Counterlogicals as Counterconventionals

Abstract

We develop and defend a new approach to counterlogicals. Non-vacuous counterlogicals, we argue, fall within a broader class of counterfactuals known as counterconventionals. Existing semantics for counterconventionals (developed by Einheuser (Philosophical Studies, 127(3), 459–482 (2006)) and (Kocurek et al. Philosophers’ Imprint, 20(22), 1–27 (2020)) allow counterfactuals to shift the interpretation of predicates and relations. We extend these theories to counterlogicals by allowing counterfactuals to shift the interpretation of logical vocabulary. This yields an elegant semantics for counterlogicals that avoids problems with the usual impossible worlds semantics. We conclude by showing how this approach can be extended to counterpossibles more generally.

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Acknowledgements

We are grateful to Bob Beddor, Kelly Gaus, Jens Kipper, Daniel Nolan, Dave Ripley, Rachel Rudolph, Alex Sandgren, Zeynep Soysal, James Walsh, and two anonymous referees for their helpful comments. This paper was presented at the Richard Wollheim Society (2018), the Melbourne Logic Seminar (2018), the Central APA (2019), the Cornell Workshop in Linguistics & Philosophy (2019), the Australian National University (2020), the faculty reading group at the National University of Singapore (2019), and Zeynep Soysal’s hyperintensionality seminar at the University of Rochester (2020). We are grateful to the audience members of all these venues for their feedback.

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Appendix

Appendix

In this A, we establish that the impossible worlds semantics and the expressivist semantics generate the same logic over Ł, i.e., that \(\vDash _{\textsf {i}} = \vDash _{\exp }\). To do this, we establish the following:

Theorem 1

  1. 1.

    For any expressivist model \(\mathcal {E} = \langle W,f \rangle \) and any xIW, there is a impossible worlds model \(\mathcal {E}^{\textsf {i}} = \langle W^{\textsf {i}},P^{\textsf {i}},f^{\textsf {i}},V^{\textsf {i}} \rangle \) and a wWi such that for all ϕ ∈Ł:

    $$ \begin{array}{@{}rcl@{}} \mathcal{E},x \Vdash_{\exp} \phi & \quad{\Leftrightarrow}\quad \mathcal{E}^{\textsf{i}},w \Vdash_{\textsf{i}} \phi. \end{array} $$

    When xCIW, we can take wPi.

  2. 2.

    For any impossible worlds model \(\mathcal {I} = \langle W,P,f,V \rangle \) and any wW, there is a expressivist model \(\mathcal {I}^{\exp } = \langle W^{\exp },f^{\exp } \rangle \) and an \(x \in I_{W^{\exp }}\) such that for all ϕ ∈Ł:

    $$ \begin{array}{@{}rcl@{}} \mathcal{I},w \Vdash_{\textsf{i}} \phi & \quad{\Leftrightarrow}\quad \mathcal{I}^{\exp},x \Vdash_{\exp} \phi. \end{array} $$

    When wP, we can take \(x \in CI_{W^{\exp }}\).

Corollary 1

For any \({\Gamma } \mathrel {\subseteq } \L \) and ϕ ∈Ł, \({\Gamma } \vDash _{\textsf {i}} \phi \) iff \({\Gamma } \vDash _{\exp } \phi \).

It is easiest to establish Theorem 1(a) first.

Proof Proof (Theorem 1(a))

Suppose first that xCIW. Define \(\mathcal {E}^{\textsf {i}} = \langle W^{\textsf {i}},P^{\textsf {i}},f^{\textsf {i}},V^{\textsf {i}} \rangle \) as follows:

  • Wi = IW

  • Pi = W ×{cx}

  • for each yWi and \(X \mathrel {\subseteq } W^{\textsf {i}}\), fi(X,y) = f(X,y)

  • for each yPi, Vi(p,y) = 1 iff wycy(p)

  • for each yPi, Vi(ϕ,y) = 1 iff \(\mathcal {E},y \Vdash _{\exp } \phi \).

Clearly, \(\mathcal {E}^{\textsf {i}}\) is a impossible worlds model and xPi. It suffices to show that for any ϕ and any yIW:

$$ \begin{array}{@{}rcl@{}} \mathcal{E},y \Vdash_{\exp} \phi & \quad {\Leftrightarrow}\quad \mathcal{E}^{\textsf{i}},y \Vdash_{\textsf{i}} \phi. \end{array} $$

If yPi, then by construction, \(\mathcal {E}^{\textsf {i}},y \Vdash _{\textsf {i}} \phi \) iff Vi(ϕ,y) = 1 iff \(\mathcal {E},y \Vdash _{\exp } \phi \). If yPi, then we proceed by induction. The atomic case holds by definition of Vi. The other cases are straightforward since cy = cx is classical and since Pi = W ×{cx}.

