2.1 Modality: Absolute vs. Relative

It is customary and, in fact, the contemporary orthodoxy to represent modality, in particular, necessity and possibility, in terms of possible worlds. Modal claims (necessity and possibility claims) are assessed not only by what is actually the case but by reference to the ‘space’ of possible worlds. We assume that there is a set of possible worlds each representing a way things could be. We then evaluate modal claims as follows.

• A necessity claim, $\square A$ ( $A$ is necessarily the case), is true iff $A$ is true at all relevant possible worlds.

• A possibility claim, $◊A$ ( $A$ is possibly the case), is true iff $A$ is true at some relevant possible world.

Depending on what counts as relevant, we may place a constraint on the ‘space’ of possible worlds. By varying the constraint, we would generate different (normal) modal logics such as C. I. Lewis’ S4 and S5 (Reference LewisC. I. Lewis (1918)) – at least, we would generate the semantics that are sound and complete with respect to C. I. Lewis’ (normal) modal logics. What is important here is that an assessment of the truth values of modal claims depends not only on what the worlds are like but also how they relate to each other. Hence, the notions of necessity and possibility used in the semantics for evaluating modal claims is a relative notion. A world may be necessary or possible relative to a given world.

However, it is also part of the contemporary orthodoxy that a set of possible worlds used in evaluating modal claims is not an unstructured body of ‘dots’ or ‘points’. It is more like a structured web. The necessity or possibility of a claim is taken to depend on the structure of the set of worlds which is, in turn, taken to have a necessary structure. Importantly, this structure is often assumed to be pluriversal in that the set of worlds as a whole has a necessary structure. This is one of the reasons why many take the structure of the space of possible worlds to support the inferences associated with S4 (or stronger) like $\square p\models \square \square p$ and $◊◊p\models ◊p$ (or the axioms associated with S4 like $\square p\to \square \square p$ and $◊◊p\to ◊p$ ).

When impossible worlds are included, one may still hold that what counts as impossible is part of the necessary structure of the whole modal space.Footnote ⁵ This means that if something is impossible given the whole space of worlds, it is impossible in some absolute sense. Roughly speaking, the absolutists will contend that the line between the impossible and the possible does not depend on which world we are evaluating a modal claim from; rather, the line is drawn by the overall structure of the modal space.

This absolute sense of possibility and impossibility holds even in the context of counterfactuals. A counterfactual is a conditional whose antecedent concerns what may not actually be but could be.Footnote ⁶ The standard semantics to account for counterfactuals (Reference KratzerKratzer (1977), Reference LewisD. Lewis (1973), Reference Stalnaker and RescherStalnaker (1968)) evaluates them in terms of the most similar or closest worlds where the antecedent is true. This means that it requires the worlds or the points of evaluations to be ordered in terms of similarity or closeness relation between worlds. As has been claimed by D. Lewis and others, this relation is context sensitive. That is, there is no one fixed or absolute way to order the worlds. Nevertheless, it might be thought that, once the worlds are ordered in a context, the division between possible and impossible worlds might be drawn by the overall structure given by the ordering. Hence, a world may be impossible in an absolute sense even if the overall structure is context sensitive.

2.2 Non-normal Worlds

When C. I. Lewis introduced modal logics, he introduced not only S4 and S5, so called normal modal logics, but also non-normal modal logics such as S2 and S3.Footnote ⁷ These systems are weaker than more familiar normal modal systems and, as a result, modality is treated in a way that may be unfamiliar to contemporary audience.

The primary characteristic of the non-normal systems is the failure of necessitation: it is not always the case that if $\models A$ then $\models \square A$ . In a normal system, a necessity claim, $\square A$ , is evaluated in terms of the truth value of $A$ at all relevant worlds. So, if $A$ is true at all worlds (and, thus, $\models A$ ), then $A$ is true at all relevant worlds. So $\square A$ is true at all worlds. Thus, $\models \square A$ . Hence, in a normal system, if $\models A$ then $\models \square A$ .

In order to capture the failure of necessitation semantically, Reference Kripke, Addison, Henkin and TarskiKripke (1965) introduced non-normal worlds where $\square A$ fails to hold for any $A$ . (By the interchangeability of $\square \neg$ and $\neg ◊$ which holds in non-normal systems, $◊A$ is true for every $A$ at a non-normal world.) Validity, $\models$ , is defined in terms of truth preservation at all normal worlds.Footnote ⁸ Since $B\vee \neg B$ is true at every world for any $B$ , $\models B\vee \neg B$ . At a non-normal world, however, $\square (B\vee \neg B)$ fails to hold. Hence, it is not the case that $\models \square (B\vee \neg B)$ . Non-normal worlds, thus, capture the failure of necessitation.

The idea of non-normal worlds was generalised by Reference Routley, Meyer and LeblancRoutley & Meyer (1973) in the semantics for relevant logics (which is an extension of the semantics for First Degree Entailment (FDE) introduced by Reference Routley and RoutleyRoutley & Routley (1972)). In relevant logics, $A\to \left(B\to B\right)$ should come out invalid because $A$ and $B\to B$ are irrelevant (they do not share any propositional variables). This means that there must be a way for $B\to B$ to fail to be true while $A$ is true in the semantics. However, $B\to B$ is a logical truth in (most) relevant logics. So, there must be a way for a logical truth to fail in order to invalidate $A\to \left(B\to B\right)$ . Routley & Meyer used non-normal worlds to achieve this effect.

The way that Routley & Meyer invalidated $A\to \left(B\to B\right)$ is as follows. At a non-normal world, the truth value of a conditional, $A\to B$ , is not determined by the truth values of $A$ and $B$ at the same world. It is determined by those values at worlds that are different but related in a ternary manner. What is important here is not the ternary relations as such but the fact that the antecedent and the consequent of a conditional are evaluated at different worlds. That allows $B\to B$ to fail to be true at a non-normal world. So, the introduction of non-normal worlds was vital for the development of relevant logics.Footnote ⁹

2.3 Entailment Statements

In order to talk about the logical characteristics of a world that contains logical impossibility, we assume that there are logical laws that hold at a world and that we have a language to express them. These assumptions need to be shown to be cogent (Priest (202+)). In this Element, however, we take these assumptions to be in place.

When logical laws hold at a world, we say that there is a set of entailment schemas that represent the laws. An entailment schema takes the form: $A_1,A_2,\dots \models B_1,B_2,\dots$ where $A_i$ and $B_i$ are meta-variables meaning that they are atomic propositions or complex formulas.Footnote ¹⁰ If $A_1,A_2,\dots \models B_1,B_2,\dots$ holds at $w$ , then $B_i$ takes a ‘good’ value for some $i$ at $w$ when $A_i$ takes a ‘good’ value for all $i$ at $w$ . Alternatively, if $A_i$ takes a ‘good’ value for all $i$ at $w$ , then $B_i$ does not take a ‘bad’ value for any $i$ at $w$ . What count as ‘good’ and ‘bad’ values depends on the logic in question. But, in general, a ‘good’ value is the one that is preserved in a valid inference and a ‘bad’ value is one that forms part of a counterexample to a valid inference. For illustrative purposes, we say that if $A$ takes a good value, it is true, and if $A$ takes a bad value, it is non-true. We simply take truth and non-truth to represent good and bad values understood in terms of validity and counterexample respectively. What exactly truth and non-truth amount to is a contentious question but we need not settle it here.Footnote ¹¹

Then, an entailment statement is an instance of an entailment schema.Footnote ¹² It specifies a valid inference according to a logical law. For instance, if $A\wedge B\models A$ is a logical law, its instance such as $p\wedge q\models p$ is an entailment statement. For simplicity, we assume that the language for the statements is propositional. We also assume that the language for the entailment statements does not contain a conditional. This is to avoid the issue that comes up in the context of relevant logics, especially in the context of nested relevant conditionals.

Counterexamples to an entailment schema $A_1,A_2,\dots \models B_1,B_2,\dots$ at a world $w$ are $p_1,p_2,\dots$ and $q_1,q_2,\dots$ that hold at $w$ such that $p_i$ and $q_i$ are propositions (or whatever the truth bearers are) and that $p_i$ and $q_i$ can be used to instantiate $A_1,A_2,\dots \models B_1,B_2,\dots$ where all the premises are true and the conclusions non-true. For instance, let’s assume that Identity ( $A\models A$ ) holds at $w$ . Suppose that $p$ is both true and non-true. Then, an instance of $A\models A$ , $p\models p$ , takes you from truth to non-truth. So, if $A\models A$ holds at $w$ , $p$ is a counterexample to Identity at  $w$ .Footnote ¹³

2.4 Logical Humean Supervenience

We assume that a world $w$ consists of a set of entailment statements that specify valid inferences and a set of (non-logical) facts that hold at $w$ . For the purposes of this Element, we do not need to go into answering what facts are. We just take them to be represented by propositions. So, if $A_1,A_2,\dots \models B_1,B_2,\dots$ is a logical law that holds at a world, we let, for instance, $p_1,p_2,\dots \models q_1,q_2,\dots$ represent a valid inference and $p_i$ , $q_i$ , … represent facts that hold at the world. If an entailment schema $A_1,A_2,\dots \models B_1,B_2,\dots$ has counterexamples at a world $w$ , there are instances of it such that those instances have premises which are true at $w$ and conclusions that are non-true at $w$ . However, if the entailment schema holds at $w$ , then $B_1,B_2,\dots$ must be true for some $B_i$ when $A_i$ s are all true. So, the presence of counterexamples implies that some instances of the laws of logic (expressed by entailment statements) do not obey the laws.

