New foundations for counterfactuals

Abstract

Philosophers typically rely on intuitions when providing a semantics for counterfactual conditionals. However, intuitions regarding counterfactual conditionals are notoriously shaky. The aim of this paper is to provide a principled account of the semantics of counterfactual conditionals. This principled account is provided by what I dub the Royal Rule, a deterministic analogue of the Principal Principle relating chance and credence. The Royal Rule says that an ideal doxastic agent’s initial grade of disbelief in a proposition $A$, given that the counterfactual distance in a given context to the closest $A$-worlds equals $n$, and no further information that is not admissible in this context, should equal $n$. Under the two assumptions that the presuppositions of a given context are admissible in this context, and that the theory of deterministic alethic or metaphysical modality is admissible in any context, it follows that the counterfactual distance distribution in a given context has the structure of a ranking function. The basic conditional logic V is shown to be sound and complete with respect to the resulting rank-theoretic semantics of counterfactuals.

Appendix: Logical considerations

Let us first consider system C2 from Stalnaker (1968). Let ${\mathcal {L}}_0$ be the smallest set that contains a given countable set of propositional variables $PV$ and is closed under the classical connectives $\lnot , \wedge , \vee$, and $\supset$. Let ${\mathcal {L}}_1$ be the smallest set containing ${\mathcal {L}}_0$ and $\alpha \,\square \!\!\!\rightarrow \beta$ for any two $\alpha ,\beta$ from ${\mathcal {L}}_0$. Let $\mathcal L$ be the smallest set that contains ${\mathcal {L}}_1$ and is closed under the classical connectives. Finally, let ${\mathcal {L}}^+$ be the smallest set containing $PV$ (or any of the above mentioned languages) that is closed under the classical connectives $\lnot , \wedge , \vee , \supset$ plus $\,\square \!\!\!\rightarrow$.

So the language $\mathcal{L}$ is built up form a countable set of propositional variables in the usual way, with the only exception that $\alpha \,\square \!\!\!\rightarrow \gamma$ is a well-formed formula if and only if $\alpha$ and $\gamma$ are well-formed formulae and do not contain an occurrence of $\,\square \!\!\!\rightarrow$. $A,B,C,\ldots$ are the sets of models in which $\alpha ,\beta ,\gamma ,\ldots$ are true. Ranking functions are defined on some finitary/$\sigma$-/complete field over the set of models of $\mathcal L, M_\mathcal{L}$.

The rules of inference of C2 are Modus Ponens and Necessitation, both of which preserve validity.

$$\begin{aligned}&\mathbf{RI \ 1} \vdash \alpha ,\quad \vdash \alpha \supset \beta \quad \Rightarrow \quad \vdash \beta \\&\mathbf{RI \ 2} \vdash \alpha \quad \Rightarrow \quad \vdash \Box \alpha . \end{aligned}$$

The axiom schemata of C2 are St 1–St 7, all of which are valid except for Conditional Excluded Middle St 5 and Weak Centering St 6.

$$\begin{aligned}&\mathbf{St \ 1} {\hbox {All \ tautologies \ are \ axioms.}}\\&\mathbf{St \ 2} \Box \left( \alpha \supset \gamma \right) \supset \left( \Box \alpha \supset \Box \gamma \right) \\&\mathbf{St \ 3} \Box \left( \alpha \supset \gamma \right) \supset \left( \alpha \,\square \!\!\!\rightarrow \gamma \right) . \end{aligned}$$

St 1 is trivially validated, because classical logic is presupposed. As to St 2 and St 3, suppose $\Box \left( \alpha \supset \gamma \right)$ is true at $w_u$ in $c$ relative to $r_u$. This means that $\lnot \left( \alpha \supset \gamma \right) \,\square \!\!\!\rightarrow \left( \alpha \supset \gamma \right)$ is true at $w_u$ in $c$ relative to $r_u$, i.e. $\left( A\cap \overline{C}\right) ^{r_{uc}}\subseteq \overline{A}\cup C$. Since $\left( A\cap \overline{C}\right) ^{r_{uc}}\subseteq A\cap \overline{C}$, this can only happen if $\overline{A}\cup C=Mod_\mathcal{L}$, i.e. $A\subseteq C$. For St 2 suppose that $\Box \alpha$ is true at $w_u$ in $c$ relative to $r_u$. This means $\overline{A}^{r_{uc}}\subseteq A$ and since $\overline{A}^{r_{uc}}\subseteq \overline{A}$, this can only happen if $A=Mod_\mathcal{L}$. Consequently $C=Mod_\mathcal{L}$ and hence $\overline{C}^{r_{uc}}\subseteq \overline{C}=\emptyset$ and hence $\overline{C}^{r_{uc}}\subseteq C$, which means that $\Box \gamma$ is true at $w_u$ in $c$ relative to $r_u$. For St 3 we have $A^{r_{uc}}\subseteq A\subseteq C$, which means that $\alpha \,\square \!\!\!\rightarrow \gamma$ is true at $w_u$ in $c$ relative to $r_u$.

