Supervenience arguments under relaxed assumptions

Abstract

When it comes to evaluating reductive hypotheses in metaphysics, supervenience arguments are the tools of the trade. Jaegwon Kim and Frank Jackson have argued, respectively, that strong and global supervenience are sufficient for reduction, and others have argued that supervenience theses stand in need of the kind of explanation that reductive hypotheses are particularly suited to provide. Simon Blackburn’s arguments about what he claims are the specifically problematic features of the supervenience of the moral on the natural have also been influential. But most discussions of these arguments have proceeded under the strong and restrictive assumptions of the S5 modal logic. In this paper we aim to remedy that defect, by illustrating in an accessible way what happens to these arguments under relaxed assumptions and why. The occasion is recent work by Ralph Wedgwood, who seeks to defend non-reductive accounts of moral and mental properties together with strong supervenience, but to evade both the arguments of Kim and Jackson and the explanatory challenge by accepting only the weaker, B, modal logic. In addition to drawing general lessons about what happens to supervenience arguments under relaxed assumptions, our goal is therefore to shed some light on both the virtues and costs of Wedgwood’s proposal.

Appendices

Appendix A.1: proof that ii and SS^Abs are equivalent

We begin by noting that SS^Abs can be broken up into the following two parts (i.e., that it is equivalent to their conjunction):

$$ {\text{SS}}_{{_{ 1} }}^{\text{Abs}} \quad \forall {\text{w}}_{{:{\text{R}}(@,{\text{w}})}} \forall {\text{v}}_{{:{\text{R}}(@,{\text{v}})}} \forall {\text{x}}_{{ \in {\text{D}}\left( {\text{w}} \right)}} \forall {\text{y}}_{{ \in {\text{D}}\left( {\text{v}} \right)}} [\forall {\text{G}}_{{ \in {\text{B}}}} ({\text{Gx}} \equiv {\text{Gy}}) \supset \forall {\text{F}}_{{ \in {\text{A}}}} ({\text{Fx}} \supset {\text{Fy}})] $$

$$ {\text{SS}}_{ 2}^{\text{Abs}} \quad \forall {\text{w}}_{{:{\text{R}}(@,{\text{w}})}} \forall {\text{v}}_{{:{\text{R}}(@,{\text{v}})}} \forall {\text{x}}_{{ \in {\text{D}}\left( {\text{w}} \right)}} \forall {\text{y}}_{{ \in {\text{D}}\left( {\text{v}} \right)}} [\forall {\text{G}}_{{ \in {\text{B}}}} ({\text{Gx}} \equiv {\text{Gy}}) \supset \forall {\text{F}}_{{ \in {\text{A}}}} ({\text{Fy}} \supset {\text{Fx}})] $$

Observe that SS ^Abs₁ and SS ^Abs₂ are notational variants of one another (swapping x for y and w for v, and switching the order of the universal quantifiers and the biconditional). Hence, since SS^Abs is equivalent to their conjunction, it is equivalent to each. So this reduces our problem to that of showing that ii is equivalent to SS ^Abs₁ .

• ii. ∀F_∈A∀G_∈B*[◊∃x(Fx&Gx) ⊃ □∀y(Gy ⊃ Fy)]

⇔ (by possible-world equivalences for ‘□’ and ‘◊’)

• 1 ∀F_∈A∀G_∈B*[∃w_:R(@,w)∃x_∈D(w)(Fx&Gx) ⊃ ∀v_:R(@,v)∀y_∈D(v)(Gy ⊃ Fy)]

⇔ (by quantifier movement rule from predicate logic)

• 2 ∀w_:R(@,w)∀v_:R(@,v)∀F_∈A∀G_∈B*[∃x_∈D(w)(Fx&Gx) ⊃ ∀y_∈D(v)(Gy ⊃ Fy)]

⇔ (by second application of quantifier movement rule from predicate logic)

• 3 ∀w_:R(@,w)∀v_:R(@,v)∀x_∈D(w)∀y_∈D(v)∀F_∈A∀G_∈B*[(Fx&Gx) ⊃ (Gy ⊃ Fy)]

⇔ (by exportation and importation)

