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    Thesis Defense Exam Presentation Thesis Defense Exam Presentation Presentation Transcript

    • Thesis Defense Exam Presentation Development of Fuzzy Syllogistic Algorithms and Applications Distributed Reasoning Approaches Hüseyin Çakır Izmir Institude of Technology huseyincakir@iyte.edu.tr
    • Contents ● Introduction ● Research Approach ● Background ● Structural Analysis of Syllogisms ● Applications for Syllogistic Reasoning ● Conclusion 15/12/10 2
    • 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 3
    • Introduction ● The aim of the thesis was to: ● Use syllogisms as reasoning mechanism. ● Analyze the structural properties of syllogisms. ● ● ● 15/12/10 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. 4
    • 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 5
    • Research Approach ● Aim of the thesis ● Literature survey ● Development ● Application ● Conclusion 15/12/10 6
    • 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 7
    • Background/ Syllogism ● Depending on alternative placements of the objects within the premises, 4 basic types of syllogistic figures are possible. Figure Name I II III IV Major Premise Minor Premise ―――――― Conclusion MP SM ―― SP PM SM ―― SP MP MS ―― SP PM MS ―― SP Example: Figure 1 MAMMALS: MAJOR HUMANS: MINOR PRIMATES: MIDDLE ALL PRIMATES ARE MAMMALS ALL M ARE P ALL HUMANS ARE PRIMATES ALL S ARE M ----------------------------------------------------------------------------------------ALL HUMANS ARE MAMMALS ALL S ARE P 15/12/10 8
    • Background/ Syllogism ● Propositions has a number of dualistic attributes that characterize the propositions. Name Universality Positivity A Universal positive E Universal negative I Particular positive O Particular negative Example: Figure 1 - AAA ALL PRIMATES ARE MAMMALS ALL HUMANS ARE PRIMATES --------------------------------------------ALL HUMANS ARE MAMMALS 15/12/10 ALL M ARE P ALL S ARE M --------------------------------------------ALL S ARE P 9
    • 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 4 Barbara (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 10
    • 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 II III IV Major Premise Minor Premise ―――――― Conclusion 15/12/10 I MP SM ―― SP PM SM ―― SP MP MS ―― SP PM MS ―― SP 11
    • Background/ Syllogism Figure Name II III IV Major Premise Minor Premise ―――――― Conclusion 15/12/10 I MP SM ―― SP PM SM ―― SP MP MS ―― SP PM MS ―― SP 12
    • 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 III FIGURE IV AAA -1 AAO -1 AAE - 1 AAI - 1 ... 15/12/10 FIGURE II AAA - 2 AAO - 2 AAE - 2 AAI - 2 ... AAA - 3 AAO - 3 AAE - 3 AAI - 3 ... AAA - 4 AAO - 4 AAE - 4 AAI - 4 ... 13
    • Background/ Syllogism ● Invalid syllogisms are also one of the most important issue of syllogisms. ● Affirmative conclusion from a negative premise. – ● Existential fallacy. – ● Conclusion I or O while premiss is E or A.[Ex: AAI] Fallacy of exclusive premises. – 15/12/10 Conclusion A or I while premiss is E or O.[Ex: AEA] Two negative premisses. [Ex: EEA] 14
    • Background/ Syllogism ● Fallacy of the undistributed middle. – ● Illicit major/minor. – ● Middle term must be distributed in at least one premiss. 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 P ALL M ARE P (A) Disributed Undistributed ALL M ARE NOT P (E) Distributed Distributed SOME M ARE P (I) 15/12/10 Subject M Undistributed Undistributed SOME M ARE NOT P (O) Undistributed Distributed 15
    • 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 16
    • 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 17