If a world contains a counterexample, what we call Logical Humean Supervenience (LHS), fails at that world:

We are conceiving of a world to consist of a set of entailment statements and a set of (non-logical) facts. An entailment statement is an instance of a law of logic. We can think of it as representing the logical ‘qualities’ of the (non-logical) facts that hold at the world. From a logical point of view, a world consists of those ‘qualities’. So a world is ‘a vast mosaic of local matters of particular fact’ and nothing more, to use the phrase of Reference LewisD. Lewis (1986: ix). When a law of logic holds at a world, it supervenes on the ‘arrangement of [logical] qualities’ (Reference LewisD. Lewis (1986: ix)).

Now, LHS ensures that all the worlds are logically ‘well-behaved’. If LHS is in place, there is no world such that, for instance, $p$ and $q$ are true but $p\wedge q$ is non-true while Conjunction Introduction ( $A,B\models A\wedge B$ ) holds at that world. So, there would not be any world where it is raining and the grass is wet but it is non-true that it is raining and the grass is wet. LHS rules out such a world. Hence, the presence of counterexamples often signals a schism between the logical laws and what happens at a world. We will see in Sections 3–6 that LHS plays a crucial role in separating different accounts of impossible worlds (or the impossibility of impossible worlds).

2.5 Non-classical Logics

Classical logic is a logic that was devised by Frege, Russell, and Hilbert at the turn of the twentieth century. A non-classical logic is a logic that does not validate some of the laws of classical logic. This means that a set of non-classical laws may not correspond to the set of classical laws. We will be using one kind of non-classical logics as an example in this Element. That is paraconsistent logic. This is because a contrast can be seen rather sharply when we compare classical logic and paraconsistent logic.Footnote ¹⁴ Classical logic is a logic that validates ex contradictione quodlibet (ECQ): $A,\neg A\models B$ for any $A$ and $B$ . Classically, everything (that can be expressed by the given language) is a valid consequence of a contradiction. This means that, classically, if a contradiction holds at a world, such a world is a trivial world, that is, a world where everything (expressible in the language) holds. A paraconsistent logic is a logic that does not validate ECQ. This means that, paraconsistently, even if a world contains a contradiction, it may not be a trivial world. If a world is trivial, that is not because everything is a consequence of a contradiction. Hence, a paraconsistent logic is a logic that can circumvent a contradiction from spreading everywhere (hence, non-explosive).Footnote ¹⁵

3.1 An Analogy to Physics

Reference PriestPriest (1992) motivates the characterisation of impossibility in terms of difference by an analogy to the laws of physics (or nature).

Priest then calls such worlds ‘logically impossible worlds’:

Having suggested that impossible worlds are logically different worlds and appealing to this analogy to the laws of physics, Priest does not step through the analogy to show what lessons can be drawn from it. In order to judge the adequacy of the analogy and, in turn, the characterisation of impossible worlds as logically different worlds, we must unpack the analogy.

The analogy seems to go as follows. It is a law of physics (or nature) at the actual world that no one can travel faster than the speed of light. So it may be a physical possibility that Usain Bolt keeps breaking world records but a physical impossibility that he accelerates past the speed of light (at least, this is what the analogy assumes). However, at a physically impossible world, the laws of physics at that world may allow Usain Bolt to accelerate indefinitely and run faster than the speed of light. The physically impossible Olympics would be a spectacular event (though mostly unobservable). This is the case even if he does not break any records at that world, or run faster than the speed of light at that world, etc.Footnote ¹⁷ The thought is that the laws of physics at that impossible world allow Usain Bolt to accelerate past the speed of light even if he may not do so at that world but that the laws of physics at the actual world do not make this allowance. That impossible world and our world must, thus, be governed by different physical laws.

Analogously, a logically impossible world is a world where the laws of logic are different, in particular, from the actual world. If classical logic holds at the actual world, then a world where a paraconsistent logic holds is a logically different world. Thus, if classical logic holds at the actual world, a world where paraconsistent logic holds is an impossible world.

But what makes these logically (and physically) different worlds impossible? In the case of physical impossibility, the laws of physics (or nature) at the actual world dictate that no one can travel faster than the speed of light. Relative to the actual laws of physics, then, it is a physical impossibility that Usain Bolt arrives at the end of the track before light gets there. Similarly, if the actual world is a classical world, relative to the actual laws of logic, it is a logical impossibility that ECQ is invalid.

To make this point clear, consider the following analogy which reflects Priest’s more recent thought about the issue.Footnote ¹⁸ Let’s suppose that ‘all lumps of gold are less than 100 kg’ is contingently true at the actual world. Now imagine a world that is exactly the same as the actual world, except that there is a law of physics (or nature) to the effect that all lumps of gold more than or equal to 100 kg explode. Since there is a law that holds in that world but does not hold in the actual world, it is a physically different world. In such a physically different world, there being a 101 kg stable lump of gold is impossible. However, it is possible that there is a stable 101 kg lump of gold at the actual world even though there is, as a matter of contingent fact, no such lump. In such a case, the only difference between the two worlds is the difference of the laws of physics (or nature). So, what explains the possibility and impossibility of a stable 101 kg lump of gold is the difference of the laws of physics (or nature) at the two worlds. Similarly, so the analogy goes, ECQ being invalid is impossible at a classical world but it is possible at a paraconsistent world, even if there are no true contradictions at either world. Given that those two worlds may be the same except the difference in their laws, it is difference that is the mark of impossibility, or so it is argued.

3.2 Is Difference the Mark of Impossibility?

There are three things that should be noted about this characterisation of impossibility and the analogy to the laws of physics (or nature). First, the analogy makes use of the relative notion of possibility and impossibility. Whether Usain Bolt accelerating past the speed of light is possible or impossible is described as relative to a world: it is possible relative to a world where there is no law to the effect that he cannot run faster than the speed of light but impossible relative to the actual world and its laws. Indeed, the absolute sense of difference, difference-full-stop, does not make sense; difference is always difference from something. Thus, if we characterise impossible worlds in terms of difference, impossibility is given a modal characterisation: describing a world as impossible requires a reference to other worlds, or other points of evaluation, in modal space.Footnote ¹⁹

Second, physically or logically different worlds are individuated by different sets of laws. On this story, what makes a world impossible is not what happens at the world but the impossibility of the laws of physics (or nature) in comparison to a different set of laws. Similarly, what makes a paraconsistent world impossible (assuming that the actual world is classical) is the laws of logic that hold at the world irrespective of the behaviour of the instances of the laws at that world. So, the analogy characterises the impossibility of impossible worlds in terms of some set of laws themselves being impossible.

Third (which is a consequence of the second), the notion of difference applies to the laws and it is the laws that determine what the worlds are like. In the case of physically different worlds, the difference between the actual world and the physically different world appealed to in the analogy concerns a law that no one can travel faster than the speed of light. Moreover, at the actual world where there is a law to the effect that no one can travel faster than the speed of light, no one indeed travels faster than the speed of light. But Usain Bolt may beat the speed of light in a physically different world where there is no such law. So, the analogy suggests that laws supervene on what happens at the world. For instance, if Usain Bolt were to run faster than the speed of light, it would have to be in the physically impossible world rather than in the actual world that he would do so. Analogously, the measure of difference in the context of logic is the difference of the laws of logic. The analogy compares a classical world and a paraconsistent world without any consideration of what instances of the laws might obtain at those worlds. So, the analogy presupposes LHS; that is, that there is no mismatch between the laws and their instances that obtain at any of the worlds.

3.3 What Difference Does Difference Make?

The three points discussed in the previous section seem to come in conflict with other desiderata identified in the literature. In this section, we will discuss two such desiderata. We will not resolve them or use them to argue against difference as the mark of impossibility here; however, they should be kept in mind when we consider applications of impossible worlds in Sections 7–8.

(1) Despite the fact that the analogy makes use of the relative sense of impossibility, the notion of difference in play presupposes the absolute sense of impossibility. Consider all of the physically different worlds which are characterised in terms of the difference of the laws of physics (or nature) from those of the actual world. If we metaphorically think of these worlds as scattered around the space of all physically different worlds, the space represents all the ways in which the laws of physics (or nature) might or might not be centred around the actual world. Among them are impossible worlds such as the one where there is no law to the effect that no one can run faster than the speed of light. This means that, given the universe of all the physically possible worlds centred around the actual world, some regions of the universe are conceived to contain impossible worlds. Thus, the notion of impossibility in play is absolute: a world is impossible if it resides in the region of the whole modal space which is identified as impossible.