• 4 ∀w_:R(@,w)∀v_:R(@,v)∀x_∈D(w)∀y_∈D(v)∀F_∈A∀G_∈B*[(Gx&Gy) ⊃ (Fx ⊃ Fy)]

⇔ (reversing quantifier movement rule from predicate logic)

• 5 ∀w_:R(@,w)∀v_:R(@,v)∀x_∈D(w)∀y_∈D(v)[∃G_∈B*(Gx&Gy) ⊃ ∀F_∈A(Fx ⊃ Fy)]

⇔ (by definition of B* (the set of B-maximal properties))

• 6 ∀w_:R(@,w)∀v_:R(@,v)∀x_∈D(w)∀y_∈D(v)[∀G_∈B(Gx ≡ Gy) ⊃ ∀F_∈A(Fx ⊃ Fy)].$\quad \square$

It is a trivial corollary of the equivalence of ii and SS^Abs that for any B model M, M⊨SS^Abs just in case M⊨ii.

Appendix A.2: proof that for all B models M, M⊨SS^WR just in case M⊨i

As with SS^Abs, we begin by noting that SS^WR can be broken up into the following two parts (i.e., that it is equivalent to their conjunction):

$$ {\text{SS}}_{1}^{\text{WR}} \quad \forall {\text{w}}_{{:{\text{R}}(@,{\text{w}})}} \forall {\text{v}}_{{:{\text{R}}({\text{w}},{\text{v}})}} \forall {\text{x}}_{{ \in {\text{D}}\left( {\text{w}} \right)}} \forall {\text{y}}_{{ \in {\text{D}}\left( {\text{v}} \right)}} [\forall {\text{G}}_{{ \in {\text{B}}}} ({\text{Gx}} \equiv {\text{Gy}}) \supset \forall {\text{F}}_{{ \in {\text{A}}}} ({\text{Fx}} \supset {\text{Fy}})] $$

$$ {\text{SS}}_{ 2}^{\text{WR}} \quad \forall {\text{w}}_{{:{\text{R}}(@,{\text{w}})}} \forall {\text{v}}_{{:{\text{R}}({\text{w}},{\text{v}})}} \forall {\text{x}}_{{ \in {\text{D}}\left( {\text{w}} \right)}} \forall {\text{y}}_{{ \in {\text{D}}\left( {\text{v}} \right)}} [\forall {\text{G}}_{{ \in {\text{B}}}} ({\text{Gx}} \equiv {\text{Gy}}) \supset \forall {\text{F}}_{{ \in {\text{A}}}} ({\text{Fy}} \supset {\text{Fx}})] $$

Unlike SS^Abs, however, SS ^WR₁ and SS ^WR₂ are not equivalent. (This is why we show only that in B models, M⊨i just in case M⊨SS^WR.) So we’ll proceed by first showing that for any world @, SS ^WR₁ holds at @ just in case i. does. So if M⊨SS^WR, then M⊨i, giving us one direction, and if M⊨i, then M⊨SS ^WR₁ , giving us one half of the other.

$$ {\text{SS}}_{1}^{\text{WR}} \quad \forall {\text{w}}_{{:{\text{R}}(@,{\text{w}})}} \forall {\text{v}}_{{:{\text{R}}({\text{w}},{\text{v}})}} \forall {\text{x}}_{{ \in {\text{D}}\left( {\text{w}} \right)}} \forall {\text{y}}_{{ \in {\text{D}}\left( {\text{v}} \right)}} [\forall {\text{G}}_{{ \in {\text{B}}}} ({\text{Gx}} \equiv {\text{Gy}}) \supset \forall {\text{F}}_{{ \in {\text{A}}}} ({\text{Fx}} \supset {\text{Fy}})] $$

⇔ (by the definition of B* (the set of B-maximal properties))

• 1 ∀w_:R(@,w)∀v_:R(w,v)∀x_∈D(w)∀y_∈D(v)[∃G_∈B*(Gx&Gy) ⊃ ∀F_∈A(Fx ⊃ Fy)]

⇔ (by quantifier movement rule from predicate logic)