    • Background/ Formal Representation ● The terms in a proposition are related to each other in four different ways. (Set-Theoretic App.) Operator  Proposition  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 Set-Theoretic Representation of Logical Cases 18
    • 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. Possibility Probability 15/12/10 19
    • Background/ Application Areas ● Data mining ● Object-oriented programming ● Semantic Web ● Artificial Intelligence/ Reasoning 15/12/10 20
    • 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 21
    • Structural Analysis of Syllogisms Example: 11 distinct set situations ... ... 15/12/10 41 Set relationships 22
    • Structural Analysis of Syllogisms M P g f a+e a+b a+c d S 15/12/10 23
    • 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 24
    • 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 1 1 1 1 1 15/12/10 0 0 25
    • Structural Analysis of Syllogisms Example: Figure 1 - AAA ALL PRIMATES ARE MAMMALS ALL HUMANS ARE PRIMATES --------------------------------------------ALL HUMANS ARE MAMMALS ALL M ARE P ALL S ARE M --------------------------------------------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 15/12/10 26
    • 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 0 1 0 0 15/12/10 1 1 27
    • 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 15/12/10 28
    • 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 29
    • 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 validateSomeAre DISPLAY valid and invalid cases of the mood VALIDATE 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 30
    • DETERMINE MOOD Structural Analysis of Syllogisms FIGURE 1,2,3,4 PROPOSITION A,E,I,O GENERATE 41 POSSIBLE SET COMBINATIONS SET RELATIONSHIPS INTO ARRAY VALIDATE EVERY PROPOSITION 15/12/10 31
    • 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 32
    • 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 33
    • 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 34
    • Structural Analysis of Syllogisms 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 |-25mood[3]: | 1 | 0 |-25mood[5]: | 1 | 2 |-29mood[6]: | 1 | 2 |-21mood[14]: | 1 | 7 |-21mood[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 35
    • Structural Analysis of Syllogisms 22 21 20 19 18 17 16 15 14 13 12 11 valid invalid 10 9 8 7 6 5 4 3 2 1 0 1 0 15/12/10 3 2 5 4 7 6 9 8 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 36
    • 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 37
    • 15/12/10 Valids for Figure 1: Mood[1]: AAA mood[3]: AAI Mood[11]: AII mood[18]: EAE Mood[20]: EAO Mood[28]: EIO Mood[26]: *EIE Mood[9]: *AIA mood[20]: mood[11]: mood[1]: mood[24]: mood[19]: mood[15]: mood[7]: mood[2]: mood[52]: mood[35]: mood[5]: mood[60]: mood[51]: mood[36]: mood[27]: mood[21]: mood[12]: mood[43]: mood[26]: mood[50]: mood[47]: mood[33]: mood[53]: mood[44]: mood[10]: mood[59]: mood[25]: mood[29]: mood[58]: mood[45]: ● mood[42]: mood[57]: Structural Analysis of Syllogisms Change in conclusion: (Figure 1) 25 20 15 10 valid invalid valid 5 invalid 0 38
    • 15/12/10 Valids for Figure 2: Mood[6]: AEE Mood[8]: AEO Mood[16]: AOO Mood[18]: EAE Mood[20]: EAO Mood[28]: EIO Mood[14]: *AOE Mood[26]: *EIE mood[20]: mood[16]: mood[6]: mood[24]: mood[19]: mood[12]: mood[5]: mood[3]: mood[51]: mood[35]: mood[11]: mood[55]: mood[49]: mood[36]: mood[27]: mood[21]: mood[2]: mood[43]: mood[26]: mood[59]: mood[47]: mood[33]: mood[64]: mood[44]: mood[10]: mood[53]: mood[25]: mood[29]: mood[57]: mood[45]: ● mood[42]: mood[61]: Structural Analysis of Syllogisms Change in conclusion: (Figure 2) 25 20 15 10 valid invalid valid 5 invalid 0 39
