Peirce's thesis: logical analysis and ontological consequences

Abstract

C. Peirce put forward the hypothesis that any relations can be reduced to relations whose arity does not exceed three. This hypothesis is closely related to the basic categories of his phenomenology.

In the paper we provide semantic analysis and detailed proof of the following two results.

1. Any first-order theory can be represented in a language containing a finite set of one-place functional symbols. A philosophically important consequence of this is the same expressive power of relational and functional languages and corresponding ontologies.

2. Any first-order theory can be represented in a language containing a single three-place predicate symbol $ U $ and a finite set of individual constants. This confirms Peirce's conjecture about the fundamental role that three-place relationships play. A language whose descriptive symbols contain only individual constants and the only three-place predicate turns out to be a universal language for representing any first-order theories.

There is a structural similarity between the three-place predicate $ U(e,u,x) $ and the three-place predicate $ UM(e,u,x) $, which represents the universal Turing machine. Both can be considered as containers of one-place functions. From this point of view, the predicate $ U $ is an extension of the predicate $ UM $ to any, and not only effectively computable functions.
Proved theorems force us to take a fresh look at many physical theories.

In the theory of relativity, the four-dimensional Minkowski space-time is considered as the fundamental structure, which can be represented as a four-place predicate $ S(x,y,z,t) $. It~follows from the theorem on the existence of the predicate $ U(e,u,x) $ that the four-dimensional Minkowski space-time is not a fundamental physical structure, since it can be reduced to a single three-place relation.