Thesis Defense Exam Presentation

Thesis Defense Exam PresentationDevelopment of Fuzzy Syllogistic Algorithms and Applications Distributed Reasoning Approaches

Hüseyin ÇakırIzmir Institude of Technology

[email protected]

15/12/10 2

Contents● Introduction● Research Approach● Background● Structural Analysis of Syllogisms● Applications for Syllogistic Reasoning● Conclusion

15/12/10 3

Introduction● A syllogism is a logical argument in which

conclusion can be inferred from two other premises.Example:

ALL PRIMATES ARE MAMMALS <<major premiss>>ALL HUMANS ARE PRIMATES <<minor premiss>>---------------------------------------------ALL HUMANS ARE MAMMALS <<conclusion>>

15/12/10 4

Introduction● The aim of the thesis was to:

● Use syllogisms as reasoning mechanism.● Analyze the structural properties of syllogisms. ● Introduce the fuzzy syllogisms, which helps giving

possibilistic values to syllogistic propositions.● Verify the truth of the approach with applications.● Discuss the possibble application areas and

drawbacks of syllogistic reasoning.

15/12/10 5

Introduction● Computational logic can be used to model

syllogistic reasoning, originally developed by Aristotle some 2.300 years ago.

● By modelling syllogisms, it is possibble to analyze the stuctural properties of syllogisms and syllogistic search space.

15/12/10 6

Research Approach● Aim of the thesis● Literature survey● Development● Application● Conclusion

15/12/10 7

Background/ Syllogism● The origin of the logic studies known goes

among ancient Babylonian, Greeks, Indian, Chiese and Islamic cultures.

● Aristotle's theory suggests that in some cases the answer (conclusion) is predictable based on earlier answers which called premisses.

Example:ALL PRIMATES ARE MAMMALS <<major premiss>>ALL HUMANS ARE PRIMATES <<minor premiss>>---------------------------------------------ALL HUMANS ARE MAMMALS <<conclusion>>

15/12/10 8

Background/ Syllogism● Depending on alternative placements of the

objects within the premises, 4 basic types of syllogistic figures are possible.

Example: Figure 1 MAMMALS: MAJOR HUMANS: MINOR PRIMATES: MIDDLEALL PRIMATES ARE MAMMALS ALL M ARE PALL HUMANS ARE PRIMATES ALL S ARE M--------------------------------------------- --------------------------------------------- ALL HUMANS ARE MAMMALS ALL S ARE P

Figure Name I II III IVMajor PremiseMinor Premise――――――Conclusion

MPSM――SP

PMSM――SP

MPMS――SP

PMMS――SP

15/12/10 9

Background/ Syllogism● Propositions has a number of dualistic

attributes that characterize the propositions.

Example: Figure 1 - AAAALL PRIMATES ARE MAMMALS ALL M ARE PALL HUMANS ARE PRIMATES ALL S ARE M--------------------------------------------- --------------------------------------------- ALL HUMANS ARE MAMMALS ALL S ARE P

Name Universality Positivity

A Universal positive

E Universal negative

I Particular positive

O Particular negative

15/12/10 10

Background/ Syllogism● The letters A, E, I, O have been used since the

medieval schools and to memorise valid moods mnemonic names used as follows:

Figure 1 Figure 2 Figure 3 Figure 4Barbara (AAA) Cesare (EAE) Datisi (AII) Calemes (AEE)Celarent (EAE) Camestres

(AEE)Disamis (IAI) Dimatis (IAI)

Darii (AII) Festino (EIO) Ferison (EIO) Fresison (EIO)Ferio (EIO) Baroco (AOO) Bocardo (OAO) Calemos (AEO)Barbari (AAI) Cesaro (EAO) Felapton (EAO) Fesapo (EAO)Celaront (EAO) Camestros

(AEO)Darapti (AAI) Bamalip (AAI)

15/12/10 11

Background/ Syllogism● Aristotle had specified the first three figures.

The 4th figure was discovered in the middle age.

● The first proposition consist of a quantified relationship between the objects M and P, the second proposition of S and M, the conclusion of S and P.


MPSM――SP

PMSM――SP

MPMS――SP

PMMS――SP

15/12/10 12

Background/ Syllogism


MPSM――SP

PMSM――SP

MPMS――SP

PMMS――SP

15/12/10 13

Background/ Syllogism● Since the proposition operator may have 4

values, 64 syllogistic moods are possible for every figure and 256 moods for all 4 figures in total.