$$\begin{aligned} \mathbf{St \ 4} \diamond \alpha \supset \left( \left( \alpha \,\square \!\!\!\rightarrow \beta \right) \supset \lnot \left( \alpha \,\square \!\!\!\rightarrow \lnot \beta \right) \right) . \end{aligned}$$

Suppose $\diamond \alpha$ and $\alpha \,\square \!\!\!\rightarrow \beta$ are true at $w_u$ in $c$ relative to $r_u$, i.e. $A^{r_{uc}}\not \subseteq \overline{A}$ and $A^{r_{uc}}\subseteq B$. It follows that $\emptyset \ne A^{r_{uc}}\cap A\subseteq A^{r_{uc}}\subseteq B$, and hence that $A^{r_{uc}}\not \subseteq \overline{B}$, i.e. that $\lnot \left( \alpha \,\square \!\!\!\rightarrow \lnot \beta \right)$ is true at $w_u$ in $c$ relative to $r_u$.

$$\begin{aligned} \mathbf{St \ 5} (\mathbf{Conditional \ Excluded \ Middle}) \left( \alpha \,\square \!\!\!\rightarrow \left( \beta \vee \gamma \right) \right) \supset \left( \alpha \,\square \!\!\!\rightarrow \beta \right) \vee \left( \alpha \,\square \!\!\!\rightarrow \gamma \right) . \end{aligned}$$

Conditional Excluded Middle St 5 is not valid. Consider the language $\mathcal L$ over the two propositional variables $p$ and $q$. $Mod_\mathcal{L}=\left\{ w_{pq},w_{p\lnot q},w_{\lnot pq},w_{\lnot p\lnot q}\right\}$. Let $c\left( u\right) =Mod_\mathcal{L}\times \left\{ R\right\}$ for all $u\in Mod_\mathcal{L}\times R$, and let $u^*=\left( w_{u^*},r_{u^*}\right)$ with $w_{u^*}=w_{pq}$ and $r_{u^*c}\left( \left\{ w\right\} \right) =0$ for all $w\in Mod_\mathcal{L}$. $p\,\square \!\!\!\rightarrow q\vee \lnot q$ is true at $w_{u^*}$ in $c$ relative to $r_{u^*}$, but both $p\,\square \!\!\!\rightarrow q$ and $p\,\square \!\!\!\rightarrow \lnot q$ are false at $w_{u^*}$ in $c$ relative to $r_{u^*}$. This is so because $P^{r_{u^*c}}=P$ and $Q^{r_{u^*c}}=Q$ and $\overline{P}^{r_{u^*c}}=\overline{P}$ and $\overline{Q}^{r_{u^*c}}=\overline{Q}$, and hence $P^{r_{u^*c}}\subseteq Q\cup \overline{Q}=Mod_\mathcal{L}$ while $P^{r_{u^*c}}\not \subseteq Q$ and $P^{r_{u^*c}}\not \subseteq \overline{Q}$.

• St 6 (Weak Centering) $\left( \alpha \,\square \!\!\!\rightarrow \beta \right) \supset \left( \alpha \supset \beta \right)$.

