Bimodal Logics with Contingency and Accident

Abstract

Contingency and accident are two important notions in philosophy and philosophical logic. Their meanings are so close that they are mixed up sometimes, in both daily life and academic research. This indicates that it is necessary to study them in a unified framework. However, there has been no logical research on them together. In this paper, we propose a language of a bimodal logic with these two concepts, investigate its model-theoretical properties such as expressivity and frame definability. We axiomatize this logic over various classes of frames, whose completeness proofs are shown with the help of a crucial schema. The interactions between contingency and accident can sharpen our understanding of both notions. Then we extend the logic to a dynamic case: public announcements. By finding the required reduction axioms, we obtain a complete axiomatization, which gives us a good application to Moore sentences.

Appendix: The Proofs of Propositions 28 and 29

To show Proposition 28, we show that in K4^∇∙, (1) Δϕ→ΔⁿΔ^mϕ (i.e. Δϕ→Δ^n+mϕ), (2) Δϕ→Δ^l∘^kϕ, (3) Δϕ→∘^jΔⁱϕ, (4) Δϕ→∘^h∘^gϕ (i.e. Δϕ→∘^h+gϕ) are all provable, where $1\leq n,m,l,k,j,i,h,g\in \mathbb {N}$. The proofs are shown by induction on n to g. The base case of each item follows from Proposition 27. We only need to consider the inductive step.

For (1), by induction hypothesis, we have ⊩Δϕ→Δ^n+mϕ, to show that ⊩Δϕ→Δ^n+m+ 1ϕ. For this, we have the following proof sequences:

$$\begin{array}{lll} (i)&{\Delta}{\phi}{\to}{\Delta}^{n+m}{\phi}&\text{IH}\\ (ii)&{\Delta}^{n+m}{\phi}{\to} {\Delta}^{n+m + 1}{\phi}&\text{Proposition~23}\\ (iii)&{\Delta}{\phi}{\to}{\Delta}^{n+m + 1}{\phi}&(i)(ii) \end{array} $$

For (2), by induction hypothesis, we have ⊩Δϕ→Δ^l∘^kϕ. We need to show that ⊩Δϕ→Δ^l+ 1∘^kϕ and ⊩Δϕ→Δ^l∘^k+ 1ϕ. For this, we have the following proof sequences:

$$\begin{array}{lll} (i)&{\Delta}{\phi}{\to}{\Delta}^{l}{\circ}^{k}{\phi}&\text{IH}\\ (ii)&{\Delta}^{l}{\circ}^{k}{\phi}{\to}{\Delta}^{l + 1}{\circ}^{k}{\phi}&\text{Proposition~23}\\ (iii)&{\Delta}{\phi}{\to}{\Delta}^{l + 1}{\circ}^{k}{\phi}&(i)(ii) \end{array} $$

$$\begin{array}{lll} (i)&{\Delta}\circ{\phi}{\to}{\Delta}^{l}{\circ}^{k + 1}{\phi}&\text{IH}\\ (ii)&{\Delta}{\phi}{\to}{\Delta}\circ{\phi}&\text{Proposition~27}\\ (iii)&{\Delta}{\phi}{\to}{\Delta}^{l}{\circ}^{k + 1}{\phi}&(i)(ii) \end{array} $$

For (3), by induction hypothesis, we have ⊩Δϕ→∘^jΔⁱϕ. We need to show that ⊩Δϕ→∘^j+ 1Δⁱϕ and ⊩Δϕ→∘^jΔ^i+ 1ϕ. Note that ∘ (ϕ→ψ) ∧∘ϕ ∧ ϕ→∘ ψ (denoted (⋆)) is a validity, thus also a theorem of K4^∇∙. Then we have the following proof sequences:

$$\begin{array}{lll} (i)&{\Delta}{\phi}{\to}{\Delta}{\Delta}{\phi}&\text{Proposition~23}\\ (ii)&{\Delta}{\Delta}{\phi}{\to}{\circ}^{j}{\Delta}^{i + 1}{\phi}&\text{IH}\\ (iii)&{\Delta}{\phi}{\to}{\circ}^{j}{\Delta}^{i + 1}{\phi}&(i)(ii) \end{array} $$

