Russell's Theories of Events and Instants from the Perspective of Point-Free Ontologies in the Tradition of the Lvov-Warsaw School

Abstract

We classify two of Bertrand Russell's theories of events within the point-free ontology. The first of such approaches was presented informally by Russell in ‘The World of Physics and the World of Sense’ (Lecture IV in Our Knowledge of the External World of 1914). Based on this theory, Russell sketched ways to construct instants as collections of events. This paper formalizes Russell's approach from 1914. We will also show that in such a reconstructed theory, we obtain all axioms of Russell's second theory from 1936 and all axioms of Thomason's theory of events from 1989. Russell's work certainly influenced the works of Stanisław Leśniewski, his student Alfred Tarski, and Czesław Lejewski – prominent members of the Lvov-Warsaw School (LWS). We see our work in the tradition of the research of Leśniewski and Tarski. Building on the technical tools developed in this environment and in the spirit of the traditional research of the LWS, we engage here, in particular, with two classic works by Russell on fundamental ontology.

Keywords:

• Point-free ontology
• Russell's theory of events
• Thomason's theory of events
• Tarski's geometry of solids
• Lvov-Warsaw School

Acknowledgments

I am grateful to the anonymous referees of this journal for their helpful comments on an earlier version of this paper.

Notes

1 We will provide pages of the reprint of Russell 1936 in Russell 1956.

2 For any binary relation $R$, $R_1$a and $R_2$ on $\mathsf{\text{U}}$ we define the following relations on $\mathsf{\text{U}}$. The product of $R_1$ and $R_2$ is the relation $R_1\cap R_2$ such that for all $x,y\in \mathsf{\text{U}}$: $x{R}_1\cap R_{2}y$ iff $x{R}_{1}y\wedge x{R}_{1}y$. The difference between $R_1$ and $R_2$ is the relation $R_1\setminus R_2$ such that for all $x,y\in \mathsf{\text{U}}$: $x{R}_1\setminus R_{2}y$ iff $x{R}_{1}y\wedge \mathrm{\neg }x{R}_{1}y$. The complement of R is the relation $\overline{R}$ such that: $x\overline{R}y$ iff $\mathrm{\neg }xRy$. So $R_1\setminus R_2=R_1\cap {\overline{R}}_2$. The converse relation of R is the relation $\breve{R}$ such that: $x\breve{R}y$ iff $yRx$. So $R=\breve{\stackrel{˘}{R}}$. The relative product of $R_1$ and $R_2$ is the relation $R_1\mid R_2$ such that: $x{R}_1\mid R_{2}y$ iff ${\mathrm{\exists }}_{u\in \mathsf{\text{U}}}(x{R}_{1}u\wedge u{R}_{2}y)$. So $(R_1\mid R_2)\stackrel{˘}{}=\breve{R_2}\mid \breve{R_1}$.

3 We have adopted the following convention for marking some formulas. There may be a label to the right of a given formula to indicate what the formula is saying. Furthermore, if the label appears to the right of a given formula, it means that it is assumed as an axiom of a formalized theory, and the given digit indicates the next number of the axiom. We have adopted as axioms only those formulas that are not derivable from other premises.

4 Russell more formally expresses the same at the beginning of Russell 1936.

5 Model 4 in Appendix 2 formally presents Anderson's diagram. See also model 5 in this appendix.

6 Grzegorczyk's theory from 1960 is examined in detail in Gruszczyński and Pietruszczak 2018, 2019.

Additional information

Funding

Work on this paper has been supported by the National Science Centre, Poland [grant number 2021/43/B/HS1/03187].