1 Introduction
A counterfactual conditional (or simply a counterfactual) is a conditional statement of the form “If antecedent were the case, then consequent would be the case”, where the antecedent is usually assumed to be false. Counterfactuals have been studied in different fields, such as linguistics, artificial intelligence, and philosophy. The logical analysis of counterfactuals is rooted in the work of Lewis [Reference Lewis16, Reference Lewis17] and Stalnaker [Reference Stalnaker and Rescher29, Reference Stalnaker and Thomason30] who have introduced what has become the standard semantics for counterfactual conditionals based on particular Kripke models (called sphere models) equipped with a similarity relation among the possible worlds. In Lewis’s language, a counterfactual is formalized as a formula of the kind 
which is intended to mean that if
$\varphi $
were the case, then
$\psi $
would be the case. Lewis develops a hierarchy of logics meant to deal with different kinds of conditionals that have had a notable impact, and are quite well-known and studied; surprisingly, the literature on these logics (quite vast: Lewis’s book counts thousands of citations at the present date) essentially lacks a logico-algebraic treatment.
The reader needs not be reminded that the role of algebra has been pivotal in the formalization and understanding of correct reasoning; indeed, modern logic really flourishes with the rise of the formal methods of mathematical logic, which moves its first steps with George Boole’s intuition of using the symbolic language of algebra as a mean to formalize how sentences connect together via logical connectives [Reference Boole7]. More recently, the advancements of the discipline of (abstract) algebraic logic have been one of the main drivers behind the surge of systems of nonclassical logics in the 20th century, in particular via the notion of equivalent algebraic semantics of a logic introduced by Blok and Pigozzi [Reference Blok and Pigozzi6]. In this framework, the deductions of a logic are fully and faithfully interpreted by the semantical consequence of the related algebras, and powerful bridge theorems allow one to study properties of the logics in the corresponding algebraic framework and vice versa [Reference Font10].
While a few works present a semantics in terms of algebraic structures for Lewis’s conditional logics [Reference Nute23, Reference Rosella, Flaminio and Bonzio26, Reference Segerberg28, Reference Weiss31], the results therein are either partial or fall outside the framework of the abstract algebraic analysis (however see [Reference Paoli24] for a logico-algebraic work on a conditional logic outside of Lewis’s framework). On a different note, the proof-theoretic perspective on Lewis’s conditional logics is instead more developed, in particular it is carried out in a series of recent works [Reference Girlando, Negri and Sbardolini11, Reference Girlando12, Reference Negri and Sbardolini21]. However, although the research on Lewis’s conditional logics has been and still is very prolific, a foundational work that carries Lewis’s hierarchy within the realm of the well-developed discipline of (abstract) algebraic logic is notably missing in the literature; the present manuscript aims at filling this void.
To this end, we start by considering Lewis’s logics as consequence relations, instead of just sets of theorems; this brings us to consider two different kinds of derivation, depending on whether the deductive rules are applied only to theorems (giving a relatively weaker calculus) or to all derivations (i.e., yielding a stronger calculus). We stress that this distinction, although relevant, is often blurred in the literature. As it is the case for modal logic (see [Reference Blackburn, de Rijke and Venema4, Reference Wen32]), these two choices turn out to correspond to considering two different consequence relations on the intended sphere models: a local and a global one; the latter, to the best of our knowledge, has not been considered in the literature. The strong completeness of the local consequence with respect to the weaker calculi essentially follows from Lewis’s work; we provide an analogous result for the global consequence relation with respect to the stronger calculi (Theorem 3.23). It is worth stressing that the axiomatizations we introduce here are novel, and simpler than the original ones in [Reference Lewis17].
Inspired by some results connecting modal operators and Lewis’s counterfactuals (see [Reference Lewis17, Reference Williamson, Hale and Hoffmann33]), our work unveils a deep relationship between Lewis’s logics and modal logic, which will guide the groundwork of this investigation. Specifically, we demonstrate how several model-theoretic techniques commonly used in standard Kripke semantics for modal logic can be successfully applied to Lewis’s sphere semantics, thanks to a modal operator
$\Box $
that can be term-defined in the language. This allows us to prove, for example, a deduction theorem for the strong calculus (Theorem 3.25), whereas the weak calculus is known to have the classical deduction theorem [Reference Lewis17].
One should however keep in mind that the binary operator 
does not straightforwardly inherit the plethora of results on modal operators; for the reader more expert on the algebraic perspective, the models are not Boolean algebras with an operator in the usual sense (see [Reference Jónsson and Tarski14]), since 
is not additive on both arguments (more precisely, it only distributes over infima on the right) and it cannot be recovered from a unary modal operator.
The other main results show that the stronger calculi, associated with the global consequence relation, are strongly algebraizable in the sense of Blok–Pigozzi (Corollary 4.7), i.e., there is a class of algebras axiomatized by means of equations (a variety of Boolean algebras with an extra binary operator 
) that serve as the equivalent algebraic semantics; the weaker calculi, associated with the local consequence relation, are shown to not be algebraizable in general (there is no class of algebras whose consequence relation “corresponds” to the deduction of the logic), but they are the logics preserving the degrees of truth of the same algebraic models (Corollaries 4.19 and 4.22). Therefore, the same class of algebras can be meaningfully used to study both versions of Lewis’s logics; precisely, we have strong completeness of both calculi with respect to the algebraic models. We also initiate the study of the structure theory of the algebraic models; interestingly, we demonstrate that the congruences of the algebras, which are in one-one correspondence with the deductive filters inherited by the logics, can be characterized by means of the congruences of their modal reducts (Proposition 4.12 and Corollary 4.13).
2 Lewis’s logics: New axiomatizations
This section lays out the groundwork for a logico-algebraic study of the hierarchy of logics introduced by Lewis in his seminal book [Reference Lewis17] to reason with counterfactuals conditionals, and their intended models, i.e., sphere models. All the logics in the hierarchy are axiomatic extensions of the system
$\mathtt {V}$
, which according to Lewis is the “weakest system that has any claim to be called the logic of conditionals” [Reference Lewis16, p. 80]; therefore our investigation starts with
$\mathtt {V}$
, and all our results will carry through its axiomatic extensionsFootnote
1
. In particular, we will give a new and simpler axiomatization of
$\mathtt {V}$
with respect to the original ones [Reference Lewis17]; we will take the counterfactual connective as a primitive symbol in the language, and we will distinguish between two different consequence relations: a weak one, where the rules of the calculus only apply to theorems (which is the one usually considered in the literature), and a strong one, where the rules instead apply to all deductions. We will see in the next section that these two choices correspond to considering two different consequence relations over sphere models: a local and a global one, in analogy with the case of modal logic.
While often in the literature
$\mathtt {V}$
is presented as a set of theorems, we are interested in studying logics as consequence relations; let us be more precise. Given an algebraic language
$\mathcal {L}$
, we denote the set of its formulas built from a denumerable set of variables by
$Fm_{\mathcal {L}}$
and the corresponding algebra of formulas by
${\mathbf {Fm}}_{\mathcal {L}}$
. A consequence relation on
${\mathbf {Fm}}_{\mathcal {L}}$
is a relation
$\vdash \, \subseteq \mathscr{P}(Fm_{\mathcal {L}}) \times Fm_{\mathcal {L}}$
(and we write
$\Sigma \vdash \gamma $
for
$(\Sigma , \gamma ) \in \,\vdash $
) such that:
-
1. if
$\alpha \in \Gamma $
then
$\Gamma \vdash \alpha $
; -
2. if
$\Gamma \vdash \delta $
for all
$\delta \in \Delta $
and
$\Delta \vdash \beta $
, then
$\Gamma \vdash \beta $
.
We call substitution any endomorphism of
${\mathbf {Fm}}_{\mathcal {L}}$
(i.e., any function on itself that respects all operations);
$\vdash $
is substitution invariant (also called structural) if
$\Gamma \vdash \alpha $
implies
$\{\sigma (\gamma ): \gamma \in \Gamma \} \vdash \sigma (\alpha )$
for each substitution
$\sigma $
. Finally,
$\vdash $
is finitary if
$\Gamma \vdash \alpha $
implies that there is a finite
$\Gamma ' \subseteq \Gamma $
such that
$\Gamma ' \vdash \alpha $
. A logic
$\mathtt {L}$
is an ordered pair
$\mathtt {L} = (\mathcal {L}, \vdash _{\mathtt {L}})$
given by a language
$\mathcal {L}$
and a substitution-invariant consequence relation
$\vdash _{\mathtt {L}}$
on
${\mathbf {Fm}}_{\mathcal {L}}$
; in this work we will actually only be interested in finitary logics.
Let us now fix the language 
to be the one obtained from a denumerable set of variables and expanding the language of propositional classical logic
$\{\land , \lor , \to , 0, 1\}$
with a binary connective 
, where 
should be read as
“if it were the case that
$\varphi $
, then it would be the case that
$\psi $
”.
As usual, one can define further classical connectives by:
$\neg x : = x \to 0, \;\; x \leftrightarrow y : = (x \to y) \land (y \to x).$
The following derived connectives will be also considered:
Notation 2.1. We denote with 
the algebra of 
-formulas over a fixed denumerable set of variables
$Var$
.
We will hence distinguish two different logics,
$\mathtt {GV}$
and
$\mathtt {LV}$
, which arise depending on whether we apply the rules of Lewis calculus only to theorems (for the weaker logic
$\mathtt {LV}$
) or to all derivations (for the stronger logic
$\mathtt {GV}$
). We remark that this distinction, although significant, is often blurred in the literature. The two systems
$\mathtt {GV}$
and
$\mathtt {LV}$
share the same axioms, that is, given 
we have:
-
(L0) the reader’s favorite Hilbert-style axioms of classical logicFootnote 2 ;
-
(L1)
; -
(L2)
; -
(L3)
; -
(L4)
;
;
;
;
;Moreover, both
$\mathtt {GV}$
and
$\mathtt {LV}$
satisfy modus ponens:
-
(MP)
$\varphi , \varphi \to \psi \vdash \psi $
.
While
$\mathtt {GV}$
satisfies the following rule involving the counterfactual implication:
-
(C)
,
,
$\mathtt {LV}$
satisfies the following weaker version of the rule:
-
(wC)
$ \vdash \varphi \to \psi $
implies .
.Definition 2.2. The logic
$\mathtt {GV}$
is given by the pair 
, where
$\vdash _{\mathtt {GV}}$
is the smallest finitary consequence relation satisfying all axioms (L1)–(L4), (MP), and (C). The logic
$\mathtt {LV}$
is given by 
, where
$\vdash _{\mathtt {LV}}$
is the smallest finitary consequence relation satisfying all axioms (L1)–(L4), (MP), and (wC).
Lewis’s conditional logics are the axiomatic extensions of the above systems with the axioms:
-
(W)
(weak centering) -
(C)
(centering) -
(N)
$\vdash \square \varphi \to \Diamond \varphi $
(normality) -
(T)
$\vdash \square \varphi \to \varphi $
(total reflexivity) -
(S)
(Stalnakerian) -
(U)
$\vdash (\Diamond \varphi \to \square \Diamond \varphi ) \wedge (\square \varphi \to \square \square \varphi )$
(uniformity) -
(A)
$\vdash ((\varphi \preccurlyeq \psi )\to \square (\varphi \preccurlyeq \psi )) \wedge ((\varphi \prec \psi )\to \square (\varphi \prec \psi ))$
(absoluteness)
(weak centering)
(centering)
(Stalnakerian)
We indicate a certain system in the family of Lewis’s conditional logics by just juxtaposing to
$\mathtt {GV}$
or
$\mathtt {LV}$
the corresponding letter for axioms. For instance,
$\mathtt {LVCA}$
indicates the axiomatic extension of the logic
$\mathtt {LV}$
with the axioms
$\mathtt {C}$
and
$\mathtt {A}$
. Among these axiomatic extensions, it is worth mentioning the system
$\mathtt {LVC}$
which is considered by Lewis to be the “correct logic of counterfactual conditionals” [Reference Lewis16], while
$\mathtt {LVCS}$
essentially corresponds to Stalnaker’s logic of conditionals [Reference Stalnaker and Rescher29, Reference Stalnaker and Thomason30].
