A graded semantics for counterfactuals

Abstract

This article presents an extension of Lewis’ analysis of counterfactuals to a graded framework. Unlike standard graded approaches, which use the probabilistic framework, we employ that of many-valued logics. Our principal goal is to provide an adequate analysis of the main background notion of Lewis’ approach—the one of the similarity of possible worlds. We discuss the requirements imposed on the analysis of counterfactuals by the imprecise character of similarity and concentrate in particular on robustness, i.e., the requirement that small changes in the similarity relation should not significantly change the truth value of the counterfactual in question. Our second motivation is related to the logical analysis of natural language: analyzing counterfactuals in the framework of many-valued logics allows us to extend the analysis to counterfactuals that include vague statements. Unlike previous proposals of this kind in the literature, our approach makes it possible to apply gradedness at various levels of the analysis and hence provide a more detailed account of the phenomenon of vagueness in the context of counterfactuals. Finally, our framework admits a novel way of avoiding the Limit Assumption, keeping the core of Lewis’ truth condition for counterfactuals unchanged.

Appendices

Appendix

1 The many-valued logic of Łukasiewicz

Fuzzy logics form an intensively studied and well-understood family of many-valued logics, designed especially for handling gradable (i.e., capable of being more or less true) propositions. For an introduction to formal fuzzy logic see, e.g., Hájek (1998) or Běhounek et al. (2011). In this paper, we restrict our attention to Łukasiewicz fuzzy logic (e.g., Hájek 1998; Cignoli et al. 1999; DiNola and Leuştean 2011).

The standard semantics of infinite-valued Łukasiewicz logic (${\mathrm L}$) evaluates formulae in the real unit interval [0, 1]. The truth value 1 is usually understood as the full truth of the proposition; the truth value 0 as the full falsity; and the intermediate truth values as the degrees of (partial) truth of gradable propositions. Łukasiewicz logic, thus, generalizes the Boolean logic of bivalent (or, in the terminology of fuzzy logic, crisp) propositions by admitting a gradual change between the truth and falsity of the propositions. The propositional connectives of ${\mathrm L}$ are interpreted truth-functionally, with the following truth functions on [0, 1]:

$$ \begin{aligned} x\rightarrow y&= \min (1-x+y,1)\\ x\vee y&= \max (x,y)\\ x\wedge y&= \min (x,y)\\ x\mathbin { \& }y&= \min (0,x+y-1)\\ \lnot x&= 1-x \end{aligned}$$

This entails, in particular, the following values of implication:

$$\begin{aligned} \text {If}\quad&x&\le y,&\quad \text {then}\quad&(x\rightarrow y)&=1\\&x&=1&(x\rightarrow y)&=y \\&y&=0&(x\rightarrow y)&=1 - x. \end{aligned}$$

Notice that Łukasiewicz logic possesses two conjunctive connectives: the lattice conjunction $\wedge$, interpreted as the minimum, and the residuated conjunction $\mathbin { \& }$, which is non-idempotent: ${x\mathbin { \& }x}<x$ if $0<x<1$. We will also need the following connective, which (in the standard semantics) crisply asserts the full truth of its argument:²⁸

$$\begin{aligned} {\triangle }x= & {} {\left\{ \begin{array}{ll} 1 &{} \text {if }x=1 \\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

To save some parentheses, we will assume that the unary connectives ${\lnot },{\triangle }$ have the largest and ${\mathrel {\rightarrow }},{\mathrel {\leftrightarrow }}$ the smallest priority in formulae.