Weak Centering St 6 is not valid. Let $\mathcal L, M_\mathcal{L}$, and $c$ be as before, and let $u^*=\left( w_{u^*},r_{u^*}\right)$ with $w_{u^*}=w_{p\lnot q}$ and $r_{u^*c}\left( \left\{ w_{p\lnot q}\right\} \right) =1$ and $r_{u^*c}\left( \left\{ w\right\} \right) =0$ for all $w\in Mod_\mathcal{L}$ with $w\ne w_{p\lnot q}$. $p\,\square \!\!\!\rightarrow q$ is true at $w_{u^*}=w_{p\lnot q}$ in $c$ relative to $r_{u^*}$, but $p\supset q$ is false at $w_{u^*}=w_{p\lnot q}$ in $c$ relative to $r_{u^*}$. This is so because $P^{r_{u^*c}}=\left\{ w_{pq}\right\}$ and $Q=\left\{ w_{pq},w_{p\lnot q}\right\}$, and hence $P^{r_{u^*c}}\subseteq Q$ while $\left\{ w_{p\lnot q}\right\} \not \in P\supset Q=\overline{P}\cup Q$.

• St 7 $\left( \left( \alpha \,\square \!\!\!\rightarrow \beta \right) \wedge \left( \beta \,\square \!\!\!\rightarrow \alpha \right) \right) \supset \left( \left( \alpha \,\square \!\!\!\rightarrow \gamma \right) \supset \left( \beta \,\square \!\!\!\rightarrow \gamma \right) \right)$.

Suppose $\left( \alpha \,\square \!\!\!\rightarrow \beta \right) \wedge \left( \beta \,\square \!\!\!\rightarrow \alpha \right)$ is true at $w_u$ in $c$ relative to $r_u$, i.e. $A^{r_{uc}}\subseteq B$ and $B^{r_{uc}}\subseteq A$. This implies $A^{r_{uc}}=\left( A\cap B\right) ^{r_{uc}}$ and $B^{r_{uc}}=\left( A\cap B\right) ^{r_{uc}}$, and so $A^{r_{uc}}=B^{r_{uc}}$. It follows that $B^{r_{uc}}\subseteq C$ if $C^{r_{uc}}\subseteq C$, i.e that $\left( \alpha \,\square \!\!\!\rightarrow \gamma \right) \supset \left( \beta \,\square \!\!\!\rightarrow \gamma \right)$ is true at $w_u$ in $c$ relative to $r_u$.

Strong Centering St 8 is a consequence of St 1–7 and RI 1–2.

• St 8 (Strong Centering) $\left( \alpha \wedge \gamma \right) \supset \left( \alpha \,\square \!\!\!\rightarrow \gamma \right)$.

Strong Centering St 8 is not valid. Let $\mathcal L, M_\mathcal{L}, c$, and $u^*$ be as in the countermodel to Conditional Excluded Middle St 5. $p\wedge q$ is true at $w_{u^*}=w_{pq}$ in $c$ relative to $r_{u^*}$, but $p\,\square \!\!\!\rightarrow q$ is not, because $P^{r_{u^*c}}=P\not \subseteq Q$.

In Lewis’ (1973) terminology C2 is the system VCS. VCS results from the basic conditional logic V by adding St 5, St 6, and St 8. Lewis’ “official logic of counterfactuals” (Lewis 1973, p. 132) is the system VC, which results from V by adding Weak Centering St 6 and Strong Centering St 8. The system VW results from V by adding Weak Centering St 6.

The system V consists of the rules of inference R 1–3 and the axiom schemata L 1–5, where $\equiv$ is the material biconditional, $\Box \alpha$ is defined as $\lnot \alpha \,\square \!\!\!\rightarrow \alpha$, and $\diamond \alpha$ is defined as $\lnot \Box \lnot \alpha$.

• R1 $\vdash \alpha ,\quad \vdash \alpha \supset \beta \quad \Rightarrow \quad \vdash \beta$

• R2 $\vdash \beta _1\wedge \cdots \wedge \beta _n\supset \gamma \quad \Rightarrow \quad \left( \alpha \,\square \!\!\!\rightarrow \beta _1\right) \wedge \cdots \wedge \left( \alpha \,\square \!\!\!\rightarrow \beta _n\right) \supset \left( \alpha \,\square \!\!\!\rightarrow \gamma \right)$

• R3 $\vdash \alpha \equiv \beta \quad \Rightarrow \quad \vdash \gamma \equiv \gamma \left[ \beta /\alpha \right] ,$

where $\gamma \left[ \alpha /\beta \right]$ results from $\gamma$ by substituting $\alpha$ for some occurrence of $\beta$ in $\gamma$.

• L 1 Truth-functional tautologies.