It is clear from the definition that
$\mathtt {GV}$
is a stronger deductive system than
$\mathtt {LV}$
, i.e.,:
Lemma 2.3. For any set
$\Gamma \cup \{\varphi \}$
of 
-formulas,
$\Gamma \vdash _{\mathtt {LV}} \varphi $
implies
$\Gamma \vdash _{\mathtt {GV}} \varphi $
.
While
$\mathtt {GV}$
is strictly stronger than
$\mathtt {LV}$
, e.g., the latter does not satisfy (C), they do have the same theorems; the following proof is standard.
Theorem 2.4.
$\mathtt {GV}$
and
$\mathtt {LV}$
have the same theorems.
Proof. It follows from Lemma 2.3 that if a formula
$\varphi $
is a theorem of
$\mathtt {LV}$
, it is also a theorem of
$\mathtt {GV}$
; we show the converse. Let
$\varphi $
be a theorem of
$\mathtt {GV}$
, we show by induction on the length of the proof that
$\varphi $
is a theorem of
$\mathtt {LV}$
. The base case is for
$\varphi $
being an axiom, thus the thesis holds given that
$\mathtt {GV}$
and
$\mathtt {LV}$
share the same axioms. Assume now that
$\varphi $
is obtained by an application of a rule of
$\mathtt {GV}$
, i.e., either by modus ponens or (C). But such rule is applied to theorems or axioms of
$\mathtt {GV}$
, that by inductive hypothesis are theorems of
$\mathtt {LV}$
; therefore, one can obtain the same conclusion by applying (MP) or (wC).
We will now see that the axiomatization we have given is equivalent to the one given by Lewis in [Reference Lewis16]; with respect to the latter, we have added axioms (L4) and removed the denumerable set of rules describing “deductions within conditionals”. Let us present the latter in the two versions, the strong ones:
for each
$n \in \mathbb {N}, n \geq 1$
, and the weaker versions:
for each
$n \in \mathbb {N}, n \geq 1$
. We start by noting that the monotonicity of 
on the consequent can be shown to be a consequence of the axioms.
Lemma 2.5. The following holds for all 
-formulas
$\varphi , \psi , \gamma $
:
-
1.
.
.Proof. Observe first that by the axioms and rules of classical logic, it holds that
$ \vdash _{\mathtt {GV}} \psi \leftrightarrow (\psi \land (\psi \lor \gamma ))$
. Therefore, by using (C) and (L4) we get
from which we can derive 
, which concludes the proof.
Moreover:
Lemma 2.6. Consider a logic 
satisfying the axioms of classical logic, (MP), and (DWC
$_2$
). Then:
-
1.
; -
2.
; -
3.
.
;
;
.Proof. For (1), from (DWC
$_2$
) we get:
(2) is a direct consequence of (1). For (3) we have the following:
which, given that
$\vdash ' \psi \leftrightarrow (\psi \land \psi )$
, proves the claim.
Proposition 2.7. Consider a logic 
satisfying the axioms of classical logic and (MP). The following are equivalent.
-
1.
$\vdash '$
satisfies (L1)–(L4) and (C); -
2.
$\vdash '$
satisfies (L1)–(L3) and the rule (DWC
$_2$
).
The same holds replacing (C), and (DWC
$_2$
) with their weaker versions (wC), and (wDWC
$_2$
).
Proof. Let us first show that (1) implies (2), i.e., we derive the rule (DWC
$_2$
) using (L4) and (C). Using (C) we obtain that 
using now (L4) we get
which is exactly (DWC
$_2$
).
Conversely, let us prove that (DWC
$_2$
) implies (C) and (L4); (C) can be derived by (DWC
$_2$
) by setting
$\varphi _1 = \varphi _2 := \varphi $
; for (L4), by the axioms and rules of classical logic and (DWC
$_2$
) we get:
moreover, one can use monotonicity of 
(Lemma 2.5) and the fact that classically equivalent formulas can be substituted in the consequent of 
(Lemma 2.6) and obtain
and therefore 
, which shows (L4).
The proof for the statement involving the weaker rules goes similarly, with the proviso that every derivation starts from theorems and axioms.
The next theorem shows that both the alternative axiomatizations we have introduced are equivalent to the one presented by Lewis’s in [Reference Lewis16].
Theorem 2.8. Consider a logic 
satisfying the axioms of classical logic and (MP). The following are equivalent.
-
1.
$\vdash '$
satisfies (L1)–(L3) and (DWC
$_n$
) for all
$n \in \mathbb {N}$
; -
2.
$\vdash '$
satisfies (L1)–(L3) and (DWC
$_2$
); -
3.
$\vdash '$
coincides with
$\vdash _{\mathtt {GV}}$
.
The same holds replacing (DWC
$_2$
), (DWC
$_n$
), and
$\vdash _{\mathtt {GV}}$
with their weakened versions (wDWC
$_2$
), (wDWC
$_n$
), and
$\vdash _{\mathtt {LV}}$
.
Proof. (2) and (3) are equivalent by Proposition 2.7; moreover, it is obvious that (1) implies (2); let us show the converse. (DWC
$_1$
) follows from (DWC
$_2$
) by setting
$\varphi _1 = \varphi _2$
. Now, consider
$n \in \mathbb {N}, n \geq 2$
, then with (DWC
$_2$
) we obtain immediately:
which using the fact that (L4) holds by Proposition 2.7, yields that (DWC
$_n$
) holds for all
$n \geq 2$
. In order to show that (DWC
$_0$
) holds, we first observe that 
is a theorem, indeed by (L4) we get
where in the derivations we used (L1) and substitution of classically equivalent formulas in the consequent of 
(Lemma 2.6). Finally, (DWC
$_0$
) is then a consequence of applying (DWC
$_2$
) with
$\varphi _1 = \varphi _2 := 1$
,
$\psi := \psi $
,
$\gamma := \varphi $
.
The proof for the weaker calculus is completely analogous.
Lastly, let us observe that both
$\mathtt {GV}$
and
$\mathtt {LV}$
satisfy the substitution of logical equivalents, in the following sense.
Proposition 2.9. The following hold for any 
:
-
1.
, -
2.
, -
3.
and .
,
,
and 
.Proof. First, notice that (1) follows by (C). Moreover, since (DWC
$_1$
) holds by Theorem 2.8, one gets
and its symmetric copy; thus it follows 
which via (L2) gives (2). Finally, (3) follows from the previous points, given that
$\mathtt {GV}$
and
$\mathtt {LV}$
have the same theorems (Theorem 2.4).
3 Sphere models
Lewis bases his interpretation of the counterfactual connective 
on a neighbourhood-style semantics. The intuitive idea to evaluate the connective 
is that 
is true at some world
$w$
if in the closest worlds to
$w$
in which
$\varphi $
is true, also
$\psi $
is true. This results in the definition of what Lewis calls a “variably strict conditional”, where the word “variably” stresses the fact that to evaluate counterfactuals with different antecedents at some world
$w$
, one might need to evaluate the corresponding classical implication in different worlds; from another point of view, this also means that, in general, a counterfactual 
does not arise as some
$\Box (\varphi \to \psi )$
, for some modality
$\Box $
. Lewis formalizes this intuition by assigning to each possible world
$w$
a nested set of spheres, which are subsets of possible worlds, meant to describe a similarity relationship with
$w$
; the smaller is the sphere to which a world
$w'$
belongs, the closer, and therefore more similar, it is to
$w$
.
In this section we will introduce two different consequence relations over sphere models: a local and a global one, in complete analogy with the case of modal logic. This parallel will continue and guide the investigation throughout the rest of this section. In particular, we will use the tool of generated submodels, borrowed from modal logic (see [Reference Blackburn, de Rijke and Venema4]), and apply it to sphere models to first characterize the global consequence relation via the local one, secondly to prove a deduction theorem, and finally to prove the strong completeness of the global consequence with respect to the strong version of the presented Hilbert-style calculus. Let us now be more precise.
Definition 3.1. A sphere model
$\mathcal {M}$
is a tuple
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
where:
-
1.
$W$
is a set of possible worlds; -
2.
$\mathcal {S}: W\to \mathscr{P}(\mathscr{P}(W))$
is a function assigning to each
$w \in W$
a non-emptyFootnote
3
set
$\mathcal {S}(w)$
of nested subsets of
${W}$
, i.e., for all
$X, Y \in \mathcal {S}(w)$
, either
$ X\subseteq Y$
or
$Y\subseteq X$
. -
3.
$v: Var \to \mathscr{P}(W)$
is an assignment of the variables to subsets of
$W$
, extending to all -formulas as follows:
-formulas as follows: 
Given a sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
, and a set of 
-formulas
$\Gamma $
, we set:
$$ \begin{align} w \Vdash \Gamma \text{ iff } w \in \bigcap\{v(\gamma):\gamma \in \Gamma\}; \end{align} $$
$$ \begin{align} \mathcal{M}\Vdash \Gamma \text{ iff for all } w \in W, w\Vdash \Gamma. \end{align} $$
Notation 3.2. If
$\Gamma = \{\gamma \}$
, we drop the parentheses and write
$w \Vdash \gamma $
(or
$\mathcal {M} \Vdash \gamma $
) instead of
$w \Vdash \{\gamma \}$
(or
$\mathcal {M} \Vdash \{\gamma \}$
). Moreover, since in what follows we will be dealing with some submodels, it is sometimes convenient to stress to which universe a world belongs; given a sphere model
$\mathcal {M} = (W, \mathcal {S}, v)$
,
$w \in W$
, we then write
$$ \begin{align} \mathcal{M}, w \Vdash \varphi \text{ iff } w \Vdash \varphi. \end{align} $$
The theorems of
$\mathtt {GV}$
and
$\mathtt {LV}$
(which are the same by Theorem 2.4) are exactly the set of formulas true in all sphere models, i.e., the set of 
-formulas
$\varphi $
such that
$\mathcal {M} \Vdash \varphi $
for all sphere models
$\mathcal {M}$
(see Theorem 3.8 below).
3.1 Local and global semantics
We now introduce two natural notions of semantical consequence, in close analogy with the local and global consequence relations of modal logic, and we will see by the end of this section that they are sound and (strongly) complete with respect to
$\mathtt {LV}$
and
$\mathtt {GV}$
respectively.
Definition 3.3. Let
$\mathfrak {S}$
be a class of sphere models.
-
1. We define the global
$\mathfrak {S}$
-consequence relation on sphere models as:
$\Gamma \vDash _{\mathfrak {S},g} \varphi $
if and only if for all sphere models
$\mathcal {M} \in \mathfrak {S}$
,
$\mathcal {M}\Vdash \Gamma $
implies
$\mathcal {M}\Vdash \varphi $
. -
2. We define the local
$\mathfrak {S}$
-consequence relation on sphere models as:
$\Gamma \vDash _{\mathfrak {S},\ell } \varphi $
if and only if for all sphere models
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle \in \mathfrak {S}$
and all
$w \in W$
,
$w\Vdash \Gamma $
implies
$w\Vdash \varphi $
.
Notation 3.4. When
$\mathfrak {S}$
is the class of all sphere models we write
$\vDash _{g}$
for
$\vDash _{\mathfrak {S},g}$
and
$\vDash _{\ell }$
for
$\vDash _{\mathfrak {S},\ell }$
.
The following is a direct consequence of the definitions.
Theorem 3.5. Given any 
-formula
$\varphi $
,
$\vDash _{g} \varphi $
if and only if
$\vDash _{l} \varphi $
.
Lewis [Reference Lewis17] considers the classes of sphere models corresponding to the main axiomatic extensions of the system
$\mathtt {V}$
; those classes are listed in the following definition:
Definition 3.6. Let
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
be a sphere model.
-
1.
$\mathcal {M}$
is normal if for each
$w \in W$
,
$\bigcup \mathcal {S}(w)\neq \emptyset $
. -
2.