The tautologies of propositional Łukasiewicz logic are the formulae that evaluate to 1 under all [0, 1]-evaluations of propositional letters. The set of propositional tautologies of Łukasiewicz logic can be axiomatized, e.g., by the following Hilbert-style axiomatic system for the primitive connectives $\mathrel {\rightarrow },{\lnot },{\triangle }$:²⁹

$$\begin{aligned}&({\mathrm L}1)&\quad&\varphi \mathrel {\rightarrow }(\psi \mathrel {\rightarrow }\varphi )&\qquad&\text {(weakening)} \\&({\mathrm L}2)&(\varphi \mathrel {\rightarrow }\psi )\mathrel {\rightarrow }((\psi \mathrel {\rightarrow }\chi )\mathrel {\rightarrow }(\varphi \mathrel {\rightarrow }\chi ))&\text {(suffixing)} \\&({\mathrm L}3)&({\lnot }\varphi \mathrel {\rightarrow }{\lnot }\psi )\mathrel {\rightarrow }(\psi \mathrel {\rightarrow }\varphi )&\text {(negative contraposition)} \\&({\mathrm L}4)&((\varphi \mathrel {\rightarrow }\psi )\mathrel {\rightarrow }\psi )\mathrel {\rightarrow }((\psi \mathrel {\rightarrow }\varphi )\mathrel {\rightarrow }\varphi )&\text {(commutativity of}\, \mathbin {\vee }) \\&({\triangle }1)&({\lnot }{\triangle }\varphi \mathrel {\rightarrow }{\triangle }\varphi )\mathrel {\rightarrow }{\triangle }\varphi&\text {(excluded middle for}\, {\triangle }) \\&({\triangle }2)&{\triangle }(\varphi \mathrel {\rightarrow }\psi )\mathrel {\rightarrow }({\triangle }\varphi \mathrel {\rightarrow }{\triangle }\psi )&\text {(K for}\, {\triangle }) \\&({\triangle }3)&{\triangle }\varphi \mathrel {\rightarrow }\varphi&\text {(T for}\, {\triangle }) \\&({\triangle }4)&{\triangle }\varphi \mathrel {\rightarrow }{\triangle }{\triangle }\varphi&\text {(4 for}\, {\triangle }) \end{aligned}$$

The derivation rules are modus ponens (from $\varphi$ and $\varphi \mathrel {\rightarrow }\psi$ derive $\psi$) and ${\triangle }$-necessitation (from $\varphi$ derive ${\triangle }\varphi$). The remaining connectives are defined as follows:

$$ \begin{aligned} \varphi \mathbin { \& }\psi&\equiv {\lnot }(\varphi \mathrel {\rightarrow }{\lnot }\psi ) \\ \varphi \mathbin {\wedge }\psi&\equiv \varphi \mathbin { \& }(\varphi \mathrel {\rightarrow }\psi ) \\ \varphi \mathbin {\vee }\psi&\equiv (\varphi \mathrel {\rightarrow }\psi )\mathrel {\rightarrow }\psi \\ \varphi \mathrel {\leftrightarrow }\psi&\equiv (\varphi \mathrel {\rightarrow }\psi )\mathbin {\wedge }(\psi \mathrel {\rightarrow }\varphi ) \\ \top&\equiv \varphi \mathrel {\rightarrow }\varphi \\ \bot&\equiv {\lnot }\top \end{aligned}$$

The aforementioned axiomatic system is sound and complete w.r.t. the standard [0, 1]-semantics of ${\mathrm L}_{\triangle }$: a formula is provable iff it is a tautology of ${{\mathrm L}_{\triangle }}$. It should be noted that the standard [0, 1]-valued semantics is not the only sound and complete algebraic semantics for Łukasiewicz logic. The algebras for which ${{\mathrm L}_{\triangle }}$ is sound are called ${\mathrm {MV}_{\triangle }}$-algebras (see, e.g., Hájek 1998, Sects. 2.4, 3.2; Běhounek et al. 2011, Sects. 1.3, 2.2.1). The logic ${{\mathrm L}_{\triangle }}$ is also sound and complete w.r.t. the class of all ${\mathrm {MV}_{\triangle }}$-algebras, as well as w.r.t. the class of all linearly ordered ${\mathrm {MV}_{\triangle }}$-algebras (where the ordering is determined by the lattice connectives $\mathbin {\vee },\mathbin {\wedge }$). The interval [0, 1] equipped with the aforementioned truth functions of connectives is called the standard ${\mathrm {MV}_{\triangle }}$-algebra and denoted by $[0,1]_{{{\mathrm L}_{\triangle }}}$; other ${\mathrm {MV}_{\triangle }}$-algebras are called non-standard. Non-standard ${\mathrm {MV}_{\triangle }}$-algebras provide algebraic semantics for Łukasiewicz logic in which the degrees can have abstract, non-numerical values.