Now suppose xCIW. Then we can define \(\mathcal {E}^{\textsf {i}}\) as above except now we take Pi = CIW. Then \(\mathcal {E},x \Vdash _{\exp } \phi \) iff \(\mathcal {E}^{\textsf {i}},x \Vdash _{\textsf {i}} \phi \) by construction of Vi. □

Theorem 1(a) is not terribly surprising in retrospect. All it says is that anything that is i-valid is also \(\exp \)-valid. But i-validity is pretty weak without further constraints. One way to make that clear is to observe that, as far as the logic is concerned, counterfactuals behave exactly like distinct atomic sentences.

Definition 1

An Ł-formula is an S5-formula if it does not contain . An Ł-formula is a counterfactual if its main connective is .

Proposition 1

Let \({\mathscr{M}} = \langle P,i \rangle \) be an S5-model (where \(i(p) \mathrel {\subseteq } P\) for all p ∈At) and let \({\Phi }:{P}\rightarrow \wp ({\mathscr{L}})\) map every wP to a set Φw of counterfactuals. Then there is an impossible worlds model \(\mathcal {I} = \langle {W,P,f,V}\rangle \) such that for any wP:

  1. 1.

    if ϕ is an S5-formula, then \(\mathcal {I},w \Vdash _{\textsf {i}} \phi \) iff \({\mathscr{M}},w \Vdash _{\textbf {S5}} \phi \)

  2. 2.

    if ψ is a counterfactual, then \(\mathcal {I},w \Vdash _{\textsf {i}} \psi \) iff ψ ∈Φw.

Proof

WLOG, we may assume that P is disjoint from Ł and from (Ł × P). Define \(\mathcal {I} = \langle P \cup \L \cup (\L \times P),P,f,V \rangle \), where:

  • for each p ∈At and wP, V (p,w) = 1 iff wi(p)

  • for each ϕ ∈Ł and α ∈Ł, V (ϕ,α) = 1 iff α = ϕ

  • for each ϕ ∈Ł and 〈α,w〉∈ (Ł × P), V (ϕ,〈α,w〉) = 1 iff

  • f is any selection function with the following property: if X ∩Ł = {α} and wP, then f(X,w) = {〈α,w〉}.

It is easy to establish (i) by induction. As for (ii), note that \(\llbracket {\alpha }\rrbracket ^{\mathcal {I}} \cap \L = \{\alpha \}\), so \(f(\llbracket {\alpha }\rrbracket ^{\mathcal {I}},w) = \{\langle {\alpha ,w}\rangle \}\). Hence, iff \(\mathcal {I},\langle {\alpha ,w}\rangle \Vdash _{\textsf {i}} \beta \), i.e., V (β,〈α,w〉) = 1, which holds iff . □

Corollary 2

Let 𝜃 be any consistent S5-formula, and let 𝜃 be the result of simultaneously uniformly substituting one or more atomic sentences in 𝜃 for distinct counterfactuals. Then 𝜃 is i-satisfiable.

Proof

Let \({q}_1, \dots , {q}_n\) be the atomics in 𝜃 that are substituted for distinct counterfactuals \({\psi }_1, \dots , {\psi }_n\) resulting in 𝜃. Since 𝜃 is consistent, it is S5-satisfiable. Let \({\mathscr{M}},w \Vdash _{\textbf {S5}} \theta \). For each \(v \in W^{{\mathscr{M}}}\), define:

$$ \begin{array}{@{}rcl@{}} {\Phi}_v \mathrel{:=} \{\psi_{i} | \mathcal{M},v \Vdash_\textbf{S5} q_i\} \end{array} $$

By Proposition 1, this guarantees us an S5-equivalent impossible worlds model \(\mathcal {I}\) such that \(\mathcal {I},v \Vdash _{\textsf {i}} \psi \) iff ψ ∈Φv where ψ is a counterfactual. Moreover, in this model, . And since \(\mathcal {I},w \Vdash _{\textsf {i}} \theta \), it follows that \(\mathcal {I},w \Vdash _{\textsf {i}} \theta ^{*}\). □

Corollary 2 effectively says that there are no non-trivial valid inferences governing counterfactuals in the impossible worlds semantics: any inference with counterfactuals that’s i-valid is already S5-valid.