FIGURE I FIGURE II FIGURE III FIGURE IVAAA -1AAO -1AAE - 1AAI - 1...

AAA - 2AAO - 2AAE - 2AAI - 2...

AAA - 3AAO - 3AAE - 3AAI - 3...

AAA - 4AAO - 4AAE - 4AAI - 4...

15/12/10 14

Background/ Syllogism● Invalid syllogisms are also one of the most

important issue of syllogisms.● Affirmative conclusion from a negative premise.

– Conclusion A or I while premiss is E or O.[Ex: AEA]● Existential fallacy.

– Conclusion I or O while premiss is E or A.[Ex: AAI]● Fallacy of exclusive premises.

– Two negative premisses. [Ex: EEA]

15/12/10 15

Background/ Syllogism● Fallacy of the undistributed middle.

– Middle term must be distributed in at least one premiss.● Illicit major/minor.

– No term can be distributed in conclusion which is not distributed in premiss.

● Fallacy of necessity.– Exactly three terms, used in same sense.

Statement Subject M Subject P

ALL M ARE P (A) Disributed Undistributed

ALL M ARE NOT P (E) Distributed Distributed

SOME M ARE P (I) Undistributed Undistributed

SOME M ARE NOT P (O)

Undistributed Distributed

15/12/10 16

Background/ Reasoning● The syllogism is part of deductive reasoning,

where facts are determined by combining existing statements, in contrast to inductive reasoning.

15/12/10 17

Background/ Formal Representation● Formal representation of syllogisms can be

made by using several approaches:● Euler Diagram Representation● Venn Diagram Representation● Linear Representation● ...

15/12/10 18

Background/ Formal Representation● The terms in a proposition are related to each

other in four different ways. (Set-Theoretic App.)Operator Proposition Set-Theoretic Representation of Logical Cases

A All S are P

E All S are not P

I Some S are P

O Some S are not P

15/12/10 19

Background/ Fuzzy Logic● Fuzzy logic is reasoning that is approximate

rather than accurate. (opposite of crisp logic)● Fuzzy logic variables can have a truth value

that ranges between 0 and 1.

PossibilityPossibility

Probability

15/12/10 20

Background/ Application Areas● Data mining● Object-oriented programming● Semantic Web● Artificial Intelligence/ Reasoning

15/12/10 21

Structural Analysis of Syllogisms● For three symmetrically intersecting sets there

are in total 11 possible sub-sets in a Venn diagram.

● If symmetric set relationships are relaxed and the three sets are named, for instance with the syllogistic terms P, M and S, then 41 set relationships are possible.

15/12/10 22

Structural Analysis of Syllogisms

Example:

...

...11 distinct setsituations

41Setrelationships

15/12/10 23

Structural Analysis of SyllogismsM P

S

a+e

a+ca+b

g f

d

15/12/10 24

Structural Analysis of Syllogisms● 9 distinct relationships exists between the three

sets P, M and S. ● For instance P∩M is mapped onto 1=a+e and

P-M is mapped onto 4=f+b.

Sub-Set Number 1 2 3 4 5 6 7 8 9

Arithmetic Relation

a+e a+c a+b f+b f+e g+c g+e d+b d+c

Syllogistic Case P∩M M∩S S∩P P-M P-S M-P M-S S-M S-P

15/12/10 25


Sub-Set Number 1 2 3 4 5 6 7 8 9

Arithmetic Relation



#21 1 0 0 0 1 1 1 1 1

15/12/10 26

Structural Analysis of SyllogismsExample: Figure 1 - AAA

ALL PRIMATES ARE MAMMALS ALL M ARE PALL HUMANS ARE PRIMATES ALL S ARE M--------------------------------------------- --------------------------------------------- ALL HUMANS ARE MAMMALS ALL S ARE P

Sub-Set Number 1 2 3 4 5 6 7 8 9

Arithmetic Relation



0 0 0

15/12/10 27

Structural Analysis of Syllogisms● Valid Stiuations:

Sub-Set Number 1 2 3 4 5 6 7 8 9

Arithmetic Relation



#25 1 1 1 1 1 0 1 0 0

15/12/10 28

Structural Analysis of Syllogisms● The above homomorphism represents the

essential data structure of the algorithm for deciding syllogistic moods.

Arithmetic Relation


#1 1 1 1 1 1 0 1 0 0

#2

... ... ... ... ... ... ... ... ... ...

