Actual Issues for Relevant Logics

Abstract

In this paper, I motivate the addition of an actuality operator to relevant logics. Straightforward ways of doing this are in tension with standard motivations for relevant logics, but I show how to add the operator in a way that permits one to maintain the intuitions behind relevant logics. I close by exploring some of the philosophical consequences of the addition.

It is common enough in work on relevant logics using Routley-Meyer frames to talk about a real world. This is naturally construed as the formal representative of the actual world. It is a natural step, though so far untaken, to add an actuality operator, \(\mathbb{A}\), to provide a way to incorporate this idea into the object language for relevant logics. Some straightforward ways of doing this are, surprisingly, incompatible with the guiding intuitions behind relevant logics. In this paper, I will explore options for adding an actuality operator to relevant logics and some of the consequences of doing so.

In addition to the inherent logical interest of adding an actuality operator to relevant logics, there are some philosophical consequences of this work that are worth noting. There is much work on two-dimensional semantics and metaphysics developed against the background of classical modal logic. If one wants to explore the idea that a relevant logic can be one’s general purpose logic, then there is a natural opening for developing non-classical semantic and metaphysical theories. Yet, without the groundwork developed here, one cannot begin to pursue two-dimensional semantics and metaphysics against a relevant logical background. As we will see in the final section of this paper, there are substantive questions about how to develop such theories. Even if one is not a proponent of relevant logics, one might be interested in the development of non-classical metaphysics for the distinctions that it draws, which may pass unnoticed in the more familiar classical setting.

The picture of actuality that results in the context of relevant logics differs from the more familiar, classical two-dimensional picture in a number of ways. First, while in the classical setting of \(\sf{S5}\) there is a natural candidate for the logic of actuality, no similarly overwhelmingly natural candidate emerges in the relevant setting. There are some plausible candidate logics, but choosing among them requires further work. Second, it will emerge that the natural view of actuality for the relevant logician, and the relevance-inclined metaphysician, does not fit neatly with either an indexical or an absolute view of actuality. Finally, it raises some issues for actualism in ontology, as it complicates the notion of being an actual object.

The paper will begin with a brief introduction to relevant logics (§1). In §2, I will motivate the addition of an actuality operator, \(\mathbb{A}\), to relevant logics. In §3, I will argue against defining an actuality operator out of the resources available in standard relevant logics. Then I will examine two natural ways of interpreting actuality on Routley-Meyer frames and argue that they are not satisfactory (§4). In §5, I will propose a new way to interpret the actuality operator that fits with relevant logics and discuss options for strengthening the basic logic. Following that, I will discuss some of the philosophical issues that arise from my proposal (§6). Finally, in the Appendix I will prove some of the technical results that arise in the course of the discussion.

1. Introduction to Relevant Logics

In this section, I will provide the basic background on relevant logics, their frame semantics, and axiom systems. Relevant logics are non-classical logics requiring a strong connection between the antecedent and consequent of valid conditionals.[1] Part of the guiding intuition of relevant logics, at least in the early development by Anderson and Belnap (1975), was that the antecedent and consequent of a valid conditional had to be relevant to each other, in the sense that one had to really use the antecedent in obtaining consequent.[2] While the exact demarcation of the connection was left somewhat open, a necessary condition on the connection is that the two formulas share a propositional variable.[3] Both \(\mathrm{p}\rightarrow(\mathrm{q}\rightarrow \mathrm{p})\) and \(\mathrm{p}\rightarrow(\mathrm{q}\rightarrow \mathrm{q})\) exemplify the sort of irrelevance rejected by relevant logics; in the former, \(\mathrm{q}\) appears to be idle and in the latter, \(\mathrm{p}\) has nothing much to do with \(\mathrm{q}\rightarrow \mathrm{q}\). Relatedly, implications in relevant logics are stricter than the material conditional, defined out of negation and disjunction, and so principles that are “too material” will also be rejected, such as \((\mathrm{p}\land \mathrm{q})\rightarrow(\mathrm{p}\rightarrow \mathrm{q})\).[4] The fact that \(\mathrm{p}\) and \(\mathrm{q}\) are true together, accidentally as it were, falls short of establishing that \(\mathrm{p}\) implies \(\mathrm{q}\), in the sense of \(\mathrm{p}\rightarrow \mathrm{q}\). These guiding intuitions will play a role in the later sections of this paper.

