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A significance-based account of ceteris paribus counterfactuals

Avgerinos Delkos, Marianna Girlando

TL;DR

This paper develops a significance-based account of ceteris paribus counterfactuals by updating centered sphere models according to the relative significance of formulas in the cp-set. It introduces literal weights $\mathbf{w}_{x}(A)$ and a disagreement update $\mathcal{M}^{\Gamma,x}$ that orders nearby worlds by implausibility across the most significant cp-set formulas. The main result shows that validity in the resulting logics $VW^{d}$ and $VC^{d}$ matches Lewis' standard logics $VW$ and $VC$, establishing soundness and completeness. The framework resolves issues with counting-based priorizations, handles nested cp-modalities, and points to extensions such as other cp-set constraints and impossible worlds.

Abstract

When evaluating a counterfactual statement, it is often convenient to specify conditions that ought to be kept unchanged. Formally, this can be done by associating to each counterfactual a ceteris paribus set of formulas, specifying the facts that "ought to be kept unchanged". Ceteris paribus counterfactuals originate in the debate between D. Lewis and Fine in the 1970s, and have been captured in formal accounts. However, these accounts are merely based on 'counting' formulas, and can yield counterintuitive results. In this paper, we develop a novel approach to evaluate ceteris paribus counterfactuals at (weakly) centered sphere models, by taking into account the 'significance' of formulas that ought to be kept unchanged. Hypothetical states that keep the most significant formulas unchanged will be prioritized in the evaluation of a counterfactual. We show that the resulting notion of validity coincides with theoremhood in Lewis' conditional logics VC or VW.

A significance-based account of ceteris paribus counterfactuals

TL;DR

This paper develops a significance-based account of ceteris paribus counterfactuals by updating centered sphere models according to the relative significance of formulas in the cp-set. It introduces literal weights and a disagreement update that orders nearby worlds by implausibility across the most significant cp-set formulas. The main result shows that validity in the resulting logics and matches Lewis' standard logics and , establishing soundness and completeness. The framework resolves issues with counting-based priorizations, handles nested cp-modalities, and points to extensions such as other cp-set constraints and impossible worlds.

Abstract

When evaluating a counterfactual statement, it is often convenient to specify conditions that ought to be kept unchanged. Formally, this can be done by associating to each counterfactual a ceteris paribus set of formulas, specifying the facts that "ought to be kept unchanged". Ceteris paribus counterfactuals originate in the debate between D. Lewis and Fine in the 1970s, and have been captured in formal accounts. However, these accounts are merely based on 'counting' formulas, and can yield counterintuitive results. In this paper, we develop a novel approach to evaluate ceteris paribus counterfactuals at (weakly) centered sphere models, by taking into account the 'significance' of formulas that ought to be kept unchanged. Hypothetical states that keep the most significant formulas unchanged will be prioritized in the evaluation of a counterfactual. We show that the resulting notion of validity coincides with theoremhood in Lewis' conditional logics VC or VW.
Paper Structure (3 sections, 1 theorem, 1 figure)

This paper contains 3 sections, 1 theorem, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. From weight to significance

Key Result

proposition thmcounterproposition

For $A,B$ literals, if $B \mathrel{ \ooalign{ \raisebox{.2ex}{$\blacktriangleleft$}\cr$≤$\cr }}_{x} A$ then $\bar{A} \mathrel{ \ooalign{ \raisebox{.2ex}{$\blacktriangleleft$}\cr$≤$\cr }}_{x}\bar{B}$.

Figures (1)

  • Figure 1: Left: Meaning of atomic formulas and system of spheres $S(x)$ within $\mathcal{M} = \langle W, S, v\rangle$, for $W = \{x, z_1, z_2, y_1, y_2\}$, $S(z_i)=\{\{z_i\}\}$ and $S(y_i)=\{\{y_i\}\}$ for $i \in\{1,2\}$, and $S(x) = \{\{x\},\{x,z_1\},\{x,z_1,z_2\},\{x,z_1,z_2,y_1\},\{x,z_1,z_2,y_1,y_2\}\}$. Moreover, $v(p) = \{z_1, z_2, y_1, y_2\}$; $v(e_1) = v(e_2) = \{z_1, z_2\}$; and $v(l) = v(h) = \{y_1, y_2\}$. Right: Updated systems of spheres $S^{\Gamma,~\!\!x}(x)$ and $S^{\Sigma,~\!\!x}(x)$. Refer to Def. \ref{['def:update']} and \ref{['ex:running 2']}.

Theorems & Definitions (5)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • proposition thmcounterproposition