#41

15/12/10 29

Structural Analysis of Syllogisms● The pseudo code of the algorithm for

determining the true and false cases of a given moods is based on selecting the possible set relationships for that mood, out of all 41 possible set relationships.

15/12/10 30


Pseudocode:DETERMINE mood READ figure number {1,2,3,4} READ with 3 proposition ids {A,E,I,O}GENERATE 41 possible set combinations with 9 relationships into an array SetCombi[41,9]={{1,1,1,1,1,1,1,1,1}, ..., {0,1,0,0,1,1,1,1,1}}VALIDATE every proposition with either validateAllAre, validateAllAreNot, validateSomeAreNot or validateSomeAreDISPLAY valid and invalid cases of the moodVALIDATE mood validateAllAre(x,y) //all M are P if(x=='M' && y=='P')CHECK the sets suitable for this mood in setCombi if 1=1 and 2=0 then add this situation as valid if(setCombi[i][0]==1 && setCombi[i][1]==0)//similar for validateAllAreNot(), validateSomeAre(),validateSomeAreNot()

15/12/10 31


FIGURE 1,2,3,4

PROPOSITION A,E,I,ODET

ERM

INE

MO

OD

GENERATE 41 POSSIBLE SET COMBINATIONS

SET RELATIONSHIPS INTO ARRAY

VALIDATE EVERY PROPOSITION

15/12/10 32

Structural Analysis of Syllogisms● Statistics gained from the algorithm mentioned

in previous section.● This algorithm provides some beneficial

statistics about syllogisms which enables understanding the structural behaviours of syllogisms.

15/12/10 33

Structural Analysis of Syllogisms● According to the model there exists 11 distinct

relations among Venn Diagrams that provide determining syllogisms.

● Every mood has 0 to 21 true and 0 to 21 false cases, which is a real subset of the 41 distinct cases.

15/12/10 34

Structural Analysis of Syllogisms● For any given figure the total number of all true

cases is equal to all false cases, ie 328 true and 328 false cases.

● For all 4 syllogistic figures the total number of 4 x 2 x 328 = 2624 cases.

15/12/10 35

Structural Analysis of SyllogismsMOOD # of valids # of invalids valid cases-------------------------------------------------------------------mood[2]: | 0 | 1 |mood[4]: | 0 | 1 |mood[10]: | 0 | 6 |mood[17]: | 0 | 1 |mood[19]: | 0 | 1 |mood[25]: | 0 | 7 |mood[1]: | 1 | 0 |-25-mood[3]: | 1 | 0 |-25-mood[5]: | 1 | 2 |-29-mood[6]: | 1 | 2 |-21-mood[14]: | 1 | 7 |-21-mood[49]: | 2 | 6 |-5—10-…-------------------------------------------------------------------

TOTAL NUMBER OF VALID SUBSETS FOR THIS FIGURE:328

TOTAL NUMBER OF INVALID SUBSETS FOR THIS FIGURE:328

TOTAL NUMBER OF SUBSETS FOR THIS FIGURE:656

-------------------------------------------------------------------

15/12/10 36


01

23

45

67

89

1011

1213

1415

1617

1819

2021

2223

2425

2627

2829

3031

3233

3435

3637

3839

4041

4243

4445

4647

4849

5051

5253

5455

5657

5859

6061

6263

6465

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

validinvalid

15/12/10 37

Structural Analysis of Syllogisms● Reducing fallacies:

Rule 1, “convert E into O since the information in O also contains the information in E”.Rule 2 , “convert A into I since the information in A also contains the information in I”.

15/12/10 38

Structural Analysis of Syllogisms● Change in conclusion: (Figure 1)

moo

d[57

]:

moo

d[42

]:

moo

d[45

]:

moo

d[58

]:

moo

d[29

]:

moo

d[25

]:

moo

d[59

]:

moo

d[10

]:

moo

d[44

]:

moo

d[53

]:

moo

d[33

]:

moo

d[47

]:

moo

d[50

]:

moo

d[26

]:

moo

d[43

]:

moo

d[12

]:

moo

d[21

]:

moo

d[27

]:

moo

d[36

]:

moo

d[51

]:

moo

d[60

]:

moo

d[5]

:

moo

d[35

]:

moo

d[52

]:

moo

d[2]

:

moo

d[7]

:

moo

d[15

]:

moo

d[19

]:

moo

d[24

]:

moo

d[1]