$$\begin{array}{lll} (i)&{\Delta}{\phi}{\to}{\circ}^{j}{\Delta}^{i}{\phi}&\text{IH}\\ (ii)&\circ({\Delta}{\phi}{\to}{\circ}^{j}{\Delta}^{i}{\phi})&(i),\text{R2}\\ (iii)&\circ({\Delta}{\phi}{\to}{\circ}^{j}{\Delta}^{i}{\phi})\land\circ{\Delta}{\phi}\land{\Delta}{\phi}{\to}{\circ}^{j + 1}{\Delta}^{i}{\phi}&(\star)\\ (iv)&\circ{\Delta}{\phi}\land{\Delta}{\phi}{\to}{\circ}^{j + 1}{\Delta}^{i}{\phi}&(ii),(iii),\text{MP}\\ (v)&\circ{\Delta}{\phi}&\text{A4-1}\\ (vi)&{\Delta}{\phi}{\to}{\circ}^{j + 1}{\Delta}^{i}{\phi}&(iv)(v) \end{array} $$

For (4), by induction hypothesis, we have ⊩Δϕ→∘^h+gϕ, to show that ⊩Δϕ→∘^h+g+ 1ϕ. For this, we have the following proof sequences:

$$\begin{array}{lll} (i)&{\Delta}\circ{\phi}{\to}{\circ}^{h+g + 1}{\phi}&\text{IH}\\ (ii)&{\Delta}{\phi}{\to}{\Delta}\circ{\phi}&\text{Proposition~27}\\ (iii)&{\Delta}{\phi}{\to}{\circ}^{h+g + 1}{\phi}&(i)(ii) \end{array} $$

This concludes the proof of Proposition 28.

To show Proposition 29, we show that in K4^∇∙, (1) ⊩Δϕ→(Δ^l∘^k)^mϕ and (2) ⊩Δϕ→(∘^jΔⁱ)ⁿϕ, where $1\leq n,m,l,k,j,i\in \mathbb {N}$. We proceed by induction on m and n, respectively.

• Base step, i.e. m = 1 and n = 1. This follows directly from Proposition 28.

• Inductive step. Assume by induction hypothesis that the proposition holds for the cases m and n, respectively. We need to show the cases m + 1 and n + 1, respectively, i.e., ⊩Δϕ→(Δ^l∘^k)^m+ 1ϕ and ⊩Δϕ→(∘^jΔⁱ)^n+ 1ϕ.

  The proofs are as follows.

  $$\begin{array}{lll} (i)&{\Delta}^{l + 1}{\circ}^{k}{\phi}{\to}({\Delta}^{l}{\circ}^{k})^{m}{\Delta}^{l}{\circ}^{k}{\phi}&\text{IH}\\ (ii)&{\Delta}{\phi}{\to} {\Delta}^{l + 1}{\circ}^{k}{\phi}&\text{Proposition~28}\\ (iii)&{\Delta}{\phi}{\to}({\Delta}^{l}{\circ}^{k})^{m + 1}{\phi}&(i)(ii) \end{array} $$
  $$\begin{array}{lll} (i)&{\Delta}{\circ}^{j}{\Delta}^{i}{\phi}{\to}({\circ}^{j}{\Delta}^{i})^{n}{\circ}^{j}{\Delta}^{i}{\phi}&\text{IH}\\ (ii)&{\Delta}^{i + 1}{\phi}{\to} {\Delta}{\circ}^{j}{\Delta}^{i}{\phi}&\text{Proposition~28}\\ (iii)&{\Delta}{\phi}{\to}{\Delta}^{i + 1}{\phi}&\text{Proposition~28}\\ (iv)&{\Delta}{\phi}{\to}({\circ}^{j}{\Delta}^{i})^{n + 1}{\phi}&(i)-(iii) \end{array} $$