First-order Łukasiewicz logic ${{\mathrm L}\forall _{\triangle }}$ extends the propositional syntax in the usual way. Semantically, in a model for a given first-order language $\mathcal L$ over an ${\mathrm {MV}_{\triangle }}$-algebra L (standardly, $L=[0,1]_{{\mathrm L}_{\triangle }}$) and the domain of discourse U, the n-ary predicate symbols of $\mathcal L$ are interpreted as functions $U^n\rightarrow L$, assigning degrees from L to each n-tuple of objects from U. The n-ary function symbols of $\mathcal L$ are interpreted as in classical logic, by functions $U^n\rightarrow U$. The truth value of a formula $\varphi$ in a model M under an evaluation e of individual variables will be denoted by $\left\| {\varphi }\right\| _e^M$. The interpretation of the quantifiers $\forall$ and $\exists$ is, respectively, that of the infimum and the supremum of the truth values of all instances of the quantified formula:

$$\begin{aligned} \left\| {(\forall {x})\varphi }\right\| _e^M&= \inf _{a\in U}\left\| {\varphi }\right\| _{e[x\mapsto a]}^M \\ \left\| {(\exists {x})\varphi }\right\| _e^M&= \sup _{a\in U}\left\| {\varphi }\right\| _{e[x\mapsto a]}^M \end{aligned}$$

where $e[x\mapsto a]$ assigns a to x and otherwise coincides with e.

Unlike propositional Łukasiewicz logic, the standard [0, 1]-semantics of its first-order variant is not axiomatizable without infinitary rules. Nevertheless, its general and linear semantics, i.e., the logic of safe³⁰ models over all (linearly ordered) ${\mathrm {MV}_{\triangle }}$-algebras, can be axiomatized by extending the propositional axiomatic system by the derivation rule of generalization (from $\varphi$ derive $(\forall {x})\varphi$) and the following axioms for quantifiers:

$$\begin{aligned}&({\mathrm L}\forall 1)&\quad&(\forall {x})\varphi (x)\mathrel {\rightarrow }\varphi (t) \\&({\mathrm L}\forall 2)&(\forall {x})(\psi \mathrel {\rightarrow }\varphi )\mathrel {\rightarrow }(\psi \mathrel {\rightarrow }(\forall {x})\varphi ) \end{aligned}$$

where t is a term substitutable for x in $\varphi$ and $\psi$ does not contain free x. The existential quantifier is defined in Łukasiewicz logic by $(\exists {x})\varphi \equiv {\lnot }(\forall {x}){\lnot }\varphi$. For more details on first-order Łukasiewicz logic see, e.g., Hájek (1998, Sect. 5.4); Běhounek et al. (2011, Sect. 5.1).

2 Proofs

In this appendix, we collect the proofs of the claims from the main text.

Corollary 2 (p. 18) In every fuzzy counterfactual frame, the fuzzy relations ${\preccurlyeq _w},{\approx _w}$ satisfy the following properties, for all $w,u,v,v',{v''\in W}$:

• 1. Reflexivity and symmetry: $({u\preccurlyeq _wu})=({u\approx _wu})=1$ and $({u\approx _wv})=({v\approx _wu})$.