One of the main features of relevant logics is that they reject the axiom of Weakening, \(\mathrm{A}\rightarrow(\mathrm{B}\rightarrow \mathrm{A})\), and its kin.[5] For the purposes of this paper, a relevant logic is a logic between the logics \(\sf{B}\) and \(\sf{R}\), which will be defined shortly. Much of the focus of the paper will be on the Routley-Meyer frame semantics for relevant logics, although I will provide axioms for some logics under consideration below.[6] The basic logics will be in the vocabulary \(\{\rightarrow, \land, \sim , \sf{t}\}\), with formulas built out of these connectives and a countable set of atoms, \(\sf{At}\). I will adopt the usual definitions for disjunction, \(\mathrm{A\lor \mathrm{B}\mathbin{=_{Df}}\sim\! \! (\sim\! \! \mathrm{A}\; \land\sim\! \! \mathrm{B})}\), and the biconditional, \(\mathrm{\mathrm{A}\leftrightarrow \mathrm{B}\mathbin{=_{Df}}(\mathrm{A}\rightarrow \mathrm{B})\land(\mathrm{B}\rightarrow \mathrm{A})}\). To cut down on parentheses, unary connectives will bind tightest, followed by conjunction and disjunction, which bind more tightly than the conditional and biconditional. The language will be extended with a unary operator, \(\mathbb{A}\), later.

2. Actuality

As mentioned at the outset, it is not hard to find examples, in the relevant logic literature, of people talking about the real, or actual, world in Routley-Meyer frames.[15] This is particularly the case in early work on Routley-Meyer frame semantics, where the frames were slightly different. An RMg-frame is a tuple \(\langle{\mathrm{K}},{\mathrm{N}},{\mathrm{g}},{^*},{\mathrm{R}}\rangle\) where \(\langle{\mathrm{K}},{\mathrm{N}},{^*},{\mathrm{R}}\rangle\) is a Routley-Meyer frame and \(\mathrm{g}\in \mathrm{N}\).[16] Let us look at some quotations, in which emphasis has been added. We begin at the beginning, with the early work of Routley and Meyer.

Since the introduction of Routley-Meyer models, there have been questions about the philosophical interpretation of the models. As indicated by the preceding quotations, in their early papers, Routley and Meyer talked about the distinguished world as the real world. This is naturally taken to be a candidate for representing the actual world, although they do not put it in quite those terms, and they did not add an operator to refer to that world in the object language. In slightly later work with Plumwood and Brady, they call the distinguished world “factual”, again suggesting it be taken to represent the actual world.[17]

It was not just Routley and Meyer who talked about the real world of the models. Other authors, such as Slaney and Giambrone, did as well. Slaney and Giambrone are discussing reduced models, which require a bit of explanation. In this paper, I have adopted a form of Routley-Meyer model that has became standard, one that has a set of normal worlds, \(\mathrm{N}\), rather than just a single normal world, \(\mathrm{g}\). Reduced models are those which adhere more closely to the original presentation of Routley-Meyer models, using singleton sets of normal worlds rather than a potentially larger set of normal worlds. Truth in a model is defined with respect to the single normal world of the model. These models have some technical niceties. As indicated by the following quotations, there is also a philosophical viewpoint that seems allied to the reduced models. On this view, the single normal world of each model represents the real world, with the rest of the worlds doing the semantic work of ensuring that the conditionals are properly relevant.

Giambrone (1992) responds to Slaney (1987), but there is no disavowal of the idea that the distinguished world is the real one. Indeed, Giambrone reiterates the point.

The final quotation, from Fuhrmann, is particularly of interest because he had, at the time of publication, recently finished his PhD thesis at ANU, which was then a hotbed of relevant logic activity.[18] I take it that the intuition he describes is one that was in the air at the time.