Without this absolute sense of impossibility, it is hard to see how there is any impossibility involved in the analogy. To see this, consider the second analogy about the lumps of gold. After the actual world is described as a world where ‘all lumps of gold are less than 100 kg’ is contingently true, a physically different world is introduced as representing a way in which things could be. As Priest says, ‘there are possible worlds where the laws of logic are different’ (Reference PriestPriest (1992: 292)). So it is introduced as a possible world relative to the actual world. The only way to count this physically different world as impossible is to introduce the absolute sense of impossibility and claim that it is impossible because it occupies a certain region of the modal space where impossible worlds reside.

(2) Reference NolanNolan (1997) holds the following principle in advocating impossible worlds:

If $A$ is possible, there is at least one possible world where $A$ holds. So, Nolan’s principle is equivalent to

Reference NolanNolan (1997) thinks of this principle as unrestricted comprehension.

Reference PriestPriest (2016a) adopts a stronger principle:

Priest’s principle is stronger as it says not only about the worlds where $A$ holds but also the worlds where $A$ fails.Footnote ²² Priest calls this principle the ‘Primary Directive’ and presents it as the ‘leading principle of impossible-world semantics’ (Reference PriestPriest (2016a: 2653)).Footnote ²³

We call Priest’s principle the unrestricted comprehension principle (because that sounds more like a principle). Given unrestricted comprehension, there will be at least one world where $A$ holds but there is at least one world where $A$ fails even though the laws of logic that hold at those worlds are the same. So, at some worlds, there will be a discrepancy between the laws and what happens at a world (including the instances of a given set of laws). Hence, according to the unrestricted comprehension principle, LHS must fail in that the laws do not supervene on their instances.

In order to rule out a world where there is such a mismatch between the laws and their instances, one might reject unrestricted comprehension principle. In fact, Priest (202+) suggests exactly this:

The exclusion of the facts about the laws of logic from constituting worlds is a way of constraining the laws that hold at worlds. This allows logical qualities of (non-logical) facts to always match or obey the laws in that it allows us to exclude the worlds where the laws and their instances come apart. This modification would restore LHS.

However, removing the worlds at which instances do not match the laws from the picture raises a tension in the dialectic. One motivation for introducing impossible worlds is to account for hyperintensional phenomena where necessary equivalents can be distinguished. Restricting the worlds so that the laws and their instances match at all worlds imposes a kind of necessary structure over the whole modal space which the introduction of impossible worlds is meant to counteract in the context of hyperintentionality. So, excluding the worlds where the laws and their instances come apart undermines the very motivation for introducing impossible worlds. Hence, if we are to account for hyperintensionality generally, we should not rule out worlds at which the laws and their instances come apart and, thus, we should reject LHS.Footnote ²⁵

Thus, characterising impossible worlds as logically different worlds conflicts with some of the desiderata that the introduction of impossible worlds is meant to accommodate. Whether or not this means that impossible worlds should not be characterised as logically different worlds is a question we leave to the side for now. However, the above discussion shows that such a characterisation cannot be motivated solely by the ‘intuitive’ appeal of an analogy to physics (or nature).

4.1 Anarchy

One common way to understand openness follows the ‘motto … that at an impossible world, anything can happen’ (Reference PriestPriest (2016b: 190)). Given that the focus here is on logical impossibility, following this motto, open worlds are understood as worlds where anything happens with respect to logic. A logic specifies valid inferences and separates them from invalid ones. So, a world where anything happens is a world where logical operations are unsystematic. That is, at an open world, all logical operations behave arbitrarily. An open world understood in this way is, thus, a world where assignments of truth values to atomic and complex sentences are all arbitrary. This means that, at an open world, the evaluation of $A$ is independent of that of $B$ for any $A$ and $B$ . This is the kind of open worlds popularised by Reference PriestPriest (2005). But it was Reference RantalaRantala (1982a, Reference Rantala1982b) who first introduced them to address the problem of logical ominiscience in the context of epistemic logic. Let’s call worlds of this kind Rantala worlds and see Rantala worlds in action.Footnote ³⁴

Consider a Rantala frame which is a structure $⟨W,N,R⟩$ where $W$ is the set of worlds, $N\subseteq W$ is the set of normal worlds (then, $W-N$ is the set of Rantala worlds) and $R\subseteq W\times W$ is a binary relation on worlds. Once a Rantala frame is equipped with an evaluation function, we have a Rantala model $⟨W,N,R,\nu ⟩$ . At normal worlds, atomic sentences are assigned truth values by the evaluation function and complex sentences are evaluated recursively in the standard manner. However, at a Rantala world, all sentences, whether atomic or complex, are assigned a truth value by the evaluation function directly. This allows an evaluation of $p$ to be independent of that of $p\vee q$ , for instance. So, $p\vee q$ may be non-true even if $p$ is true. Then, to generate a logic using Rantala models, validity is defined in terms of truth-preservation at all normal worlds in all Rantala models.

As we saw before, open worlds were introduced to reject various closure principles such as the closure principle for a knowledge operator $K$ :

In order to reject this, one can provide the following counter-model within a Rantala frame that is modified to apply to epistemic context: where $N=\left\{w_1\right\}$ (Reference Berto and JagoBerto & Jago (2019: 113)) as in Figure 1. (If a sentence, whether atomic or complex, does not appear in the graphic representation of the model, it is non-true.) Since $p$ is true at $w_2$ , $Kp$ is true at $w_1$ (where $K$ works just like $\square$ ). Also, given that validity is defined in terms of truth preservation at all normal worlds, $p\models p\vee q$ . But $p\vee q$ is non-true at $w_2$ . So $K(p\vee q)$ is non-true at $w_1$ . Hence the closure principle in question is invalidated.

Figure 1

One way to describe a Rantala world (or worlds introduced to invalidate various closure principles) is that it is logically anarchic (Reference Berto and JagoBerto & Jago (2019: § 5.4)). At a Rantala world, all sentences, whether atomic or complex, are assigned truth values by the evaluation function directly. This means that complex sentences are evaluated just like atomic sentences. A consequence of this is that logical operations are not recursive. In fact, there is nothing systematic about any of the logical operators. For instance, the evaluation of $p\vee q$ is independent of that of $p$ . So, at a Rantala world, there is no systematic relationship between $p\vee q$ and $p$ .

Now, anarchy is something that holds at (or in) a world or worlds. If an assignment of truth values is arbitrary, it is at a world that logical operations are arbitrary. In other words, it is the logical operations intrinsic to a world that are anarchic.Footnote ³⁵ This means that an anarchic world is open in a restrictive sense: an anarchic world is open when the world is not closed under any intrinsic logic (a logic that holds at that world).

4.2 Nihilism

Another class of worlds that might be regarded as open are logically nihilistic worlds. Logically nihilistic worlds are worlds at which no logic holds. Using our terminology, this means that there are no entailment schemas that hold at (or in) that world. If a world is logically nihilistic in this way, there is no logic intrinsic to the world. So a logically nihilistic world is open because there is no logic intrinsic to the world under which the world is closed. Given that there is no logic that holds at (or in) a logically nihilistic world, there is no internal perspective from which to close the world. Thus, a logically nihilistic world is an open world. (Depending on how to define closure, a logically nihilistic world may also be closed because it is closed under any logic, assuming that we allow ‘any logic’ to be satisfied vacuously.Footnote ³⁶ But the focus here is on openness.)

4.3 Silence

Anarchy and nihilism are notions of openness that concern logics that hold at worlds. It restricts which logics are in play when characterising openness. There is, however, a way to understand the notion of openness that is non-restrictive but which does not fall into the trap of implying that there are no open worlds at which something is true. The idea is to appeal to silent worlds (Tanaka (202+)).

Let’s assume that $p$ and $q$ are propositions that are part of the language and $A\models B$ (where $A$ and $B$ are meta-variables) is an entailment schema that expresses a law of logic. $A\models B$ does not have to be intrinsic to a silent world: it may be an entailment schema that holds at the actual world but does not hold at the silent world, for instance. For $p$ and $q$ , however, we assume that they represent facts that obtain at a world. There may be some $p$ and $q$ such that $\models$ is undefined for them at a world. This is the case when there is no logical relation between $p$ and $q$ according to $\models$ . In such a case, we say that $\models$ is silent about $p$ and $q$ at the world. Then, a world $w$ is silent if there are some propositions of the language, $p$ and $q$ , representing facts obtaining at $w$ such that $\models$ is silent about them.

A silent world is like a situation which is incomplete with respect to the assignments of truth values (Reference BarwiseBarwise (1989), Reference Barwise and PerryBarwise & Perry (1983)). For instance, the political situation in Australia does not say anything about the political situation in New Zealand. The truth value of the proposition expressing the political situation in New Zealand is undefined in a political situation in Australia. In this respect, a situation is silent about the truth values of some propositions. However, there is an important difference between a situation and a silent world. In a situation, it is the assignment of truth values of propositions that is undefined; whereas, in a silent world, it is the consequence relation between propositions that is undefined. This difference becomes important in considering logical impossibility.

