Abstract
We develop and defend a new approach to counterlogicals. Nonvacuous 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.
For any expressivist model \(\mathcal {E} = \langle W,f \rangle \) and any x ∈ I_{W}, 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 w ∈ W^{i} 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 x ∈ CI_{W}, we can take w ∈ P^{i}.

2.
For any impossible worlds model \(\mathcal {I} = \langle W,P,f,V \rangle \) and any w ∈ W, 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 w ∈ P, 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 x ∈ CI_{W}. Define \(\mathcal {E}^{\textsf {i}} = \langle W^{\textsf {i}},P^{\textsf {i}},f^{\textsf {i}},V^{\textsf {i}} \rangle \) as follows:

W^{i} = I_{W}

P^{i} = W ×{c_{x}}

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

for each y ∈ P^{i}, V^{i}(p,y) = 1 iff w_{y} ∈ c_{y}(p)

for each y∉P^{i}, V^{i}(ϕ,y) = 1 iff \(\mathcal {E},y \Vdash _{\exp } \phi \).
Clearly, \(\mathcal {E}^{\textsf {i}}\) is a impossible worlds model and x ∈ P^{i}. It suffices to show that for any ϕ and any y ∈ I_{W}:
If y∉P^{i}, then by construction, \(\mathcal {E}^{\textsf {i}},y \Vdash _{\textsf {i}} \phi \) iff V^{i}(ϕ,y) = 1 iff \(\mathcal {E},y \Vdash _{\exp } \phi \). If y ∈ P^{i}, then we proceed by induction. The atomic case holds by definition of V^{i}. The other cases are straightforward since c_{y} = c_{x} is classical and since P^{i} = W ×{c_{x}}.
Now suppose x∉CI_{W}. Then we can define \(\mathcal {E}^{\textsf {i}}\) as above except now we take P^{i} = CI_{W}. Then \(\mathcal {E},x \Vdash _{\exp } \phi \) iff \(\mathcal {E}^{\textsf {i}},x \Vdash _{\textsf {i}} \phi \) by construction of V^{i}. □
Theorem 1(a) is not terribly surprising in retrospect. All it says is that anything that is ivalid is also \(\exp \)valid. But ivalidity 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 Łformula is an S5formula if it does not contain . An Łformula is a counterfactual if its main connective is .
Proposition 1
Let \({\mathscr{M}} = \langle P,i \rangle \) be an S5model (where \(i(p) \mathrel {\subseteq } P\) for all p ∈At) and let \({\Phi }:{P}\rightarrow \wp ({\mathscr{L}})\) map every w ∈ P to a set Φ_{w} of counterfactuals. Then there is an impossible worlds model \(\mathcal {I} = \langle {W,P,f,V}\rangle \) such that for any w ∈ P:

1.
if ϕ is an S5formula, then \(\mathcal {I},w \Vdash _{\textsf {i}} \phi \) iff \({\mathscr{M}},w \Vdash _{\textbf {S5}} \phi \)

2.
if ψ is a counterfactual, then \(\mathcal {I},w \Vdash _{\textsf {i}} \psi \) iff ψ ∈Φ_{w}.
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 w ∈ P, V (p,w) = 1 iff w ∈ i(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 w ∈ P, 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 S5formula, and let 𝜃^{∗} be the result of simultaneously uniformly substituting one or more atomic sentences in 𝜃 for distinct counterfactuals. Then 𝜃^{∗} is isatisfiable.
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 S5satisfiable. Let \({\mathscr{M}},w \Vdash _{\textbf {S5}} \theta \). For each \(v \in W^{{\mathscr{M}}}\), define:
By Proposition 1, this guarantees us an S5equivalent impossible worlds model \(\mathcal {I}\) such that \(\mathcal {I},v \Vdash _{\textsf {i}} \psi \) iff ψ ∈Φ_{v} 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 nontrivial valid inferences governing counterfactuals in the impossible worlds semantics: any inference with counterfactuals that’s ivalid is already S5valid.
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 Łformulas 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 S5model and let \({\Phi }:{P}\rightarrow \wp ({\mathscr{L}})\) map every w ∈ W to a set Φ_{w} of counterfactuals. Then there is a expressivist model \(\mathcal {E} = \langle W,f \rangle \) and a classical hyperconvention c such that for any w ∈ W:

(i)
if ϕ is an S5formula, then \(\mathcal {E},w,c \Vdash _{\exp } \phi \) iff \({\mathscr{M}},w \Vdash _{\textbf {S5}} \phi \)

(ii)
if ψ is a counterfactual, then \(\mathcal {E},w,c \Vdash _{\exp } \psi \) iff ψ ∈Φ_{w}.
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 w_{0} ∈ W. Let \({h}:{{\mathscr{L}}}\rightarrow {W  \{w_{0},w\}}\) be a bijection. We’ll write w_{ϕ} in place of h(ϕ) throughout. Now, let \({\Gamma } \subseteq \L \). Define the hyperconvention c_{Γ} as follows (where and \(\circ \in \{\wedge ,\vee ,\rightarrow \}\)):
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 c_{Γ}. 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 c_{Γ}. 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 ∈Γ^{α}, 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 S5formula, 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 w ∈ P. Let:
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 w∉P. 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 = 〈w_{0},c_{Γ}〉. □
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Kocurek, A.W., Jerzak, E.J. Counterlogicals as Counterconventionals. J Philos Logic (2021). https://doi.org/10.1007/s10992020095816
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Keywords
 Counterlogicals
 Counterconventionals
 Counterpossibles
 Logical expressivism
 Impossible worlds
 Hyperconventions