Sequence and Consequence

2.1 Order models for C2

There are many model-theoretic semantics for conditional connectives in the literature, generalizing Kripke’s possible-worlds semantics for modal logic to conditional languages. Our focus in this paper will be on order models for conditionals, which were introduced by Lewis (1973) as models for his logic (which is strictly weaker than C2). Order models turn out to be particularly intuitive for the study of C2 and its strengthenings. In particular, we can model C2 with order models where each world is associated with a well-ordering of worlds. As we will see, sequence models can be naturally viewed as a special case of well-order models, making it easy to see that their logic includes C2.

A Kripke model equips a set of possible worlds with a binary accessibility relation R, representing relative necessity and possibility: p is necessary at w just in case p is true throughout the worlds accessible from w, which we write R(w), and p is possible at w just in case p is true somewhere in R(w). An order model is like a Kripke model but with additional structure: in addition to a set of worlds R(w), an order model associates each world w with an ordering  of R(w). We pronounce as ‘u is closer to w than v’. In such a model, assuming there are some closest p-worlds to w, $p>q$ is true at w just in case all of these worlds are q-worlds. $p>q$ is also (‘vacuously’) true at w when there aren’t any p-worlds in R(w).¹¹

Lewis conceived of closeness in terms of similarity: means that x is more similar to w than y is (in whatever respects turn out to be relevant). But using order models does not commit us to a similarity-based interpretation of the order functions, any more than Kripke semantics for the modal operators commits us to any particular theory of necessity and possibility. Thus, skeptics of similarity-based approaches to conditionals (a group in which we include ourselves) have no special reason to object to the use of order models. Indeed, in modelling strengthenings of C2, we will need to impose conditions on order models which would be completely implausible if closeness had to be interpreted as similarity, so insofar as the strengthenings are well-motivated, they will add to the already strong case against similarity-based approaches (see §4.1).In the order models characteristic of C2, each ordering is a well-order: that is, a transitive, connected, asymmetric, well-founded relation on R(w). This means that we can restate the order semantics for conditionals in terms of the unique closest antecedent world, if there is one: $p>q$ is true at w just in case q is true at the first p-world in , or there are no p-worlds in R(w). This uniqueness assumption guarantees that the controversial CEM axiom holds in the model; logics without CEM (like Lewis’s) can be obtained by relaxing this assumption. However, we will not further discuss such models in this paper, so by ‘order model’ we will always mean a well-order model.¹²

Let us lay this all out more formally, and introduce some standard terminology which will be helpful:

Definition 2.1. An order frame is a pair , where W is a non-empty set and is a function which takes any $w\in W$ to a strict well-order on some subset of W such that whenever , $w=x$ or .

We can read off an accessibility relation from an order frame: the worlds accessible from w are those that w strictly precedes in the ordering induced at w, together with w itself:¹³

As we will see, R plays the same role with respect to the defined $\mathrm{\square }$ as accessibility relations usually do in Kripke models. When it is clear which order frame is under discussion, we will always use R for the corresponding accessibility relation.

Definition 2.2. An order model is a triple where is an order frame and $V:\mathit{\text{At}}\times W\to \{0,1\}$ is a valuation function.

We will write $V(p_k)$ for $\{w:V(p_k,w)=1\}$ and V(w) for $\{p_k:V(p_k,w)=1\}$.

Definition 2.3. When is an order model, its denotation function is the function such that for any atom p_k and sentences p,q:

For readability, relativization of $⟦\cdot ⟧$ to a model is usually left implicit.

As usual, we can define a pointed order model as a pair of an order model with a world from its set of worlds, i.e. a pair such that $w\in W$ and is an order model; when is a pointed order model, we say that it is based on . p is true in a pointed order model $⟨w,ℳ⟩$ just in case $w\in {⟦p⟧}^{ℳ}$. When $\mathrm{\Gamma }\subseteq {ℒ}_{>}$, we can also speak of $\mathrm{\Gamma }$ being true at a pointed model to mean that all its elements are. We also write $w,ℳ⊩p$ when p is true in $⟨w,ℳ⟩$; when $ℳ$ is implicit from the context, we write simply $w⊩p$. For brevity we sometimes talk about p being true at every model in a given class; by this we mean true at every pointed model based on a model in that class. Two pointed models are equivalent just in case they verify exactly the same sentences of ${ℒ}_{>}$.

A standard induction on formulae shows:

Theorem 2.4. C2 is sound for order models: that is, ${⊢}_{\text{C2}}p$ implies that p is true in every pointed order model.

We also have a corresponding completeness result: every sentence that is true in every pointed order model is a theorem of C2. Equivalently, whenever p is consistent in C2 (that is, ${⊬}_{\text{C2}}\neg p$) it is true in some pointed order model. In fact, we can show something stronger: C2 is complete with respect to the class of finite order models:

Theorem 2.5. If $p$ is true in every finite pointed order model, then ${⊢}_{\text{C2}}p$.

One corollary is that C2 is decidable. Since every non-theorem is false in some finite pointed order model, and we can effectively enumerate all the finite pointed order models (up to isomorphism), we can test for non-theoremhood by searching through the finite pointed order models for a countermodel; this provides an effective decision-procedure when run in parallel with a proof search.

This soundness and completeness theorem also allows us to prove our earlier assertion about the modal fragment of C2:

Theorem 2.6. When $p$ is a sentence in the modal fragment of ${ℒ}_{>}$, ${⊢}_{\text{C2}}p$ iff ${⊢}_{\text{KT}}p$.