$\mathcal {M}$
is totally reflexive if for each
$w \in W$
,
$w \in \bigcup \mathcal {S}(w)$
. -
3.
$\mathcal {M}$
is weakly centered if for each
$w \in W$
,
$w \in S$
for each nonempty
$S \in \mathcal {S}(w)$
, and there is a nonempty
$S \in \mathcal {S}(w)$
. -
4.
$\mathcal {M}$
is centered if for each
$w \in W$
,
$\{w\} \in \mathcal {S}(w)$
. -
5.
$\mathcal {M}$
is Stalnakerian if for any
$w \in W$
, and any -formula
$\varphi $
such that
$v(\varphi ) \cap \bigcup S(w) \neq \emptyset $
, there is some
$S \in S(w)$
and
$y \in W$
such that
$v(\varphi ) \cap S = \{y\}$
. -
6.
$\mathcal {M}$
is locally uniform if for any
$w \in W$
and
$v \in \bigcup \mathcal {S}(w)$
,
$\bigcup \mathcal {S}(w)=\bigcup \mathcal {S}(v)$
. -
7.
$\mathcal {M}$
is locally absolute if for any
$w \in W$
and
$v \in \bigcup \mathcal {S}(w)$
,
$\mathcal {S}(w)=\mathcal {S}(v)$
. -
8.
$\mathcal {M}$
is weakly trivial if for each
$w \in W$
,
$W$
is the only non-empty member of
$\mathcal {S}(w)$
. -
9.
$\mathcal {M}$
is trivial if
$W$
is a singleton
$\{w\}$
, and
$\mathcal {S}(w)=\{\emptyset , \{w\}\}$
.
-formula 
Notice that if
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
is centered,
$\{w\}$
is the smallest (nonempty) sphere of
$S(w)$
for all
$w \in W$
. Stalnakerian sphere models are complete with respect to Stalnaker’s logic of conditionals [Reference Lewis17, Reference Stalnaker and Rescher29].
Notation 3.7. It will be convenient to set the following notation. Given an axiomatic extension of
$\mathtt {LV}$
by
$\Sigma $
, where
$\Sigma $
is a subset of the axioms
$\{\mathtt {W}, \mathtt {C}, \mathtt {N}, \mathtt {T}, \mathtt {S}, \mathtt {U}, \mathtt {A}\}$
, we denote by
$\mathfrak {S}_{\Sigma }$
the corresponding class of sphere models, defined by the corresponding conditions in Definition 3.6.
In particular then, we denote the class of spheres that are: normal by
$\mathfrak {S}_{\mathtt {N}}$
, totally reflexive by
$\mathfrak {S}_{\mathtt {T}}$
, weakly centered by
$\mathfrak {S}_{\mathtt {W}}$
, centered by
$\mathfrak {S}_{\mathtt {C}}$
, Stalnakerian by
$\mathfrak {S}_{\mathtt {S}}$
, locally uniform by
$\mathfrak {S}_{\mathtt {U}}$
, absolute by
$\mathfrak {S}_{\mathtt {A}}$
. Moreover, Lewis shows that weak triviality corresponds to the combination of axioms
$\mathtt {WA}$
thus we denote the class by
$\mathfrak {S}_{\mathtt {WA}}$
, and triviality is analogously linked to
$\mathtt {CA}$
so the corresponding class of models is
$\mathfrak {S}_{\mathtt {CA}}$
.
Lewis demonstrates in [Reference Lewis17, §6.1] that the logic
$\mathtt {LV}$
and its extensions by the axioms
$\{\mathtt {W}, \mathtt {C}, \mathtt {N}, \mathtt {T}, \mathtt {S}, \mathtt {U}, \mathtt {A}\}$
are sound and complete with respect to the corresponding classes of sphere models with the local consequence relation. This means that the theorems in a logic precisely correspond to the validities over the associated class of sphere models. Actually, Lewis’s proof can be straightforwardly extended to show strong completeness, i.e., to consider derivations instead of just theorems. Indeed, Lewis proves in particular the following fact. Given any logic
$\mathtt {LV}\Sigma $
, for
$\Sigma \subseteq \{\mathtt {W}, \mathtt {C}, \mathtt {N}, \mathtt {T}, \mathtt {S}, \mathtt {U}, \mathtt {A}\}$
, and any set of 
-formulas
$\Gamma $
that is
$\mathtt {LV}\Sigma $
-consistent, i.e., such that
$\Gamma \not \vdash _{\mathtt {LV}\Sigma } 0$
, there is a sphere model
$\mathcal {M}$
in
$\mathfrak {S}_\Sigma $
such that
$\mathcal {M} \Vdash \Gamma $
.Footnote
4
This is enough to show not only completeness, but strong completeness with standard arguments (see e.g., [Reference Blackburn, de Rijke and Venema4, proposition 4.12]).
Theorem 3.8 (Strong Completeness and Soundness of the Local Consequence [Reference Lewis17]).
Consider any logic
$\mathtt {LV}\Sigma $
for
$\Sigma \subseteq \{\mathtt {W}, \mathtt {C}, \mathtt {N}, \mathtt {T}, \mathtt {S}, \mathtt {U}, \mathtt {A}\}$
. Then for all 
,
$$\begin{align*}\Gamma \vdash_{\mathtt{LV}\Sigma}\varphi \Leftrightarrow \Gamma \vDash_{\mathfrak{S}_{\Sigma, l}} \varphi.\end{align*}$$
We will show that the global consequence relation can be characterized by means of the local one. In order to do that, we introduce a useful tool. In close analogy to the case of Kripke models, we will see how to manipulate a sphere model in order to obtain a new model, preserving the satisfaction of formulas. Namely, we will prove some invariance results for Lewis’s sphere semantics of counterfactuals.
Definition 3.9. Let
$\Sigma =\langle W, \mathcal {S}, v \rangle $
and
$\Sigma '=\langle W', \mathcal {S}', v' \rangle $
be two sphere models. We say that
$\Sigma '$
is a submodel of
$\Sigma $
if and only if:
-
1.
$W' \subseteq W$
-
2.
$\mathcal {S}'$
is the restriction of
$\mathcal {S}$
to
$W'$
, i.e., for all
$w \in W'$
,
$\mathcal {S}'(w)=\mathcal {S}(w) \cap \mathscr{P}(\mathscr{P}(W'))$
. -
3.
$v'$
is the restriction of
$ v $
to
$W'$
, i.e., for any -formula
$\varphi $
,
$v'(\varphi ) = v(\varphi ) \cap W'$
.
-formula 
We now consider a special class of submodels, namely generated submodels.
Definition 3.10. Let
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
and
$\mathcal {M}'=\langle W', \mathcal {S}', v' \rangle $
be two sphere models. We say that
$\mathcal {M}'$
is a generated submodel of
$\mathcal {M}$
if
$\mathcal {M}'$
is a submodel of
$\mathcal {M}$
such that for all
$w' \in W'$
,
$\mathcal {S}'(w')=\mathcal {S}(w')$
.
In other words, in order to obtain a generated submodel of some sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
one needs to select a subset
$W' \subseteq W$
in such a way that, for each
$w' \in W'$
, all the worlds belonging to the spheres of
$w'$
also belong to
$W'$
. This particular type of generated submodel will play a key role in our analysis. Let us show how one can effectively construct one such submodel.
Consider a sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
and a subset
$X \subseteq W$
. Let us define a binary relation
$R_{\mathcal {S}} \subseteq W\times W$
as follows: for all
$w, u \in W$
,
$$ \begin{align} w \,R_{\mathcal{S}}\, u\Leftrightarrow u\in \bigcup\mathcal{S}(w) \end{align} $$
Thus,
$wR_{\mathcal {S}}u$
if and only if
$ u $
appears in the system of spheres associated with
$w$
. Now, for all
$n \in \mathbb {N}$
, we inductively define a relation
$R_{\mathcal {S}}^n \subseteq W \times W$
in the following way:
-
•
$wR_{\mathcal {S}}^0u \Leftrightarrow w=u$
-
•
$wR_{\mathcal {S}}^{n+1}u \Leftrightarrow $
there is
$z \in W$
such that
$wR_{\mathcal {S}}^nz$
and
$zR_{\mathcal {S}}u$
.
We refer to
$R_{\mathcal {S}}$
as the accessibility relation of
$\mathcal {M}$
. Intuitively, the relation
$wR_{\mathcal {S}}^n u$
indicates that there are
$ n $
steps needed to reach the world
$ u $
starting from
$w$
, where every steps is given by checking a set of spheres. Now, consider the subset
$W_X \subseteq W$
defined as follows:
$$ \begin{align} W_X= \{w \in W : uR_{\mathcal{S}}^nw \text{ for some } n \in \mathbb{N} \text{ and } u \in X\}. \end{align} $$
Namely,
$W_X$
is the set of all worlds in
$W$
that are reachable from a member of
$ X $
by a finite number of steps via the accessibility relation
$R_{\mathcal {S}}$
. We shall now define the sphere model
$$ \begin{align} \mathcal{M}_X=\langle W_X, \mathcal{S}_X, v_X \rangle \end{align} $$
where:
-
•
$\mathcal {S}_X$
is the restriction of
$\mathcal {S}$
to
$W_X$
, i.e., for all
$w \in W_X$
,
$\mathcal {S}_X(w)=\mathcal {S}(w) \cap \mathscr{P}(\mathscr{P}(W_X))$
. -
•
$v_X$
is the restriction of
$ v $
to
$W_X$
, i.e.,
$v_X(\varphi ) = v(\varphi ) \cap W_X$
for all -formulas
$\varphi $
.
-formulas 
We say that a submodel
$\mathcal {M}'$
of
$\mathcal {M}$
is smaller than another submodel
$\mathcal {M}^*$
if the domain of
$\mathcal {M}'$
is contained in the domain of
$\mathcal {M}^*$
. Then:
Lemma 3.11. Let
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
be a sphere model,
$X \subseteq W$
, and let
$\mathcal {M}_X$
be defined as above. Then
$\mathcal {M}_X$
is the smallest generated submodel of
$\mathcal {M}$
whose domain contains
$X$
.
Proof. First observe that
$\mathcal {M}_X$
is a submodel of
$\mathcal {M}$
by definition; it is also easily seen to be a generated submodel of
$\mathcal {M}$
. Indeed by the definition of
$W_X$
the following holds: if
$w \in W_X \text { and } wR_{\mathcal {S}}u, \text { then } u \in W_X$
. Equivalently,
$$\begin{align*}\text{if } w \in W_X \text{ and } u \in \bigcup\mathcal{S}(w), \text{ then } u \in W_X,\end{align*}$$
i.e.,
$\mathcal {S}_X(w)=\mathcal {S}(w) \cap \mathscr{P}(\mathscr{P}(W_X)) = \mathcal {S}_X(w)$
and then
$\mathcal {M}_X$
is a generated submodel of
$\mathcal {M}$
by definition. By the very same equivalence and the definition of
$W_X$
, it follows that if
$\mathcal {M}^*=\langle W^*, \mathcal {S}^*, v^* \rangle $
is any other generated submodel of
$\mathcal {M}$
such that
$X \subseteq W^*$
, necessarily
$W_X$
is contained in
$W^*$
. Therefore,
$\mathcal {M}_X$
is the smallest submodel of
$\mathcal {M}$
generated by
$ X $
.
Definition 3.12. Consider a sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
, and let
$X \subseteq W$
; we call submodel generated by X the smallest submodel of
$\mathcal {M}$
whose domain contains
$ X $
. Moreover, we call centered or point-generated a submodel of
$\mathcal {M}$
generated by a singleton.
Notice that by Lemma 3.11 above, the submodel of
$\mathcal {M}$
generated by
$ X $
is exactly
$\mathcal {M}_X$
. Importantly, all generated submodels preserve the validity of formulas, as the following lemma shows.
