**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.**