Proof. We rely on the well-known fact that KT is sound and complete with respect to modal models with a reflexive accessibility relation

• ⇐ Normality for $>$ gives the K axiom for $\mathrm{\square }$, and MP for $>$ gives T for $\mathrm{\square }$.

• ⇒ Suppose we have a modal models W with a reflexive accessibility relation R; we can extend this to an order model that respects R by fixing a strict well-ordering on W, and for $x\ne y$, let iff wRx and wRy and either $x=w$, or $x\ne w$ and $y\ne w$ and . If ${⊢}_{\text{C2}}p$ then by soundness p is true in every order model, and hence in every reflexive modal model, and so by completeness of KT, ${⊢}_{\text{KT}}p$.□

2.2 The failure of strong completeness

Theorem 2.5 is about individual sentences: it is equivalent to the claim that whenever p is consistent in C2 (meaning that $\neg p$ is not a theorem of C2), p is true in some pointed order model. This kind of result is sometimes called a weak completeness theorem. By contrast, a strong completeness theorem would say that for every set of sentences that is consistent in a certain logic, there is a model based on a frame in the relevant class in which every sentence in that set is true. We define consistency for sets of sentences as usual: $\mathrm{\Gamma }$ is consistent relative to $⊢$ when there is no finite conjunction p of elements of $\mathrm{\Gamma }$ such that $⊢p\to \perp$, where $\perp$ abbreviates $p_0\wedge \neg p_0$. Somewhat surprisingly, we do not have a strong completeness theorem analogous to Theorem 2.5.¹⁴

Theorem 2.7. There are C2-consistent sets $\mathrm{\Gamma }\subseteq {ℒ}_{>}$ which are not true in any pointed order model.

Proof. One such $\mathrm{\Gamma }$ is the following:

$\mathrm{\Gamma }=\{\neg ((p_i\vee p_{i+1})>p_i)\mid i\in \mathbb{N}\}$

Suppose all the members of $\mathrm{\Gamma }$ were true in some pointed order model . Consider the set of worlds $\phi$ which verify some atom in this model, i.e., $\phi ={\bigcup }_{i}V(p_i)$. We must have $\phi \cap R(w)\ne \mathrm{\varnothing }$, for otherwise the elements of $\mathrm{\Gamma }$ would all be false at w (e.g., if p₁ and p₂ are nowhere true in R(w), then $(p_1\vee p_2)>p_1$ is vacuously true at w). Since is a well-order, $\phi$ must have a least element x in . Some atom p_k is true at x, by definition of $\phi$. Now consider $(p_k\vee p_{k+1})>p_k$. This is true at w, since the first world in where $p_k\vee p_{k+1}$ is true must be x (since x is the first world in where any atom is true), where p_k is true. Hence its negation is false at w, contrary to the assumption that w verifies all the elements of $\mathrm{\Gamma }$.

Nevertheless, $\mathrm{\Gamma }$ is consistent in C2. If it were not, then by definition, it would have some inconsistent finite subset. But every non-empty finite subset $\mathrm{\Delta }\subset \mathrm{\Gamma }$ is true in a pointed order model, which given soundness (Theorem 2.4) shows that every finite subset of $\mathrm{\Gamma }$ is consistent. To see this, let p_k be the atom with the highest index of any atom which appears in $\mathrm{\Delta }$. Consider any set U with $k+1$ members, which we label $w_0,w_1,\mathrm{\dots }{w}_k$. $\mathrm{\Delta }$ is true at the pointed order model , where is any order function with just in case $n>m$, and V any valuation such that $V(p_i)=\{w_i\}$ for $i\le k$.□

The same reasoning shows that no extension of C2 in which $\mathrm{\Gamma }$ remains consistent is strongly complete for any class of order models.

It is possible, however, to formulate a notion of a “general” order frame, and hence order model, relative to which we do have strong completeness (cf. Segerberg 1989). The idea is to add to our frames an extra “propositional domain” parameter—a set of subsets of worlds, representing the allowable denotations for sentences—and only require that our orders are well-founded relative to the elements of that parameter, rather than relative to all subsets of the set of worlds.

In more detail, let a generalized order frame be a triple , where W is any non-empty set; $ℬ$ is a set of subsets of W; and is a function which takes any $w\in W$ to a linear order on a subset of W, such that whenever orders any element of $\phi \in ℬ$, $\phi$ has a first element in , and where and $ℬ$ are coordinated so that $ℬ$ is closed under the set-theoretic operations corresponding to $\wedge ,\neg$, and $>$ in order semantics (relative to ). A generalized order model is a generalized order frame equipped with a valuation $V:\mathit{\text{At}}\to ℬ$. The definition of $⟦\cdot ⟧$ that worked for order models still works in generalized models, and yields a function from ${ℒ}_{>}$ to $ℬ$. C2 is sound and strongly complete with respect to generalized order models. For example, the consistent set $\mathrm{\Gamma }$, which is not true in any pointed order model, can be true in a generalized pointed order model where the set ${\bigcup }_{i}V(p_i)$ of worlds that verify some atom is not in the propositional domain, and does not have a minimal element in .¹⁵ Completeness can be shown with a standard canonical model construction, as in Lewis 1971.¹⁶

A generalized order frame is full if $ℬ=\mathcal{P}(W)$; full generalized frames are equivalent to order frames.¹⁷

Presumably because of facts along these lines, Segerberg (1989) writes that ‘in modal logic it seems quite natural to restrict one’s interest—at least initially—to full frames. In conditional logic, studied in the present vein, this is not so.’ Nevertheless, our interest in this paper will be full order frames, since our main goal is to identify the logics of various kinds of sequence frames, which (as we will shortly see) can be viewed as special cases of (full) order frames. Hence the completeness results we prove will be weak rather than strong completeness results.

