10.4 THE BOOLEAN ARITHMETICAL RESULTS

Transcription

1 10.4 THE BOOLEAN ARITHMETICAL RESULTS The numerical detailed calculations of these 2 6 = 64 possible problems relative to each of one of the five different hypothesis described above constitutes the following five bulky computation work, each of one has 64 syllogistic cases numbered in sequence from the first, A 1 A 2 F 1 (NR. 1), to the last, O 1 O 2 F 4 (NR. 64), for each one of the five hypothesis (among others), that is: 1 st HYPOTHESIS: 1 st PREMISE is the MAJOR. The MIDDLE, SUBJECT and syllogisms are in a SYMMETRICAL SITUATION meaning a LOGICAL EQUALITY, both in comprehension and extension. (SEE APPENDIX: ANNEX 1 - "ARISTO13.XLS"). 2 nd HYPOTHESIS: 1 st PREMISE is the MAJOR. The MIDDLE, SUBJECT and syllogisms are in an ASYMMETRICAL SITUATION, meaning a LOGICAL IMPLICATION, both in comprehension and in extension (SEE APPENDIX: ANNEX 2 - "ARISTO14.XLS"). 337

2 3 rd HYPOTHESIS: 1 st PREMISE is the MINOR. The MIDDLE, SUBJECT and syllogisms are in a REVERSE situation of a LOGICAL IMPLICATION, both in comprehension and in extension (SEE APPENDIX: ANNEX 3 - "ARISTO15.XLS"). 4 th HYPOTHESIS: 1 st PREMISE is the MAJOR. The MIDDLE TERMS form part of a LOGICAL IMPLICATION, both in comprehension and extension; but the SUBJECT and the PREDICATE TERMS are in a SYMMETRICAL SITUATION, meaning a LOGICAL EQUALITY, both in comprehension and extension, for all the four possible FIGURES (special SYLLOGISTIC MOODS) (SEE APPENDIX: ANNEX 4 - "ARISTO18.XLS"). 5 th HYPOTHESIS: 1 st PREMISE is the MINOR. The MIDDLE TERMS form part of a REVERSE LOGICAL IMPLICATION both in comprehension and extension; but the SUBJECT and PREDICATE TERMS are in a SYMMETRICAL SITUATION, meaning a LOGICAL EQUALITY, both in comprehension and extension for all the four possible FIGURES (special SYLLOGISTIC MOODS) (SEE APPENDIX: ANNEX 5 - "ARISTO19.XLS"). Thus, the final confrontation among these 5x64=320 results obtained by Boolean Arithmetic Equations applied to the respective PREMISES and MOODS (or FIGURES) of the ARISTOTELIAN SYLLOGISMS by the numerical detailed calculations of the tables "ARISTO13.XLS" (1 st HYPOTHESIS), "ARISTO14.XLS" (2 nd HYPOTHESIS), "ARISTO15.XLS" (3 rd HYPOTHESIS), "ARISTO18.XLS" (4 th HYPOTHESIS) and "ARISTO19.XLS" (5 th HYPOTHESIS), are in the table named ARISTO16.XLS, annexed in the APPENDIX of this book (see APPENDIX: ANNEX 6). From the table ARISTO16.XLS annexed (See APPENDIX: ANNEX 6), we make the table ARISTO20.XLS (see APPENDIX: ANNEX 7) which has the fifteen valid results and the table ARISTO21.XLS (see APPENDIX: ANNEX 8) which has the forty-five nonvalid results Thus, after undoubtedly proving through Boolean Arithmetic Equations at least by one of that five hypothesis, it shows that there are fifteen syllogisms of the sort desired; all these VALID SOLUTION are in the following TABLE: 338

