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

Ege University

Regarding Aristotelian Logic as a Sheffer StrokeBasic Algebra

Ibrahim SENTURK and Tahsin ONER

[email protected]

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

[email protected]




References

Ege University

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





References

Ege University

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





References

Ege University

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].





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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].





References

Ege University

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.





References

Ege University

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.





References

Ege University


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

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






References

Ege University


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






References

Ege University

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:





References

Ege University

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:





References

Ege University

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:





References

Ege University

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





References

Ege University

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).





References

Ege University

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.





References

Ege University

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





References

Ege University

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)





References

Ege University

Figures

Figure II

A quantity Q1 of P are M (Major Premise)

A quantity Q2 of S are M (Minor Premise)






References

Ege University

Figures

Figure III

A quantity Q1 of M are P (Major Premise)

A quantity Q2 of M are S (Minor Premise)






References

Ege University

Figures

Figure IV

A quantity Q1 of P are M (Major Premise)

A quantity Q2 of M are S (Minor Premise)






References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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





References

Ege University

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





References

Ege University

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





References

Ege University

Remark

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





References

Ege University

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 †





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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.





References

Ege University

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





References

Ege University

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.





References

Ege University

”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

S ′

S 0





References

Ege University

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

S 0 0

p2 P ′ P

S ′ 0 0

S 0 1

p3 P ′ P

S ′ 0 1

S 0 0

p4 P ′ P

S ′ 1 0

S 0 0

p5 P ′ P

S ′ 0 1

S 0 1

p6 P ′ P

S ′ 1 0

S 0 1

p7 P ′ P

s ′ 1 1

S 0 0

p8 P ′ P

S ′ 1 1

S 0 1

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





References

Ege University

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

S 0 0

p2 P ′ P

S ′ 0 0

S 0 1

p3 P ′ P

S ′ 0 1

S 0 0

p4 P ′ P

S ′ 1 0

S 0 0

p5 P ′ P

S ′ 0 1

S 0 1

p6 P ′ P

S ′ 1 0

S 0 1

p7 P ′ P

s ′ 1 1

S 0 0

p8 P ′ P

S ′ 1 1

S 0 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




References

Ege University

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.





References

Ege University

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∑

i=1

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 .





References

Ege University

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

Y ′

Y 0

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





References

Ege University

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

R(I ) =

X ′ X

Y ′

Y 1

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

Y

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





References

Ege University

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.





References

Ege University

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

M

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

M ′

M 0





References

Ege University

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

S ′

S 0

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





References

Ege University

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:

pi =

P ′ P

M ′ 0 0

M 1 1

and pj =

S ′ S

M ′ 0 0

M 1 0





References

Ege University

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:

pl =

P ′ P

S ′ 1 1

S 0 0

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.





References

Ege University

Question

Can we generalize it for all possible bilateral diagrams?





References

Ege University

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.





References

Ege University

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.





References

Ege University

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





References

Ege University

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}





References

Ege University

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.





References

Ege University

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.





References

Ege University

Definition

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

P ′ P

M ′

M

~S ′ S

M ′

M

=

P ′ P

S ′

S





References

Ege University

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

Con

Rset(k) ~ Rset

(l) :=n⋃

j=1

t⋃i=1

r valkj∗ r valli

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

(l) .

Theorem

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





References

Ege University

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.





References

Ege University

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

n}.





References

Ege University

Theorem

Let Rset(k) = {r valk1

, . . . , r setkn} and Rset

(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

Maj ×RsetMin → Rset

Con

Rset(k)~

†Rset(l) :=

⋃n

j=1

⋃ti=1

⋃mh=1(r valkj

∗ (r varsh∗ r var

lTi)), If S exists,⋃n

j=1

⋃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.

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

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





References

Ege University

Theorem

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





References

Ege University

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

(l)

R(k) ∧ R(l) := Rset(k) ∩ Rset

(l)





References

Ege University

Question

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





References

Ege University

Theorem

< R,∨,∧ > is a distributive lattice.

Corollary

< RsetMaj,∪,∩ >, < Rset

Min,∪,∩ > and < RsetCon,∪,∩ > are

distributive lattices.





References

Ege University

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

Rset(k) � Rset

(l) :⇔ Rset(k) ⊆ Rset

(k).

Theorem

Rset is partially ordered by the binary relation �.





References

Ege University

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

and

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





References

Ege University

Definition

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

Lemma

< R,∨,∧, 0, 1 > is a bounded lattice.

Corollary

< RsetMaj,∪,∩, 0, 1 >, < Rset

Min,∪,∩, 0, 1 > and< Rset

Con,∪,∩, 0, 1 > are bounded lattices.





References

Ege University

Lemma

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

Corollary

< RsetMaj,∪,∩,c , 0, 1 >, < Rset

Min,∪,∩,c , 0, 1 > and< Rset

Con,∪,∩,c , 0, 1 > are ortholattices.

Lemma

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

Corollary

< RsetMaj,∪,∩,c , 0, 1 >, < Rset

Min,∪,∩,c , 0, 1 > and< Rset

Con,∪,∩,c , 0, 1 > are orthomodular lattices.