3.1 Some variations on omega-sequence models

In the previous section we introduced both $\omega$-sequence models and the more specific class of full, categorical models. Prima facie, since not every $\omega$-sequence model is a full and categorical model, one might expect that some sentences that hold in all full and categorical models do not hold in all $\omega$-sequence models. But it turns out that this is not the case: the restriction to full and categorical models makes no difference to the logic. To see why this is the case, the following fact will be useful:

Fact 3.6. Pointed $\omega$-sequence models $⟨\sigma ,W,V⟩$ and $⟨\tau ,W^{\prime },V^{\prime }⟩$ are equivalent whenever $V({\sigma }^{[k:]})=V^{\prime }({\tau }^{[k:]})$ for all $k\in \mathbb{N}$.

The proof is a routine induction on the complexity of formulae. The intuition is that all that matters in assessing the truth of a sentence at the distinguished sequence in a pointed $\omega$-sequence model is how the tails of that sequence are valued. Just as the actual identity of worlds doesn’t matter in Kripke semantics, likewise the actual identity of sequences doesn’t matter in sequence semantics.

As an immediate consequence of Fact 3.6, we have:

Fact 3.7. Any pointed $\omega$-sequence model $ℳ=⟨\sigma ,W,V⟩$ is equivalent to the pointed $\omega$-sequence model ${ℳ}_{\omega }=⟨\overrightarrow{\omega },\mathrm{tails}(\overrightarrow{\omega }),V^{\prime }⟩$ where $\overrightarrow{\omega }$ is the sequence $⟨0,1,2,3\mathrm{\dots }⟩$ of the natural numbers in their standard order (i.e. ${\mathrm{id}}_{\mathbb{N}}$), and $V^{\prime }({\overrightarrow{\omega }}^{[k:]})=V({\sigma }^{[k:]})$.

We can think of ${ℳ}_{\omega }$ as a kind of minimal representation of $ℳ$. Its underlying frame is obviously isomorphic to the order frame with domain $\mathbb{N}$ in which iff . Thus the logic of that order frame, the logic of $⟨\mathrm{tails}(\overrightarrow{\omega }),\prec ⟩$, and the logic of the class of all $\omega$-sequence frames are all the same.

The minimal representation ${ℳ}_{\omega }$ is always categorical, since no two sequences in its domain share an initial element. Fact 3.7 thus shows that requiring categoricity makes no difference to the logic.

The underlying frame of the minimal representation ${ℳ}_{\omega }$ also has another noteworthy property that is worth giving a name to:

Definition 3.8. An order frame (or model ) is linear iff there is a strict partial order $\prec$ on W such that for all iff $w⪯x\prec y$.

Given the definition of order frames, any such $\prec$ will also be a strict well-order on any principal upper set of W (any set of elements $⪰w$ for any $w\in W$).

$\omega$-sequence models need not be linear: consider, for example, the $\omega$-sequence model with domain $\{⟨0,1,0,1,\mathrm{\dots }⟩,⟨1,0,1,0,\mathrm{\dots }⟩\}$. But Fact 3.7 implies that the logic of $\omega$-sequence frames is at least as strong as the logic of linear order frames, since ${ℳ}_{\omega }$ is linear. We will see later that this inclusion is proper.¹⁸

From Fact 3.7 a number of interesting invariance facts immediately follow. First, we can extend any pointed $\omega$-sequence model to a full pointed $\omega$-sequence model in which W includes all $\omega$-sequences over P, extending the valuation to the new sequences however we please, without making any difference to what’s true in the model. The logic of full $\omega$-sequence frames is thus the same as the logic of all $\omega$-sequence frames. From the other end, we can prune any pointed $\omega$-sequence model back to the generated model in which W is just the set of all non-empty tails of the designated sequence without making a difference to what’s true in the model. Thus the logic of $\omega$-sequence frames whose domain is $\mathrm{tails}(\sigma )$ for some $\sigma$ is also the same as the logic of all $\omega$-sequence frames.

We can also use Fact 3.7 to identify some further conditions which we could, if we wished, impose on $\omega$-sequence models without making a difference to the logic. In our models, we allow the domain to include eventually cyclic sequences some of whose tails are identical: for instance, $⟨1,2,1,2,\mathrm{\dots }⟩$ is the first, third, fifth, …tail of $⟨3,1,2,1,2,\mathrm{\dots }⟩$.¹⁹ However, it would not matter if we ruled out eventually cyclic sequences, since none of the sequences in the minimal representation are eventually repeating.²⁰ For the same reason, it would make no difference if we ruled out all repetition of protoworlds within a sequence, so that (e.g.) we cannot have a sequence beginning $⟨1,2,1,\mathrm{\dots }⟩$.