In order to describe a silent world as an open world, we have to reexamine the notion of closure that often plays a crucial role in defining open worlds. When we questioned the cogency of characterising openness as amounting to not being closed under logical consequence, we used a semantic consideration. We claimed that if $A$ is true at a world, it must be true there; and so Identity ( $A\models A$ ) must hold. To then think that a world must be closed at least under Identity is to think of closure to be a semantic matter. However, the notion of closure does not have to be semantic. We might understand it in terms of the laws of logic subsuming their instances. Let $S$ be a set of entailment schemas representing the laws of logic that hold at a world. Then, we might think that a world is closed under $S$ if all the instances $s\in S$ are subsumed by $S$ at the world.

Now, let’s assume that $A\models B$ is an entailment schema. If a world $w$ is silent with respect to $p$ and $q$ , then neither $p\models q$ nor $p̸\models q$ holds at $w$ . However, $p\models q$ is an instance of $A\models B$ . So $A\models B$ does not subsume some of its instances at $w$ . Thus, if a world is silent with respect to any entailment schema, then a silent world is not closed and, hence, it is an open world. The notion of openness appealed to here is unrestrictive as a silent world is open when it is silent with respect to any entailment schema.

Just like we distinguished silent worlds from situations, we must also distinguish silent worlds from what we might call logically nihilistic worlds. A logically nihilistic world is a world where no logic holds. In our terminology, this means that it is a world where the set of entailment schemas that hold at the world is empty. The set of entailment schemas that hold at a silent world does not have to be empty, however. At a silent world, a logical law may hold even though it may fail to subsume some or all of its instances. This feature of a silent world will be important in considering such a world as an impossible world as we see below.

4.4 Openness as Impossibility?

Before examining whether or not anarchic worlds, logically nihilistic worlds and silent worlds should count as impossible worlds, we should note that there is a prima facie reason to think that openness as such cannot be the mark of impossibility. Identifying openness as the mark of impossibility sometimes comes with identifying closure as the mark of possibility as Priest (202+) does explicitly. However, closure appears not to be the mark of possibility. Consider a trivial world, $w_T$ , at which everything (every sentence or well-formed formula of the language) is true. To be more precise, let $L$ be a set of propositional language consisting of atomic sentences ( $p,q,\dots$ ) and well-formed formulas consisting of atomic sentences and logical operators ( $\neg$ , $\wedge$ , $\vee$ ). Then, at $w_T$ , for any $A\in L$ , $A$ is evaluated as true. Since everything is true at $w_T$ , reasoning in accordance with any law of logic leads from truth to truth. So $w_T$ is closed under any logic. Thus, if closure is the mark of possibility, a trivial world, $w_T$ , comes out as possible.Footnote ³⁷ However, a trivial world seems and is widely thought to be impossible. For instance, Reference NolanNolan (1997: 544) suggests that a world where everything is true is ‘one of the most absurd situations conceivable’ and Reference Weber and OmoriWeber & Omori (2019: 976) call such a world ‘the ultimate impossibility’. If the trivial world is impossible, closure cannot be the mark of possibility. If defining impossible worlds in terms of openness is paired with defining possible worlds in terms of closure, there is prima facie reason to think that openness is not the defining characteristic of impossible worlds. So, how can we understand open worlds so that openness might be taken to be indicative of impossibility?

4.5 Anarchy as Impossibility?

An anarchic world is a world at which logical operations are unsystematic. It is an open world in the sense that it is not closed under any logic intrinsic to the anarchic world. Having clarified the sense in which an anarchic world is open, we can now work out how an anarchic world might be thought of as characteristially impossible. We will argue that anarchy is not sufficient nor necessary for a world to be impossible.

First, logical anarchy cannot be sufficient for impossibility. This is because an anarchic world is closed under an anarchic logic according to which $A\models B$ holds for any $A$ and $B$ . This is the case even though it is not plausible to think that such a logic is ‘correct’, that is, it is not plausible that it is the actual logic.Footnote ³⁸ This means that openness cannot be sufficient for the impossibility of a world.

Second, logical anarchy isn’t necessary for impossibility. Consider a world which we call a Mortensen world. At a Mortensen world, everything is possible (Reference MortensenMortensen (1989)). Given that everything is possible, arbitrary assignments of truth values to atomic and complex sentences must also be possible at a Mortensen world. So, if a Mortensen world is possible, an anarchic world is possible. Hence, an anarchic world is not necessarily an impossible world as it may be possible.

Perhaps, however, anarchic worlds are impossible relative to the actual world. Except Mortensen, very few think that everything is actually possible. Moreover, it might be thought that closure is a special property of the actual world. That is, the actual world might best be understood as closed such that all the worlds that are actually possible are also closed. In this case, an anarchic world might be thought to be impossible relative to the actual world. This line of thought assumes that an anarchic world is different from the actual world. So, it does not seem to be anarchy that is making for the impossibility of these worlds; rather, it is difference. Moreover, this line of thought relies on the actual world that is extrinsic to the anarchic world in question. Hence, there does not seem to be anything about a world’s being anarchic that, in itself, is a necessary condition on its being impossible.

Lastly, if there is anything impossible about a logically anarchic world, that seems to be because it contains counterexamples to some set of logical laws. In the next section, we will describe such a world as a world containing logical violations. For now, we will show that a logically anarchic world contains counterexamples to some laws of logic.

Recall that, at a logically anarchic world, the evaluation of $p\vee q$ is independent of that of $p$ . So, $p\vee q$ may be non-true even though $p$ (or $q$ ) is true. However, at a non-anarchic, normal world, the evaluation of $p\vee q$ may depend on the evaluations of $p$ (and $q$ ). At least, this is the case in a Rantala model. Given that validity is defined in terms of truth-preservation at all normal worlds in all (Rantala) models, Disjunction Introduction ( $A\models A\vee B$ ) may be valid (though this depends on how disjunction behaves at normal worlds). This example shows that what is characteristic of logically anarchic worlds is that they contain counterexamples. The characteristic feature of a logically anarchic world is not that it is open but that it contains counterexamples. Thus, if there is anything impossible about a logically anarchic world, that is because it contains counterexamples to the logical laws that hold at the world.

4.6 Nihilism as Impossibility?

Is there any reason to think that an open world understood as logically nihilistic world is impossible? There seems to be a reason to think that a logically nihilistic world is possible (although this does not necessarily rule out such a world being impossible as well). Logical nihilism is often motivated in the context of the debate between logical monism and logical pluralism. Logical monism is the view that there is only one ‘correct’ logic and logical pluralism is the view that there are at least two ‘correct’ or ‘admissible’ logics.Footnote ³⁹ If logical pluralism is a possible view, then, even if it turns out to be false, it raises the possibility that the number of ‘correct’ or ‘admissible’ logics could not only go up from one but could also go down, that is, down to zero. So, logical pluralism seems to make it possible that there is no logic. In other words, logical pluralism, if it is a possible view, leads to the possibility of logical nihilism.Footnote ⁴⁰ If this line of argument works, a logically nihilistic world is possible. Hence, it is not clear that an open world understood as a logically nihilistic world should necessarily count as impossible.

4.7 Silence as Impossibility?

Characterising anarchy as the mark of impossibility seems to face difficulty as we saw above. What about silent worlds? Is there a sense in which silent worlds should count as impossible?

To answer this question, consider LHS: the laws of logic that hold at a world supervene on their instances at that world. One important feature of a silent world is that LHS fails with respect to a set of logical laws. At a silent world, $p\models q$ may not hold for some $p$ and $q$ even if $A\models B$ holds. Because the world may be silent about $p\models q$ (and $p̸\models q$ ), the instances of the law $A\models B$ , for instance, $p\models q$ , may not cooperate with the law. Hence, LHS does not necessarily hold.

Interestingly, however, this does not mean that there are counterexamples to the laws of logic within that world. When we introduced LHS, we characterised its failure in terms of the existence of counterexamples to a set of laws. At a silent world, however, that is not how LHS fails. That the world does not say anything about the consequence relation between $p$ and $q$ does not mean that $p\models q$ takes you from a good value to a bad value. So, LHS fails not because there are counterexamples to a set of logical laws but because a silent world is silent about the instances of the laws.