• 2. Congruence: If ${v\equiv _wv'}$, then $({u\preccurlyeq _wv})=({u\preccurlyeq _wv'})$ as well as $({v\preccurlyeq _wu})=({v'\preccurlyeq _wu})$.

• 3. Compatibility with $\le _w$:

  If $v\le _w{v'\le _wv''}$, then $({v\approx _wv''})\le ({v\approx _wv'})$ and $({v\approx _wv''})\le ({v'\approx _wv''})$.

Proof

• 1. Reflexivity and symmetry are immediate by the linearity of $\preccurlyeq _w$ and the definition of $\approx _w$.

• 2. By assumption, $({v\preccurlyeq _wv'})=({v'\preccurlyeq _wv})=1$. Thus, $({u\preccurlyeq _wv})=({u\preccurlyeq _wv})\mathbin { \& }1=({u\preccurlyeq _wv})\mathbin { \& }({v\preccurlyeq _wv'})\le ({u\preccurlyeq _wv'})$ by transitivity; and analogously for the converse.

• 3. By assumption, $({v\preccurlyeq _wv'})=({v'\preccurlyeq _wv''})=({v\preccurlyeq _wv''})=1$. Consequently, $({v\approx _wv''})=({v''\preccurlyeq _wv})=({v'\preccurlyeq _wv''})\mathbin { \& }({v''\preccurlyeq _wv})\le ({v'\preccurlyeq _wv})=({v\approx _wv'})$; the second claim is proved analogously.

$\square$

Proposition 1 (p. 20) Let $F=(W,\{ {\le _w} \}_{w\in W})$ be a counterfactual frame of Definition 1. Then there is a distance-based counterfactual frame $\Phi$ on W such that its similarity orderings $\le _w$ coincide with those of F.

Proof

Let $D_w=W/{\equiv _w}$ for every $w\in W$, i.e., $D_w=\{ [u]_{\equiv _w} \mid {u\in W} \}$, where $[u]_{\equiv _w}$ are the equivalence classes $[u]_{\equiv _w}=\{ {v\in W} \mid ({v\le _wu})\mathbin { \& }({u\le _wv)} \}$. Set $[u]_{\equiv _w}\unlhd _w[v]_{\equiv _w}$ iff ${u\le _wv}$, for all ${u,v\in W}$ (Corollary 2.2 ensures that $\unlhd _w$ is well-defined), and let $\delta _w(u)=[u]_{\equiv _w}$ for all ${u\in W}$. Then $\Phi$ obviously satisfies the conditions of Definition 6 and by definition, $u\le _wv$ iff $[u]_{\equiv _w}\unlhd _w[v]_{\equiv _w}$ iff $\delta _w(u)\unlhd _w\delta _w(v)$. $\square$

Proposition 2 (p. 22) Let $\Phi$ be a distance-based fuzzy counterfactual frame of Definition 7. Then $\bigl (W,\{ {\preccurlyeq _w} \}_{w\in W}\bigr )$ is a fuzzy counterfactual frame in the sense of Definition 5, where $\preccurlyeq _w$ are the graded orderings of worlds in $\Phi$ as defined in Definition 7.

Proof

Fuzzy transitivity and linearity of $\preccurlyeq _w$ follow directly from the corresponding properties of Corollary 3. The strict minimality of w in $\preccurlyeq _w$ is proved as follows, using Corollary 3.5 and the properties of the distance $0_w$ ensured by Definition 6: $({v\preccurlyeq _ww})=1$ iff iff ${\delta _w(v)\unlhd _w\delta _w(w)}$ iff ${\delta _w(v)\unlhd _w0_w}$ iff ${\delta _w(v)=0_w}$ iff ${v=w}$. $\square$

Proposition 3 (p. 22) Let $F=\bigl (W,\{ {\preccurlyeq _w} \}_{w\in W}\bigr )$ be a fuzzy counterfactual frame of Definition 5. Then there is a distance-based fuzzy counterfactual frame $\Phi$ on W such that its graded similarity orderings $\preccurlyeq _w$ coincide with those of F.