With all this in hand, it is worth noting from the opposite direction that some approaches which bear a close resemblance to $\omega$-sequence semantics have logics that are very different from the logics we consider, and indeed are orthogonal to or weaker than C2, rather than strengthening C2. Two noteworthy recent examples are the approach of Bacon (2015), who develops a version of sequence semantics which gives up Reciprocity; and that of Goldstein and Santorio (2021), which marries finite sequence semantics with the domain semantics from Yalcin (2007). We will set aside these approaches, as well as other variants whose logic is orthogonal to C2, focusing instead on semantics corresponding to logics that extend C2.

4.1 Evaluating Flattening

As we mentioned above, one way to interpret order models is as representing relative similarity between worlds. From the point of view of that interpretation, it is no accident that Flattening fails in those models: Flattening is in clear tension with that interpretation of order models.

Schematically, similarity-based theories of the conditional predict Flattening can fail because the most similar pq-world(s) to actuality need not be the most similar pq-world(s) to the p-world(s) most similar to actuality. For a simple concrete example of this, consider a variation on a toy example of Lewis’s (1973: 20) involving a line L. Lewis’s similarity-based intuition was that, given a world w where L has length n, if x and y are otherwise exactly alike, except that in x the length of L is closer to n than it is in y, then x is more similar to w than y is. Now suppose that L is in fact 10 inches long, and compare (1a) and (1b):

(1a) and (1b) instantiate the two sides of Flattening (in the rule-based formulation) since not being strictly between 8–13 inches entails not being strictly between 8–11 inches. But, if we interpret conditionals via similarity, in particular with Lewis’s simple assumption above, then (1a) should be true while (1b) is false. The world x most similar to actuality where the line isn’t strictly between 8 and 11 inches is one where it’s 11 inches. The world y most similar to x where the line isn’t strictly between 8–13 inches is one where it’s 13 inches. So (1a) is true. By contrast, the world most like actuality where the line isn’t between 8 and 13 inches is one where it’s 8 inches, so (1b) is false. (For a counterexample in the opposite direction, change ‘it would have been 13 inches’ to ‘it would have been 8 inches’.)

Is this counterexample convincing? We find it difficult to detect a clear divergence between (1a) and (1b), except by doggedly holding in mind the Lewisian interpretation of ‘if p…’ as a proxy for ‘in the world most similar to actuality where $p\mathrm{\dots }$’. Of course, a defender of a similarity-based view could claim that we simply fail to clearly see a contrast which does exist here. But they would need a story about why we make an error here, whereas we have clear intuitions about many other subtle judgments recorded in the literature on conditionals. Barring such a theory, the apparent validity of Flattening might provide a new argument in the battery of well-known arguments against similarity theories of conditionals.

Setting aside the baggage of similarity, we can try to evaluate Flattening on its own terms, by considering pairs of sentences that would be logically equivalent according to Flattening and seeing whether they in fact seem equivalent. It seems to us that the results of this exercise speak in favor of Flattening. For instance, compare these pairs:

These feel pairwise equivalent. At best, the (b)-variants feel redundant; the first antecedent feels like it’s doing nothing. This is not explained by order semantics, according to which the two variants have logically orthogonal meanings. But this intuition is explained (given standard theories of redundancy) if Flattening is valid, since then the (b)-variants are equivalent to their consequents.

By relying on the rule form of Flattening, we can formulate test pairs that feel somewhat less clunky:

Again, these feel pairwise equivalent. We have checked many instances of Flattening, in both the indicative and subjunctive mood, and have not found clear counterexamples.

To be sure, there are superficial counterexamples to Flattening involving tense and anaphora:

But it seems implausible that these are counterexamples to Flattening, and more plausible that the felt inequivalence in these pairs arises from different indexing of tense/anaphora in the two pairs. This is a somewhat delicate issue, involving questions about the representation of context-sensitivity that are beyond our scope. But it is worth noting that if we accept these as counterexamples to Flattening, then we also have to accept that there are counterexamples to the very widely accepted ‘Contraction’ principle $p>(p>q)\leftrightarrow p>q$. For the following also feel pairwise inequivalent:

Thus Flattening seems, from the point of view of natural language, in at least as good prima facie standing as Contraction (a theorem of C2 as well as many weaker logics of conditionals, though one that must notoriously be rejected by non-classical logicians attempting to maintain a naïve theory of truth). This is a strong position to be in.

However, there are reasons to be wary about these appearances. Flattening is a “cautious” cousin of the following well-known axiom schema:

Import-Export (IE)                               $p>(q>r)\leftrightarrow \mathit{\text{pq}}>r$

The only difference between Flattening and Import-Export is that, in Flattening, p recurs in the antecedent of the conditional consequent on the left-hand side, so we have $p>(\mathit{\text{pq}}>r)$, rather than $p>(q>r)$ as in IE. From a logical point of view, this small difference is crucial, for, as Dale (1974, 1979) and Gibbard (1981) showed, adding IE to C2 (or, indeed, to many weaker conditional logics) collapses $>$ to the material conditional, that is, results in a logic that validates:

Materialism                                            $p>q\leftrightarrow (p\to q)$

Materialism is, however, widely rejected in the literature on conditionals, as we noted above; see Edgington 1995 for many arguments against it. For one brief argument, consider the claim that no tree is deciduous if it keeps its leaves through the winter. According to materialism this entails that every tree keeps its leaves through the winter, since the negation of $p\to q$ entails p. But this is obviously wrong.²¹ For another argument, note that $(p\to q)\vee (\neg p\to r)$ is a theorem, but disjunctions of conditionals don’t always strike us that way; there is little temptation to think that (9) is true:

To be sure, there are some defenders of Materialism (e.g. Jackson 1979; Williamson 2020). But most view it as an untenable hypothesis. That makes the Dale/Gibbard collapse result a challenge to the co-tenability of Import-Export with C2, and it is important to ask whether this challenge extends to Flattening.