Regardless of how it fails, the failure of LHS might be thought to point towards impossibility. If LHS fails, there are instances of the laws of logic that do not accord the laws. This is the case even though there are no counterexamples and, thus, there are no instances that go against the laws. However, it might be thought that the laws of logic subsume their instances by the very nature of the laws. So, there is a sense in which, from the point of view of the laws, it is impossible for the instances of the laws to not align with the laws. If a silent world is impossible, that is because some instances of the laws of logic come apart from the laws.Footnote ⁴¹

5.1 Non-normal Worlds

Non-normal worlds were introduced to capture the semantics for non-normal modal logics such as S2 and S3 as well as various relevant logics (see § 2.2). In the literature on impossible worlds, they are often appealed to in motivating and demystifying impossible worlds. A non-normal world is a world where $\square A$ is non-true even if $A$ is true at all worlds ( $\models A$ ) and, thus, $\square A$ is true at all worlds ( $\models \square A$ ) or a world where $B\to B$ is non-true even if it is a logical truth ( $\models B\to B$ ). This feature of non-normal worlds has been glossed as worlds where the laws of logic fail. For instance, Reference PriestPriest (1992) describes non-normal worlds as follows:

If we take a logical truth to express a law of logic, a non-normal world is described as a world where a law of logic may fail. This observation about non-normal worlds has then been developed in terms of logical difference (Reference PriestPriest (1992, Reference Priest2008, 202+)) or openness (Reference Berto and JagoBerto & Jago (2019)).

However, it is not clear that the above analysis provides an adequate description of non-normal worlds. It is misleading to say that the laws of logic may fail at a non-normal world. We can see this clearly in the case of relevant logics. In (most) relevant logics, $B\to B$ is a logical truth (and, thus, expresses a law of logic) even in the presence of a non-normal world where $B\to B$ is non-true. So it is not quite correct to say that ‘logical truths [the laws of logic] may fail’ (Reference Berto and JagoBerto & Jago (2019)).

Instead, what goes on at a non-normal world is that it may provide a counterexample to a law of logic. In the case of relevant logic, at a non-normal world, for some $p$ , $p\to p$ may be non-true. But this does not necessarily invalidate $B\to B$ as it is a law of logic. Nevertheless, if $p\to p$ is non-true, such $p$ serves as a counterexample to the logical law $B\to B$ . So the law of logic does not fail but it is violated. What happens at a non-normal world is, thus, that a law of logic may be violated. Hence, a non-normal world is a world that may contain logical violations of some logical laws.

5.2 How Do Violations Account for Impossibility?

A world that contains logical violations has distinctive features that are absent from logically different worlds and open worlds. In this section, we will examine three such features. Articulating those features illuminate how logical violations account for impossibility.

5.2.3 Impossibility Is Something That Happens

As the failure of LHS indicates, an impossible world, under the characterisation in terms of logical violations, is a world where impossibility happens. The notion of violations concerns not only logical laws but also (non-logical) facts as the presence of counterexamples can reveal discrepancies between the laws and facts. So, impossibility is not always a matter of logical laws being somehow impossible. Impossibility, if we understand it in terms of violations, is not given by the structure of the logical space. Rather, it is something that happens at a world (Tanaka (202+)).

However, what counts as impossible is relative to a set of logical laws. So, the question of whether a world is impossible or not has no fixed answer, since the answer depends on a set of logical laws with respect to which we are characterising the world. Thus, if we understand impossibility in terms of the presence of logical violations, we shouldn’t think of impossible worlds as the worlds that need to be added to the set of all worlds in the way that Reference PriestPriest (2005) conceives of them as is shown by Figure 2 (Reference PriestPriest (2005: 22, Figure 1.1)):Footnote ⁴² where, in Priest’s terminology, impossible worlds are the non-normal worlds that are used for the semantics for non-normal modal systems and for the semantics for relevant logics, and possible and impossible worlds are considered to be closed.Footnote ⁴³ Instead, logical violations allow us to understand impossible worlds as worlds that are already there when all the worlds are delivered by the comprehension principle for worlds, which are then categorised as possible or impossible with respect to this or that set of laws of logic.

Figure 2

6.1 Difference and Openness

A logically different world is a world whose laws of logic are different (especially from those that hold at the actual world). An open world, on the other hand, does not necessarily hold a different set of logical laws from the ones that hold at the actual world. For instance, at a logically anarchic world, the laws of logic may be the same as those that hold at the actual world. It is just that there may be some instances of the laws that do not obey the laws (see § 4.5). Hence, difference and openness are not equivalent in the sense that a logically different world and an open world may be two different worlds.

However, Priest (202+: § 1) takes it as the ‘guiding thought’ that ‘a possible world is one that is closed under logical consequence and an impossible world is one that is not’. In trying to understand what exactly that thought comes down to, he suggests to think about logical impossibility in analogy with physical impossibility. He then provides an analogy like the ones that we saw in § 3 (Priest (202+: § 5)). Thus, for Priest, the notions of difference and openness are tightly intertwined.

It is not clear why Priest tightly connects difference and openness since he does not spell it out. However, there are conditions under which difference amounts to openness. As we saw in § 4, one way a world is open is for there to be no laws of logic at the world, that is, a world is a logically nihilistic world. If a world is logically nihilistic, there are no laws of logic intrinsic to the world under which it can be closed. Hence, such a world is open. But this is one way in which a world is logically different. To see this, we note that logically different worlds are worlds where different sets of logical laws hold at those worlds. In the analogy to physical impossibility, it is assumed that a physically different world has the laws of physics (or nature) that hold at or in that world. These laws do not ‘spill’ over to other worlds: the effects of the laws are all contained within the world. This is why a physically different world may be a world where the physically impossible Olympics may take place. But, any world whose set of the logical laws is different from, for instance, the actual world, no matter how small the difference is, still counts as a logically different world. And, one set of logical laws that is different from, for instance, the set of logical laws holding at the actual world, is an empty one. So, when the set of logical laws that hold at a world is empty, that world is logically different. Hence, an open world may count as logically different world if it is a logically nihilistic world.

6.2 Difference and Violation

Similarly, even though difference is not generally equivalent to violation, there are conditions under which difference counts as a violation. Consider a world $w_1$ where the laws of logic are classical. At $w_1$ , ECQ is valid ( $A,\neg A\models B$ holds at $w_1$ ). Now consider another world $w_2$ where the laws of logic are those of LP (Logic of Paradox, Reference PriestPriest (1979)). So, at $w_2$ , ECQ is invalid ( $A,\neg A\models B$ does not hold at $w_2$ ). They are, thus, worlds where different logical laws hold. That is, they are logically different worlds.

Now, LP is a sub-logic of classical logic in the sense that all of LP theorems are also theorems of classical logic. This means that worlds which contain no violations of classical logic will not contain violations of LP. However, worlds in which LP holds and which contain no violations of LP laws may contain violations of the classical laws. Hence, despite the fact that $w_1$ (where the classical laws hold) and $w_2$ (where the laws of LP hold) are logically different, $w_1$ does not contain violations with respect to $w_2$ as there are no counterexamples of LP laws at $w_1$ . Thus, logically different worlds do not always contain violations of each other’s laws. So, logical difference comes apart from the presence of logical violations.

Despite the fact that difference is not equivalent to violations, there are some situations where difference counts as a violation. We can illustrate the conditions under which difference constitutes violations by providing a schema for a modal semantics that makes room for the distinctions between difference and violation. This will help us see the relationships between difference and violation.

Consider the set of all worlds.Footnote ⁴⁶ We assume unrestricted comprehension principle following Reference NolanNolan (1997) and Reference PriestPriest (2016a). So, for any set of statements $\Gamma$ (including entailment schemas), there is at least one world where $\Gamma$ is true. To this, we introduce an ‘accessibility’ relation that tracks the absence of violation, v-accessibility. We say that $w_1$ v-accesses $w_2$ just in case $w_2$ contains no violations of the laws of logic that hold at $w_1$ . Conversely, if $w_1$ does not v-access $w_2$ then $w_2$ contains at least one violation of the laws of logic that hold at $w_1$ .

Within this schema, we can define logical difference. A world $w_1$ is logically different from $w_2$ iff either there is some world that is v-accessible from $w_1$ that is not v-accessible from $w_2$ or that is v-accessible from $w_2$ but not from $w_1$ . For instance, Figure 3 (where the arrows mark v-accessibility) represents a situation where $w_1$ and $w_2$ are logically different. That is, logically different worlds v-access different worlds; different things count as logical violations according to logically different worlds. This schema does not provide a complete theory of (logical) impossibility; however, it does allow us to visualise how logical difference relates to violations and in what way logical difference involves violations.Footnote ⁴⁷

Figure 3

6.3 Openness and Violation

If a world is open, it does not necessarily contain a violation. An open world is one that is not closed under logical consequence. There are two ways in which a world can be open while not containing a violation. First, if a world is logically nihilistic and open, there are no laws of logic that hold at that world. Then, there is no set of laws that is violated. Second, if a consequence relation is silent about some propositions that hold at a world, then it does not say anything about the validity between those propositions. So there are no instances of a law of logic involving those propositions that can serve as counterexamples. Hence, if a world is silent about some instances of the logical laws, it does not necessarily contain a violation. Hence, an open world does not necessarily contain a violation. So, in general, openness comes apart from violations.

However, as we have seen, if a world is logically anarchic, it may contain a violation of the logical laws. At a logically anarchic world, the operation of $\vee$ , for instance, is not recursive. So $p\vee q$ is evaluated independently of $p$ and $q$ . Then, it may not be that $p\models p\vee q$ as $p\vee q$ may be non-true even if $p$ (or $q$ ) is true. This is the case even if $A\models A\vee B$ is a law of logic. In such a case, the instances of the law may come apart from the law. If a world is logically anarchic in this way, it contains counterexamples to the law of logic. Hence, a logically anarchic world may contain violations of the laws of logic.