• L 2 $\alpha \,\square \!\!\!\rightarrow \alpha$

• L 3 $\Box \gamma \supset \left( \alpha \,\square \!\!\!\rightarrow \gamma \right)$

• L 4 $\lnot \left( \alpha \,\square \!\!\!\rightarrow \lnot \beta \right) \supset \left( \left( \alpha \wedge \beta \,\square \!\!\!\rightarrow \gamma \right) \equiv \left( \alpha \,\square \!\!\!\rightarrow \left( \beta \supset \gamma \right) \right) \right)$.

V is sound and complete with respect to the rank-theoretic semantics of counterfactuals. To state and prove this result some technical terminology has to be introduced, though I will ignore contexts in order to avoid unnecessary complications. $\left( W,{\mathcal {A}}_W,R,\Omega ,\varphi \right)$ is a rank-theoretic model for $\mathcal L$ just in case $W$ is a non-empty set of factual worlds, ${\mathcal {A}}_W$ is an algebra over $W$ such that $\varphi \left( \alpha \right) \in {\mathcal {A}}_W\times \left\{ R\right\}$ for every $\alpha$ from ${\mathcal {L}}_0, R$ is a set of ranking functions $r:{\mathcal {A}}_W\rightarrow \mathbb {N}\cup \left\{ \infty \right\} , \Omega \subseteq W\times R$ is such that for each $w\in W$ there is at least one $r\in R$ such that $(w,r)\in \Omega$, and $\varphi :{\mathcal {L}}\rightarrow \wp \left( \Omega \right)$ is an interpretation function such that for all $\alpha ,\beta$ from $\mathcal L$:

• 1. $\varphi \left( p\right) \in \left( \wp \left( W\right) \times \left\{ R\right\} \right) \cap \Omega$ if $p\in PV$

• 2. $\varphi \left( \lnot \alpha \right) =\Omega \setminus \varphi \left( \alpha \right)$

• 3. $\varphi \left( \alpha \wedge \beta \right) =\varphi \left( \alpha \right) \cap \varphi \left( \beta \right)$

• 4. $\varphi \left( \alpha \,\square \!\!\!\rightarrow \beta \right) =\left\{ u=\left( w_u,r_u\right) \in \Omega :f\left( \varphi \left( \alpha \right) \right) ^{r_{u}}\times R\subseteq \varphi \left( \beta \right) \right\}$,

where $f\left( \varphi \left( \alpha \right) \right) =\left\{ w\in W:\exists r:\left( w,r\right) \in \varphi \left( \alpha \right) \right\}$.

$\left( W,{\mathcal {A}}_W,R,\Omega ,\varphi \right)$ is a Humean rank-theoretic model for ${\mathcal {L}}^+$ just in case $W$ is a non-empty set of factual worlds, ${\mathcal {A}}_W$ is an algebra over $W$ such that $\varphi \left( \alpha \right) \in {\mathcal {A}}_W$ for every $\alpha$ from ${\mathcal {L}}^+, R$ is a set of ranking functions $r:{\mathcal {A}}_W\rightarrow N\cup \left\{ \infty \right\} , \Omega \subseteq W\times R$ is such that for each $w\in W$ there is exactly one $r=r_w\in R$ such that $\left( w,r_w\right) \in \Omega$, and $\varphi :{\mathcal {L}}^+\rightarrow \wp \left( W\right)$ is an interpretation function such that for all $\alpha ,\beta$ from ${\mathcal {L}}^+$:

• 1. $\varphi \left( p\right) \in \wp \left( W\right)$ if $p\in PV$

• 2. $\varphi \left( \lnot \alpha \right) =W\setminus \varphi \left( \alpha \right)$

• 3. $\varphi \left( \alpha \wedge \beta \right) =\varphi \left( \alpha \right) \cap \varphi \left( \beta \right)$

• 4. $\varphi \left( \alpha \,\square \!\!\!\rightarrow \beta \right) =\left\{ w\in W:\varphi \left( \alpha \right) ^{r_{w}}\subseteq \varphi \left( \beta \right) \right\}$.