The answer is that it does not. For we have already seen that Flattening is valid in $\omega$-sequence models, while Materialism is not. For instance, in the $\omega$-sequence model generated from $\sigma =⟨1,2,1,2,\mathrm{\dots }⟩$, where p is false at $\sigma$ and true at $⟨2,1,2,1,\mathrm{\dots }⟩$, while q is false at both sequences, the material implication $p\to q$ is true at $\sigma$ while the conditional $p>q$ is false. Nor does Flattening lead to any other troubling form of triviality, as $\omega$-sequence models show. Moreover, at least one error theory of the apparent validity of IE precisely relies, in part, on the validity of Flattening (Mandelkern 2024). So validating Flattening may turn out to be a key stepping stone towards explaining the apparent validity of IE.

A final relevant observation is that there are compelling counterexamples to IE in the case of subjunctive conditionals: e.g. the sentences in (10) can intuitively diverge in meaning (Etlin 2008).

A standard desideratum in the theory of conditionals is to give a unified theory of indicative and subjunctive conditionals: there is one word ‘if’ which can express both conditionals, depending on the mood of the rest of the sentence. So we have positive reason not to validate IE as a matter of the logic of ‘if’, and instead to explain its apparent validity for indicatives, and lack thereof for subjunctives, as arising from the interaction of the meaning of ‘if’ with mood. But matters are different for Flattening, which appears valid for both indicatives and subjunctives. Indeed, the pairs which look like counterexamples to IE for subjunctive conditionals still look equivalent when we change them to instantiate Flattening:

In sum, despite their superficial similarity, Flattening and IE have very different statuses vis-à-vis the theory of conditionals; reasonable arguments against IE do not extend to Flattening.

Stepping back, we think the relationship of Flattening to Import-Export is reminiscent of the relationship of the following pairs of principles:

Cautious Transitivity                           $(p>q)\wedge (\mathit{\text{pq}}>r)\to (p>r)$

Transitivity                                           $(p>q)\wedge (q>r)\to (p>r)$

Cautious Monotonicity                        $(p>\mathit{\text{qr}})\to (\mathit{\text{pr}}>q)$

Monotonicity                                        $(p>q)\to (\mathit{\text{pr}}>q)$

The “incautious” principles Transitivity and Monotonicity are widely rejected (Stalnaker 1968); and, just as for Import-Export, adding either of them to C2 results in Materialism. By contrast, the “cautious” principles are widely accepted, and are theorems of C2 and of many other popular conditional logics. In this light, we might naturally regard Flattening as a plausible cautious cousin of the implausible, incautious Import-Export principle.

Nevertheless, we do not want to suggest that the case for the validity of Flattening is anything like watertight. The strongest reason for worry we see involves the fact that, as we noted, Flattening implies the 4 and H principles for the $\mathrm{\square }$ defined in terms of $>$. This is a potential warning sign, since there are well known arguments against the 4 principle for many seemingly relevant interpretations of $\mathrm{\square }$, and many of these arguments also extend to the H principle. Williamson (2000) and Dorr, Goodman, and Hawthorne (2014) argue against 4 on an interpretation where $\mathrm{\square }$ means ‘a is in a position to know that…’. This suggests that 4 might also fail for the epistemic ‘must’ if its meaning is related to that of ‘know’; and the arguments can also be adapted to directly use epistemic ‘must’. Insofar as the $\mathrm{\square }$ defined in terms of $>$ interpreted as an indicative conditional is equivalent to, or otherwise intimately connected to, the epistemic modal, these considerations may also threaten the 4 axiom for that $\mathrm{\square }$. Meanwhile, when we turn to the notion of nomic necessity—which might be thought to be identical to the defined $\mathrm{\square }$ on some counterfactual interpretations of $>$—we find influential forms of Humeanism which motivate rejection of at least H and perhaps also 4. On the ‘best system’ theory of laws (Lewis 1994), being a nomic necessity is being entailed by whatever collection of true axioms achieves the best balance of simplicity and strength. On this picture, there could be a complex world where there are two jointly inconsistent simple propositions both of which are false but nomically possible, and (because of their simplicity) such that necessarily, if they are true, they are nomically necessary. This is inconsistent with the connectedness of nomic accessibility, and thus with H. (It is also arguable that Humeans should reject 4 for nomic necessity, though we will not go into that here.) While it is not so plausible that nomically necessary truths are counterfactually necessary on every interpretation of counterfactuals, one might think that there are some salient interpretations of counterfactuals that involve “holding the laws fixed” in such a way that these Humean worries would carry over to 4 and H for the defined $\mathrm{\square }$. There are also potential reasons for doubting both 4 and H for metaphysical modality, which many take to be equivalent to $\mathrm{\square }$ defined in terms of counterfactuals: Salmon (2005) rejects 4 for metaphysical necessity in order to solve certain puzzles of Tolerance (though see Dorr, Hawthorne, and Yli-Vakkuri 2021 for an alternative approach to those puzzles which preserves 4); meanwhile, Bacon (2020) and Bacon and Dorr (2024) endorse 4, but are sympathetic to versions of “combinatorialism” on which metaphysical modality would not obey H.²²

We will not here undertake to evaluate these arguments against 4 and H for various familiar interpretations of necessity, or adjudicate the question to what extent they carry over to the $\mathrm{\square }$ defined in terms of ‘if’. If we accept these arguments (for that defined $\mathrm{\square }$), we will have to reject Flattening, and develop an error theory of its apparent validity. But the appearances favoring Flattening are quite strong, so the difficulty of this task should not be underestimated. In any case, we think it’s clear that the logic C2.F has strong prima facie appeal as (at least part of) the logic of the natural language conditional.

