Principles of Construction of Structural Descriptions of Formal Theories Sets

Шиян Т.А. Принципы построения структурных описаний множеств формальных теорий // Смирновские чтения. 4 Международная конференция. М., 2003. С. 185-187.
Shiyan T.A. Principles of Construction of Structural Descriptions of Formal Theories Sets // Smirnov's Readings. 4-th Conference. Moscow, 2003. P. 185-187.

The evolution of logic led to formulating great number of calculi, formal theories and another theory-similar objects. It caused a developing of different methods of comparing of formal theories: method of comparing by deductive potentiality, method of imbedding operations, method of definitional extensions.

But bivalent comparing of theories is not very useful, when we study more multiple sets of formal theories. There are some methods of structural descriptions of sets of formal theories for overcome this weakness. These methods deal with constructing of different structures (trees, lattices etc.) under formal theories. Bivalent comparing, in this way, become the test of presence of connections between formal theories.

Structural descriptions may be visual presented by oriented graphs. These graphical representations have, for not very multiple sets, great mnemonic and heuristic potentialities.

There are two main foundations for comparing of formal theories: by deductive power (deals with sets of theorems) and by expressive power (deals with sets of predicates). Comparing of formal theories by deductive power induce strict order and lattice structures. Foundations of investigations in the field were put by A. Tarsky. Comparing of formal theories by expressive power induce nonstrict order, formal theories unite in classes of equivalence, and lattice structures are appearing under the classes.

Building of structural descriptions of sets of formal theories may be used as method of historical-logical (historical-mathematical) investigations. I have seen formal syllogistics described in Russian literature in 20 last yeas. These are about 50 systems, formulated in more than 10 different formal languages. The results of the studies will be presented in my talk in the sign of graphs of theories.

Methods of structural descriptions may be applied in methodology of sciences for system analysis of scientific knowledge. In this way appearance some new questions: about correlation of structural descriptions done under different base; about expanding of structural descriptions under another theory-similar objects; about quantitative valuations of distance between formal theories; etc.

© Shiyan T.A., 2003.