Every Humean rank-theoretic model ${\mathcal {M}}^+=\left( W^+,{\mathcal {A}}_{W^+},R^+,\Omega ^+,\varphi ^+\right)$ for ${\mathcal {L}}^+$ can be reduced to a rank-theoretic model ${\mathcal {M}}=\left( W,{\mathcal {A}}_W,R,\Omega ,\varphi \right)$ for ${\mathcal {L}}$ as follows. Let $W=W^+$ and ${\mathcal {A}}_W={\mathcal {A}}_{W^+}\cap \left\{ \varphi ^+\left( \alpha \right) :\alpha \in {\mathcal {L}}_0\right\}$. Take any $r^+_w\in R^+$ and restrict it to ${\mathcal {A}}_W$, i.e. $r_w\left( A\right) =r^+_w\left( A\right)$ if $A\in {\mathcal {A}}_W$ and undefined otherwise. This gives us $R$. Let $\Omega$ be the set of all pairs $\left( w,r_w\right)$ with $r_w$ resulting from $r^+_w$ as indicated. Since for each $w\in W^+=W$ there is exactly one $r^+_w\in R^+$ it immediately follows that there is at least one $r_w\in R$ for each $w\in W$. Finally, let $\varphi \left( \alpha \right) =\varphi ^+\left( \alpha \right) \times R$ for $\alpha \in {\mathcal {L}}$ to obtain $\varphi :{\mathcal {L}}\rightarrow \wp \left( \Omega \right)$. It is routine to check that $\alpha \in {\mathcal {L}}$ is true in the Humean rank-theoretic model ${\mathcal {M}}^+$ just in case $\alpha$ is true in the rank-theoretic model ${\mathcal {M}}$. Consequently each set $L$ of sentences from $\mathcal L$ that has a Humean rank-theoretic model also has a rank-theoretic model.

Soundness is easily checked—especially since we are ignoring contexts. R 1 is RI 1. R 3 is trivial, because classical logic is presupposed. As to R 2, suppose $\vdash \beta _1\wedge \cdots \beta _n\supset \gamma$, i.e. $B_1\cap \cdots \cap B_2\subseteq C$. If $\left( \alpha \,\square \!\!\!\rightarrow \beta _1\right) \wedge \cdots \wedge \left( \alpha \,\square \!\!\!\rightarrow \beta _n\right)$ is true at $w_u$ relative to $r_u$, then $A^{r_u}\subseteq B_1$, and $\ldots$, and $A^{r_u}\subseteq B_n$ and so $A^{r_u}\subseteq B_1\cap \cdots \cap B_n\subseteq C$, which means that $\alpha \,\square \!\!\!\rightarrow \gamma$ is true at $w_u$ relative to $r_u$, for any universe $u=\left( w_u,r_u\right)$.

L1 is St1. L2 follows from St1, RI2, and St3. L3 follows from St3. As to L4, suppose $\lnot \left( \alpha \lnot \,\square \!\!\!\rightarrow \beta \right)$ is true at $w_u$ relative to $r_u$. This means that $A^{r_u}\not \subseteq \overline{B}$. Suppose first that $\alpha \wedge \beta \,\square \!\!\!\rightarrow \gamma$ is true at $w_u$ relative to $r_u$. This means that $\left( A\cap B\right) ^{r_u}\subseteq C$. If $w_u\in A^{r_u}\cap \overline{B}$, then $w_u\in \overline{B}\cup C$. If $w_u\in A^{r_u}\cap B$, then $w_u\in \left( A\cap B\right) ^{r_u}\subseteq C$. Therefore $A^{r_u}\subseteq \overline{B}\cup C$, which means that $\alpha \,\square \!\!\!\rightarrow \left( B\supset \gamma \right)$ is true at $w_u$ relative to $r_u$. Now suppose $\alpha \,\square \!\!\!\rightarrow \left( \beta \supset \gamma \right)$ is true at $w_u$ relative to $r_u$. This means that $A^{r_u}\subseteq \overline{B}\cup C$. If $w\in \left( A\cap B\right) ^{r_u}\cap A^{r_u}$, then $w_u\in A^{r_u}\cap B$, and so $w_u\in C$. If $w_u\in \left( A\cap B\right) ^{r_u}\cap \overline{A^{r_u}}$, then $r_u\left( A\right) <r_u\left( A\cap B\right)$ and so $A^{r_u}\cap B=\emptyset$, contradicting $A^{r_u}\not \subseteq \overline{B}$. Therefore $\left( A\cap B\right) ^{r_u}\subseteq C$, which means that $\alpha \wedge \beta \,\square \!\!\!\rightarrow \gamma$ is true at $w_u$ relative to $r_u$.

