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Thesis Defense Exam PresentationDevelopment of Fuzzy Syllogistic Algorithms and Applications Distributed Reasoning Approaches
Hüseyin ÇakırIzmir Institude of Technology
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Contents● Introduction● Research Approach● Background● Structural Analysis of Syllogisms● Applications for Syllogistic Reasoning● Conclusion
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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>>
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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.
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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.
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Research Approach● Aim of the thesis● Literature survey● Development● Application● Conclusion
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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>>
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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
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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
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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)
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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.
Figure Name I II III IVMajor PremiseMinor Premise――――――Conclusion
MPSM――SP
PMSM――SP
MPMS――SP
PMMS――SP
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Background/ Syllogism
Figure Name I II III IVMajor PremiseMinor Premise――――――Conclusion
MPSM――SP
PMSM――SP
MPMS――SP
PMMS――SP
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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...
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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]
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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
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Background/ Reasoning● The syllogism is part of deductive reasoning,
where facts are determined by combining existing statements, in contrast to inductive reasoning.
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Background/ Formal Representation● Formal representation of syllogisms can be
made by using several approaches:● Euler Diagram Representation● Venn Diagram Representation● Linear Representation● ...
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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
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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
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Background/ Application Areas● Data mining● Object-oriented programming● Semantic Web● Artificial Intelligence/ Reasoning
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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.
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Structural Analysis of Syllogisms
Example:
...
...11 distinct setsituations
41Setrelationships
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Structural Analysis of SyllogismsM P
S
a+e
a+ca+b
g f
d
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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
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Structural Analysis of Syllogisms
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
#21 1 0 0 0 1 1 1 1 1
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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
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
0 0 0
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Structural Analysis of Syllogisms● Valid Stiuations:
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
#25 1 1 1 1 1 0 1 0 0
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Structural Analysis of Syllogisms● The above homomorphism represents the
essential data structure of the algorithm for deciding syllogistic moods.
Arithmetic Relation
a+e a+c a+b f+b f+e g+c g+e d+b d+c
#1 1 1 1 1 1 0 1 0 0
#2
... ... ... ... ... ... ... ... ... ...
#41
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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.
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Structural Analysis of Syllogisms
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()
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Structural Analysis of Syllogisms
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
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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.
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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.
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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.
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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
-------------------------------------------------------------------
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Structural Analysis of Syllogisms
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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”.
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Structural Analysis of Syllogisms● Change in conclusion: (Figure 1)
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Valids for Figure 1:Mood[1]: AAAmood[3]: AAIMood[11]: AIImood[18]: EAEMood[20]: EAOMood[28]: EIOMood[26]: *EIEMood[9]: *AIA
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Structural Analysis of Syllogisms● Change in conclusion: (Figure 2)
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Valids for Figure 2:Mood[6]: AEEMood[8]: AEOMood[16]: AOOMood[18]: EAEMood[20]: EAOMood[28]: EIOMood[14]: *AOEMood[26]: *EIE
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Structural Analysis of Syllogisms● Change in conclusion: (Figure 3)
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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
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Structural Analysis of Syllogisms● Change in conclusion: (Figure 4)
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]:
moo
d[4]
:
moo
d[8]
:
moo
d[28
]:
0
2
4
6
8
10
12
14
16
18
20
validinvalidvalidinvalid
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
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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.
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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].
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Structural Analysis of Syllogism
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Structural Analysis of Syllogism
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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
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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
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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
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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
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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
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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.
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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-
...
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Applications for Syllogistic Reasoning● Application 2:
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Applications for Syllogistic Reasoning● Application 3:
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Applications for Syllogistic Reasoning● Application 4:
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Applications for Syllogistic Reasoning● Application 5:
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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.
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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.
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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.