If we do accept C2.F, then we might naturally hope to develop some intuition about its semantics which might help to explain its validity. As we saw above, similarity-based approaches will not work here. Indeed, even the more abstract terminology of “closeness” may be rather misleading, since it suggests an underlying metric structure, and no ordinary metric space has the property that when $X\subseteq Y$, the closest element of X to a given point is the closest element of X to the closest element of Y to that point. Of course, if “closeness” is just a label for whatever plays the role of the order function in order semantics, a closeness-based semantics is consistent with the validity of C2.F; but this does not suggest anything we could think of as an explanation. We think this is an important challenge, to which we do not have a satisfactory answer; we leave it as an open problem for proponents of C2.F.²³

4.2 The logic C2.F

As we have already asserted, the logic of $\omega$-sequence frames is not exhausted by C2.F. To see why this is the case, and to get some intuition for what is missing, it will be useful to introduce a different class of order frames with respect to which C2.F is complete as well as sound: the flat order frames.

Definition 4.1. Order frame is

• – semi-flat iff whenever , $y\in R(x)$ and for all z such that or $z\in R(x)\backslash R(w)$, .

• – flat iff it is semi-flat and transitive.

(Note that in a transitive frame, the case where and $z\in R(x)\backslash R(w)$ cannot arise, since guarantees $x\in R(w)$ and hence $R(x)\subseteq R(w)$.) In other words: a flat order frame is a transitive frame where, whenever x is accessible from w, orders all the worlds that come after x in in just the same way they were ordered in . By contrast, worlds that came before x in may occur at any position in , or not be accessible from x at all. See Figure 2 for an illustration of a flat and non-flat order frame.

Figure 2: Illustrations of a flat order function (left) and a non-flat (because not semi-flat) order function (right) in frames with worlds {1,2,3,4}.

Linear order frames, whose order function can be derived from some unrelativized $\prec$ via iff $w⪯x\prec y$, are always flat: they are obviously transitive, and if , then $w⪯x\prec y\prec z$ and so $y\in R(x)$ and . But not all flat frames are linear: for example, the one on the left of Figure 2 is not.

We can show that Flattening is valid on all flat order frames. In fact, flatness characterizes Flattening, in the sense that the order frames on which Flattening is valid are exactly the flat ones. Since we already know that Flattening is equivalent to the combination of Cautious Importation with Crashing Cautious Exportation, and that the latter is equivalent to 4 (which is characterized by transitivity), it suffices to prove the following lemma:

Lemma 4.2. Cautious Importation is valid on an order frame iff it is semi-flat.

Proof

• ⇒ Suppose .

  • (a) If $y\notin R(x)$, let $V(p)=\{x,y\},V(q)=\{y\},V(r)=\mathrm{\varnothing }$.

  • (b) If $y\in R(x)$ and for some z, but $y{\nless }_{x}z$, then:

    • · if $z\notin R(x)$, let $V(p)=\{x,z\},V(q)=\{z\},V(r)=\mathrm{\varnothing }$;

    • · if , let $V(p)=\{x,y,z\},V(q)=\{y,z\},V(r)=\{z\}$.

  • (c) if $y\in R(x)$ and for some z, $z\in R(x)\backslash R(w)$ but $y{\nless }_{x}z$, let $V(p)=\{x,z,y\},V(q)=\{y,z\},V(r)=\{z\}$.

      In each case, Cautious Importation will fail at w.

• ⇐ Suppose is semi-flat, and consider any V and $w\in W$ such that $w⊩p>(\mathit{\text{pq}}>r)$. If there is no pq-world in R(w), or the first p-world in is a pq-world, then $w⊩\mathit{\text{pq}}>r$ and we are done. Otherwise, let x be the first p-world in and let y be the first pq-world in . Since , $y\in R(x)$, so R(x) contains a pq-world; let u be the first pq-world according to . We know $u⊩r$, so it suffices to show $y=u$. We cannot have since then y would not be the first pq-world in . And we also cannot have or $u\notin R(w)$, since in either case semi-flatness would yield , meaning that u would not be the first pq-world in . The only remaining possibility is that $y=u$.□

Since Flattening is equivalent (modulo C2) to 4 together with Cautious Importation, it follows that Flattening is characterized by flatness:

Theorem 4.3. Flattening is valid on an order frame iff is flat.

We can also formulate frame conditions that characterize Cautious Exportation and Crashing Cautious Importation; the combination of these conditions is also equivalent to flatness, since the conjunction of the axioms is equivalent to Flattening.²⁴

With this characterization result in hand, we can turn to soundness and completeness results for C2.F. The right-to-left direction of Theorem 4.3 says that C2.F is sound for flat order frames: that is, all the theorems of C2.F are valid on every flat order frame. However, completeness is another matter: a characterization result like Theorem 4.3 does not entail a completeness result. Abstractly, a characterization result for a logic L against a background class of frames $ℱ$ specifies the subset ${ℱ}_{\text{L}}$ of $ℱ$ such that for all $F\in ℱ$, $F\in {ℱ}_{\text{L}}$ iff L is valid on F. However, it is possible that L is not complete with respect to ${ℱ}_{\text{L}}$, when there is some sentence p which is valid on every $F\in {ℱ}_{\text{L}}$ but is not a theorem of L: i.e., when the set ${ℱ}_{\text{L}}$ is “too small” to find a countermodel to every non-theorem of L.