3 NUMBER SYLLOGISM LATIN MNEMONIC PROVED BY 1/1 <A1 A2 A3> F1 barbara 1 st and 4 th HYPOTHESIS. 6/2 <A1 E2 E3> F2 camestres 1 st HYPOTHESIS 8/3 <A1 E2 E3> F4 camenes 1 st HYPOTHESIS 9/4 <A1 I2 I3> F1 daraptis 1 st and 3 rd HYPOTHESIS 11/5 <A1 I2 I3> F3 datisis 1 st and 4 th HYPOTHESIS. 14/6 <A1 O2 O3> F2 baroko 1 st HYPOTHESIS 17/7 <E1 A2 E3> F1 celarent 1 st, 2 nd and 4 th HYPOTHESIS 18/8 <E1 A2 E3> F2 cesare 1 st, 2 nd and 4 th HYPOTHESIS 25/9 <E1 I2 O3> F1 ferio 1 st and 4 th HYPOTHESIS 26/10 <E1 I2 O3> F2 festino 1 st and 4 th HYPOTHESIS 27/11 <E1 I2 O3> F3 feriso 1 st and 4 th HYPOTHESIS 28/12 <E1 I2 O3> F4 fresison 1 st and 4 th HYPOTHESIS 35/13 <I1 A2 I3> F3 disamis 1 st, 3 rd and 5 th HYPOTHESIS 36/14 <I1 A2 I3> F4 dimaris 1 st HYPOTHESIS 51/15 <O1 A2 O3> F3 bokardo 1 st and 5 th HYPOTHESIS TABLE 24 Using the same words used by Hilbert & Ackermann [05], we can observe the following: However, our method has not yielded all the Aristotelian syllogisms. Rather, the four syllogisms, Darapti, Bamalip 1, Felapton, and Fesapo, are missing from the list just obtained. This discrepancy is due to the fact that the meaning of the universal affirmative statement ( All A is B ), traditional since Aristotle, is not fully consistent with our interpretation of the formula X Y. According to Aristotle the sentence All A is B is valid only when there are objects which are A. Our deviation from Aristotle in this respect is justified by the mathematical applications of logic, in which the Aristotelian interpretation would not be useful. In fact the following TABLE, shows the four results obtained in the page 5 of the table ARISTO20.XLS, which corresponds to the above mentioned by Hilbert: NUMBER SYLLOGISM LATIN MNEMONIC NOT PROVED 3/1 st <A1 A2 i3> F1 darapti (WITHOUT DEMONSTRATION) 4/2 nd <A1 A2 I3> F4 bramantip 2 (WITHOUT DEMONSTRATION) 19/3 rd <E1 A2 O3> F3 felapton (WITHOUT DEMONSTRATION) 20/4 th <E1 A2 O3> F4 fesapo (WITHOUT DEMONSTRATION) TABLE 25 But, there are other five Aristotelian syllogisms, considered as a second solution of the problem, which appear as NOTE in the pages 1 and 2 of the table ARISTO20.XLS whose NOTES we present in the following TABLE: 1 Also bramantip 339

4 SYLLOGISM LATIN MNEMONIC SOLUTION NOTE/(I) <A1 A2 I3> F1 barbari (NOT FOUND) NOTE/(II) <A1 E2 O3> F2 camestrop (NOT FOUND) NOTE/(III) <A1 E2 O3> F4 camenop (NOT FOUND) NOTE/(IV) <E1 A2 O3> F1 celaront (NOT FOUND) NOTE/(v) <E1 A2 O3> F2 cesaro (NOT FOUND) TABLE 26 It is very important to observe that these former nine case (four from TABLE 25 and five from TABLE 26), have always both premises universal (affirmative or negative) and the conclusion is particular (affirmative or negative). In Carney & Scheer [06], we find the following set of six rules as conditions of validity traditionally stated: Rule 1: The middle term must be distributed at least one once. Rule 2: No term undistributed in the premises may be distributed in the conclusion. Rule 3: If both premises are negative (EE, EO, OE, or OO), no conclusion is possible. Rule 4: If one premises is negative, the conclusion must be negative. Rule 5: If neither premise is negative, the conclusion must be affirmative (A or I) Rule 6: If both premises are universal (AA, EA, or AE), the conclusion cannot be particular (I or O). That is: If a syllogism satisfy all these six rules, it is regarded as valid. If it fails to satisfy one or more of these rules, it is invalid. If a rule does not apply, the rule is satisfied. Thus, all these nine referred syllogisms disobey clearly the above Rule 6. But now we have a question of principles: The results obtained is a consequence of the application of the Boolean Arithmetical Equations; for this reason its results serve to prove and justify the truthfulness of these rules, as we can see if one makes a confronting examination with the results of those fifteen VALID SOLUTION of TABLE 24: From the table ARISTO16.XLS annexed, we make the table ARISTO21.XLS which has the results of the forty-five syllogisms where INVALID SOLUTIONS were found. But, it is very important to point out one of this forty-five cases of INVALID SOLUTION mathematically demonstrated, exactly the first of page 1 (NR. 2) of the table ARISTO21.XLS. It corresponds to the syllogistic mood of the FIGURE 2, and has the first and the second premises universal affirmatives, <A1 A2 M3> F2, where the 2 Also bamalip 340

5 CONCLUSION M3 does not exist (neither A3, E3, I3, nor O3) in any of those five hypothesis adopted. Now, we can use practically the same Boole s words [02] concerning to the present essay: The mathematical condition in question, therefore, - the irreducibility of the final equation to the form 0 = 0" (or in ours Boolean Arithmetic, [0000] 2.{2;X 32 X 31 } = 0), adequately represents the logical condition of there being no middle term, or common medium of comparison, in the given premises. I am not aware that the distinction occasioned by the presence or absence of middle term, is the strict sense here understood, has been noticed by logicians before. The distinction, though real and deserving attention, is indeed by no means an obvious one, and it would have been unnoticed in the present instance but for the peculiarity of its mathematical expression. What appears to be novel in the above case is the proof of the existence of combinations of premises in which there is absolutely no medium of comparison. When such a medium of comparison, or true middle term, does exist, the condition that its quantification in both premises together shall exceed the quantification as a single whole, has been ably and clearly shewn by Professor De Morgan to be necessary to lawful inference (Cambridge Memoirs, Vol. VIII. Part 3). And this is undoubtedly the true principle of the Syllogism, viewed from the standing-point of Arithmetic. Indeed, after studying thoroughly the syllogistic system, Lukasiewicz [03] ends his study with the following unforgettable words: With the solution of this problem the main researches on the syllogistic of Aristotle goes through. Only rest a problem, or better, a mysterious point which bide one s explanation: for withhold all the false expressions of the system it is necessary and sufficient to withhold axiomatically only one false expression, that is, the syllogistic mood of the second figure with universal affirmative premises and conclusion particular affirmative. It s not exist any other appropriate expression for such ending. The explanation of this curious logic fact perhaps might to conduct to new discovery in the field of logic. 341

