Marko Ursic: MATRICES OF LOGOS (1987)

Logico-Philosophical Essays And Studies

(Abstracts in English)

1. Paradoxes and Hierarchies. The main subject of this introductory essay is the question of logical paradoxes. Syntactical (»logical«) and semantical paradoxes are considered, and two classical methods of avoiding paradoxes are outlined, i.e. the hierarchy of logical types (Russell) and the hierarchy of languages (Tarski). In contrast to these »orthodox« approaches Kripke's »unorthodox« solution of Liar's paradox is presented, and, on the basis of Kripke's theory of truth (1975) a sort of »radical interpretation« is suggested. The principal point of my modest proposal concerning paradoxes is the idea that »truth-value gaps« should be interpreted as »points of transcendence« which denote limits of logic and in the same time place formal logics into philosophical, even metaphysical contexts. Paradoxes are not just meaningless, they are »gateless gates«.

2. An Attempt of Returning to Contents. The paper is a presentation of the main points of C. I. Lewis' modal logic, especially of his idea of the strict implication. The difference between material and strict implication is outlined, and the introduction of strict implication is interpreted as an attempt to take into account »logical contents« of axiomatic systems and formalization in general. The answer whether such an attempt is possible or not is left open, the discussion on this topic is going to be continued in the next chapter.

3. Intensional Implications. The principal approach is comparative: in context of the question of implication this paper compares extensional (Łukasiewicz, Tarski) and intensional (C. I. Lewis, Anderson & Belnap) variants of modal logics. The first section deals with extensional and intensional languages in connection with the possibility of reduction of some classical meta-language concepts (deducibility, entailment etc.) into the object-language of the calculus itself. The second section presents Lewis' motives for introducing the strict implication, and after that – starting from Tarski's extensional definition of possibility – compares extensional vs. intensional approaches, and states the paradoxes of extensionally defined alethic modalities in analogy with paradoxes of the material implication. The third section mentions some other variants of intensional (relevant) implications.

4. Logic and Determinism. This is the largest essay in the present collection, subtitled »A Study of Łukasiewicz' many-valued logics in connection with the interpretation of Aristotle's problem of future contingents«. The main topic of this study is a presentation, an interpretation and a re-actualization of Łukasiewicz' many-valued approaches to the philosophical problem of determinism. In the first section the historical (Aristotelian) background of the problem is outlined and the concept of contingency is formally defined. In the second section some principles of the »basic modal logic« are discussed and formalized in two ways: axiomatically and by matrix method; an example of matrix-verification in the four-valued system is included. The third section discusses the concept of possibility vs. the concept of contingency in their formal presentations. The fourth section introduces the question of future contingents and determinism, in the fifth three-valued logics are discussed, in the sixth Prior's three-valued concept of alternative (in connection with »basic laws of logic«), in the seventh we are dealing with four-valued system, in the eight the discussion of »finite causal sequences« is taking place, and in the last section some philosophical dimensions of many-valued logics are developed and a possible frame of reference for solving the problem of determinism is proposed.

5. Bringing About the Past. On the basis of some papers of Michael Dummett in his book Truth and Other Enigmas (1978) this essay discusses the problem whether it is possible that in some special cases effect precedes and acts upon its cause, i.e. if something lake retroactive causality exists. From such a possibility it could follow a possibility of bringing about the past, of changing personal and historical memory-data etc. In the first section Dummett's concept of the controversy between realism and anti-realism is outlined, the second section deals with five special cases of a hypothetical retroactive causality, in the third section some problems and dilemmas in connection with historical memory and the thesis of several »possible histories« are discussed.

6. De futuris contingentibus in the Medieval Scholastic Philosophy. This is a historical overview of the problem of future contingents in works and thoughts of the main medieval philosophers: Anselm of Canterbury, Peter Abelard, Peter Lombard, Thomas Aquinas, Duns Scotus, Pseudo-Scot and William Ockham. Some theo-philosophical problems concerning pre-destination and free will are discussed.

7. Aristotle's Modal Conversions. In this study two of Aristotle's modal conversions are formally analysed from the standpoint of modern formal logic (in Łukasiewicz' manner): 1) »complementary conversion« (so named by David Ross) and 2) conversion of contingent universal-negative premisses. The paper presents some new results concerning the modal conversions, it shows some already known inconsistencies of Aristotle's modal logic, and it discusses the formal issues from the meta-logical point, stating some analogies with well-known paradoxes and difficulties with iterations of modal prefixes.

8. The Decision Problem and Liar's paradox. This is a formal presentation and analysis of the famous Gödel's discovery of the undecidable self-referential sentence G which demonstrates the principal incompleteness of (sufficiently rich) formal systems. For the purpose of presentation of Gödel's paradox I have chosen a variant which is similar to Kneale's presentation in his Development of Logic and to Tarski, Mostowski and Robinson's analysis in their work Undecidable Theories. (This chapter is a formal introduction to the topics discussed in the next chapter.)

9. The Limits of Isomorphism. This essay is a longer review of D. R. Hofstadter's famous book Gödel, Escher, Bach, An Eternal Golden Braid (1979). It is not just a review, since it develops on Hofstadter's themas several new »counterpoints«. In the first section, the essay begins with an interpretation of Escher's graphic work Print Gallery; I try to include in this analysis the point (topos) of the »mind's eye« (that means: »I«), the irreductible point which, as I argue, cannot be reduced to any program/algorithm. In the second section I discuss the method of isomorphism; the weaving of »Indra's net« (as Hofstadter says), especially with respect to the three main themes of the GEB: Gödel, Escher, Bach. In the third section some principal problems concerning recursivity are discussed, and the question whether some kind of creativity and »freedom« of computers (or so­me artificial intelligence) is possible. This discussion goes on in the fourth section where I argue against Hofstadter and with J. R. Lucas that an isomorphism between human brain (consciousness, mind) and machine can­not be achieved in principle because of the essential self-reference and self-reflexiveness of human thought which cannot be reduced to any recursive program/process. In the last section, the paper terminates with Hofstad­ter's »U-theme«, with zen-koans as »gateless gates«.

10. Anumana Inference in the Indian System Nyaya and the Comparison with Greek Logic. Classical Indian logic was mostly developing in two philosophical systems: Nyaya and Buddhism. In the fast paper in this volume, I present and discuss nyaya-logic, especially its principal inference – anumana, a sort of Indian five-step »syllogism«, Anumana logical structure is analysed in comparison with Aristotle's syllogisms and stoic schemas of inference. Special stress is put on a »curious« singular fact in the middle of anumana, which is a crossing between deductive and inductive reasoning: nyaya-logic offers an original solution of the question what is the relation between singular and universal, which is one of the main questions of philosophy of all times, both in the East and in the West.