This shows that the system V on the language $\mathcal L$ is sound with respect to the rank-theoretic semantics of counterfactuals and, consequently, that the system V on the language ${\mathcal {L}}^+$ is sound with respect to the Humean rank-theoretic semantics of counterfactuals.

Let us now establish completeness.¹³ It follows from Lewis (1973, Chap. 6) that each V-consistent set $L^+$ of sentences from ${\mathcal {L}}^+$ has a model ${\mathcal {M}}^*=\left( W,\left( \$_w\right) _{w\in W},\varphi \right)$, where for each $w\in W, \$_w\subseteq \wp \left( W\right)$ is a set of spheres (around $w$), i.e. $\$_w$ is nested and closed under arbitrary unions and intersections. Since V has the finite model property with respect to Lewis’ semantics, we can assume $W$ to be finite. For each $w\in W, \$_w$ is of the form $S_0\cup S_1\cup \cdots \cup S_{w_n}$, where any two $S_i$ and $S_j$ are disjoint and $\bigcup _{0\le i\le j}S_i$ is the $j$-th sphere (around $w$). For $w'\in W$, define $r_w\left( \left\{ w'\right\} \right) =i$ if $w'\in S_i$, and let $r_w\left( A\right) =\min \left\{ r_w\left( \left\{ w'\right\} \right) :w'\in A\right\}$. Each $r_w$ defined in this way is a regular ranking function on $\wp \left( W\right)$. Let ${\mathcal {A}}_W=\wp \left( W\right) , R=\left\{ r_w:w\in W\right\}$, and $\Omega =\left\{ \left( w,r_w\right) :w\in W\right\}$. ${\mathcal {M}}^+=\left( W,{\mathcal {A}}_W,R,\Omega ,\varphi \right)$ is a Humean rank-theoretic model. For $\alpha \in {\mathcal {L}}^+$ let $j_\alpha$ be the smallest number (if any) such that $\varphi \left( \alpha \right) \cap S_{j_\alpha }$ is non-empty. By definition $\varphi \left( \alpha \right) ^{r_w}=S_{j_\alpha }\cap \varphi \left( \alpha \right)$. Therefore $\alpha \in L^+$ is true in $w$ in ${\mathcal {M}}^*$ in Lewis’ sense just in case $\alpha$ is true in $w$ in ${\mathcal {M}}^+$ in our sense.

Now suppose $L$ is a V-consistent set of sentences from ${\mathcal {L}}$. Close $L$ under the classical connectives $\lnot , \wedge , \vee , \supset$ plus $\,\square \!\!\!\rightarrow$ to obtain a set of sentences $L^+$ from ${\mathcal {L}}^+$. By the above, $L^+$ has a Humean rank-theoretic model ${\mathcal {M}}^+$. Reduce ${\mathcal {M}}^+$ to obtain a rank-theoretic model ${\mathcal {M}}$ for ${\mathcal {L}}$ as shown above. This completes the proof for

Theorem 2

V restricted to $\mathcal L$ is sound and complete with respect to the rank-theoretic semantics of counterfactuals. V on the full language ${\mathcal {L}}^+$ is sound and complete with respect to the Humean rank-theoretic semantics of counterfactuals.

Call a (Humean) rank-theoretic model $\left( W,{\mathcal {A}}_W,R,\Omega ,\varphi \right)$ weakly/strongly centered just in case each $r_u\in R$ is such that $r_u\left( \left\{ w\right\} \right) =0$ if/iff $w=w_u$. Then it holds that

Theorem 3

VW restricted to $\mathcal L$ is sound and complete with respect to the weakly centered rank-theoretic semantics of counterfactuals. VW on the full language ${\mathcal {L}}^+$ is sound and complete with respect to the weakly centered Humean rank-theoretic semantics of counterfactuals.

Theorem 4

VC restricted to $\mathcal L$ is sound and complete with respect to the strongly centered rank-theoretic semantics of counterfactuals. VC on the full language ${\mathcal {L}}^+$ is sound and complete with respect to the strongly centered Humean rank-theoretic semantics of counterfactuals.