6 Now this is what happened, with the INVALID SOLUTION mathematically demonstrated on page 1 (NR. 2) of the table ARISTO21.XLS referred above, which has, as we have seen, the first and the second universal affirmative premises, <A1 A2 M3> F2, and the CONCLUSION M3 will not be neither A3, E3, I3, nor O3, in any of those five hypothesis adopted. This study is only a mathematical application of the principles of Boolean Arithmetical Equations to the system of Aristotelian Syllogisms. However, it is possible to use this important branch of human knowledge to make a mathematical connection between LOGIC and COMPUTATION. Perhaps, this fundamental knowledge in Boolean Arithmetic might be used as a set-up in software computing, as Boolean Algebra is used presently in the Ultra Very Large Semiconductor Integrated (UVLSI) circuits of the hardware computing. In the APPENDIX of this book, we have the following ANNEXES above mentioned: ANNEX 1 1 st HYPOTHESIS: 1 st PREMISE is the MAJOR. The MIDDLE, SUBJECT and PREDICATE TERMS of the four FIGURES (special SYLLOGISTIC MOODS) of the syllogisms are in a SYMMETRICAL SITUATION meaning a LOGICAL EQUALITY, both in comprehension and extension. ("ARISTO13.XLS"). ANNEX 2 2 n HYPOTHESIS: 1 st PREMISE is the MAJOR. The MIDDLE, SUBJECT and PREDICATE TERMS of the four FIGURES (special SYLLOGISTIC MOODS) of the syllogisms are in an ASYMMETRICAL SITUATION, meaning a LOGICAL IMPLICATION, both in comprehension and extension ( ARISTO14.XLS"). ANNEX 3 3 rd HYPOTHESIS: 1 st PREMISE is the MINOR. The MIDDLE, SUBJECT and syllogisms are in a REVERSE situation of a LOGICAL IMPLICATION, both in comprehension and extension ("ARISTO15.XLS"). ANNEX 4 4 th HYPOTHESIS: 1 st PREMISE is the MAJOR. The MIDDLE TERMS form part of a LOGICAL IMPLICATION, both in comprehension and extension; but the SUBJECT and the PREDICATE TERMS are in a SYMMETRICAL SITUATION, meaning a LOGICAL EQUALITY, both in comprehension and extension, for all the four possible FIGURES (special SYLLOGISTIC MOODS) ("ARISTO18.XLS"). 342

7 ANNEX 5 5 th HYPOTHESIS: 1 st PREMISE is the MINOR. The MIDDLE TERMS form part of a REVERSE LOGICAL IMPLICATION both in comprehension and extension; but the SUBJECT and PREDICATE TERMS are in a SYMMETRICAL SITUATION, meaning a LOGICAL EQUALITY, both in comprehension and extension for all the four possible FIGURES (special SYLLOGISTIC MOODS) ("ARISTO19.XLS"). ANNEX 6 ARISTO16.XLS (16 pages) ANNEX 7 ARISTO20.XLS (5 pages) ANNEX 8 ARISTO21.XLS (12 pages) BIBLIOGRAPHY [01] De Morgan, A. - Formal Logic: or, The Calculus of Inference, Necessary and Probable, London :Taylor and Walton, Booksellers and Publishers to University College, 28, Upper Gover Streey, 1947, p [02] Boole, G. - - The Mathematical Analysis of Logic Being an essay towards a calculus of deductive reasoning, Cambridge: MacMillan, Barklay, & MacMillan; London: George Bell, 1847; Reprinted by Oxford Basil Blackwell, 1965, p.11, 13 and 43. [03] Lukasiewicz, J., Aristotle s Syllogistic from the Stand point of Modern Formal logic, Clarendom Press, Oxford, The tag referred is the free translation to the English of the last paragraph of the Spanish version of La Silogistica de Aristoteles desde el punto de vista de la Logica Formal Moderna, Editorial Tecnos, Madrid, 1977, pp. 69 and 76. [04] Gardner, C. K., 1958, Logic machines and diagrams, McGraw-Hill Book Company, Inc., New York Toronto London, p.33. [05] Hilbert D. & Ackermann, W., Principles of Mathematical Logic, Chelsea Publishing Company, New York, 1950, p. 53. [06] Carney, J. D., & Scheer, R. K., Fundamentals of Logic, The MacMillan Company, New York / Collier-MacMillan Limited, London, 1964, p