Lemma 3.13. Let
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
be a sphere model, and let
$\mathcal {M}'=\langle W', \mathcal {S}', v' \rangle $
be a generated submodel of
$\mathcal {M}$
. The following holds for all
$w \in W'$
, and all 
-formulas
$\varphi $
:
$$\begin{align*}\mathcal{M}, w \Vdash \varphi \Leftrightarrow \mathcal{M}', w \Vdash \varphi \end{align*}$$
Proof. The statement can be easily proved by induction on the construction of the formula
$\varphi $
. In particular, the base case where
$\varphi $
is a variable and the inductive cases given by the classical connectives (i.e.,
$\varphi = \psi * \gamma $
for
$* \in \{\land , \lor , \to \}$
) directly follow from the fact that
$v'$
is the restriction of
$ v $
to
$W'$
. The inductive case 
follows from the fact that
$v'$
is the restriction of
$ v $
to
$W'$
and that
$\mathcal {S}'(w) = \mathcal {S}(w)$
for all
$w \in W'$
.
Moreover, the following is a direct consequence of the definitions.
Lemma 3.14. All the classes of spheres in Definition 3.6 are closed under generated submodels.
Before delving into the relationship between local and global consequences, we provide a first application of generated submodels. Lewis in [Reference Lewis17] considers three additional classes of sphere models, that we have not included in Definition 3.6; let us consider them now.
Definition 3.15. Let
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
be a sphere model, then:
-
1.
$\mathcal {M}$
is uniform if for any
$w, v \in W$
,
$\bigcup \mathcal {S}(w)=\bigcup \mathcal {S}(v)$
. -
2.
$\mathcal {M}$
is absolute if for any
$w, v \in W$
,
$\mathcal {S}(w)=\mathcal {S}(v)$
. -
3.
$\mathcal {M}$
is universal if for each
$w \in W$
,
$\bigcup \mathcal {S}(w)=W$
.
We denote by
$\mathfrak {S}_{\mathtt {U}^+}$
the class of uniform models, by
$\mathfrak {S}_{\mathtt {A}^+}$
the class of absolute models, and by
$\mathfrak {S}_{\mathtt {UT}}$
the class of universal models.
As the reader can easily check, uniformity implies local uniformity, and absoluteness implies local absoluteness. Lewis observes in [Reference Lewis17] that the validity of formulas does not change between the classes of, respectively, locally uniform and uniform models, and locally absolute and absolute ones. We demonstrate that such classes of models are indistinguishable also from the point of view of derivations.
Proposition 3.16. The following hold for any 
:
-
1.
$\Gamma \vDash _{\mathfrak {S}_{\mathtt {U}, l}} \varphi \Leftrightarrow \Gamma \vDash _{\mathfrak {S}_{\mathtt {U}^+, l}} \varphi $
and
$\Gamma \vDash _{\mathfrak {S}_{\mathtt {U}, g}} \varphi \Leftrightarrow \Gamma \vDash _{\mathfrak {S}_{\mathtt {U}^+, g}} \varphi $
-
2.
$\Gamma \vDash _{\mathfrak {S}_{\mathtt {A}, l}} \varphi \Leftrightarrow \Gamma \vDash _{\mathfrak {S}_{\mathtt {A}^+, l}} \varphi $
and
$\Gamma \vDash _{\mathfrak {S}_{\mathtt {A}, g}} \varphi \Leftrightarrow \Gamma \vDash _{\mathfrak {S}_{\mathtt {A}^+, g}} \varphi $
Proof. We prove (1), the proof of (2) being similar. Let us focus on the local consequence first. The
$(\Rightarrow )$
direction is straightforward since every uniform models is also locally uniform. For the
$(\Leftarrow )$
direction, we reason by contraposition and assume that
$\Gamma \nvDash _{\mathfrak {S}_{\mathtt {U}, l}} \varphi $
, namely there is a locally uniform sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
and a
$w \in W$
such that
$w \Vdash \Gamma $
and
$w \nVdash \varphi $
. Now, consider the submodel generated by
$\{w\}$
,
$\mathcal {M}_w=\langle W_w, \mathcal {S}_w, v_w \rangle $
and observe that all
$x \in W_w$
are such that
$wR^n_{\mathcal {S}} x$
for some
$n \in \mathbb {N}$
, by Definition 3.12. We prove by induction on
$ n $
that for all
$x \in W_w$
such that
$wR^n_{\mathcal {S}}x$
,
$\bigcup \mathcal {S}(x)=\bigcup \mathcal {S}(w)$
. For the base case, if
$wR^0_{\mathcal {S}}x$
we have that
$x=w$
by definition of
$R^0_{\mathcal {S}}$
, and then clearly
$\bigcup \mathcal {S}(x)=\bigcup \mathcal {S}(w)$
. For the inductive step, by inductive hypothesis we have that for all
$ y $
such that
$wR^n_{\mathcal {S}}y$
,
$\bigcup \mathcal {S}(w)=\bigcup \mathcal {S}(y)$
. Assume that
$w R^{n+1}_{\mathcal {S}}x$
; then by definition of
$R^{n+1}_{\mathcal {S}}$
, we have that
$wR^n_{\mathcal {S}}y R_{\mathcal {S}} x$
for some
$ y $
, and so
$x \in \bigcup \mathcal {S}(y)$
. Since the original model
$\mathcal {M}$
is locally uniform, we have that
$\bigcup \mathcal {S}(y)=\bigcup \mathcal {S}(x)$
, which yields that
$\bigcup \mathcal {S}(w)= \bigcup \mathcal {S}(x)$
. This proves that
$\mathcal {M}_w$
is uniform. Moreover, by Lemma 3.13, we have that
$\mathcal {M}_w, w \Vdash \Gamma $
and
$\mathcal {M}_w, w \nVdash \varphi $
, hence
$\Gamma \nvDash _{\mathfrak {S}_{\mathtt {U}^+, l}} \varphi $
. The proof for the global consequence proceeds analogously.
Additionally, as Lewis [Reference Lewis17, p.120] himself noted, the class of universal sphere models corresponds to the class of models that are both uniform and totally reflexive. Therefore, for the purpose of the present work, it is sufficient to confine our attention to the class of models presented in Definition 3.6. Nonetheless, in light of the previous observations, is important to note that all of our results can be easily extended to include these other classes of models as well.
After this brief digression, we can continue toward the main focus of this section. Before presenting the characterization of the global consequence relation by means of the local consequence, we need another technical result. Observe that, given a sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
and
$w \in W$
, 
if and only if
$\bigcup \mathcal {S}(w) \cap v(\neg \varphi )=\emptyset $
, or equivalently
$\bigcup \mathcal {S}(w)\subseteq v(\varphi )$
. Recall that 
. It is then straightforward that, given a sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
,
$\Box $
can be characterized by means of the relation
$R_{\mathcal {S}}$
defined in (4) above.
Lemma 3.17. Let
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
be a sphere model; given any
$w \in W$
and 
-formula
$\varphi $
, the following are equivalent:
-
1.
$w \Vdash \Box \varphi $
; -
2.
$\bigcup \mathcal {S}(w)\subseteq v(\varphi )$
; -
3.
$wR_{\mathcal {S}}u$
implies
$u\Vdash \varphi $
.
One can then easily show that
$\Box $
is a modal operator, in the following sense.
Proposition 3.18. The following hold for all 
:
-
1.
$\vDash _g \Box (\varphi \to \psi ) \to (\Box \varphi \to \Box \psi )$
; -
2.
$\varphi \vDash _g \Box \varphi $
; -
3.
$\vDash _l \varphi $
implies
$\vDash _l \Box \varphi $
; -
4.
$\vDash _g \Box (\varphi \land \psi ) \leftrightarrow (\Box \varphi \land \Box \psi )$
.
Proof. Let us start with (1); consider any sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
, and let
$w \in W$
. Suppose
$w \Vdash \Box (\varphi \to \psi )$
, i.e., by Lemma 3.17
$\bigcup \mathcal {S}(w)\subseteq v(\varphi \to \psi )$
. Therefore, if
$w \Vdash \Box \varphi $
, or equivalently
$\bigcup \mathcal {S}(w)\subseteq v(\varphi )$
, it follows that
$\bigcup \mathcal {S}(w)\subseteq v(\psi )$
; applying Lemma 3.17 again, we get that
$w \Vdash \Box \psi $
, which proves the claim.
Let us now prove (2); one needs to show that for all sphere models
$\mathcal {M} = (W, S, v)$
,
$\mathcal {M} \Vdash \varphi $
implies
$\mathcal {M} \Vdash \Box \varphi $
, which is an easy consequence of Lemma 3.17, since if
$\mathcal {M} \Vdash \varphi $
, then for every
$w$
,
$w \in v(\varphi )$
. (3) can be proved analogously, while (4) follows from the previous points.
Let us define inductively an operator
$\Box ^n$
, that iterates
$\Box $
, for
$n\in \mathbb {N}$
:
$$ \begin{align} \Box^0\varphi:=\varphi;\qquad \Box^{n+1}\varphi:=\; \Box\Box^{n}\varphi. \end{align} $$
We are now ready to characterize the connection between local and global consequence relations.
Theorem 3.19. Let
$\mathfrak {S}$
be a class of sphere models closed under generated submodels. For all sets of 
-formulas
$\Gamma $
and 
-formula
$\varphi $
the following are equivalent:
-
1.
$\Gamma \vDash _{\mathfrak {S},g} \varphi $
; -
2.
$\{\Box ^n\gamma : n \in \mathbb {N}, \gamma \in \Gamma \}\vDash _{\mathfrak {S},l} \varphi $
.
Proof. We verify that (2) implies (1) by contraposition. Assume
$\Gamma \nvDash _{\mathfrak {S},g} \varphi $
; i.e., there is a sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle \in \mathfrak {S}$
such that
$w \Vdash \gamma $
for all
$w \in W$
,
$\gamma \in \Gamma $
, and for some
$u \in W$
,
$u \nVdash \varphi $
. By the definition of
$\Box ^n$
and Lemma 3.17, it follows that
$w \Vdash \Box ^n \gamma $
for all
$\gamma \in \Gamma , n \in \mathbb {N}$
and
$w \in W$
. Thus in particular
$u \Vdash \Box ^n\gamma $
for all
$\gamma \in \Gamma , n \in \mathbb {N}$
, but
$u \nVdash \varphi $
. Therefore
$\{\Box ^n\gamma \mid n \in \mathbb {N}\text { and } \gamma \in \Gamma \}\nvDash _{\mathfrak {S},l} \varphi $
.
We now prove that (1) implies (2), again by contraposition; assume that
$\{\Box ^n\gamma \mid n \in \mathbb {N} \text { and } \gamma \in \Gamma \}\nvDash _{\mathfrak {S},l} \varphi $
. Thus, there is a sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle \in \mathfrak {S}$
and
$x \in W$
such that
$x \Vdash \Box ^n\gamma $
for all
$n \in \mathbb {N}$
and
$\gamma \in \Gamma $
but
$x \nVdash \varphi $
. Consider the submodel generated by
$\{x\}$
,
$\mathcal {M}_x=\langle W_x, \mathcal {S}_x, v_x \rangle $
, where
$$ \begin{align*}W_x = \{w \in W: x R^n_{\mathcal{S}}w \text{ for some } n \in \mathbb{N}\}.\end{align*} $$
By Lemma 3.11,
$\mathcal {M}_x$
is a sphere model and it is in
$\mathfrak {S}$
since
$\mathfrak {S}$
is closed under generated submodels by assumption; moreover, by Lemma 3.13 we have that for all
$w \in W_x$
$$ \begin{align*}\mathcal{M}, w \Vdash \varphi \text{ if and only if } \mathcal{M}_x, w \Vdash \varphi.\end{align*} $$
Hence, in particular,
$\mathcal {M}_x, x \Vdash \Box ^n\gamma $
for all
$n \in \mathbb {N},\gamma \in \Gamma $
but
$\mathcal {M}_x, x \nVdash \varphi $
. We now prove that for all
$w \in W_x$
,
$w \Vdash \Gamma $
, which will conclude the proof by showing that
$\mathcal {M}_x \Vdash \Gamma $
but
$\mathcal {M}_x \not \Vdash \varphi $
(since
$\mathcal {M}_x, x \nVdash \varphi $
). By definition, all elements
$w \in W_x$
are such that
$x R^m_{\mathcal {S}} w$
for some
$m \in \mathbb {N}$
; we show by induction on
$ k $
that
$x R^k_{\mathcal {S}} w$
implies
$w \Vdash \Box ^n \gamma $
for all
$n \in \mathbb {N}, \gamma \in \Gamma $
.