These quotations, taken together, give an indication that the idea that the distinguished world does, or at least can, represent the actual world was relatively common in work on relevant logics.[19] The idea that a model can contain a distinguished world that represents the actual world has been studied in the context of classical modal logic.[20] There are no formal problems with talking about actuality, via a unary operator, \(\mathbb{A}\), in classical modal logic.[21] The logic for actuality, \(\sf{S5A}\), of Crossley and Humberstone (1977) adds the following axioms and rule to classical \(\sf{S5}\), with the rule \(\sf{(Nec)}\), from \(\mathrm{A}\) to infer \(\Box \mathrm{A}\).

• \(\mathbb{A}(\mathbb{A} \mathrm{A}\supset \mathrm{A})\)[22]
• \(\mathbb{A}(\mathrm{A}\supset \mathrm{B})\supset(\mathbb{A} \mathrm{A}\supset \mathbb{A} \mathrm{B})\)
• \(\mathbb{A} \mathrm{A}\equiv \sim\! \! \mathbb{A}\sim\! \! \mathrm{A}\)
• \(\Box \mathrm{A}\supset\mathbb{A} \mathrm{A}\)
• \(\mathbb{A} \mathrm{A}\supset \Box\mathbb{A} \mathrm{A}\)

The frames for this logic add a distinguished world, \(\mathrm{g}\), to a Kripke frame with a universal accessibility relation. The axioms for \(\mathbb{A}\) are enough to ensure that the operator commutes with the truth-functional connectives. This can be reproduced in the relevant logical context, although there appear to be some options as to what the logic should be, which I will explore in §5 and §6.2.

In addition to there being no formal barrier to studying actuality in classical modal logic, there are many benefits from having such an operator.[23] The addition of actuality to classical modal logics connects to the rich field of two-dimensional semantics.[24] Apart from the attempt to formalize some of the ideas from the motivating quotations above, there would be value in taking steps towards non-classical, and specifically relevant, theories of actuality and two-dimensional semantics.[25]

The question I will explore for the remainder of the paper is how one can make good on the idea of talking about actuality, formally, through the addition of an actuality operator to relevant logics while maintaining something of the relevance intuitions that many find attractive about relevant logics. I will consider three options before settling on the preferred approach of this paper. For the remainder of the paper, the language will include a unary operator, \(\mathbb{A}\).

3. Why Not the Ackermann Constant?

The first option eschews the addition of anything new to the familiar frames in favor of trying to define actuality in terms of the Ackermann constant, \(\sf{t}\). The intuitive interpretation of \(\sf{t}\) is the conjunction of all logical truths (Anderson & Belnap 1975: 342). The interpretation of \(\sf{t}\) in a model is the set \(\mathrm{N}\) of normal worlds, worlds at which the truths of logic hold. Plausibly, the actual world is normal in that sense, so one might think that there could be a connection between an actuality operator and \(\sf{t}\). This approach would offer one main benefit over the other approaches to be considered. That benefit is that it does not require the addition of new logical machinery, instead using the rich resources of relevant logics to define the new operator. One would simply adopt an axiom, \(\mathbb{A} \mathrm{A}\leftrightarrow \phi(\mathrm{A})\), where \(\phi\) is some propositional context involving \(\sf{t}\). At least, that would be the benefit, were this approach to work out.

There are two natural candidates for defining \(\mathbb{A}\): \(\sf{t}\land\underline{\phantom{\mathrm{A}}}\) and \(\sf{t}\rightarrow\underline{\phantom{\mathrm{A}}}\). I will take these in turn. The conjunctive version has some apparently unpalatableconsequences. For example, it has the consequence that \(\mathbb{A} \mathrm{p}\rightarrow (\mathrm{q}\rightarrow \mathrm{q})\) in the class of all RM-frames. It is perfectly understandable and welcome that \(\sf{t}\land \mathrm{p}\rightarrow (\mathrm{q}\rightarrow \mathrm{q})\) is valid, as \(\sf{t}\land \mathrm{p}\rightarrow \sf{t}\) and \(\sf{t}\rightarrow (\mathrm{q}\rightarrow \mathrm{q})\) are both theorems of \(\sf{B}\). It is, however, intuitively wrong that it actually being the case that \(\mathrm{p}\) should imply, in the strong sense of the relevant conditional, any old logical truth, such as \(\mathrm{q}\rightarrow \mathrm{q}\). This greatly strains—if not outright violates—relevance intuitions. After all, \(\mathrm{p}\) does not imply \(\mathrm{q}\rightarrow \mathrm{q}\), and the addition of `actually' does not seem to establish the appropriate sort of connection between \(\mathrm{p}\) and \(\mathrm{q}\rightarrow \mathrm{q}\). I will, then, set the conjunctive option aside.

