rset

Maj, RsetMin and Rset

Con, respectively.

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Indeed, the possible conclusion is an image of possible premisesunder a mapping.

Definition

Universes of values sets of major premises, minor primises, andconclusions are denoted by Rset

Maj, RsetMin and Rset

Con, respectively.

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Let Rset(k) be an element of Rset

Maj and Rset(l) be an element of Rset

Min.The main problem is what the conclusion of these premises is. Insyllogistic, we have some patterns which are mentioned in Table 2and Table 3 above. Now, we explain them by using bilateraldiagrams with an algebraic approach.

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Definition

The syllogistic mapping, denoted by ~, is a mapping which givesus the conclusion of the major and the minor premises as below:

P ′ P

~S ′ S

P ′ P

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Theorem

Let Rset(k) = {r valk1

, . . . , r setkn} and Rset

(l) = {r vall1, . . . , r vallt

} two setscorresponding to the Major and the Minor premises. Then~ : Rset

Maj ×RsetMin → Rset

Rset(k) ~ Rset

(l) :=n⋃

t⋃i=1

r valkj∗ r valli

is the conclusion of the premises Rset(k) and Rset

Theorem

A syllogism is valid if and only if it is provable in SLCD.

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Theorem

(l) = {r vall1, . . . , r vallt

} two setscorresponding to the Major and the Minor premises. Then~ : Rset

Rset(k) ~ Rset

(l) :=n⋃

t⋃i=1

r valkj∗ r valli

Theorem

A syllogism is valid if and only if it is provable in SLCD.

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For conditional valid forms, we need an addition rule which is”Some X are X”. We can use above Theorem by taking intoconsideration this rule.

Definition

Let SLCD† be the formal system which is obtained from SLCD byaddition of the rule:

` IXXto show the validity of strengthened formulas.

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For conditional valid forms, we need an addition rule which is”Some X are X”. We can use above Theorem by taking intoconsideration this rule.

Definition

Let SLCD† be the formal system which is obtained from SLCD byaddition of the rule:

` IXXto show the validity of strengthened formulas.

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Definition

Let R(k) be the bilateral diagram presentation of the premise. Thetransposition of a premise is the symmetric positions with respectto the main diagonal. It is shown by Trans(R(k)).

Trans : Rset → Rset ,

Rset(k) → Trans(Rset

(k)) = {r valkT

1, . . . , r setkT

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Theorem

(l) = {r vall1, . . . , r vallt

} be two setsto correspond to the Major and the Minor premises values sets andRset

(s) = {r vals1, . . . , r setsm } be set to correspond to the constant set

values which means ”Some S are S”, ”Some M are M” and ”SomeP are P”. Then ~† : Rset

Rset(k)~

†Rset(l) :=

⋃ti=1

⋃mh=1(r valkj

∗ (r varsh∗ r var

lTi)), If S exists,⋃n

⋃ti=1

⋃mh=1(r valkj

∗ (r varli∗ r varsh

)), If M exists,⋃nj=1

⋃ti=1

⋃mh=1((r varsh

∗ r valkTj

) ∗ r varli), If P exists.

(l) under theconditions S exists, M exists or P exists.

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Theorem

A strengthened syllogism is valid if and only if it is provable inSLCD†.

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In this section, we construct a Sheffer stroke basic algebra oncategorical syllogisms by means of set theoretical representation oftheir bilateral diagrams. At first, we define ∧ (meet) and ∨ (join)operators on the set of numbers corresponding to possible form ofbilateral diagrams.

Definition

Let R(k) and R(l) be elements of R. Then the definitions of binaryjoin and meet operations are as follows:

R(k) ∨ R(l) := Rset(k) ∪ Rset

R(k) ∧ R(l) := Rset(k) ∩ Rset

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In this section, we construct a Sheffer stroke basic algebra oncategorical syllogisms by means of set theoretical representation oftheir bilateral diagrams. At first, we define ∧ (meet) and ∨ (join)operators on the set of numbers corresponding to possible form ofbilateral diagrams.

Definition

Let R(k) and R(l) be elements of R. Then the definitions of binaryjoin and meet operations are as follows:

R(k) ∨ R(l) := Rset(k) ∪ Rset

R(k) ∧ R(l) := Rset(k) ∩ Rset

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Question

Which type algebraic structures are constructed by means of settheoretical representation of Aristotelian Logic?

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Theorem

< R,∨,∧ > is a distributive lattice.

Corollary

< RsetMaj,∪,∩ >, < Rset

Min,∪,∩ > and < RsetCon,∪,∩ > are

distributive lattices.

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Now, we define an order relation on Rset as follows:

Rset(k) � Rset

(l) :⇔ Rset(k) ⊆ Rset

Theorem

Rset is partially ordered by the binary relation �.

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Now, we define an order relation on Rset as follows:

Rset(k) � Rset

(l) :⇔ Rset(k) ⊆ Rset

Theorem

Rset is partially ordered by the binary relation �.

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Let (Rset ,�) be a poset. The greatest element of Rset is{0, 1, . . . , 15}, denoted by 1 and the least element is ∅, denoted by0. We notice again that 0 and 0 are different from each other. LetRk be any element of R. Then we have

R(k) ∧ 0 = Rset(k) ∩ ∅ = ∅ = 0

R(k) ∨ 1 = Rset(k) ∪ {0, 1, . . . , 15} = {0, 1, . . . , 15} = 1.

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Definition

The complement of R, denoted by Rc , Rc = {0, 1, 2, . . . , 15} \ R.

< R,∨,∧, 0, 1 > is a bounded lattice.

Corollary

< RsetMaj,∪,∩, 0, 1 >, < Rset

Min,∪,∩, 0, 1 > and< Rset

Con,∪,∩, 0, 1 > are bounded lattices.

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Definition

Corollary

< RsetMaj,∪,∩, 0, 1 >, < Rset

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Definition

Corollary

< RsetMaj,∪,∩, 0, 1 >, < Rset

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< R,∨,∧,c , 0, 1 > is an ortholattice.

Corollary