-
• If
$k = 0$
, we get
$w = x$
and thus by assumption
$x \Vdash \Box ^n\gamma $
for all
$\gamma \in \Gamma ,n\in \mathbb {N}$
. -
• Assume that the inductive hypothesis holds for
$ k $
, we show it for
$k+1$
. Suppose
$x R^{k+1}_{\mathcal {S}}w$
, i.e., by definition of
$R^{k+1}_{\mathcal {S}}$
, there is some
$z \in W_x$
such that
$x \,R^{k}_{\mathcal {S}}\,z \,R_{\mathcal {S}}\,w$
. By inductive hypothesis
$z \Vdash \Box ^n \gamma $
for all
$n\in \mathbb {N}, \gamma \in \Gamma $
. Thus Lemma 3.17 implies that also
$w \Vdash \Box ^n \gamma $
for all
$n\in \mathbb {N}, \gamma \in \Gamma $
.
Therefore, we have shown that, in particular, all elements
$w \in W_x$
are such that
$w \Vdash \Box ^0 \gamma = \gamma $
for all
$\gamma \in \Gamma $
, which concludes the proof.
We will now use the last result to prove a deduction theorem and a strong completeness result for the global consequence relation.
3.2 Completeness and deduction theorem
Lewis proves soundness and (strong) completeness with respect to sphere models of what we called the local consequence relation with respect to the logic
$\vdash _{\mathtt {LV}}$
(Theorem 3.8); moreover he proves a deduction theorem for the local consequence with respect to the classical implication.
Theorem 3.20 (Deduction Theorem of the local consequence).
For all 
the following holds:
$\Gamma , \psi \vDash _l \varphi $
if and only if
$\Gamma \vDash _l \psi \to \varphi $
.
In this subsection we prove the analogous results for the stronger deductive systems and the corresponding global consequence relations; but first, some technical results.
Lemma 3.21. Consider any 
-formula
$\varphi $
, then
$\varphi \vdash _{\mathtt {GV}} \Box ^n\varphi $
for all
$n \in \mathbb {N}$
.
Proof. The claim is easily shown by induction on
$ n $
; indeed the case
$n = 0$
is obvious, and the inductive case is given by one application of (DWC
$_0$
): 
, which holds for
$\mathtt {GV}$
by Proposition 2.7 and Theorem 2.8.
Proposition 3.22. Let
$\mathtt {L}$
be any axiomatic extension of
$\mathtt {GV}$
. For all 
the following are equivalent:
-
1.
$\Gamma \vdash _{\mathtt {L}} \varphi $
; -
2.
$\{\Box ^n\gamma \mid \gamma \in \Gamma \text { and }n \in \mathbb {N}\} \vdash _{\mathtt {L}} \varphi $
. -
3. There exist a finite subset
$\Gamma _0 \subseteq \Gamma $
and
$n_0 \in \mathbb {N}$
such that
$\{\Box ^n\gamma \mid \gamma \in \Gamma _0 \text { and }n\leq n_0\} \vdash _{\mathtt {L}} \varphi $
.
Proof. The fact that (1) implies (2) is obvious, since
$\Gamma \subseteq \{\Box ^n\gamma \mid \gamma \in \Gamma \text { and }n \in \mathbb {N}\}$
. For the converse, let us assume that
$\{\Box ^n\gamma \mid \gamma \in \Gamma \text { and }n \in \mathbb {N}\} \vdash _{\mathtt {L}} \varphi $
; By Lemma 3.21, we have that
$\Gamma \vdash _{\mathtt {L}} \Box ^n \gamma $
for all
$\gamma \in \Gamma $
and
$n \in \mathbb {N}$
, and thus also
$\Gamma \vdash _{\mathtt {L}} \varphi $
. Lastly, (2) and (3) are equivalent since
$\vdash _{\mathtt {L}}$
is a finitary consequence relation.
We now have all the ingredients to prove our completeness result.
Theorem 3.23 (Soundness and strong completeness of the global consequence).
Let
$\Sigma $
be a subset of the axioms
$\{\mathtt {W}, \mathtt {C}, \mathtt {N}, \mathtt {T}, \mathtt {S}, \mathtt {U}, \mathtt {A}\}$
. For all subsets 
,
$$ \begin{align*}\Gamma \vdash_{\mathtt{GV}\Sigma} \varphi \Leftrightarrow \Gamma \vDash _{\mathfrak{S}_{\Sigma, g}} \varphi.\end{align*} $$
Proof. The soundness follows from the facts that: (MP) and (C) are easily seen to be sound with respect to sphere models, and the axioms of
$\mathtt {GV}\Sigma $
are the same axioms of
$\mathtt {LV}\Sigma $
, which is sound with respect to the same class of models for the local consequence relation (Theorem 3.8), and the latter has the same valid formulas as the global one (Theorem 3.5).
We prove completeness by contraposition; assume
$\Gamma \nvdash _{\mathtt {GV}\Sigma } \varphi $
. By Proposition 3.22, we have that
$\{\Box ^n \gamma \mid \gamma \in \Gamma \text { and } n \in \mathbb {N}\} \nvdash _{\mathtt {GV}\Sigma } \varphi $
. By the fact that all deductions of
$\vdash _{\mathtt {LV}\Sigma } $
are deductions of
$\vdash _{\mathtt {GV}\Sigma }$
(Lemma 2.3), it follows that
$\{\Box ^n \gamma \mid \gamma \in \Gamma \text { and } n \in \mathbb {N}\} \nvdash _{\mathtt {LV}\Sigma } \varphi $
. By the strong completeness of
$\vdash _{\mathtt {LV}\Sigma } $
in Theorem 3.8, we get that
$\{\Box ^n \gamma \mid \gamma \in \Gamma \text { and }n \in \mathbb {N}\} \nvDash _{\mathfrak {S}_{\Sigma , l}} \varphi $
. Theorem 3.19 then yields that
$\Gamma \nvDash _{\mathfrak {S}_{\Sigma , g}} \varphi $
and the proof is complete.
We will now show another relevant fact, i.e., that the global consequence relation has a deduction theorem. However, it generally does not have the classical deduction theorem, as the following example shows.
Example 3.24. By Lemma 3.21 and the strong completeness in Theorem 3.23,
$\varphi \vDash _g \Box \varphi $
for any 
-formula
$\varphi $
. However, it is easily seen that in general
$\nvDash _g \varphi \to \Box \varphi $
. Consider the following sphere model
$\mathcal {M} = (W, \mathcal {S}, v)$
such that
-
•
$W = \{w_1, w_2\}$
; -
•
$\mathcal {S}(w_1) = \{\{w_2\}\}$
;
$\mathcal {S}(w_2) = \{w_2, \{w_1, w_2\}\}$
; -
•
$ v $
is such that it maps a propositional variable
$ p $
to
$v(p) = \{w_1\}$
.
Note that then
$v(\Box p) = \emptyset $
, and therefore
$w \Vdash p$
but
$w \nVdash \Box p$
; hence
$\mathcal {M} \nVdash p \to \Box p$
. One can easily adapt this same example for any class of sphere models that allow at least two different worlds in
$W$
(thus, except for the trivial class of models).
Nonetheless:
Theorem 3.25 (Deduction Theorem of the global consequence).
For all 
, the following are equivalent:
-
1.
$\Gamma , \psi \vDash _{g} \varphi $
, -
2. there is
$n \in \mathbb {N}$
such that
$\Gamma \vDash _{g} (\bigwedge \limits _{m \leq n} \Box ^m\psi )\to \varphi $
,
Proof. Let us start by proving that (1) implies (2); assume
$\Gamma , \psi \vDash _{\mathfrak {S},g} \varphi $
. By Theorem 3.19, this is equivalent to the fact that
$$ \begin{align*}\{\Box^n \gamma: \gamma \in \Gamma, n \in \mathbb{N}\} \cup\{\Box^n\psi: n \in \mathbb{N}\} \vDash _{\mathfrak{S},l} \varphi\end{align*} $$
By the deduction theorem for the local consequence, and the fact that this is finitary (given that it is strongly complete with respect to a finitary logic by Theorem 3.8), it follows that there is
$n_0 \in \mathbb {N}$
$$ \begin{align*}\{\Box^n \gamma: \gamma \in \Gamma, n \in \mathbb{N}\} \vDash _{\mathfrak{S}, l} \left(\bigwedge_{k \leq n_0} \Box^k\psi \right) \to \varphi.\end{align*} $$
Using Theorem 3.19 again, we have that
$\Gamma \vDash _{\mathfrak {S},g} (\bigwedge _{k \leq n_0} \Box ^k\psi ) \to \varphi $
.
We now show that (2) implies (1); assume that there is a
$n \in \mathbb {N}$
such that
$\Gamma \vDash _{\mathfrak {S},g} (\bigwedge _{m \leq n} \Box ^m\psi )\to \varphi $
. Equivalently, for all sphere models
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle \in \mathfrak {S}$
, we have that if
$\mathcal {M}\Vdash \gamma $
for all
$\gamma \in \Gamma $
, then
$\mathcal {M}\Vdash (\bigwedge _{m \leq n} \Box ^m\psi )\to \varphi $
. Consider then a sphere model
$\mathcal {M} = (W, \mathcal {S}, v)$
such that
$\mathcal {M}\Vdash \gamma $
for all
$\gamma \in \Gamma $
and
$\mathcal {M} \Vdash \psi $
. By assumption, we then have that for any world
$w \in W$
,
$w \Vdash \psi $
and
$w \Vdash (\bigwedge _{m \leq n} \Box ^m\psi )\to \varphi $
. By Proposition 3.18, this implies that for all
$w \in W$
,
$w \Vdash \bigwedge _{m \leq n} \Box ^m\psi $
. Therefore, by modus ponens, we have that for all
$w \in W$
,
$w \Vdash \varphi $
as well; i.e.,
$\mathcal {M} \Vdash \varphi $
. Hence
$\Gamma , \psi \vDash _{\mathfrak {S},g} \varphi $
.
By the strong completeness in Theorem 3.23, the theorem above can also be read as follows:
Corollary 3.26. For all 
,
$\Gamma , \psi \vdash _{\mathtt {GV}} \varphi $
if and only if
$\Gamma \vdash _{\mathtt {GV}} (\bigwedge \limits _{m \leq n} \Box ^m\psi )\to \varphi $
for some
$n \in \mathbb {N}$
.
We are now ready to proceed with our investigation toward an algebraic study of Lewis’s hierarchy of logics for variably strict conditionals.
4 Algebraic semantics
In this section we show that the stronger calculi are algebraizable in the sense of Blok–Pigozzi, and we study the equivalent algebraic semantics, given by varieties of Boolean algebras with an extra operator 
. Moreover, we show that such algebras give a semantics for the weaker logics as well, in the sense that the latter are the logics preserving the degrees of truth of the algebras. For the sake of the reader, we first recall the basics of the Blok–Pigozzi machinery [Reference Blok and Pigozzi6], that connects algebraizable logics with their equivalent algebraic semantics. For the omitted details we refer to [Reference Blok and Pigozzi6, Reference Font10].
Let us set some notation; Roman bold letters will be used to represent algebras, while the corresponding Roman standard letters will denote their underlying domains. For instance, if
$\mathbf {A}$
is an algebra, then the symbol
$ A $
will refer to its domain. Given an algebraic language
$\mathcal {L},$
recall that we write
${\mathbf {Fm}}_{\mathcal {L}}$
for its algebra of formulas written over a denumerable set of variables. An equation of the language
$\mathcal {L}$
(or an
$\mathcal {L}$
-equation for short) is a pair
$(p,q)$
of
$\mathcal {L}$
-formulas (i.e., elements of
$Fm_{\mathcal {L}}$
) that we write suggestively as
$p \approx q$
. We write
$Eq_{\mathcal {L}}$
for the set of all
$\mathcal {L}$
-equations. A quasi-equation of
$\mathcal {L}$
is a first-order formula of the form
$\&_{i = 1}^n p_i \approx q_i \Rightarrow p \approx q$
where
$\{p_i \approx q_i: i = 1, \ldots , n\} \cup \{p \approx q\}\subseteq Eq_{\mathcal {L}}$
, and
$\&$
and
$\Rightarrow $
are, respectively, first order conjunction and implication. It is understood that this expression also covers the case of an empty antecedent, so that equations can be seen as particular cases of quasi-equations.