And indeed, it is well known that there are normal modal logics which are not complete with respect to the class of modal frames that characterize them: in other words, certain formulae that are not theorems of the logic are nevertheless valid on every Kripke frame (Fine 1974; Thomason 1974; van Benthem 1978; see Holliday and Litak 2019 for a helpful recent discussion). So we cannot simply assume that a characterization result yields a corresponding completeness result. Moreover, the fact that C2.F, like many extensions of C2, is not strongly complete for any class of order frames (which we can show analogously to Theorem 2.7) means that the standard canonical model method for proving completeness will not work for these logics.

With that said, we do in fact have a completeness result for C2.F: in the appendix, we show that C2.F is weakly complete for flat order frames. Indeed, we show that it is weakly complete for finite such frames:

Theorem 4.4. C2.F is weakly complete with respect to finite flat order frames.

Because of its restriction to finite frames (and the fact that flatness is a decidable property of finite frames), this result has the corollary that C2.F is decidable.

Since all linear order frames are flat, the fact that C2.F is sound for flat order frames immediately implies that it is sound for linear order frames. Since not all flat order frames are linear, one might guess that the logic of linear order frames would be stronger than C2.F. But in fact the logics turn out to be the same:

Theorem 4.5. C2.F is sound and weakly complete with respect to linear order frames.

Both Theorem 4.4 and Theorem 4.5 follow from a third completeness result for C2.F which we come to in §6 (namely, Theorem 6.2). That result is based on a generalization of $\omega$-sequence frames which replaces $\omega$-sequences with sequences of arbitrary length, while keeping the other definitions fixed.

That generalization is natural to explore when we start from $\omega$-sequence semantics. However, for fans of C2.F but not C2.FS, the resulting semantics might be a hard sell: it is at least somewhat difficult to motivate the explanatory force of a highly theoretical notion like transfinite sequences for modeling the meaning of ‘if’. Similarly, our characterization of C2.F yielded a complex frame condition (flatness) which is not terribly easy to get an independent grip on. By contrast, the definition of linear order frames is extremely simple, and, we think, provides the clearest semantic picture of the difference between C2.F and C2. C2 is sound and complete for order frames whose order functions endow every world with a separate well-order which need have no relation to that of any other world. By contrast, C2.F is sound and complete for a semantics based on a single master order: each world’s order is just the result of deleting all its predecessors from the master order. The less constrained order semantics for C2 has, in our view, more prima facie philosophical plausibility, so this new perspective does not yet explain in some deeper sense why C2.F would be the correct logic for the conditional. But it may help get a grip on what that question amounts to.

One noteworthy corollary of the soundness direction of Theorem 4.3 is that the two modal schemas 4 and H we mentioned in §4 in fact exhaust the purely modal content of C2.F. Here are the two axioms again:

4                                    $\mathrm{\square }p\to \mathrm{\square }\mathrm{\square }p$

H                                    $\mathrm{◇}p\wedge \mathrm{◇}q\to \mathrm{◇}(p\wedge \mathrm{◇}q)\vee \mathrm{◇}(q\wedge \mathrm{◇}p)$

The modal logic that adds these two axiom schemas to KT is called S4.3, and it is sound and complete (indeed strongly complete) for modal frames with a reflexive, transitive, and connected accessibility relation. It is also weakly complete for finite such frames (Bull 1966), which we use to show:

Theorem 4.6. When $p$ belongs to the modal fragment of ${ℒ}_{>}$, ${⊢}_{\text{C2.F}}p$ iff ${⊢}_{\text{S4.3}}p$.

Proof. We already established the right-to-left direction in §4.²⁵ For the left-to-right direction, suppose p is false in a finite reflexive, transitive, and connected pointed model $⟨v,⟨W,R,V⟩⟩$. Then (by a standard induction, relying on the transitivity of R) p is also false in the generated submodel $⟨v,⟨R(v),R_v,V_v⟩⟩$ (where R_v and V_v are respectively R and V restricted to R(v)). Next we give a recipe to transform $⟨v,⟨R(v),R_v,V_v⟩⟩$ into an equivalent pointed flat order model , yielding a flat order model where p is also false.

To construct , choose a strict linear order $\prec$ on R(v) such that $x\prec y$ implies xRy (the existence of which is guaranteed by the connectedness of R). Then set iff $y\ne x$, $y\ne w$, wRxRy, and either $x=w$ or $x\prec y$.

The accessibility relation induced by obviously agrees with R_v (relying on the transitivity of R and the fact that $x\prec y$ implies xRy). It is obviously irreflexive. It is well-founded by the finitude of R(v). Transitivity of follows from the transitivity of $\prec$ and R. And it is connected: when $w\in R(v)$: for any distinct $x,y\in R(w)$, if $x=w$ or $y=w$ then or respectively; otherwise by the totality of $\prec$ either $x\prec y$ or $y\prec x$ and hence or , respectively.

