A Case for Modal Fragmentalism

Abstract

The idea of fragmentalism has been proposed by Kit Fine as a non-standard view of tense realism. This paper examines a modal version of the view, called modal fragmentalism, which combines genuine realism and realism of modality. Modal fragmentalism has been recently discussed by Iaquinto. But unlike Iaquinto, who primarily focused on possibilities de re, in this paper, we focus on expressions of possibilities de dicto. We argue that the chief idea of modal realism should be that different worlds are distinguished not just in terms of how things are differently with respect to each world, but also in terms of how things could have been differently with respect to each world. This demands a realism-oriented semantics for suppositional contents, and more specifically, for conditionals. By deploying a multidimensional semantics for conditionals, we show that there are good reasons to consider modal fragmentalism as a serious approach in metaphysics, which shares many similarities with the fragmentalism of tense.

Appendix

Language

For a set ${\mathscr{A}}$ of atomic sentences p,q,r,…, and a set Ω of nominals w₁,w₂,…, the formulas in language ${\mathscr{L}}$ are defined by the following inductive syntax rule:

$$\phi :: = \bot\ |\ p\ |\ \mathsf{w_{1}} \ |\ \neg\phi\ |\ \phi\wedge\phi\ |\ \mathsf{At\ w_{1}},\ \phi $$

Language ${\mathscr{L}}^{\rightarrow }$ is an extension on ${\mathscr{L}}$, added with the operator → for conditionals, such that ${\mathscr{L}}^{\rightarrow } \supseteq {\mathscr{L}}$, and for any $\phi ,\psi \in {\mathscr{L}}$, $\phi \rightarrow \psi \in {\mathscr{L}}^{\rightarrow }$.

Model

Our multidimensional model M is a tuple 〈W,P,F,V 〉 such that

• 1. W is a finite, non-empty set of points.

• 2. $P\subseteq {\mathscr{P}}(W)$ is a set of (non-suppositional) propositions such that

  • (2a) ∅∈ P

  • (2b) If A ∈ P, then W∖A ∈ P

  • (2c) If A ∈ P and B ∈ P, then A ∩ B ∈ P.

• 3. We call F a set of fragments, or equally, a suppositional space, which can be generated as follows: If there are a total of n propositions A₁,A₂,…,A_n, we let each proposition represent a unique supposition. Then, the suppositional space is the cartesian product F = W × A₁ ×… × A_n.

  Now for each f ∈ F, we can equivalently write it as a pair f = 〈w,w_x〉, composed of a point w (the ‘actual world’) and a suppositional vector $\mathbf {w}_{x} = \langle w_{A_{1}}, w_{A_{2}}, {\ldots } , w_{A_{n}}\rangle$ indicating a complete series of ‘counteractual worlds’. (We call $w_{A_{i}}$ a counteractual world for A_i such that $w_{A_{i}}$ is a point at which A_i is true under supposition.)

• 4. V is valuation function: ${\mathscr{A}}\cup \varOmega \mapsto {\mathscr{P}}(W)$, such that for all atoms $p\in {\mathscr{A}}$, V (p) is a proposition, and for all nominals w_i ∈Ω, V (w_i) is a singleton subset of W, and for any w ∈ W, ∃w_j ∈Ω such that w ∈ V (w_j).

Semantics

The semantic clauses relative to a model M and a fragment 〈w,w_x〉 are given as follows:²⁹ (Relativization to M will be dropped to ease readability.)

• (1) $\llbracket p\rrbracket ^{M,\langle w,\mathbf {w}_{x}\rangle }=1$ iff w ∈ V (p).

• (2) $\llbracket \neg \phi \rrbracket ^{\langle w,\mathbf {w}_{x}\rangle }=1$ iff $\llbracket \phi \rrbracket ^{\langle w,\mathbf {w}_{x}\rangle }=0$.

• (3) $\llbracket \phi \wedge \psi \rrbracket ^{\langle w,\mathbf {w}_{x}\rangle }=1$ iff $\llbracket \phi \rrbracket ^{\langle w,\mathbf {w}_{x}\rangle }=1$ and $\llbracket \psi \rrbracket ^{\langle w,\mathbf {w}_{x}\rangle }=1$.

• (4) $\llbracket \mathsf {w_{1}}\rrbracket ^{\langle w,\mathbf {w}_{x}\rangle } =1$ iff w ∈ V (w₁).

• (5) $\llbracket \mathsf {At\ w_{1}},\ \phi \rrbracket ^{\langle w,\mathbf {w}_{x}\rangle }=1$ iff $\llbracket \phi \rrbracket ^{\langle w',\mathbf {w}_{x}\rangle }=1$ for w^′∈ V (w₁).

Specifically, we write ⟦ϕ⟧^w = 1 if for any suppositional vector w_x we always have $\llbracket \phi \rrbracket ^{\langle w,\mathbf {w}_{x}\rangle }=1$. Then, the formula ϕ can be seen as expressing a non-suppositional proposition under M: |ϕ| = {w : ⟦ϕ⟧^w = 1}. For a sentence ϕ such that |ϕ| = W, we write it as ⊤ and its negation ⊥. The semantics for conditional sentences can be given as follows:

• (6) $\llbracket \phi \rightarrow \psi \rrbracket ^{\langle w,\mathbf {w}_{x}\rangle }=1$ iff

  • (i) either |ϕ| = ∅

  • (ii) or |ϕ|≠∅ and $\llbracket \mathsf {At\ w_{i}},\ \psi \rrbracket ^{\langle w,\mathbf {w}_{x}\rangle }=1$, where w_i nominates the counteractual |ϕ|-world in w_x.

Logical Consequence

To begin with, we define the truth in a model M as follows:

• (a) M⊧ϕ iff there is a fragment 〈w,w_x〉 s.t. $\llbracket \phi \rrbracket ^{M,\langle w,\mathbf {w}_{x}\rangle }=1$.

  The validity of an inference as:

• (b) Σ⊧ϕ iff for any M, if M⊧Σ then M⊧ϕ.

According to our definitions we can make the following observations: The first is that two contradictory sentences can be both true in a model:

• (o1) For some ϕ and M, M⊧ϕ and M⊧¬ϕ.

But no model will ever make their conjunction true:

• (o2) For any ϕ and M, M⊮ϕ ∧¬ϕ.

This follows from a more general feature of our interpretation of fragmentalism (which agrees with Torrengo and Iaquinto (2020)), namely that the following rule of conjunction elimination holds:

• (o3) ϕ ∧ ψ⊧ϕ,ψ.

But the rule of conjunction introduction does not generally hold:

• (o4) ϕ,ψ⊮ϕ ∧ ψ.

This is a desired result since ϕ and ψ might be two statements that are true at each of their respective fragments but are nevertheless never both true at a single fragment.³⁰