An assignment is a homomorphism (i.e., a function which commutes with all the operations) from the algebra of formulas
${\mathbf {Fm}}_{\mathcal {L}}$
to some
$\mathcal {L}$
-algebra
${\mathbf {A}}$
. An
$\mathcal {L}$
-algebra
${\mathbf {A}}$
satisfies an
$\mathcal {L}$
-equation
$p \approx q$
with an assignment
$ h $
(and we write
${\mathbf {A}}, h \vDash p \approx q$
) if
$h(p) = h(q)$
in
${\mathbf {A}}$
. An
$\mathcal {L}$
-equation
$p \approx q$
is valid in
${\mathbf {A}}$
(and we write
${\mathbf {A}} \vDash p \approx q$
) if for all assignments
$ h $
to
${\mathbf {A}}$
,
${\mathbf {A}}, h \vDash p \approx q$
; if
$\Sigma $
is a set of
$\mathcal {L}$
-equations then
${\mathbf {A}} \vDash \Sigma $
if
${\mathbf {A}} \vDash \sigma $
for all
$\sigma \in \Sigma $
. An
$\mathcal {L}$
-equation
$p \approx q$
is valid in a class of
$\mathcal {L}$
-algebras
$\mathsf {K}$
, and we write
$\mathsf {K} \vDash p \approx q$
or
$\vDash _{\mathsf {K}} p \approx q$
, if
${\mathbf {A}} \vDash p \approx q$
for all
${\mathbf {A}} \in \mathsf {K}$
. With respect to quasi-equations, an
$\mathcal {L}$
-algebra
${\mathbf {A}}$
satisfies an
$\mathcal {L}$
-quasi-equation
$\&_{i = 1}^n p_i \approx q_i \Rightarrow p \approx q$
with an assignment
$ h $
if
$h(p_i) = h(q_i)$
for all
$i = 1, \ldots , n$
implies
$h(p) = h(q)$
; the other notions of validity extend to quasi-equations in the obvious way.
Moreover, given any set of
$\mathcal {L}$
-equations
$\Sigma \cup \{p \approx q\}$
, and any class of
$\mathcal {L}$
-algebras
$\mathsf {K}$
, we write
$$ \begin{align*}\Sigma \vDash _{\mathsf {K}} p \approx q\end{align*} $$
if for every algebra
${\mathbf {A}} \in \mathsf {K}$
, and any assignment
$ h $
to
${\mathbf {A}}$
, if
$h(p') = h(q')$
for all
$p' \approx q' \in \Sigma $
, then
$h(p) = h(q)$
.
$\Sigma \vDash _{\mathsf {K}} \Delta $
, for
$\Sigma , \Delta $
sets of
$\mathcal {L}$
-equations, is interpreted as
$\Sigma \vDash _{\mathsf {K}} \delta $
for all
$\delta \in \Delta $
.
$\vDash _{\mathsf {K}}$
is called the equational consequence relative to
$\mathsf {K}$
. We write 
as a shortening of
$\Sigma \vDash _{\mathsf {K}} \Delta $
and
$\Delta \vDash _{\mathsf {K}} \Sigma $
.
Intuitively, in order to establish the algebraizability of a logic
$\mathtt {L}$
with respect to a class of algebras
$\mathsf {K}_{\mathtt {L}}$
over the same language
$\mathcal {L}$
, one wants to be able to fully and faithfully interpret the consequence relation of
$\mathtt {L}$
into the equational consequence relative to
$\mathsf {K}_{\mathtt {L}}$
.
Let us be precise. Fix a language
$\mathcal {L}$
; a transformer from formulas to (sets of) equations is any function
$\tau $
mapping each
$\mathcal {L}$
-formula to a set of
$\mathcal {L}$
-equations,
$\tau : Fm_{\mathcal {L}} \to \mathscr{P}(Eq_{\mathcal {L}})$
. This extends to sets of formulas by setting, for any set of formulas
$\Gamma \subseteq Fm_{\mathcal {L}}$
,
$\tau (\Gamma ) = \bigcup _{\gamma \in \Gamma }\tau (\gamma )$
. Similarly, a transformer of equations into (sets of) formulas is a function
$\rho $
mapping each
$\mathcal {L}$
-equation into a set of
$\mathcal {L}$
-formulas,
$\rho : Eq_{\mathcal {L}} \to \mathscr{P}(Fm_{\mathcal {L}})$
, that extends to sets of equations by taking unions. In particular, one wants to consider structural transformers: a transformer is structural when it commutes with substitutions.
A logic
$\mathtt {L}$
is algebraizable when there is a class of algebras
$\mathsf {K}$
and structural transformers
$\tau , \rho $
(from formulas into equations and from equations into formulas, respectively) such that the following conditions are satisfied, for all
$\Gamma \cup \{\varphi \} \subseteq Fm_{\mathcal {L}}$
and
$\mathcal {L}$
-equation
$p \approx q$
:
$$ \begin{align} \Gamma \vdash_{\mathtt{L}} \varphi\quad\text{iff}\quad \tau(\Gamma) \vDash_{\mathsf {K}_{\mathtt{L}}} \tau(\varphi) \end{align} $$
The transformers
$\tau $
and
$\rho $
are said to witness the algebraizability of
$\mathtt {L}$
with respect to the class
$\mathsf {K}$
. The equations
$E(x) := \tau (x)$
are called the defining equations and the formulas in
$\Delta (x,y):= \rho (x \approx y)$
are called the equivalence formulas. Given
$\mathtt {L}$
an algebraizable logic, its equivalent algebraic semantics is the largest class of algebras
$\mathsf {K}$
such that
$\mathtt {L}$
is algebraizable with respect to it. In particular, if
$\mathtt {L}$
is algebraizable with respect to a quasivariety (i.e., a class of models of a set of quasi-equations)
$\mathsf {K}$
,
$\mathsf {K}$
is the equivalent algebraic semantics of
$\mathtt {L}$
, and every finitary logic has a quasivariety as its equivalent algebraic semantics. When the equivalent algebraic semantics can be axiomatized by equations (i.e., it is a variety), we say that a finitary logic is strongly algebraizable.
Example 4.1. Classical logic is (strongly) algebraizable with respect to the variety of Boolean algebras, as testified by
$\tau (x) = \{x \approx 1\}$
and
$\rho (x \approx y) = \{x \leftrightarrow y\}$
.
While the conditions (8) and (9) above are necessary and sufficient to show the algebraizability of a logic, in some cases there are easier ways to check whether a logic is algebraizable. In fact, many of the well-known algebraizable logics belong to the class of implicative logics, that is, they have a well-behaved binary connective
$\to $
which allows to show that (8) and (9) hold.
Definition 4.2. An implicative logic is a logic
$\mathtt {L}$
in a language
$\mathcal {L}$
with a binary term
$\to $
such that:
-
1.
$\vdash x \to x$
-
2.
$x \to y, y \to z \vdash x \to z$
-
3.
$x_1 \to y_1, \ldots , x_n \to y_n, y_1 \to x_1, \ldots , y_n \to x_n \vdash \lambda (x_1, \ldots , x_n) \to \lambda (y_1, \ldots , y_n)$
for each term
$\lambda \in \mathcal {L}$
of arity
$n> 0$
-
4.
$x, x \to y \vdash y$
-
5.
$x \vdash y \to x$
.
Classical logic is an example of an implicative logic. In an implicative logic that does not have a constant
$1$
that is a theorem, one can always define
$1:= x \to x$
for a fixed variable
$ x $
, and
$1$
is a theorem by the above definition.
Theorem 4.3 [Reference Font10].
All implicative logics are algebraizable, with defining equation
$\tau (x) := \{x \approx x \to x\}$
and equivalence formulas
$\Delta (x, y) := \{x\to y, y\to x\}$
. If there is a constant
$1$
that is a theorem,
$\tau (x) := \{x \approx 1\}$
. If the logic is finitary, the quasivariety that is its equivalent algebraic semantics can be presented by the equations and quasi-equations that result by applying the transformation
$\tau $
to the axioms and rules of any Hilbert-style presentation of the logic.
From the more strictly algebraic perspective, in a class of algebras that is the equivalent algebraic semantics of a logic
$\mathtt {L}$
over a language
$\mathcal {L}$
, congruences are in one-one correspondence with the deductive filters induced by the logic. An
$\mathtt {L}$
-deductive filter
$ F $
of an algebra
${\mathbf {A}}$
is a subset of the domain of
${\mathbf {A}}$
that is closed under the deductions of the logic
$\mathtt {L}$
; that is, for every
$\Gamma \cup \{\varphi \} \subseteq Fm_{\mathcal {L}}$
, if
$\Gamma \vdash _{\mathtt {L}} \varphi $
, for every homomorphism
$ f $
from
${\mathbf {Fm}}_{\mathcal {L}}$
to
${\mathbf {A}}$
, if
$f[\Gamma ] \subseteq F$
, then
$f(\varphi ) \in F$
.
Theorem 4.4 [Reference Font10, Theorem 3.51].
Let
$\mathtt {L}=(\mathcal {L}, \vdash )$
be a finitary logic with equivalent algebraic semantics a quasivariety
$\mathsf {K}$
, and let
${\mathbf {A}}$
be an
$\mathcal {L}$
-algebra. Then the
$\mathcal {L}$
- deductive filters of
${\mathbf {A}}$
are in bijection with the
$\mathsf {K}$
-relative congruences of
${\mathbf {A}}$
.
We mention that the
$\mathtt {L}$
-deductive filters of the algebras of formulas are the theories of the logic
$\mathtt {L}$
. In particular, if
$\mathtt {L}$
is an implicative logic and
$\mathsf {K}_{\mathtt {L}}$
its equivalent algebraic semantics, for every
${\mathbf {A}}$
in
$\mathsf {K}_{\mathtt {L}}$
the correspondence between congruences and deductive filters is given by the following maps:
$$ \begin{align*}\theta \mapsto F_\theta = \{a \in A: (a, a \to a) \in \theta\}, \qquad F \mapsto \theta_F =\{(a, b): a \to b, b \to a \in F\}.\end{align*} $$
where
$\theta $
is any congruence of
${\mathbf {A}}$
and
$ F $
is any deductive filter of
${\mathbf {A}}$
. If there is a constant
$1$
in the language of
$\mathtt {L}$
that is a theorem, as it is the case for classical logic, then congruences are totally determined by their
$1$
-blocks, i.e., the first map above becomes:
$$ \begin{align*}\theta \mapsto F_\theta = \{a \in A: (a, 1) \in \theta\} = 1/\theta.\end{align*} $$
This means that
$\mathsf {K}_{\mathtt {L}}$
is ideal-determined with respect to
$1$
, and so in particular also 1-regular (
$1/\theta = 1/\gamma $
implies
$\theta = \gamma $
for all congruences
$\theta ,\gamma $
of every algebra
${\mathbf {A}} \in \mathsf {K}_{\mathtt {L}}$
), see [Reference Aglianò and Ursini2, Reference Gumm and Ursini13].
4.1 Global equivalent algebraic semantics
The algebraizability of
$\mathtt {GV}$
follows from the fact that it is an implicative logic.
Theorem 4.5. Let
$\mathtt {L}$
be any axiomatic extension of
$\mathtt {GV}$
. Then
$\mathtt {L}$
is an implicative logic.
Proof. We need to show that the conditions of Definition 4.2 hold; (3) follows from Proposition 2.9 and the others follow from the fact that
$\to $
is a Boolean implication.