    • 15/12/10 Valids for Figure 3: Mood[3]: AAI Mood[11]: AII Mood[20]: EAO Mood[28]: EIO Mood[35]: IAI Mood[52]: OAO Mood[1]: *AAA Mood[9]: *AIA Mood[18]: *EAE Mood[26]: *EIE Mood[33]: *IAA Mood[50]: *OAE mood[35]: mood[20]: mood[3]: mood[36]: mood[24]: mood[19]: mood[8]: mood[4]: mood[40]: mood[6]: mood[1]: mood[55]: mood[31]: mood[22]: mood[17]: mood[13]: mood[9]: mood[43]: mood[26]: mood[54]: mood[37]: mood[30]: mood[53]: mood[44]: mood[10]: mood[48]: mood[29]: mood[63]: mood[58]: mood[45]: ● mood[42]: mood[57]: Structural Analysis of Syllogisms Change in conclusion: (Figure 3) 25 20 15 10 valid invalid 5 valid invalid 0 40
    • 15/12/10 Valids for Figure 4: Mood[3]: AAI Mood[4]: AAO Mood[6]: AEE Mood[8]: AEO Mood[20]: EAO Mood[28]: EIO Mood[35]: IAI Mood[1]: *AAA Mood[2]: *AAE Mood[18]: *EAE Mood[26]: *EIE Mood[33]: *IAA mood[28]: mood[8]: mood[4]: mood[52]: mood[36]: mood[24]: mood[19]: mood[12]: mood[5]: mood[1]: mood[18]: mood[56]: mood[39]: mood[27]: mood[21]: mood[15]: mood[38]: mood[60]: mood[37]: mood[30]: mood[44]: mood[10]: mood[63]: mood[53]: mood[49]: mood[47]: mood[25]: mood[9]: mood[61]: ● mood[57]: mood[45]: mood[42]: Structural Analysis of Syllogisms Change in conclusion: (Figure 4) 20 18 16 14 12 10 8 valid 6 invalid 4 valid 2 invalid 0 41
    • 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 42
    • 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 43
    • Structural Analysis of Syllogism 15/12/10 44
    • Structural Analysis of Syllogism 15/12/10 45
    • Structural Analysis of Syllogism ● Certainly Not: 8 7 6 5 4 INVALID VALID 3 2 1 0 EIA - 2 EIA - 1 EIA - 4 EIA - 3 15/12/10 AIE - 3 AIE - 1 OAA - 3 AOA - 2 EAA - 3 AAE - 1 EAA - 1 AEA - 2 EAA - 2 AAA - 4 AEA - 4 IAE - 3 IAE - 4 AAE - 3 EAA - 4 AAO - 1 EAI - 1 AEI - 2 EAI - 2 AAE - 4 AEI - 4 46
    • Structural Analysis of Syllogism ● Unlikely: 25 20 15 INVALID 10 VALID 5 0 OOA - 1 IIE - 4 IOA - 3 IIA - 2 IOE - 2 EOA - 2 OEE - 4 EOA - 4 IAE - 2 OEE - 1 IEA - 1 OAE - 3 EIE - 3 OAE - 2 EEE - 2 AEE - 1 OOA - 2 IIE - 1 OOE - 1 IOA - 4 IOA - 1 OIA - 4 OEA - 2 EOA - 3 OAA - 1 AOA - 1 OEE - 3 IEA - 4 IEE - 1 IEE - 4 EAA - 1 EEA - 4 AEE - 3 15/12/10 47
    • Structural Analysis of Syllogism Uncertain: ● 3,5 3 2,5 2 1,5 1 0,5 0 AIA - 1 15/12/10 AIO - 1 AIA - 3 AIO - 3 AOA - 3 AOO - 3 48
    • Structural Analysis of Syllogism Likely: ● 25 20 15 INVALID 10 VALID 5 0 OOO - 3 III - 4 IOO - 2 OOI - 1 OOI - 4 EOO - 3 OAO - 4 OEO - 4 IAI - 2 IEO - 1 OAI - 1 EOI - 4 IEI - 3 EEO - 2 EEO - 1 AAO - 3 OIO - 1 III - 1 OII - 2 IIO - 3 OII - 3 EOO - 1 AOO - 4 OEI - 4 OEO - 1 IAO - 4 IEO - 4 EOI - 3 EII - 2 AAI - 2 EEI - 4 OAO - 2 EAI - 3 15/12/10 49
    • Structural Analysis of Syllogism ● Certainly 8 7 6 5 4 INVALID VALID 3 2 1 0 EIO - 2 EIO - 1 15/12/10 EIO - 4 EIO - 3 AII - 3 AII - 1 OAO - 3 IAI - 3 IAI - 4 AOO - 2 EAO - 3 AAA - 1 EAE - 1 AEE - 2 EAE - 2 AAI - 4 AEE - 4 AAI - 3 EAO - 4 AAI - 1 EAO - 1 AEO - 2 EAO - 2 AAO - 4 AEO - 4 50
    • Applications for Syllogistic Reasoning ● During this study various applications developed to check validty of algorithm. ● ● ● 15/12/10 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. 51
    • 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 |-25mood[3]: | 1 | 0 |-25mood[5]: | 1 | 2 |-29mood[6]: | 1 | 2 |-21mood[14]: | 1 | 7 |-21... 15/12/10 52
    • Applications for Syllogistic Reasoning ● Application 2: 15/12/10 53
    • Applications for Syllogistic Reasoning ● Application 3: 15/12/10 54
    • Applications for Syllogistic Reasoning ● Application 4: 15/12/10 55
    • Applications for Syllogistic Reasoning ● Application 5: 15/12/10 56
    • 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 57
    • 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 58
    • 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. 15/12/10 59