7.1 Counterpossibles

A counterpossible is a kind of counterfactual conditional whose antecedent involves impossibility. The following conditionals are often-cited examples of counterpossibles (Reference NolanNolan (1997)):

1. (1) If Hobbes had (secretly) squared the circle, all sick children in the mountains of South America at the time would have cared.

2. (2) If Hobbes had (secretly) squared the circle, all sick children in the mountains of South America at the time would not have cared.

Given that it is impossible to square a circle, these conditionals have impossible antecedents and they are, thus, counterpossibles.

The standard semantics for counterfactuals of Reference KratzerKratzer (1977), Reference LewisD. Lewis (1973) and Reference Stalnaker and RescherStalnaker (1968) evaluate both of the above counterpossibles as true. According to that semantics, a counterfactual is true iff the consequent is true at all possible worlds closest to the actual world satisfying the antecedent. But if a counterfactual has an impossible antecedent, there is no possible world where the antecedent is true. So if we apply this semantics for counterfactuals to evaluate the above counterpossibles, they are both vacuously true.

Whatever the ‘intuition’ one may have about the sick children in the mountains of South America, evaluating both counterpossibles to come out true looks implausible.Footnote ⁴⁸ The consequents of those counterpossibles are contradictory. So it is hard to make sense of treating them as logically equivalent.

The easiest way to accommodate this line of thought is to supplement the possible world based semantics for counterfactuals with impossible worlds.Footnote ⁴⁹ Then, since there may be a world, that is, an impossible world, where the antecedent is true, the evaluation of a counterpossible is nonvacuous. The truth value of the conditional depends on the truth value of the consequent at the closest worlds to the actual world where the antecedent is true.

A simple nonvacuist semantics for counterpossibles can be presented as follows (adapted from Reference Berto, French, Priest and RipleyBerto et al. (2018)). Given a propositional language including connectives $\neg$ , $\wedge$ , $\vee$ , (the counterfactual conditional) and modal operators $\square$ and $◊$ , let $\Pi$ be the set of propositional parameters and $\Phi$ be the set of formulas. An interpretation is a tuple $⟨W,P,R_A,\nu ⟩$ where

• $W$ is the set of all worlds, possible and impossible

• $P\subset W$ is the set of possible worlds (and $I=W-P$ is the set of impossible worlds)

• $R_A\subset W\times W$ is a binary relation on $W$ for every $A\in \Phi$

• $\nu$ is an evaluation function

$R_A$ can be understood in terms of the closest world in which $A$ is true. The notion of closeness or similarity is not required to be any different from the standard account except that impossible worlds have to be part of the closeness or similarity relations.

The evaluation function, $\nu$ , assigns 1 or 0 to every propositional parameter $p\in \Pi$ at every $w\in W$ ( ${\nu }_w\left(p\right)=1$ or ${\nu }_w\left(p\right)=0$ ) and to every formula $A\in \Phi$ at every impossible world ( ${\nu }_w\left(A\right)=1$ or ${\nu }_w\left(A\right)=0$ ). This means that, at an impossible world, $\nu$ assigns 1 or 0 directly to every formula. The truth conditions for formulas at a possible world, $w\in P$ , are standard:

• ${\nu }_w\left(\neg A\right)=1$ iff ${\nu }_w\left(A\right)=0$ (or it is not the case that ${\nu }_w\left(A\right)=1$ )

• ${\nu }_w(A\wedge B)=1$ iff ${\nu }_w\left(A\right)=1$ and ${\nu }_w\left(B\right)=1$

• ${\nu }_w(A\vee B)=1$ iff ${\nu }_w\left(A\right)=1$ or ${\nu }_w\left(B\right)=1$

• ${\nu }_w\left(\square A\right)=1$ iff for every $w^{\prime }\in P$ , ${\nu }_{w^{\prime }}\left(A\right)=1$

• ${\nu }_w\left(◊A\right)=1$ iff for some $w^{\prime }\in P$ , ${\nu }_{w^{\prime }}\left(A\right)=1$

• iff for every $w^{\prime }\in W$ such that $R_{A}w{w}^{\prime }$ , ${\nu }_{w^{\prime }}\left(B\right)=1$

Validity is defined in terms of truth-preservation at all possible worlds of all interpretations:

This provides a basic nonvacuist account of counterpossibles. As usual, stronger systems can be obtained by adding various constraints on $R_A$ .

In this account, the extensional connectives behave classically and validity is defined in terms of truth-preservation at all possible worlds. So, impossible worlds do not play any role in validity. This means that the nonvacuist account presented above behaves just like the standard vacuist account of standard counterfactuals in most cases. A difference between the two is the evaluation of counterpossibles. Consider (1) and (2). There may be an impossible world where Hobbes has squared the circle. So there may be a world where the antecedents of (1) and (2) are true. Such a world may not be contradictory concerning all sick children in the mountains of South America caring about Hobbes’ accomplishment. So the truth values of (1) and (2) may come apart. If one’s ‘intuition’ is such that either (1) or (2) is false, then this nonvacuist account can accommodate it.

In order to understand the kind of impossible worlds appealed to in this nonvacuist account of counterpossibles, recall that, at an impossible world, the evaluation function, $\nu$ , assigns a truth value directly to every formula. So, the standard logical relation between $p$ and $p\vee q$ is lost at some impossible world. Given that $p$ and $p\vee q$ are different formulas, $\nu$ assigns a truth value to them individually. It may be that $p$ is true while $p\vee q$ is not. Hence, logical operations are anarchic at an impossible world. An impossible world used in the nonvacuist account of counterpossibles presented above is, thus, an open world.

We will see that other kinds of impossible worlds have been used to account for the variations of counterpossibles. We will see that the accounts of counterlogicals and countermathematicals proposed in the literature can be analysed as appealing to impossible worlds understood as logically or mathematically different worlds or worlds that violate the logical or mathematical laws. Before we examine these accounts, however, it is important to investigate how open worlds as impossible worlds are used in the nonvacuist account of counterpossibles. Such an investigation illuminates how impossible worlds aka. open worlds are understood.

In order to emphasise that their nonacuist account of counterpossibles can ‘recapture’ everything or almost everything that a classical vacuist account of counterfactuals demands, Reference Berto, French, Priest and RipleyBerto et al. (2018) assume what Reference NolanNolan (1997) calls the Strangeness of Impossibility Condition (SIC):

For instance, consider the following closure principle:

where $\supset$ is material conditional. This principle is part of Lewis’ system of counterfactuals but it fails in the nonvacuist account of counterpossibles. To see this, notice that an instance of Closure is: If $\models B\supset C$ then . Classically, since $\models (p\wedge \neg p)\supset (q\wedge \neg q)$ , . If $A$ is $p\wedge \neg p$ , by Reflexivity of the counterfactual conditional (), we can detach the antecedent of the material conditional to get . Classically, thus, Closure (with Reflexivity and detachment) entails that a contradiction counterfactually explodes. Given that a nonvacuist account of counterpossibles is developed to prevent such an explosion, Closure has to fail in such an account.Footnote ⁵⁰

However, if we assume SIC, the following closure principle holds:

That is, if we assume that a certain claim is possible, by SIC, we can focus on the possible worlds and basically ignore all the impossible worlds. Thus, we can recapture the classical principle Closure with a classically innocuous suppressed premise $◊A$ .

We can see that ‘classical recapture’ requires SIC, which forces a nonvacuous account of counterpossibles to consider only the possible worlds unless the actual world is actually impossible. However, there are reasons to reject SIC. Most, if not all, reasons given for rejecting SIC in the literature are metaphysical in nature. For instance, consider the impossible omissive claim (Reference BernsteinBernstein (2016: 2580)):

Proving that $2+2=5$ is impossible.Footnote ⁵¹ However, in the context of proving a mathematical truth about $2+2$ , the impossible event of the mathematician proving that $2+2=5$ is more similar (or closer) to the actual event of proving that $2+2=4$ than the possible event of a random AFL (Aussie Rule) game in which the mathematician did not participate, especially if proving that $2+2=4$ is tantamount to failing to prove that $2+2=5$ .Footnote ⁵²

There are also reasons to reject SIC that are related to how to conceive of an impossible world. If impossibility (and possibility) is (are) a contingent matter, which world count as impossible can be answered only relative to a world (or a set of logical laws) and the answer cannot be absolute (in the sense of § 2.1). In order to evaluate a counterfactual, given a set of worlds, we might order them in terms of similarity. If impossibility is contingent (in the sense appealed to in this Element), the ordering does not determine which world is impossible as what counts as impossible is determined not necessarily in terms of the actual world but in terms of a world whether actual or not. In such a case, impossibility is not necessarily a matter of similarity to the actual world as the actual world does not play a privileged role in determining which world counts as impossible. So if impossibility is a contingent matter, SIC does not hold. When we consider variations of counterpossibles, we can see how this plays out. In the next few sections, we will consider counterlogicals, countermathematicals, countermetapossibles (of various kinds) and causal counterfactuals to examine other ways to understand the involvement of impossible worlds.