The conditional form, taking \(\mathbb{A} \mathrm{A}\) to be \(\sf{t}\rightarrow \mathrm{A}\), does better on the relevance intuitions front, but the philosophical interpretation is off. The Ackermann constant, \(\sf{t}\), is the minimal logical truth, the conjunction of all logical truths for the particular logic under consideration. As a definition of actuality, being implied by pure logic seems dubious. Indeed, this just seems to be the wrong approach to contingent actual facts. Such contingencies might true along with, but not be implied by the truths of logic, as the negations of those contingencies should also, at least potentially, be true along with the truths of logic, without being implied by them. Adapting a phrase from Slaney, such contingencies seem not to be logic’s business.

These points remain at the level of intuition, but the intuitions against the conditional definition of actuality can be strengthened. There are several principles that one may want to adopt to strengthen the logic of actuality, such as \(\sim\! \! \mathbb{A} \mathrm{A}\leftrightarrow \mathbb{A}\sim\! \! \mathrm{A}\) or \(\mathbb{A}(\mathrm{A}\lor \mathrm{B})\leftrightarrow \mathbb{A} \mathrm{A}\lor \mathbb{A} \mathrm{B}\). When unpacked using the definition of '\(\mathbb{A}\)', we get \(\sim\! \! (\sf{t}\rightarrow \mathrm{A})\leftrightarrow (\sf{t}\rightarrow \sim\! \! \mathrm{A})\) and \((\sf{t}\rightarrow (\mathrm{A}\lor \mathrm{B}))\leftrightarrow (\sf{t}\rightarrow \mathrm{A})\lor(\sf{t}\rightarrow \mathrm{B})\), respectively. The latter will be in inappropriate in any logic, such as \(\sf{T}\), in which excluded middle is a theorem but \((\sf{t}\rightarrow \mathrm{A})\leftrightarrow \mathrm{A}\) is not, on pain of \((\sf{t}\rightarrow \mathrm{B})\lor(\sf{t}\rightarrow \sim\! \! \mathrm{B})\) being valid for every formula, where \(\sf{t}\) need not imply either \(\mathrm{B}\) or \(\sim\! \! \mathrm{B}\), for contingent \(\mathrm{B}\). Even in weaker logics, these principles are, at best, unusual for the operator “implied by logic” whereas they are fine for the logic of actuality. The interpretation of \(\sf{t}\) is tied to the overall logic, while there are additional axioms that are plausible for governing just actuality.

When \((\sf{t}\rightarrow \mathrm{A})\leftrightarrow \mathrm{A}\) is a theorem of the logic, as is the case with \(\sf{R}\), the distinction between \(\mathbb{A} \mathrm{A}\) and \(\mathrm{A}\) collapses, as a matter of logic. Yet, it seems plausible to maintain that logic should not collapse that distinction, that logic should deal with what is sometimes called general validity, as opposed to real-world validity.[26] At the very least, which of the two forms of validity we opt for should be a choice. So, I conclude that the proposed conditional definition, \(\sf{t}\rightarrow\underline{\phantom{\mathrm{A}}}\), is out as a definition of “actually”.[27]

The objections to both options share a common core: The options are both bad things to mean by “actually”. Both options define operators that may be interesting in their own right. For example, the implicational definition provides a kind of logical necessity, along the lines discussed by Anderson and Belnap (1975: Ch.1). Neither option, however, provides a plausible, attractive thing to mean by “actually”, although their faults are different. Let us then continue to the next candidate.