Finally, is also semi-flat. We know that xRy whenever , since either $x=w$ and wRy, or $x\prec y$. So we just need to show that whenever , . (Since R is transitive, the case where $z\in R(x)\backslash R(w)$ cannot arise.) If $x=w$ this holds trivially. Otherwise, implies $x\prec y\prec z$ and hence .□

A corollary of this result is that C2.F cannot be axiomatized by adding any purely modal principles (such as 4 or H) to C2. For it is easy to see that there are non-flat order frames that are reflexive, transitive and connected: the non-flat frame in Figure 2 is an example. Such frames do not validate all of C2.F, but do validate all its purely modal theorems.

5.1 Ancestral order frames

To better understand what property of $\omega$-sequence models is responsible for the validity of Sequentiality, it is helpful to characterize the class of order frames on which C2.FS is valid. In the reasoning we just gave explaining why Restricted Sequentiality is valid over $\omega$-sequence frames, the only fact about these frames (apart from their flatness) that we relied on is that every sequence accessible from a given sequence can be reached from it in finitely many steps, where at each step we go from a sequence to its first tail. Likewise, in the reasoning we gave explaining why Sequentiality is valid, the only fact we relied on is that every sequence accessible from a given sequence has an immediate predecessor, which, in light of the well-foundedness of ordinals, is equivalent to the first property. In other words, the key property of $\omega$-sequence frames which (together with flatness) explains why they validate Sequentiality, is that every accessible world from any given world can be reached from the starting world via a succession of single steps along each world’s closeness ordering. More carefully:

Definition 5.1. Given an order frame :

• – The successor of w is w itself if $R(w)=\{w\}$, and otherwise the first world after w in .

• – x is reachable from w iff w is related to x by the transitive, reflexive closure of the successor relation.

• – Finally, the frame is ancestral iff every world that is accessible from a world is also reachable from that world.

It is easy to see that every $\omega$-sequence frame is ancestral as well as flat. And while of course not every flat ancestral order frame is an $\omega$-sequence frame, we can show that every flat ancestral frame is isomorphic to an $\omega$-sequence frame, so the logic of $\omega$-sequence frames is the logic of flat ancestral frames.

To construct an $\omega$-sequence frame isomorphic to a given flat ancestral frame , we will replace each world in W with its successor-sequence, defined as follows:

Definition 5.2. In an order frame , the successor-sequence ${\alpha }_w$ of w is the $\omega$-sequence starting with w where each element after the first is the successor of the previous element.

The set $\{{\alpha }_w\mid w\in W\}$ is obviously closed under non-empty tailhood, and thus is an $\omega$-sequence frame. A few examples should give an intuitive sense for why, when is flat and ancestral, the function mapping each world to its successor-sequence is an isomorphism from the starting frame to the corresponding $\omega$-sequence frame. Consider the frame illustrated in the center of Figure 4: replacing each member with its successor-sequence yields an $\omega$-sequence frame comprising the three sequences $⟨1,2,3,3,3\mathrm{\dots }⟩$, $⟨2,3,3,3,3\mathrm{\dots }⟩$, $⟨3,3,3\mathrm{\dots }⟩$, which is indeed isomorphic to the one we started with. Likewise, if we start with the frame on the right of Figure 4, we get the isomorphic $\omega$-sequence frame with sequences $⟨1,2,3,1,2,3\mathrm{\dots }⟩$, $⟨2,3,1,2,3,1\mathrm{\dots }⟩$, $⟨3,1,2,3,1,2\mathrm{\dots }⟩$.

Figure 4: Illustrations of a flat but not ancestral frame , and two ancestral frames . In there is no way to get from 1 to 3 by taking successor steps, since you end up stuck in a loop between 1 and 2.

By contrast, this procedure won’t work if we start with a non-ancestral order frame. To get a feel for why not, consider the flat but non-ancestral order frame in Figure 4. The successor-sequence of 1 in this order frame is $⟨1,2,1,2\mathrm{\dots }⟩$, since 1 and 2 are each other’s successors. The successor sequence of 3 is $⟨3,3,3,3\mathrm{\dots }⟩$, since 3 is its own successor. ${\alpha }_3$ is hence not accessible from (i.e., not a tail of) ${\alpha }_1$, even though 3 is accessible from 1. Hence the sequence frame constructed this way is not isomorphic to the starting frame under the successor sequence function. More generally, whenever v is accessible but not reachable from w, ${\alpha }_v$ will not be accessible from ${\alpha }_w$.

To show that flat ancestral frames are always isomorphic to the corresponding successor-sequence frame, consider any flat ancestral frame and $w\in W$. First we show that $u\in R(w)$ iff ${\alpha }_u\in R({\alpha }_w)$. The fact that F is ancestral means that every world accessible from w is reachable from it. And the fact that F is flat, and hence transitive, means every world reachable from w is accessible from it. So $u\in R(w)$ just in case u is reachable from w. But now note that, likewise, ${\alpha }_u$ is a tail of ${\alpha }_w$ iff u is reachable from w: if u is reachable from w, u will appear in ${\alpha }_w$, and the truncation of ${\alpha }_w$ at u is obviously ${\alpha }_u$; and if u is not reachable from w, u will never appear in ${\alpha }_w$, but ${\alpha }_u$ always begins with u.

Now we need to show that iff ${\alpha }_x{\prec }_{{\alpha }_w}{\alpha }_y$. To show this, it suffices to show that iff the rank of x from w is less than the rank of y from w, where we define the rank of x from w as the least n such that x is the nth element of ${\alpha }_w$.