Theorem 1(b) is harder to establish. The main issue is that while hyperconventions are allowed to redefine the semantic value of the boolean connectives, they cannot touch the semantics of . But in the impossible worlds semantics, any set of Ł-formulas is satisfied at some (perhaps impossible) world in some model, including those containing counterfactuals. Thus, if we are to establish Theorem 1(b), we need to establish the expressivist analogue of Proposition 1. Indeed, this can be done, though the proof is more involved.

Proposition 2

Let \({\mathscr{M}} = \langle W,i \rangle \) be an S5-model and let \({\Phi }:{P}\rightarrow \wp ({\mathscr{L}})\) map every wW to a set Φw of counterfactuals. Then there is a expressivist model \(\mathcal {E} = \langle W,f \rangle \) and a classical hyperconvention c such that for any wW:

  1. (i)

    if ϕ is an S5-formula, then \(\mathcal {E},w,c \Vdash _{\exp } \phi \) iff \({\mathscr{M}},w \Vdash _{\textbf {S5}} \phi \)

  2. (ii)

    if ψ is a counterfactual, then \(\mathcal {E},w,c \Vdash _{\exp } \psi \) iff ψ ∈Φw.

Proof

Since S5 is invariant under bisimulation contraction (and so, invariant under duplication of worlds), we may assume WLOG that W is infinite. We define c simply as the classical hyperconvention over W where c(p) = i(p) for all p ∈At.

We now set out to define f. Fix an arbitrary w0W. Let \({h}:{{\mathscr{L}}}\rightarrow {W - \{w_{0},w\}}\) be a bijection. We’ll write wϕ in place of h(ϕ) throughout. Now, let \({\Gamma } \subseteq \L \). Define the hyperconvention cΓ as follows (where and \(\circ \in \{\wedge ,\vee ,\rightarrow \}\)):

$$ \begin{array}{@{}rcl@{}} c_{\Gamma}(p) & =& { \begin{cases} \{w_p,w_0\} & \text{if } p \in {\Gamma} \\ \{w_p\} & \text{otherwise} \end{cases}} \\ c_{\Gamma}(\star)(X) & =& { \begin{cases} \{w_{\star\phi} ~|~ w_\phi \in X\} \cup \{w_0\} & \parbox[t]{.5\textwidth}{if $\star\phi \in {\Gamma}$ whenever $w_{\phi} \in X$} \\ \{w_{\star\phi} ~|~w_\phi \in X\} & \text{otherwise} \end{cases}} \\ c_{\Gamma}(\mathrel{\circ})(X,Y) & {=}& { {{\begin{cases} \{w_{\phi \mathrel{\circ} \psi} ~|~ w_\phi \in X \text{ and } w_\psi \in Y\} \cup \{w_0\} & \parbox[t]{.35\textwidth}{if $\phi \mathrel{\circ} \psi \in {\Gamma}$ whenever $w_{\phi} \in X$ and $w_{\psi} \in Y$} \\ \{w_{\phi \mathrel{\circ} \psi} ~|~w_\phi \in X \text{ and } w_\psi \in Y\} & \text{otherwise} \end{cases}}}} \end{array} $$

Let . Define f as follows:

Let \(\mathcal {E} = \langle W,f \rangle \). It is easy to check that (i) holds by induction. So we just need to establish (ii). First, some intermediate claims:

Claim 1

For any Γ and any ϕ,ψ: \(\mathcal {E},w_{\phi },c_{\Gamma } \Vdash \psi \) iff ϕ = ψ.

Proof

By induction. The atomic case holds by definition of cΓ. The cases for the connectives is straightforward. For the counterfactual, iff \(f(\llbracket {\alpha }\rrbracket ^{\mathcal {E}},w_{\phi },c_{{\Gamma }}) \subseteq \llbracket {\beta }\rrbracket ^{\mathcal {E}}\). By induction hypothesis, \(\langle {w_{\gamma },c_{{\Gamma }}}\rangle \in \llbracket {\beta }\rrbracket ^{\mathcal {E}}\) iff γ = β. Hence, \(\llbracket {\beta }\rrbracket ^{\mathcal {E}} \neq I_{W}\), which means \(f(\llbracket {\alpha }\rrbracket ^{\mathcal {E}},w_{\phi },c_{{\Gamma }}) \subseteq \llbracket {\beta }\rrbracket ^{\mathcal {E}}\) iff \(f(\llbracket {\alpha }\rrbracket ^{\mathcal {E}},w_{\phi },c_{{\Gamma }}) = \{\langle {w_{\beta },c_{{\Gamma }}}\rangle \}\), which holds iff . But again by induction hypothesis, \(\langle {w_{\alpha },c_{{\Gamma }}}\rangle \in \llbracket {\alpha }\rrbracket ^{\mathcal {E}}\). Thus, iff . □

Claim 2

For any Γ and any ϕ: \(\mathcal {E},w_{0},c_{\Gamma } \Vdash \phi \) iff ϕ ∈Γ.