4. Distinguished World

The second approach to adding actuality is to straightforwardly adapt the double indexing approach to actuality in classical modal logic as found in the RMg-frames. Evaluation of a formula will be done at a pair of points, \(\langle\mathrm{g},\mathrm{a}\rangle\), where \(\mathrm{g}\) is the distinguished world of the frame, which I will assume is in \(\mathrm{N}\). The truth conditions for actuality on this approach are as follows.

• \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{A}}\) iff \(\mathrm{g},{\mathrm{g}}\Vdash{\mathrm{A}}\)

Double indexing is adopted here to facilitate some later discussion and connections to other work in two-dimensional semantics. For the point of the present section, one could keep single indexing, with the actual point \(\mathrm{g}\) distinguished by the frame, and adapt the truth condition to use only the righthand point.

One needs to check that the Heredity Lemma holds in the extended language. In particular, let us check the new case, for formulas of the form \(\mathbb{A} \mathrm{B}\).

Proof. Suppose \(\mathrm{a}\leq \mathrm{b}\) and \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{B}}\). So, \({\mathrm{g}},{\mathrm{g}}\Vdash{\mathrm{B}}\). Therefore, \({\mathrm{g}},{\mathrm{b}}\Vdash{\mathbb{A} \mathrm{B}}\).

\(\Box\)

The lemma, then, holds for the extended language, but the short proof above should raise a red flag: The heredity assumption was not needed. This suggests that irrelevancy is afoot. Let us make good on that suggestion.

Lemma 3 (Ubiquity Lemma). For all RMg-models, for all \(\mathrm{a},\mathrm{b}\in \mathrm{K}\), if \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{A}}\), then \({\mathrm{g}},{\mathrm{b}}\Vdash{\mathbb{A} \mathrm{A}}\).

Proof. Let \(\mathrm{M}\) be a RMg-model and suppose \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{A}}\). This is the case iff \({\mathrm{g}},{\mathrm{g}}\Vdash{\mathrm{A}}\), which is the case iff \({\mathrm{g}},{\mathrm{b}}\Vdash{\mathbb{A} \mathrm{A}}\), as desired.

\(\Box\)

The red flags have now been joined by warning klaxons. This leads, as one might expect, to many disastrous consequences, of which I will highlight four.

Proposition 1. In the class of all RMg-frames, the following are valid.

• \(\mathrm{q}\rightarrow \mathbb{A}(\mathrm{p}\rightarrow \mathrm{p})\)
• \(\mathbb{A} \mathrm{p}\rightarrow (\mathrm{q}\rightarrow\mathbb{A} \mathrm{p})\)
• \(\mathbb{A}(\mathrm{p}\; \land \mathrm{q})\rightarrow (\mathbb{A} \mathrm{p} \leftrightarrow \mathbb{A} \mathrm{q})\)
• \(\mathbb{A} \mathrm{p}\; \land \sim\! \! \mathbb{A} \mathrm{p}\rightarrow \mathrm{q}\)

Proof. Let us begin with the first. Take a model on an arbitrary frame and suppose \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathrm{q}}\). As \(\mathrm{g}\in \mathrm{N}\), \({\mathrm{g}},{\mathrm{g}}\Vdash{\mathrm{p}\rightarrow \mathrm{p}}\), so \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A}(\mathrm{p}\rightarrow \mathrm{p})}\). Therefore, \({\mathrm{q}\rightarrow \mathbb{A}(\mathrm{p}\rightarrow \mathrm{p})}\) is valid.

For the second, take a model on an arbitrary frame, and suppose \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{p}}\). We want to show \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathrm{q}\rightarrow \mathbb{A} \mathrm{p}}\). Suppose \(\mathrm{b}\) and \(\mathrm{c}\) are such that \(\mathrm{Rabc}\) and \({\mathrm{g}},{\mathrm{b}}\Vdash{\mathrm{q}}\). By lemma 3, \({\mathrm{g}},{\mathrm{c}}\Vdash{\mathbb{A} \mathrm{p}}\). Therefore, \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathrm{q}\rightarrow\mathbb{A} \mathrm{p}}\), which suffices for the claim.

