Prehistoric Phenomena and Self-referentiality

Abstract

By terms-allowed-in-types capacity, the Logic of Proofs $\textbf{LP}$ enjoys a system of advanced combinatory terms, while including types of the form t:φ(t), which have self-referential meanings. This paper suggests a research on possible $\textbf{S4}$ measures of self-referentiality introduced by this capacity. Specifically, we define “prehistoric phenomena” in $\textbf{G3s}$, a Gentzen-style formulation of modal logic $\textbf{S4}$. A special phenomenon, namely, “left prehistoric loop”, is then shown to be necessary for self-referentiality in realizations of $\textbf{S4}$ theorems in $\textbf{LP}$.