7.2 Counterlogicals

A counterlogical is a kind of a counterpossible whose antecedent involves logical impossibility. Examples used to illustrate counterlogicals are the following:

Both examples have contradictory antecedents which are taken to be logically impossible. They are taken to be counterlogicals under the assumption that classical logic holds at the actual world. But, why are the contradictory antecedents logically impossible?

While the literature is strangely silent on this question (though the literature on counterlogicals is rather small), there are two ways to understand the logical impossibility involved in the above counterlogicals. First, according to classical logic, the Law of Non-Contradiction (LNC) ( $\models \neg (A\wedge \neg A)$ ) holds. If Saul Kripke is and is not human, then there is an instance of $A$ , say $p$ , such that $p\wedge \neg p$ is true (and non-true). So $\neg (p\wedge \neg p)$ is non-true. Hence, there is a counterexample to LNC at a world where the antecedent is true. In this way, the antecedents of the above counterlogicals are impossible in the sense that they violate a law of classical logic.Footnote ⁵³

Second, one might think that if there is a counterexample to LNC, LNC must be invalid. So, in order to evaluate a counterlogical nonvacuously, we might invoke a world where a different set of logical laws hold. If a paraconsistent logic holds at such a world, the consequent of a counterlogical may not hold there. For instance, if Sylvan’s Box is and is not empty at a world, it may not be that someone who finds it makes a space ship and sends it into orbit, as Reference PriestPriest (1997b) urges us to think, in which case the counterlogical might be thought to be not true. If this is the way to understand a counterlogical, then it is a logically different world that is specified in the antecedents.

If we are to think that the impossibility involved in counterlogicals is (logical) difference, a counterlogical seems to become not a species of counterpossibles specifically but of counterfactuals. Counterfactuals have antecedents which may not hold at the actual world. But counterfactuals needn’t have impossible antecedents. To identify a contradictory world as a logically different world is to consider a set of logical laws that may not hold at the actual world. But, just like the case of a counterfactual, there is no reason to think that any impossibility plays a role. So, if we understand a counterlogical to involve logically different worlds, there does not need to be anything counterpossible about counterlogicals.

7.3 Countermathematicals

While counterlogicals have not received much attention, their cousins countermathematicals have, at least more attention than counterlogicals. An example of countermathematicals is the following:

There are two sub-species of North American periodical cicadas. One has a life cycle of 13 years and the other has a life cycle of 17 years (Reference BakerBaker (2005)). By appealing to the countermathematicals such as above, it has been argued that the mathematical facts about 13 and 17 being prime numbers (partly) explain the lengths of their life cycles. Countermathematicals such as above have been used in such arguments.Footnote ⁵⁴

Our interest here is not with the issue of mathematical explanation that countermathematicals are put to use. Instead, our focus is on the countermathematicals that are made use of in the context of discussing the notion of explanation. There are at least two ways to analyse countermathematicals that do not seem to have been widely recognised in the relevant literature. We will present an analysis of countermathematicals and leave their import to the field of explanation for others to work out.

It has been proposed that there are three steps involved in evaluating a countermathematical (Reference BaronBaron (2020: § 3), Reference Baron, Colyvan and RipleyBaron, Colyvan, & Ripley ((2017: § 2), (Reference Baron, Colyvan and Ripley2020: § 2))):

1. 1. Holding fixed: Fix some set of facts by choosing them to be invariant under counterfactual variation.

2. 2. Twiddling: Twiddle mathematics to make the antecedent of a countermathematical true.

3. 3. Ramifying: Consider the ramification of twiddling the facts that are not fixed.

To evaluate the above countermathematical, we fix, or try to fix, as much of standard number theory as possible.Footnote ⁵⁵ We then twiddle the number 13 by assigning it the factors 2 and 6. One way to achieve this is to twiddle the multiplication function for 13. For instance, consider multiplication ${}^{*}$ that is just like the standard multiplication except that 2 multiply ${}^{*}$ 6 equals 13. We then examine the ramifications of this twiddle in the consequent of the countermathematical. Given that the consequent is about the life cycles of cicadas, this examination involves dealing with a physical or biological structure. This is where the issue of mathematical explanation becomes crucial.Footnote ⁵⁶ For our purpose, however, we only need to understand how to twiddle 13 while holding the rest (as much as possible) of number theory fixed. It is here that impossible worlds become relevant since one might think that it is impossible for 13 not to be coprime with 2 and 3.

Impossibility involved in countermathematicals is just like the impossibility involved in counterlogicals. In a counterlogical, it is assumed that classical logic holds at the actual world. If the antecedent of a counterlogical involves a contradiction that is not accommodated by classical logic, it is considered to be impossible. Similarly, if the antecedent of a countermathematical involves a piece of mathematics that is not accommodated by the standard mathematics (or arithmetic in this case), it is considered to be impossible.

Thus, there are two ways to analyse the involvement of impossibility in evaluating a countermathematical. First, when we twiddle multiplication function, we might think of this twiddling to involve the change of the mathematical principle(s) governing multiplication for 13.Footnote ⁵⁷ Given that we are holding the rest of number theory fixed, $2\times 6=12$ . But that contradicts the output of multiplication ${}^{*}$ for 2 and 6 as $2{\times }^{*}6=13$ . In order to deal with this contradiction (and other contradictions), we might think of a world where the antecedent of the countermathematical holds to be a world where a paraconsistent mathematics holds.Footnote ⁵⁸ Given that a paraconsistent mathematics may behave just like classical mathematics in a consistent context,Footnote ⁵⁹ we can simply allow those contradictions that arise explicitly from the particular twiddling of the mathematics in question but remain business as usual in doing (the rest of) mathematics.

Now, this way of twiddling involves worlds where a parconsistent mathematics holds. In evaluating our countermathematical, we can then consider a world where the antecedent holds to be a world that holds a different set of mathematical principles (or rules). In this case, we are holding fixed all of the mathematical facts except some facts about 13 but varying some of the principles. This means that we are invoking a mathematically different world to evaluate the countermathematical. If we think of a mathematically different world to be an impossible world just like a logically different world to be an impossible world, the kind of impossibility involved here ties impossibility to difference.

However, this is not the only way to understand twiddling of mathematics. So, second, we can fix mathematical principles but vary the facts about the factors of 13. We can do this by stipulating the fact about $2\times 6$ to be $13$ against the principles (or rules) for the standard multiplication function according to which $2\times 6=12$ .Footnote ⁶⁰ In this case, we are allowing some mathematical facts to violate some of the mathematical principles. For instance, if we think of Peano Arithmetic, then the fact about the primeness of 13 under this assignment serves as a counterexample to the axioms of Peano Arithmetic that concern multiplication. So, in evaluating the countermathematical, we can think of invoking a world where mathematical principles or laws are violated in order to consider a situation where the antecedent holds.

It is when we consider impossibility in terms of violation that we can iron out the unnecessary contradictions that might arise in evaluating a countermathematical. To see this, consider the countermathematical: if $1+1̸=2$ , then $2+1̸=3$ .Footnote ⁶¹ In evaluating this, we have to suppose that there is a world where $1+1̸=2$ . But if $1+1̸=2$ , then, presumably, $1+1+2̸=4$ . This contradicts our standard arithmetic according to which $1+1+2=4$ . In order to manage the proliferation of contradictions that the impossibility of $1+1̸=2$ entails, we can stipulate that the standard arithmetical principles have a counterexample according to which the fact about $1+1+2$ is not $4$ . Since we are managing facts and not principles, no systematic adjustment is required. In this way, ‘contradictions [can be] cleared from the neighbourhood of the [countermathematical] being evaluated’ (Reference BaronBaron (2020: 542)). This clearing of contradictions is possible only if the impossibility of $1+1̸=2$ is understood as a violation of the standard arithmetic.Footnote ⁶²

7.4 Countermetafactuals, Countermetalogicals, and Countermetapossibles

Even after going through some forms of counterpossibles, counterlogicals and countermathematicals that are evaluated nonvacuously by making use of impossible worlds, one may not be persuaded to add impossible worlds in addition to possible worlds in order to evaluate the counterpossibles.Footnote ⁶³ Moreover, one may insist that there is nothing wrong with evaluating all counterpossibles to be vacuously true. It might be thought that those who argue for nonvacuous evaluation of counterpossibles rely on their ‘intuitions’ to evaluate the truth values of various counterpossibles and use those intuitions to argue what the truth values of counterpossibles should be. But if one does not share the same intuitions, there may not be any reason to develop a method of nonvacuously evaluating counterpossibles (Reference WilliamsonT. Williamson (2020, Reference Williamson2021)).Footnote ⁶⁴

However, regardless of what intuition one might have about various counterpossibles, there is good reason to think that not all counterpossibles should vacuously come out true. Consider the following conditionals:

1. (3) If Classical Logic were the correct logic, it would be impossible to validly infer every proposition from a contradiction.