For the third, suppose \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A}(\mathrm{p}\land \mathrm{q})}\). By simple argument, \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{p}}\) and \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{q}}\). Suppose \(\mathrm{b}\) and \(\mathrm{c}\) are such that \(\mathrm{Rabc}\) and \({\mathrm{g}},{\mathrm{b}}\Vdash{\mathbb{A} \mathrm{p}}\). By lemma 3, \({\mathrm{g}},{\mathrm{c}}\Vdash{\mathbb{A} \mathrm{q}}\), which suffices for \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{p}\rightarrow \mathbb{A} \mathrm{q}}\). The converse conditional is similar.

For the fourth, suppose it is not valid. Then there is a counterexample model with a point \(\mathrm{a}\) such that \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{p}\; \land \sim\! \! \mathbb{A} \mathrm{p}}\) but \({\mathrm{g}},{\mathrm{a}}\not\Vdash{\mathrm{q}}\). It follows that \({\mathrm{g}},{\mathrm{a}}\Vdash{\mathbb{A} \mathrm{p}}\), so by lemma 3, \({\mathrm{g}},{\mathrm{b}}\Vdash{\mathbb{A} \mathrm{p}}\), for all \(\mathrm{b}\in \mathrm{K}\). It also follows from the assumption that \({\mathrm{g}},{\mathrm{a}}\Vdash{\sim\! \! \mathbb{A} \mathrm{p}}\), so \(\mathrm{g},{\mathrm{a}^*}\not\Vdash{\mathbb{A} \mathrm{p}}\), which is a contradiction. Thus, there is no counterexample.

\(\Box\)

It is clear, then, that the simple double indexing approach flagrantly violates relevance intuitions. Simple double indexing does not provide the sort of semantic wiggle room needed for models to falsify irrelevant conditionals. Indeed, this sort of problem may arise when new logical vocabulary is added to the basic relevant logical language. For example, suppose an identity predicate were in the language, interpreted at each point as the set of pairs \(\langle \mathrm{o},\mathrm{o} \rangle\) for each object \(\mathrm{o}\) in the domain. Then, \(\mathrm{i}=\mathrm{i}\) would be valid, but so would \(\mathrm{p}\rightarrow \mathrm{i}=\mathrm{i}\), despite the lack of connection between the antecedent and consequent. Some wiggle room is needed in order to falsify logical truths using the new vocabulary so that no irrelevancy is introduced. Fuhrmann (1990: 509) calls a formula \(\mathrm{A}\) true at all points of \(\mathrm{M}\) ubiquitous in \(\mathrm{M}\), following a suggestion of Humberstone. With the introduction of new vocabulary, one needs to avoid thereby generating ubiquitous truths.[28] Simple double indexing introduces ubiquitous truths with abandon.

While a single point won’t cut it to interpret actuality, one might think that several points would. For a given frame, a set of points \(\mathrm{X}\) is hereditarily closed iff if \(\mathrm{a}\in \mathrm{X}\) and \(\mathrm{a}\leq \mathrm{b}\), then \(\mathrm{b}\in \mathrm{X}\). Sets that are hereditarily closed are propositions.[29] Rather than a single point representing the actual world, perhaps one should instead take a proposition to represent how things actually are, with the actuality operator acting as a universal quantifier over that proposition. This move does not, in the end, make a great difference, as many of the problems highlighted above for a distinguished point can be reproduced. Yet more flexibility is required to prevent formulas of the form \(\mathbb{A} \mathrm{B}\) from being ubiquitous, if true.

5. Differing Views on Actuality

While the distinguished subset approach does not deliver an advance on the problems with the RMg-frames, it does provide a nice stepping stone to the solution adopted in this paper. The previous two options took the interpretation of actuality to be globally constant. It is that requirement that is too rigid. Rather than having a distinguished world or proposition, instead, I will opt for using a binary accessibility relation, \(\mathrm{S}\), to pick out what each point takes to be actual.[30] The relation \(\mathrm{Sab}\) holds iff \(\mathrm{a}\) takes all actual truths to hold in \(\mathrm{b}\).