:

moo

d[11

]:

moo

d[20

]:

0

5

10

15

20

25

validinvalidvalidinvalid

Valids for Figure 1:Mood[1]: AAAmood[3]: AAIMood[11]: AIImood[18]: EAEMood[20]: EAOMood[28]: EIOMood[26]: *EIEMood[9]: *AIA

15/12/10 39


moo

d[61

]:

moo

d[42

]:

moo

d[45

]:

moo

d[57

]:

moo

d[29

]:

moo

d[25

]:

moo

d[53

]:

moo

d[10

]:

moo

d[44

]:

moo

d[64

]:

moo

d[33

]:

moo

d[47

]:

moo

d[59

]:

moo

d[26

]:

moo

d[43

]:

moo

d[2]

:

moo

d[21

]:

moo

d[27

]:

moo

d[36

]:

moo

d[49

]:

moo

d[55

]:

moo

d[11

]:

moo

d[35

]:

moo

d[51

]:

moo

d[3]

:

moo

d[5]

:

moo

d[12

]:

moo

d[19

]:

moo

d[24

]:

moo

d[6]

:

moo

d[16

]:

moo

d[20

]:

0

5

10

15

20

25


Valids for Figure 2:Mood[6]: AEEMood[8]: AEOMood[16]: AOOMood[18]: EAEMood[20]: EAOMood[28]: EIOMood[14]: *AOEMood[26]: *EIE

15/12/10 40


moo

d[57

]:

moo

d[42

]:

moo

d[45

]:

moo

d[58

]:

moo

d[63

]:

moo

d[29

]:

moo

d[48

]:

moo

d[10

]:

moo

d[44

]:

moo

d[53

]:

moo

d[30

]:

moo

d[37

]:

moo

d[54

]:

moo

d[26

]:

moo

d[43

]:

moo

d[9]

:

moo

d[13

]:

moo

d[17

]:

moo

d[22

]:

moo

d[31

]:

moo

d[55

]:

moo

d[1]

:

moo

d[6]

:

moo

d[40

]:

moo

d[4]

:

moo

d[8]

:

moo

d[19

]:

moo

d[24

]:

moo

d[36

]:

moo

d[3]

:

moo

d[20

]:

moo

d[35

]:

0

5

10

15

20

25


Valids for Figure 3:Mood[3]: AAIMood[11]: AIIMood[20]: EAOMood[28]: EIOMood[35]: IAIMood[52]: OAOMood[1]: *AAAMood[9]: *AIAMood[18]: *EAEMood[26]: *EIEMood[33]: *IAAMood[50]: *OAE

15/12/10 41


moo

d[42

]:

moo

d[45

]:

moo

d[57

]:

moo

d[61

]:

moo

d[9]

:

moo

d[25

]:

moo

d[47

]:

moo

d[49

]:

moo

d[53

]:

moo

d[63

]:

moo

d[10

]:

moo

d[44

]:

moo

d[30

]:

moo

d[37

]:

moo

d[60

]:

moo

d[38

]:

moo

d[15

]:

moo

d[21

]:

moo

d[27

]:

moo

d[39

]:

moo

d[56

]:

moo

d[18

]:

moo

d[1]

:

moo

d[5]

:

moo

d[12

]:

moo

d[19

]:

moo

d[24

]:

moo

d[36

]:

moo

d[52

]:

moo

d[4]

:

moo

d[8]

:

moo

d[28

]:

0

2

4

6

8

10

12

14

16

18

20


Valids for Figure 4:Mood[3]: AAIMood[4]: AAOMood[6]: AEEMood[8]: AEOMood[20]: EAOMood[28]: EIOMood[35]: IAIMood[1]: *AAAMood[2]: *AAEMood[18]: *EAEMood[26]: *EIEMood[33]: *IAA

15/12/10 42

Structural Analysis of Syllogisms● Fuzzy Syllogisms:

● The results discussed above used same approach as in Aristotle 's, so it decides on syllogisms as valid or invalid which gives strict decisions on syllogisms either name them as true or false.

● But our objective is to utilize the full set of all 256 moods as a fuzzy syllogistic system of possibilistic arguments.

15/12/10 43

Structural Analysis of Syllogism● The truth values for every mood in form of a

truth ration between its true and false cases, so that the truth ratio becomes a real number, normalized within [0, 1].