• 2 ∀w_:R(@,w)∀v_:R(w,v)∀x_∈D(w)∀y_∈D(v)∀F_∈A∀G_∈B*[(Gx&Gy) ⊃ (Fx ⊃ Fy)]

⇔ (by exportation and importation)

• 3 ∀w_:R(@,w)∀v_:R(w,v)∀x_∈D(w)∀y_∈D(v)∀F_∈A∀G_∈B*[Fx ⊃ (Gx ⊃ (Gy ⊃ Fy))]

⇔ (reversing quantifier movement rule from predicate logic)

• 4 ∀w_:R(@,w)∀x_∈D(w)∀F_∈A[Fx ⊃ ∀G_∈B*∀v_:R(w,v)∀y_∈D(v)(Gx ⊃ (Gy ⊃ Fy))]

⇔ (second application of quantifier movement rule from predicate logic)

• 5 ∀w_:R(@,w)∀x_∈D(w)∀F_∈A[Fx ⊃ ∀G_∈B*(Gx ⊃ ∀v_:R(w,v)∀y_∈D(v)(Gy ⊃ Fy))]

⇔ (by possible-world equivalence for ‘□’)

• 6 □∀x∀F_∈A[Fx ⊃ ∀G_∈B*(Gx ⊃ □∀y(Gy ⊃ Fy))]

⇔ (by assumption that ∀x∃!G_∈B*(Gx)))

• i □∀x∀F_∈A[Fx ⊃ ∃G_∈B*(Gx&□∀y(Gy ⊃ Fy))]

What remains, then, is to show that if M⊨i, then M⊨SS ^WR₂ . We’ll show this by showing that if M⊨SS ^WR₁ , then M⊨SS ^WR₂ .

So let @ be any world in M, w be any world such that R(@,w), and v be any world such that R(w,v). To establish M⊨SS ^WR₂ , we seek to show that

$$ \forall {\text{x}}_{{ \in {\text{D}}\left( {\text{w}} \right)}} \forall {\text{y}}_{{ \in {\text{D}}\left( {\text{v}} \right)}} [\forall {\text{G}}_{{ \in {\text{B}}}} ({\text{Gx}} \equiv {\text{Gy}}) \supset \forall {\text{F}}_{{ \in {\text{A}}}} ({\text{Fy}} \supset {\text{Fx}})]. $$

Since M⊨SS ^WR₁ , we know that SS ^WR₁ holds at v—i.e., that (using a and b as bound variables over worlds)

$$ \forall {\text{a}}_{{:{\text{R}}({\text{v}},{\text{a}})}} \forall {\text{b}}_{{:{\text{R}}({\text{w}},{\text{b}})}} \forall {\text{x}}_{{ \in {\text{D}}\left( {\text{a}} \right)}} \forall {\text{y}}_{{ \in {\text{D}}\left( {\text{b}} \right)}} [\forall {\text{G}}_{{ \in {\text{B}}}} ({\text{Gx}} \equiv {\text{Gy}}) \supset \forall {\text{F}}_{{ \in {\text{A}}}} ({\text{Fx}} \supset {\text{Fy}})]. $$

Moreover, since we are assuming that M is a B model, we know that R(v,v) and R(v,w). Hence (substituting v for a and w for b and applying universal instantiation):

$$ \forall {\text{x}}_{{ \in {\text{D}}\left( {\text{v}} \right)}} \forall {\text{y}}_{{ \in {\text{D}}\left( {\text{w}} \right)}} [\forall {\text{G}}_{{ \in {\text{B}}}} ({\text{Gx}} \equiv {\text{Gy}}) \supset \forall {\text{F}}_{{ \in {\text{A}}}} ({\text{Fx}} \supset {\text{Fy}})]. $$

But this is a notational variant on what we want to show: switching x and y:

$$ \forall {\text{y}}_{{ \in {\text{D}}\left( {\text{v}} \right)}} \forall {\text{x}}_{{ \in {\text{D}}\left( {\text{w}} \right)}} [\forall {\text{G}}_{{ \in {\text{B}}}} ({\text{Gy}} \equiv {\text{Gx}}) \supset \forall {\text{F}}_{{ \in {\text{A}}}} ({\text{Fy}} \supset {\text{Fx}})] \quad \square. $$