Proof

For each $w\in W$, define $D_w,\unlhd _w,\delta _w$ as in the proof of Proposition 1, and furthermore set $\bigl ([u]_{\equiv _w}\sim _w[v]_{\equiv _w}\bigr )=(u\approx _wv)$ for all $u,v\in W$. (By Corollary 2.2, $\equiv _w$ is a congruence w.r.t. $\approx _w$, so $\sim _w$ is well-defined.) The properties of $\sim _w$ required by Definition 7 then follow straightforwardly from the corresponding properties of $\approx _w$. Moreover, . $\square$

Theorem 1 (p. 23) Let $\Phi$ be a right-connected distance-based fuzzy counterfactual frame as in Definition 7. Let M be a model over $\Phi$ and $M'$ the corresponding first-order model as in Section 4.3.

If $\left\| {\varvec{A}(v)}\right\| ^{M'}>0$ for some ${v\in W}$, then $\left\| {({ Min}_{\preccurlyeq _w}\varvec{A})(v^*)}\right\| ^{M'}>0$ for some ${v^*\in W}$.

The proof of Theorem 1 is based on a lemma known from the theory of linear fuzzy orderings, which asserts that under certain conditions, the fuzzy minima of all non-empty fuzzy sets are non-empty (Běhounek 2016, Sect. 3). Here we adapt the lemma for the setting of right-connected distance-based fuzzy counterfactual frames:

Lemma 1

Let $\Phi$ be a right-connected distance-based fuzzy counterfactual frame as in Definition 7 and $w\in W$. For any $\varvec{D}:D_w\rightarrow [0,1]$, define analogously to (8), i.e., by setting for all $d\in D_w$:

(9)

where $\mathrel {\rightarrow }$ is the truth function of Łukasiewicz implication (see Appendix 1).

If $\varvec{D}(d)>0$ for some $d\in D_w$, then for some $d^*\in D_w$.

Proof

Suppose that $\varvec{D}(d)>0$ for some $d\in D_w$. Let $D^+=\{ d'\in D_w \mid \varvec{D}(d')>0 \}$ and $d_0=\inf _{\unlhd _w}D^+$. If $\varvec{D}(d_0)>0$, then it is easy to verify by (9) that .

Assume that $\varvec{D}(d_0)=0$. Since $\sim _w$ is right-connected in $d_0=\inf _{\unlhd _w}D^+$, there is $d'\in D^+$ such that $({d'\sim _wd_0})>0$. Moreover, $d'\rhd _wd_0$, since $d_0=\inf _{\unlhd _w}D^+\notin D^+$ and $d'\in D^+$. The existence of $d^*$ such that then follows by a known theorem on fuzzy orders (Běhounek 2016, Th. 3.2).³¹$\square$

Proof of Theorem 1

Set $\varvec{D}_A(d)=\sup \{ \varvec{A}(u) \mid {u\in W},\ \delta _w(u)>0 \}$ for all ${d\in D_w}$. By assumption, $\varvec{A}(v)>0$ for some $v\in W$; thus also $\varvec{D}_A(d)>0$ for some $d\in D_w$ (namely, for $d=\delta _w(v)$). Consequently, by Lemma 1, there exists $d^*\in D_w$ such that .

By (9), ; thus, , which by the definition of $\varvec{D}_A$ implies that $d^*=\delta _w(v^*)$ for some $v^*\in W$. Let us show that $({ Min}_{\preccurlyeq _w}\varvec{A})(v^*)>0$:

For any $v'\in W$, we have that $\varvec{A}(v')\le \varvec{D}_A\bigl (\delta _w(v')\bigr )$ by the definition of $\varvec{D}_A$ and that by Definition 7. Thus, by the antitonicity of Łukasiewicz implication in the first argument (see Appendix 1),

whence

$\square$