Moreover, since
$1$
is a theorem of
$\mathtt {GV}$
(see Theorem 4.3), we get the following.
Theorem 4.6.
$\mathtt {GV}$
is algebraizable with defining equation
$\tau (x) = \{x \approx 1\}$
and equivalence formula
$\Delta (x,y) = \{ x \leftrightarrow y\}$
.
An important consequence of algebraizability is that axiomatic extensions of algebraizable logics are also algebraizable with the same equivalence formulas and defining equations. Moreover, the lattice of axiomatic extensions of the logic is dually isomorphic to the subvariety lattice of the equivalent algebraic semantics whenever the latter is a variety [Reference Font10], as it is the case here. Therefore, we obtain the algebraizability of
$\mathtt {GV}$
and all its axiomatic extensions with respect to the corresponding class of algebras.
Corollary 4.7 (Algebraizability).
Let
$\mathtt {L}$
be any axiomatic extension of
$\mathtt {GV}$
, axiomatized relatively to
$\mathtt {GV}$
by the set of axioms
$\Phi $
. Then the equivalent algebraic semantic of
$\mathtt {L}$
,
$\mathsf {K}_{\mathtt {L}}$
, is axiomatized relative to
$\mathsf {K}_{\mathtt {GV}}$
by
$\tau (\Phi ) = \{\varphi \approx 1 : \varphi \in \Phi \}$
. In particular given any set of formulas
$\Gamma $
and formulas
$\psi $
:
$$ \begin{align*}\Gamma \vdash_{\mathtt{L}} \psi\;\;\Leftrightarrow\;\; \tau(\Gamma) \vDash_{\mathsf {K}_{\mathtt{L}}} \tau(\psi).\end{align*} $$
By direct application of the Blok–Pigozzi machinery, we get an axiomatization of
$\mathsf {K}_{\mathtt {GV}}$
made of equations (coming from the axioms) and quasi-equations (coming from rules); we now show that actually equations suffice, in particular because the rules (MP) and (C) are translated to quasi-equations that already hold as a consequence of the other axioms. In other words, the equivalent algebraic semantics is a variety of algebras.
Definition 4.8. A Lewis variably strict conditional algebra, or
$\mathsf {V}$
-algebra for short, is an algebra 
where
$(C, \wedge , \vee , \to , 0, 1)$
is a Boolean algebra and 
is a binary operation such that, for all
$x, y, z \in C$
:
-
1.
-
2.
-
3.
-
4.
We denote the variety of
$\mathsf {V}$
-algebras with
$\mathsf {VA}$
.
Theorem 4.9.
$\mathsf {VA}$
is the equivalent algebraic semantics of
$\mathtt {GV}$
.
Proof. Note that since algebras in
$\mathsf {VA}$
have a Boolean algebra reduct, for
${\mathbf {C}} \in \mathsf {VA}$
, and
$x,y \in C$
,
$x \to y = 1$
iff
$x \leq 1$
; then by Corollary 4.7, the equivalent algebraic semantics of
$\mathtt {GV}$
is axiomatized by the axioms of
$\mathsf {VA}$
plus the quasi-equations:
The result then follows from the fact that the latter are easily seen to hold given the other axioms; in particular, note that the fact that 
is order-preserving on the right follows from the distributivity over the meet operation on the right: if
$x \leq y$
then
$x = x \land y$
, and so 
and so 
.
Recall that 
. Let
$\mathsf {VCA}$
be the subvariety of
$\mathsf {VA}$
further satisfying:
and
$\mathsf {VCSA}$
the subvariety of
$\mathsf {VCA}$
of Lewis conditional algebras, satisfying:
Corollary 4.10.
$\mathsf {VCA}$
and
$\mathsf {VCSA}$
are the equivalent algebraic semantics of, respectively,
$\mathtt {GVC}$
and
$\mathtt {GVCS}$
.
Let us consider again the unary connective
$\Box $
in the language as 
, and its iteration
$\Box ^n\varphi $
. We can show that, analogously to the case of modal algebras, the operator
$\Box $
can be used to characterize congruence filters.
Definition 4.11. Let
${\mathbf {A}} \in \mathsf {VA}$
; a nonempty lattice filter
$F \subseteq A$
is said to be open if
$x \in F$
implies
$\Box x \in F$
.
Proposition 4.12. Let
${\mathbf {A}} \in \mathsf {VA}$
; a nonempty lattice filter
$F \subseteq A$
is a congruence filter if and only if it is open.
Proof. The proof is based on the fact that, as a consequence of the fact that
$\mathsf {VA}$
is the equivalent algebraic semantics of
$\mathtt {GV}$
, congruence filters coincide with the deductive filters induced by the logic. In other words, for every rule
$\Gamma \vdash \varphi $
and every homomorphism
$ f $
from 
to
${\mathbf {A}}$
, if
$f[\Gamma ] \subseteq F$
, then
$f(\varphi ) \in F$
. It is clear that every deductive filter is an open lattice filter. For the converse, consider an open lattice filter
$ F $
;
$ F $
respects the axioms because it is nonempty (i.e., it contains
$1$
, where all the instances of the axioms are mapped as a direct consequence of the algebraizability result, Corollary 4.7), and it respects modus ponens since it is a lattice filter. We only need to prove that it respects the rule 
.
Suppose there is an assignment
$ h $
to
${\mathbf {A}} \in \mathsf {VA}$
such that
$h(\varphi ) = a, h(\psi ) = b, h(\gamma ) = c$
, with
$a,b,c \in A$
, and assume that
$a \to b \in F$
. From the fact that 
holds, by the deduction theorem of the global consequence (Theorem 3.25) and the strong completeness (Theorem 3.23) we obtain that there is some
$n \in \mathbb {N}$
such that
This implies that the element 
. Now, since
$a \to b \in F$
and
$ F $
is open,
$\Box ^k(a \to b) \in F$
for all
$k \in \mathbb {N}$
. Thus, since filters are closed under finitary meets, the element
$(\bigwedge _{m \leq n} \Box ^m(a \to b)) \in F$
. Since lattice filters respect modus ponens, also 
, which shows that open lattice filters respect the rule (C).
We have then shown that open filters coincide with deductive filters, and therefore with congruence filters.
We remind the reader that the proposition above describes the Gumm–Ursini ideals [Reference Gumm and Ursini13] of
$\mathsf {VA}$
, which are also the
$\mathtt {GV}$
-deductive filters of the algebras in
$\mathsf {VA}$
. The following is an interesting observation:
Corollary 4.13. Let
${\mathbf {A}} \in \mathsf {VA}$
; then the congruence filters of
${\mathbf {A}}$
are exactly the congruence filters of its modal reduct
$(A, \land , \lor , \to , \Box , 0, 1)$
.
The corollary above entails for instance that one can characterize subdirectly irreducible algebras in
$\mathsf {VA}$
by the subdirect irreducibility of their modal reduct; we leave this purely algebraic investigation of
$\mathsf {V}$
-algebras to future work.
4.2 Local algebraic semantics
In this section we focus on the weaker logic
$\mathtt {LV}$
; in particular, we will show that the latter is not algebraizable, however it can still be studied by means of
$\mathsf {V}$
-algebras. We will indeed see that
$\vdash _{\mathtt {LV}}$
coincides with the logic preserving degrees of truth of
$\mathsf {VA}$
. Let us be more precise.
We call an ordered algebra a pair
$({\mathbf {A}}, \leq )$
, where
${\mathbf {A}}$
is an algebra and
$\leq $
is a partial order on its universe. Note that all algebras with a (semi)lattice reduct can be seen as ordered algebras, and thus in particular Boolean algebras and
$\mathsf {V}$
-algebras are ordered algebras.
Definition 4.14. Let
$\mathsf {K}$
be a class of ordered algebras over a language
$\mathcal {L}$
; the logic preserving degrees of truth of
$\mathsf {K}$
, in symbols
$\mathtt {L}_{\mathsf {K}}^{\leq } = (\mathcal {L},\vdash _{\mathsf {K}}^{\leq })$
, is defined as follows: for every set
$\Gamma \cup \{\varphi \}$
of formulas in the language of
$\mathsf {K}$
,
$\Gamma \vdash _{\mathsf {K}}^{\leq } \varphi $
if and only if for all
$({\mathbf {A}}, \leq ) \in \mathsf {K}$
, and assignment
$h: {\mathbf {Fm}}_{\mathcal {L}} \to {\mathbf {A}}$
,
$a \in A$
,
$$ \begin{align*}a \leq h(\gamma) \text{ for every } \gamma \in \Gamma \;\;\Rightarrow\;\; a \leq h(\varphi).\end{align*} $$
Remark 4.15. If
$\mathsf {K}$
is an elementary class of algebras (in particular, if
$\mathsf {K}$
is a variety) with a lattice reduct, one can rephrase the above definition and say that
$$ \begin{align*}\Gamma \vdash_{\mathsf {K}}^{\leq} \varphi \;\;\text{ iff }\;\; \mathsf {K} \vDash \gamma_1 \land \ldots \land \gamma_n \leq \varphi\end{align*} $$
for some
$\{\gamma _1, \ldots , \gamma _n\} \subseteq \Gamma $
and
$n \in \mathbb {N}$
(see [Reference Moraschini, Malinowski and Palczewski20, remark 2.4]).
Example 4.16. Intuitionistic logic is the logic preserving degrees of truth of Heyting algebras, and the local consequence of the modal logic
$\mathsf {K}$
is the logic preserving degrees of truth of modal algebras.
Logics preserving the degrees of truth have been studied in generality in [Reference Font9, Reference Nowak22], and in residuated structures in [Reference Bou8]. In order to prove the analogous result for
$\mathtt {LV}$
, let us first state a useful lemma.
Lemma 4.17. For all 
-formulas
$\varphi , \psi , \gamma $
, and
$n \in \mathbb {N}$
:
-
1.
; -
2.
.
;
.Proof. We will show that the two statements hold in sphere models, which implies the claim by the completeness Theorem 3.8. We start with (1), and proceed by induction on
$ n $
. Let
$n = 0$
, we want to prove that:
consider a sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
and let
$w \in W$
. Suppose
$w \Vdash \Box (\varphi \to \psi )$
, or equivalently via Lemma 3.17
$\bigcup \mathcal {S}(w)\subseteq v(\varphi \to \psi )$
. Now, if 
, it means that there is
$S \in \mathcal {S}(w)$
such that
$\emptyset \neq S \cap v(\gamma ) \subseteq v(\varphi )$
; but since
$\bigcup \mathcal {S}(w)\subseteq v(\varphi \to \psi )$
,
$S \cap v(\varphi ) \subseteq v(\psi )$
, i.e., there is
$S \in \mathcal {S}(w)$
such that
$\emptyset \neq S \cap v(\gamma ) \subseteq v(\psi )$
, and thus 
. The inductive step follows from the fact that
$\Box $
is a modal operator, more precisely that theorems are closed under
$\Box $
and
$\Box $
distributes over the implication (Proposition 3.18). Let us now show (2), again by induction on
$ n $
; for
$n= 0$
, we prove:
Consider then a sphere model
$\mathcal {M}=\langle W, \mathcal {S}, v \rangle $
, let
$w \in W$
, and suppose
$w \Vdash \Box (\varphi \leftrightarrow \psi )$
, or equivalently by Lemma 3.17,
$\bigcup \mathcal {S}(w)\subseteq v(\varphi \leftrightarrow \psi )$
. If 
, it means that there is
$S \in \mathcal {S}(w)$
such that
$\emptyset \neq S \cap v(\varphi ) \subseteq v(\gamma )$
; but since
$\bigcup \mathcal {S}(w)\subseteq v(\varphi \leftrightarrow \psi )$
, we get that
$S \cap v(\varphi ) = S \cap v(\psi )$
, and therefore there is
$S \in \mathcal {S}(w)$
such that
$\emptyset \neq S \cap v(\psi ) \subseteq v(\gamma )$
, and thus 
. The inductive step follows again from the fact that
$\Box $
is a modal operator.