15/12/10 44

Structural Analysis of Syllogism

15/12/10 45

Structural Analysis of Syllogism

15/12/10 46

Structural Analysis of Syllogism● Certainly Not:

EIA - 1EIA - 2

EIA - 3EIA - 4

AIE - 1AIE - 3

IAE - 3OAA - 3

IAE - 4AOA - 2

AAE - 3EAA - 3

EAA - 4AAE - 1

AAO - 1EAA - 1

EAI - 1 AEA - 2

AEI - 2EAA - 2

EAI - 2AAA - 4

AAE - 4AEA - 4

AEI - 4

0

1

2

3

4

5

6

7

8

INVALIDVALID

15/12/10 47

Structural Analysis of Syllogism● Unlikely:

OOA - 2OOA - 1

IIE - 1IIE - 4

OOE - 1IOA - 3

IOA - 4IIA - 2

IOA - 1IOE - 2

OIA - 4EOA - 2

OEA - 2OEE - 4

EOA - 3EOA - 4

OAA - 1IAE - 2

AOA - 1OEE - 1

OEE - 3IEA - 1

IEA - 4OAE - 3

IEE - 1EIE - 3

IEE - 4OAE - 2

EAA - 1EEE - 2

EEA - 4AEE - 1

AEE - 3

0

5

10

15

20

25

INVALIDVALID

15/12/10 48

Structural Analysis of Syllogism● Uncertain:

AIA - 1 AIO - 1 AIA - 3 AIO - 3 AOA - 3 AOO - 30

0,5

1

1,5

2

2,5

3

3,5

15/12/10 49

Structural Analysis of Syllogism● Likely:

OIO - 1OOO - 3

III - 1III - 4

OII - 2IOO - 2

IIO - 3OOI - 1

OII - 3OOI - 4

EOO - 1EOO - 3

AOO - 4OAO - 4

OEI - 4OEO - 4

OEO - 1IAI - 2

IAO - 4IEO - 1

IEO - 4OAI - 1

EOI - 3EOI - 4

EII - 2IEI - 3

AAI - 2EEO - 2

EEI - 4EEO - 1

OAO - 2AAO - 3

EAI - 3

0

5

10

15

20

25

INVALIDVALID

15/12/10 50

Structural Analysis of Syllogism● Certainly

EIO - 1EIO - 2

EIO - 3EIO - 4

AII - 1AII - 3

IAI - 3OAO - 3

IAI - 4AOO - 2

AAI - 3EAO - 3

EAO - 4AAA - 1

AAI - 1EAE - 1

EAO - 1AEE - 2

AEO - 2EAE - 2

EAO - 2AAI - 4

AAO - 4AEE - 4

AEO - 4

0

1

2

3

4

5

6

7

8

INVALIDVALID

15/12/10 51

Applications for Syllogistic Reasoning● During this study various applications

developed to check validty of algorithm.● Mathematical applications to check validity of

algorithm and to reveal statistics about syllogism.● Application that use syllogistic reasoning in

distributed way.● Use of syllogistic reasoning in object-oriented

programming.

15/12/10 52

Applications for Syllogistic Reasoning● Application 1: Listing all valid/invalid set

situations.MOOD # of valids # of invalids valid cases-------------------------------------------------------------------mood[2]: | 0 | 1 |mood[4]: | 0 | 1 |mood[10]: | 0 | 6 |mood[17]: | 0 | 1 |mood[19]: | 0 | 1 |mood[25]: | 0 | 7 |mood[1]: | 1 | 0 |-25-mood[3]: | 1 | 0 |-25-mood[5]: | 1 | 2 |-29-mood[6]: | 1 | 2 |-21-mood[14]: | 1 | 7 |-21-

...

15/12/10 53

Applications for Syllogistic Reasoning● Application 2:

15/12/10 54


15/12/10 55


15/12/10 56


15/12/10 57

Conclusion● Mathematical properties of the whole syllogistic

system are revealed in detail including applications and statistics.

● It is believed that this thesis has two contributions to the literature, specifically to the search space of syllogisms and to the fuzzification of syllogistic values.

15/12/10 58

Conclusion● The principles that have been developed in this

thesis work can be used as a reference in developing some applications about syllogistic reasoning.

● The reason why it contributes to syllogistic reasoning field is that it shows the whole validity values for all moods in all figures.

15/12/10 59

Conclusion● A computer software, that provides the

necessary aid to the programmer as software editor can also be developed as a future work.

● This will enable the syllogistic reasoning used in applications which will make remarkable contribution to syllogistic reasoning approach.