References

Ege University

Question

Why do we construct a Sheffer stroke basic algebra on settheoretical representation of Aristotelian Logic?





References

Ege University

Our main idea

The Sheffer stroke operation has a crucial role for computersystems. To put it more explicitly, it has an useful application inchip technology as it allows to have all diods on the chip formingprocessor in a computer in a uniform manner. Hence, this ischeapher and simpler than to use different diods for other logicalconnectives such as conjunction, disjunction, negation and etc.





References

Ege University

Oner and Senturk defined an alternative signature {|} whichcomposes of only the Sheffer stroke operation[Oner T. and Senturk I., 2017]. Therefore, it is of someimportance to analyze the Sheffer stroke reduct in a general settingfor basic algebras. Now, we give the following axiomatic system forSheffer stroke reduction of basic algebras.





References

Ege University

Definition [Oner T. and Senturk I., 2017]

An algebra (A, |) of type < 2 > is called a Sheffer stroke basicalgebra if it satisfies the following identities:(SH1) (a|(a|a))|(a|a) = a,(SH2) (a|(b|b))|(b|b) = (b|(a|a))|(a|a),(SH3) (((a|(b|b))|(b|b))|(c |c))|((a|(c |c))|(a|(c |c))) = a|(a|a).





References

Ege University

Definition

Let R(k) and R(l) be elements of R. Then the definition of Shefferstroke operation is as follow:

R(k)|R(l) := (1 \ Rset(k)) ∪ (1 \ Rset

(l) )





References

Ege University

Theorem

< R, |, 0, 1 > is a Sheffer stroke basic algebra.





References

Ege University

Corollary

< RsetMaj, |, 0, 1 >, < Rset

Min, |, 0, 1 > and < RsetCon, |, 0, 1 > are

Sheffer stroke basic algebras.





References

Ege University

THANK YOU FOR YOUR ATTENTION ,





References

Ege University

Jonathan Barnes, The Complete Works of Aristotle: TheRevised Oxford Translation, ed., 1984.

Lewis Carroll, Symbolic Logic, Clarkson N. Potter, 1896.

Jan Lukasiewicz, Aristotle’s Syllogistic From the Standpoint ofModern Formal Logic, Oxford University Press, 1957.

Niittymaki, J. and E. Turunen, Traffic signal control onsimilarity logic reasoning, Fuzzy Sets and Systems, vol. 133(2003), pp. 109–131. MR 1952640.

J. Barnes, The Complete Works of Aristotle: The RevisedOxford Translation, 2th edition, Princeton/ Bollingen SeriesLXXI-2, 1984.


http://www.ams.org/mathscinet-getitem?mr=1952640




References

Ege University

J. Lukasiewicz, Aristotle’s Syllogistic From the Standpoint ofModern Formal Logic, 2th edition, Oxford University Press,1957.

L. Caroll, Symbolic Logic, 2th edition, Clarkson N. Potters,1896.

B. A.Kulik, The Logic of Natural Discourse, Nevskiy Dialekt,2001.

E. Turunen, An Algebraic Study of Peterson’s IntermediateSyllogisms, Soft Computing, vol. 18 , no. 12, (2014),pp. 2431–2444. Zbl 1330.03065 .

B. Kumova and H. Cakir, Algorithmic Decision of Syllogisms,Trends in Applied Intelligent Systems, vol. 6097, (2010),pp. 2431–2444.


http://www.emis.de/cgi-bin/MATH-item?1330.03065




References

Ege University

I. Senturk, and T. Oner, A Construction of Heyting Algebra onCategorical Syllogisms, Matematichki Bilten, vol. 40, no. 4,(2016),

I. Senturk, and T. Oner, An Algebraic Analysis of CategoricalSyllogisms by Using Carroll’s Diagrams, Filomat, (Accepted),(2018).

I. Pratt-Hartmann and L. S. Moss, On the ComputationalComplexity of the Numerically Definite Syllogistic and RelatedLogics, Review of Symbolic Logic, vol. 2, no. 4, (2009),pp. 647–683. Zbl 1166.03011 . MR 2395045.

A. E. Kulinkovich, Algorithmization of Reasoning in SolvingGeological Problems, Proceedings of the Methodology ofGeographical Sciences, Naukova Dumka, (1979), pp. 145–161.


http://www.emis.de/cgi-bin/MATH-item?1166.03011

http://www.ams.org/mathscinet-getitem?mr=2395045




References

Ege University

S. L. Kryvyi and A. V. Palagin, To the Analysis of NaturalLanguages Object, Intelligent Processing, (2009), pp. 36–43.

T. Oner and I. Senturk,The Sheffer stroke operation reducts ofbasic algebras, Open Mathematics, 2017, 15(1): 926-935.

G. Boole, An investigation of the laws of thought: on whichare founded the mathematical theories of logic andprobabilities. Dover Publications, 1854.

S. Burris, A Fragment of Boole’s Algebraic Logic Suitable ForTraditional Syllogistic Logic, Department of PureMathematics, University of Waterloo.


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

Documents