Proof

By induction. The atomic case holds by definition of cΓ. The cases for the connectives is straightforward using Claim 1 and the inductive hypothesis. For the counterfactual, iff \(f(\llbracket {\alpha }\rrbracket ^{\mathcal {E}},w_{0},c_{{\Gamma }}) \subseteq \llbracket {\beta }\rrbracket ^{\mathcal {E}}\). By Claim 1, \(\langle {w_{\gamma },c_{{\Gamma }}}\rangle \in \llbracket {\alpha }\rrbracket ^{\mathcal {E}}\) iff γ = α. So \(f(\llbracket {\alpha }\rrbracket ^{\mathcal {E}},w_{0},c_{{\Gamma }}) = \{\langle {w_{0},c_{{\Gamma }^{\alpha }}}\rangle \}\). Hence, \(\mathcal {E},w_{0},c_{{\Gamma }} \Vdash \) iff \(\mathcal {E},w_{0},c_{{\Gamma }^{\alpha }} \Vdash \beta \). But again by induction hypothesis, this holds iff β ∈Γα, i.e., . □

We are now ready to prove (ii). iff \(f(\llbracket {\alpha }\rrbracket ^{\mathcal {E}},w,c) \subseteq \llbracket {\beta }\rrbracket ^{\mathcal {E}}\). By Claim 1, \(\langle w_{\gamma },c_{{\Phi }_{w}} \rangle \in \llbracket {\alpha }\rrbracket ^{\mathcal {E}}\) iff γ = α. Hence, \(f(\llbracket {\alpha }\rrbracket ^{\mathcal {E}},w,c) = \{\langle {w_{0},c_{{\Phi }_{w}^{\alpha }}}\rangle \}\). So iff \(\mathcal {E},w_{0},c_{{\Phi }_{w}^{\alpha }} \Vdash \beta \), which by Claim 2 holds iff \(\beta \in {\Phi }_{w}^{\alpha }\), i.e., . □

Corollary 3

Let 𝜃 be any consistent S5-formula, and let 𝜃 be the result of uniformly substituting one or more atomic sentences in 𝜃 for distinct counterfactuals. Then 𝜃 is \(\exp \)-satisfiable.

Now we can establish Theorem 1(b):

Proof Proof (Theorem 1(b))

Let \(\mathcal {I} = \langle W,P,f,V \rangle \) and first let wP. Let:

$$ \begin{array}{@{}rcl@{}} {\Phi} & {=}& \{\phi ~|~ \phi \text{ is an $\textbf{S5}$-formula and } \mathcal{I},w \Vdash_\textsf{i} \phi] \\ {\Psi} & {=}& \{\phi~|~\phi \text{ is a counterfactual and } \mathcal{I},w \Vdash_\textsf{i} \phi]. \end{array} $$

By Proposition 2, there is a expressivist model \(\mathcal {I}^{\exp } = \langle W,f^{\exp } \rangle \) and a classical hyperconvention c such that \(\mathcal {I}^{\exp },w,c \Vdash _{\exp } {\Phi } \cup {\Psi }\) and if ϕ is a counterfactual not in Ψ, \(\mathcal {I}^{\exp },w,c \nVdash _{\exp } \phi \). Hence, by a simple induction, \(\mathcal {I},w \Vdash _{\textsf {i}} \phi \) iff \(\mathcal {I}^{\exp },w,c \Vdash _{\exp } \phi \).

Now let wP. Let Γ = {ϕ | V (ϕ,w) = 1} and let \(\mathcal {I}^{\exp }\) be 〈W,f〉 where f is constructed as in Proposition 2. Then by Claim 2, \(\mathcal {I}^{\exp },w_{0},c_{{\Gamma }} \Vdash _{\exp } \phi \) iff ϕ ∈Γ. Hence, we can take x = 〈w0,cΓ〉. □

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Kocurek, A.W., Jerzak, E.J. Counterlogicals as Counterconventionals. J Philos Logic (2021). https://doi.org/10.1007/s10992-020-09581-6

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Keywords

  • Counterlogicals
  • Counterconventionals
  • Counterpossibles
  • Logical expressivism
  • Impossible worlds
  • Hyperconventions