Logic for conditional strong historical necessity in branching time and analyses of an argument for future determinism

TL;DR

This paper develops a logic for conditional strong historical necessity in branching time and applies it to a nontheological version of Lavenham's argument for future determinism. It introduces contexts as sets of indefeasible ontic rules that determine acceptable timelines, and it uses an update-semantics approach to evaluate conditional necessities across these timelines. The authors provide formal languages, semantic frameworks, and an axiomatization, and show—via a running Adam/tiger example—that while all premises can be valid, the conclusion need not be, rendering the argument unsound within this framework. By contrasting with existing theories of futural necessity, the work clarifies how context-updating differs from tree-shrinking approaches and sets the stage for richer temporal logics and applications to modal-temporal reasoning.

Abstract

In this paper, we present a logic for conditional strong historical necessity in branching time and apply it to analyze a nontheological version of Lavenham's argument for future determinism. Strong historical necessity is motivated from a linguistical perspective, and an example of it is ``If I had not gotten away, I must have been dead''. The approach of the logic is as follows. The agent accepts ontic rules concerning how the world evolves over time. She takes some rules as indefeasible, which determine acceptable timelines. When evaluating a sentence with conditional strong historical necessity, we introduce its antecedent as an indefeasible ontic rule and then check whether its consequent holds for all acceptable timelines. The argument is not sound by the logic.

Figures (5)

• Figure 1: This figure indicates the situation of Example \ref{['example:tiger']}.
• Figure 2: A pointed model
• Figure 3: A contextualized pointed model
• Figure 4: A pointed model for Example \ref{['example:tiger']}
• Figure 5: Formalization of the nontheological version of Lavenham's argument in $\mathsf{ConSHN\text{-}BT}$.

Theorems & Definitions (37)

• Example 1
• Definition 1: The languages $\Phi_{\mathsf{XY}}$ and $\Phi_{\mathsf{ConSHN\text{-}BT}}$
• Definition 2: Models for $\Phi_{\mathsf{ConSHN\text{-}BT}}$
• Example 2: Pointed models
• Definition 3: Contexts
• Definition 4: Acceptable timelines by contexts
• Definition 5: Contextualized models and contextualized pointed models
• Example 3: Contextualized pointed models
• Definition 6: Semantics for $\Phi_{\mathsf{ConSHN\text{-}BT}}$
• Example 4