We are now ready to prove that
$\mathtt {LV}$
can also be studied by means of
$\mathsf {V}$
-algebras, via the following (strong) completeness result.
Theorem 4.18.
$\mathtt {LV}$
is the logic preserving degrees of truth of
$\mathsf {VA}$
; i.e., for all 
:
$$ \begin{align*}\Gamma \vdash_{\mathtt{LV}} \varphi \;\Leftrightarrow\; \Gamma \vdash^\leq_{\mathsf{VA}}\varphi.\end{align*} $$
Proof. The forward direction is a usual soundness proof; note that the axioms are mapped to
$1$
by the algebraizability result (Corollary 4.7), (MP) holds in the form
$x \land (x \to y) \leq y$
, and (wC) becomes
$x \to y = 1$
implies 
which holds in
$\mathsf {VA}$
.
For the converse, we reason by contraposition; assume
$\Gamma \not \vdash _{\mathtt {LV}} \varphi $
. Then consider the relation
$\theta $
defined as follows:
We can show that
$\theta $
is a congruence relation. In particular, while reflexivity and symmetry are trivial, transitivity follows the fact that
$\Box $
distributes over
$\to $
(Proposition 3.18); the Boolean operations are also easily shown to be respected: for
$\land $
, as a consequence of the fact that
$\Box $
distributes over
$\land $
(Proposition 3.18), and for
$\neg $
, by the observation that
$\neg \Box ^n x = \Box ^n (\neg x)$
. Let us then prove that
$\theta $
preserves the binary operation 
. Assume
$\psi \theta \gamma $
, it suffices to show that
that is to say, we need prove that:
Given the assumption that
$\Gamma \vdash _{\mathtt {LV}} \Box ^n(\psi \to \gamma )\text { and }\Gamma \vdash _{\mathtt {LV}} \Box ^n(\gamma \to \psi )$
for all
$n \in \mathbb {N}$
, and thus also
$\Gamma \vdash _{\mathtt {LV}} \Box ^{n}(\varphi \leftrightarrow \psi )$
(since
$\Box $
distributes over
$\land $
by Proposition 3.18), (i)–(iv) follow by Lemma 4.17. Therefore,
$\theta $
is a congruence, and we can consider the quotient 
; let us verify that the latter is an algebra in
$\mathsf {VA}$
. Consider the axiomatization of
$\mathsf {VA}$
, we show that for every equation
$\varepsilon \approx \delta $
appearing in it,
$(\varepsilon ,\delta )\in \theta $
. By the algebraizability of
$\mathsf {VA}$
, for each such
$\varepsilon \approx \delta $
and the completeness result of Theorem 3.23,
$\vdash _{\mathtt {GV}} \varepsilon \leftrightarrow \delta $
; thus also
$\vdash _{\mathtt {LV}}\varepsilon \leftrightarrow \delta $
(since they have the same theorems by Theorem 2.4), and then
$\vdash _{\mathtt {LV}} \Box ^n(\varepsilon \leftrightarrow \delta )$
for all
$n \in \mathbb {N}$
by the fact that theorems are closed under
$\Box $
(Proposition 3.18). Thus, each
$(\varepsilon ,\delta )\in \theta $
, and 
.
We now show that for all (nonempty) finite subsets
$\Delta \subseteq \Gamma $
,
$\Delta \not \vdash ^\leq _{\mathsf {VA}} \varphi $
. By Remark 4.15, this implies that
$\Gamma \not \vdash ^\leq _{\mathsf {VA}}\varphi $
which would complete the proof. Given that
$\Delta \vdash ^\leq _{\mathsf {VA}} \varphi $
iff
$\mathsf {VA} \vDash \bigwedge \Delta \leq \varphi $
, it is enough to show that in particular 
. Consider
$\Delta = \{\delta _1, \dots , \delta _n\} \subseteq \Gamma $
, and let
$\pi $
be the natural epimorphism 
; assume by way of contradiction that
$\pi (\delta _1)\wedge \dots \wedge \pi (\delta _n)\leq \pi (\varphi )$
; thus
$\pi ((\delta _1\wedge \dots \wedge \delta _n) \to \varphi ) = 1$
. By the definition of
$\theta $
, this implies that in particular
$\Gamma \vdash _{\mathtt {LV}} (\delta _1\wedge \dots \wedge \delta _n) \to \varphi $
. But since
$\Delta \subseteq \Gamma $
, it follows that
$\Gamma \vdash _{\mathtt {LV}} \delta _1\wedge \dots \wedge \delta _n$
; by modus ponens we would get
$\Gamma \vdash _{\mathtt {LV}} \varphi $
, a contradiction. This completes the proof.
Actually, the same proof works for any axiomatic extension of
$\mathtt {LV}$
.
Corollary 4.19. Let
$\Gamma $
be a set of formulas;
$\mathtt {LV} + \Gamma $
is the logic preserving degrees of truth of
$\mathsf {VA} + \tau (\Gamma )$
.
As a corollary of the above theorem, we get that:
Corollary 4.20. Let
$\Gamma $
be a set of formulas. For all 
,
$\vDash _{\mathsf {VA} + \tau (\Gamma )}\varphi \approx \psi $
if and only if
$\varphi \vdash _{\mathtt {LV} + \Gamma } \psi $
and
$\psi \vdash _{\mathtt {LV} + \Gamma } \varphi $
.
We are now going to conclude that the local calculus is not algebraizable. In fact, it follows from a general theory regarding (semi)lattice-based logics that have an algebraizable assertional companionFootnote 5 . We recall the relevant notions from [Reference Font10, §7].
Given a class of algebras
$\mathsf {K}$
that have a constant
$1$
in their language
$\mathcal {L}$
, we define the 1-assertional logic of
$\mathsf {K}$
as the logic
$\mathtt {L}_{\mathsf {K}}^1 = (\mathcal {L}, \vdash _{\mathsf {K}}^1)$
such that for all
$\Gamma \cup \{\varphi \} \subseteq Fm_{\mathcal {L}}$
$$ \begin{align*}\Gamma \vdash_{\mathsf {K}}^1 \varphi \;\;\Leftrightarrow\;\; \{\gamma \approx 1 :\gamma \in \Gamma\} \vDash _{\mathsf {K}} \varphi \approx 1.\end{align*} $$
Let now
$\mathsf {K}$
be a variety of algebras with semilattice reducts, and such that the associated order in algebras of
$\mathsf {K}$
has a maximum defined by some term
$1$
. The two associated logics
$\mathtt {L}_{\mathsf {K}}^{\leq }$
and
$\mathtt {L}_{\mathsf {K}}^1$
are said to be companions of each other, i.e.,
$\mathtt {L}_{\mathsf {K}}^1$
is the assertional companion of
$\mathtt {L}_{\mathsf {K}}^{\leq }$
, and the latter in turn is the semilattice-based companion of
$\mathtt {L}_{\mathsf {K}}^1$
. Moreover,
$\mathtt {L}_{\mathsf {K}}^{\leq }$
is algebraizable if and only if it coincides with
$\mathtt {L}_{\mathsf {K}}^1$
[Reference Font10, Theorem 4.2].
It is clear that
$\mathsf {K} = \mathsf {VA}$
satisfies the above assumptions; moreover, it follows from Theorem 4.6 that
$\mathtt {GV}$
is the (algebraizable) 1-assertional logic of
$\mathsf {VA}$
, and by Theorem 4.18 that
$\mathtt {LV}$
is its semilattice based companion. Since
$\mathtt {GV}$
and
$\mathtt {LV}$
do not coincide, we get at once:
Corollary 4.21.
$\mathtt {LV}$
is not algebraizable.
Given Corollary 4.19, we can actually extend this result to all the axiomatic extensions of Lewis’s logics where global and local consequence do not collapse.
Corollary 4.22. If
$\Gamma $
is a set of axioms such that
$\mathtt {LV} + \Gamma $
does not coincide with
$\mathtt {GV} + \Gamma $
, then
$\mathtt {LV} + \Gamma $
is not algebraizable.
One can actually show that all the local variably strict conditional logics considered by Lewis’s in his hierarchy are not algebraizable, since the global and local calculi do not coincide, except for
$\mathtt {LVCA}$
, which collapses to classical logic [Reference Lewis17].
Corollary 4.23. No axiomatic extension of
$\mathtt {LV}$
by the axioms in
$\{\mathtt {W}, \mathtt {C}, \mathtt {N}, \mathtt {T}, \mathtt {S}, \mathtt {U}, \mathtt {A}\}$
is algebraizable, except (the ones coinciding with)
$\mathtt {LVCA}$
. The strong and weak calculi
$\mathtt {GVCA}$
and
$\mathtt {LVCA}$
coincide, and they are both algebraizable with respect to the subvariety of
$\mathsf {VA}$
where the identity 
holds.
5 Conclusions
The main objective of this paper is to provide a logico-algebraic analysis of Lewis’s variably strict conditional logics, a subject that has been notably lacking in the literature. Our efforts have clarified several ambiguities surrounding these logics, explicitly defined and refined their properties, brought to light the deep connection with the modal logic framework, and introduced a novel general algebraic framework for their technical analysis. By doing so, this work aims to foster a fruitful synergy between a classical theme in formal philosophy and the advancements in abstract algebraic logic.
To the best of our knowledge, the model-theoretic tools and the techniques proper of the abstract algebraic logic framework we have used were not employed before to analyze these logics; these powerful tools proved instrumental in establishing several logical results, e.g., a deduction theorem, the representation of one logical consequence in terms of the other, and the strong completeness with respect to the algebraic structures for both versions of the calculi. These results collectively offer a deeper and more comprehensive understanding of the properties of Lewis’s logics and the features of their models.
Moreover, while filling a notable void in the literature, this work is just the start of a formal investigation of Lewis’s logics. The logico-algebraic machinery indeed offers a variety of tools to study logical properties from the algebraic point of view, the so-called bridge theorems. These are the core results of the field of algebraic logic; they allow one to study metalogical properties algebraically, concerning in particular the entailment of the logic, answering questions about its expressivity, or formalizing notions that can be relevant in applications [Reference Font10, Reference Kiss15, Reference Raftery25]. These allow the study of, e.g.,: the lattice of (axiomatic) extensions of the logics; definability properties, i.e., to what extent implicit properties of the logic can be made explicit [Reference Blok and Hoogland5] and interpolation properties [Reference Metcalfe, Montagna and Tsinakis19]; admissible rules [Reference Rybakov27], those that added to the logic do not change the theorems, and whether a logic is structurally complete (i.e., whether its admissible rules are also derivable) like classical logic but unlike intuitionistic logic [Reference Aglianò and Ugolini1, Reference Bergman3].
This work also lays the foundation for a deeper conceptual understanding of Lewis’s logics beyond the technical sphere. In a forthcoming work, we will focus on the fact that to properly consider infinite models one should not simply consider sets of worlds, but topological spaces; i.e., the subsets of the universe that are meant to represent the formulas (the clopen sets of the topology) play a special role. In technical terms, we will show that the variety of algebras introduced in this work enjoys a categorical duality with respect to topological spaces based on Lewis’s sphere models. From a conceptual standpoint, this will provide a fresh perspective into the similarity relationship among worlds and spheres, and the nature of the so-called limit assumption, stating the existence of most similar antecedent worlds.
Acknowledgements
We thank the two anonymous reviewers for their suggestions, which improved this work. Additionally, we are grateful to Tommaso Flaminio for the many valuable conversations on this subject.
Funding
Giuliano Rosella acknowledges financial support from the Italian Ministry of University and Research (MUR) through the PRIN 2022 grant n. 2022ARRY9N (Reasoning with hypotheses: Integrating logical, probabilistic, and experimental perspectives), funded by the European Union (Next Generation EU). Sara Ugolini acknowledges support from the Ramón y Cajal programme (RyC2021-032670-I), the Spanish project SHORE (PID2022-141529NB-C22), and the MOSAIC project (H2020-MSCA-RISE-2020 Project 101007627).
. Moreover, he uses a technique that is widely used to show completeness in modal logics, that is, he constructs the canonical model of each logic, showing that it belongs to the corresponding class of spheres. See [
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