2. (4) If LP were the correct logic, it would be impossible to validly infer every proposition from a contradiction.

These conditionals are crucial in logical disputes in which we debate about what the correct logic is or what logic holds at the actual world. If the dispute is genuine and the disputants are not talking past each other, it must be understood what it would be for those conditionals to hold. This is the case even if one is convinced that classical logic or LP is not the correct logic.Footnote ⁶⁵

From a classical perspective, it may not be impossible for LP to hold at the actual world. As we have seen before, LP is a sublogic of classical logic. So the theorems of LP form a subset of the theorems of classical logic. From a classical perspective, then, LP may be non-true since it is deficient in accounting all of the valid inferences but it is not necessarily an impossibility. However, from a perspective of LP, it is impossible for ECQ to be valid as LP invalidates it. So, from LP’s perspective, it is impossible for classical logic to hold at the actual world. From LP’s perspective, then, the antecedent of (3) is an impossibility. However, every proposition can be classically inferred from a contradiction. So, (3) should be non-true even if the antecedent may be impossible. Even though this reasoning is conducted from LP’s perspective, it is a perfectly legitimate reasoning from a classical perspective as LP is a sublogic of classical logic and so the reasoning does not involve any invalid moves from a classical perspective. Hence, even classically, not all counterpossibles should be declared to be vacuously true.

Now, the above conditionals are examples of what we call countermetalogicals (Reference Sandgren and TanakaSandgren & Tanaka (2020)).Footnote ⁶⁶ Countermetalogicals are a special kind of counterlogicals. The difference between counterlogicals and countermetalogicals turns on how logical impossibility is involved. A countermetalogical like (3) and (4) is true in case the following holds: if the logic mentioned in the antecedent holds at the actual world, then there would be no worlds, most-similar to the actual world, in which we could validly infer a certain inference mentioned in the consequent. So an evaluation of a countermetalogical requires two steps. For instance, in evaluating (3), first, we suppose that the actual laws of logic (the laws of logic that hold at the actual world) are classical. This requires a suspension of our prejudgement about what logic holds at the actual world as we are only supposing what logic holds at the actual world. Based on this supposition, we order all the worlds in terms of ‘similarity’. Then, second, we evaluate whether or not it is impossible to validly infer every proposition from a contradiction at the worlds most-similar to the actual world. Given that a countermetalogical is concerned with the correct laws of logic (the logical laws that hold at the actual world), the worlds most similar to the actual world are classical worlds in considering (3). Since ECQ is classically valid, it cannot be impossible to validly infer every proposition from a contradiction at classical worlds. Hence, (3) is non-true.

Analysed in this way, we can see that an evaluation of a countermetalogical invokes two kinds of impossibility. (i) In identifying the most-similar worlds to be classical worlds, we presuppose that there are various logically different worlds. So if it is impossible to validly infer every proposition from a contradiction, then that would be a world logically different from the actual world. (ii) But if it is impossible to validly infer every proposition from a contradiction at the most-similar classical worlds to the actual world, that must be because those worlds go against the actual laws of logic. So the impossibility embedded in the consequent appeals to logical violations. Thus, to make sense of countermetalogicals, we need to make room for two kinds of impossibility.

In contrast, an evaluation of a counterlogical does not require these two steps. In a counterlogical such as

the antecedent concerns a matter of fact. If an impossibility is involved, it is the fact expressed in the antecedent that violates the actual law. So a counterlogical invokes only the sense of impossibility tied to violation.Footnote ⁶⁷

7.5 Causal Counterfactuals

The last type of the family of counterfactuals/counterpossibles we consider is somewhat metaphysical. It concerns causal matters, in particular the connection between causation and counterfactuals under the assumption that determinism holds.Footnote ⁶⁸ There is what Reference Nolan, Beebee, Hitchcock and PriceNolan (2017) calls the deviation problem in considering counterfactuals whose antecedent represents or expresses a state that deviates from the actual state and whose consequent represents or expresses a state that obtains later than the antecedent state. It is a problem associated with answering the question: what differences do we need to accommodate in evaluating such causal counterfactuals? For instance, consider the following counterfactual:

If we are to evaluate it in terms of the closest worlds where the antecedent is true just like the standard semantics for counterfactuals in the style of Kratzer/Lewis/Stalnaker, what are the closest worlds be like where I skipped breakfast? Beside skipping breakfast, what other deviations should there be?

There are generally two ways to understand the required deviations. First, we can understand them in terms of different facts (though as close to the actual facts as possible) while maintaining the same laws. Second, we can understand the required deviations in terms of different laws of nature (at least those that pertain to causation) while fixing the (contingent) facts as close to those of the actual world as possible. The first approach involves what Reference Nolan, Beebee, Hitchcock and PriceNolan (2017) calls impossible worlds. The second approach gives rise to miracles (Reference LewisLewis (1979, Reference Lewis1981)). (No wonder no one seems happy with any proposal.) In this section, we will provide an analysis of these two approaches by appealing to different ways of understanding impossible worlds as we saw in Sections 3–6 (though we do not even attempt to settle the issue about the connection between counterfactuals and causation). We will see that both approaches are essentially impossible worlds approaches but they appeal to different kinds of impossible worlds.

Reference Nolan, Beebee, Hitchcock and PriceNolan (2017) introduces the impossible world approach to account for the closest worlds where the antecedent holds differently from the miracle approach. He specifies four conditions on the closest worlds that are relevant to evaluating the causal counterfactuals such as above. Let $w$ be a world where a causal counterfactual is true and $w^{*}$ a closest world where the antecedent is true. Then, (1) the laws of nature (or the laws of physics) that hold at $w$ also hold at $w^{*}$ , (2) all of the truths that hold before time $t$ at $w$ are also the truths that hold before $t$ at $w^{*}$ , (3) there is no proposition at $w^{*}$ that expresses that the laws of nature are violated (Reference Nolan, Beebee, Hitchcock and PriceNolan (2017: 26)). It is also assumed (4) that the facts that hold after $t$ at $w^{*}$ are constrained or governed by the laws of nature under consideration. (2) ensures that all of the truths up to $t$ hold at $w^{*}$ if they hold at $w$ . So we have worlds that have the same history up to $t$ . (3) implies that $w^{*}$ does not ‘say of itself’ that it has counterexamples. This is the case even if there are, in fact, counterexamples.Footnote ⁶⁹

Now, (1) together with (4) implies that the laws of nature (or the laws of physics) at $w^{*}$ are the same as those that hold at $w$ . This means that $w$ and $w^{*}$ are not physically different worlds as the same laws hold at both of those worlds. Nevertheless, $w^{*}$ may be an impossible world as it may contain a counterexample to the laws. For Nolan, this means that $w^{*}$ is not closed under logical consequence. In other words, it is an open world. He does not say what exactly he means by an open world. But, an open world in this sense must be a silent world (where the world is silent about the consequence relation between some facts) or an anarchic world (where the facts may behave against the laws) as some laws of nature do hold at the world.

Crucially, since $w^{*}$ has the same laws of nature as $w$ , no physically different worlds (worlds that have different laws of nature) are involved. And, because it is an open world, the mismatch between some (past) facts and the laws may not trivialise the world even when these worlds are understood from a perspective of classical logic. The advantage of this approach is that it does not involve miracles to which we now turn.

To examine the miracle approach, consider two worlds $w_0$ and $w_1$ that are exactly the same until shortly before time $t$ . That is, $w_0$ and $w_1$ have exactly the same facts until shortly before $t$ . However, shortly before $t$ , an event takes place that violates the laws of nature. This event is said to be a miracle or a ‘tiny miracle’ (Reference LewisD. Lewis (1979: 468)) because ‘whatever else a law may be, it is at least an exceptionless regularity’ (Reference LewisD. Lewis (1979: 468–469)). After the miracle, $w_0$ and $w_1$ diverge and some event may take place in $w_1$ after $t$ that does not take place in $w_0$ .

Now, Lewis explains that a miracle that takes place in $w_1$ is a violation of the laws of nature that hold at $w_0$ . This means that a miracle in $w_1$ violates the laws of nature in $w_0$ and, thus, it is a miracle relative to $w_0$ . Crucially, he claims that the laws at $w_0$ are ‘at best the almost-laws of $w_1$ ’ and that ‘[t]he laws of $w_1$ itself, if such there be, do not enter into it’ (Reference LewisD. Lewis (1979: 469)). This suggests that it is not only that different facts obtain at $w_1$ after $t$ but that different laws hold at $t$ (and, perhaps, after $t$ ). So, $w_0$ and $w_1$ are different with respect to the laws.Footnote ⁷⁰ Thus, the miracle approach is an impossible world approach given our discussion of impossible worlds in Sections 3–6. However, it invokes a different kind of impossible worlds from the impossible world approach of Nolan discussed above. A miracle world is different (from another world such as the actual world) not only with respect to factual matters but also with respect to the laws. It makes use of the notion of difference in addition to the notion of violation.