The interpretation of \(\mathrm{S}\) fits with an epistemic interpretation on the RM-frames, on which each point represents a body of information. The information may be incomplete and it may be inconsistent. Some, though perhaps not all, bodies of information talk about what is actually the case. Some of these points may be deeply, deeply confused about what is actual, but they can still take things to be actual, however wrong they turn out to be. It is this possibility, that different points can differ on what they take to be actual, that will provide the flexibility needed to break the disastrous ubiquity of actuality formulas of the previous section. In §6.1, I will return to the interpretation of the frames.

Where \(\mathrm{S}\subseteq \mathrm{K}\times \mathrm{K}\), I will use the following defined notation, following Meyer and Mares (1993).

• \(\mathrm{Sa\mathbin{=_{Df}} \{b\in K: Sab\}}\)
• \(\mathrm{R(Sa)(Sb)c\mathbin{=_{Df}} \exists u\exists v(Ruvc \land Sau\land Sbv)}\)
• \(\mathrm{S(Rab)c\mathbin{=_{Df}} \exists x(Rabx \land Sxc)}\)

Let us call an RM-frame augmented with a relation \(\mathrm{S}\subseteq \mathrm{K}\times \mathrm{K}\) an RMS-frame, provided it obeys the following constraint.

• \(\mathrm{a\leq b\Rightarrow Sb\subseteq Sa}\)

This constraint makes sense on the gloss given above for \(\mathrm{S}\), understanding \(\leq\) to be a kind of containment relation. It says that as the points say more, they may rule out options for actuality, which seems plausible.[31]

The truth condition for actuality appropriate for RMS-frames is the following.

• \(\mathrm{{a}\Vdash{\mathbb{A} A}}\) iff \(\mathrm{\forall y(Say\Rightarrow {y}\Vdash{A})}\)

Let us return again to the \(\mathbb{A} \mathrm{B}\) case of the proof of the Heredity Lemma.

Proof. Suppose \({\mathrm{a}}\Vdash{\mathbb{A} \mathrm{B}}\) and \(\mathrm{a}\leq \mathrm{b}\). Suppose \({\mathrm{b}}\not\Vdash{\mathbb{A} \mathrm{B}}\). So, there is some \(\mathrm{c}\) such that \(\mathrm{Sbc}\) and \({\mathrm{c}}\not\Vdash{\mathrm{B}}\). As \(\mathrm{a}\leq \mathrm{b}\), \(\mathrm{Sb\subseteq Sa}\), so \(\mathrm{Sac}\). But then, \({\mathrm{a}}\not\Vdash{\mathbb{A} \mathrm{B}}\), contradicting the assumption.

\(\Box\)

This time, the heredity assumption was used in a substantive way. This suggests that the blatant irrelevancies of the earlier sections will be avoided in the present setting. This idea will be borne out at the end of the Appendix.

The class of RMS-frames validates the following principles concerning actuality, in addition to the principles of the underlying logic.

• \(\sf{(\mathbb{A} \land)}\; \mathbb{A} \mathrm{A}\land\mathbb{A} \mathrm{B}\rightarrow \mathbb{A}(\mathrm{A}\land \mathrm{B})\)
• \(\sf{(RM)}\; \mathrm{A}\rightarrow \mathrm{B}\Rightarrow \mathbb{A} \mathrm{A}\rightarrow \mathbb{A} \mathrm{B}\)

All the logics under consideration entail the converse of \((\mathbb{A}\land)\), \((\mathbb{A}\land \sf{c} \mathrm{)}\), via \(\sf{(RM)}\), \(\sf{(A2)}\), and \(\sf{(R2)}\). I will use the convention that \(\sf{L.M}\) will be the logic of actuality that adds to the axioms and rules of the underlying logic \(\sf{L}\) both \(\sf{(\mathbb{A}\land )}\) and \(\sf{(RM)}\), and \(\sf{L.MX}\) will be \(\sf{L.M}\) with the addition of the principles in \(\sf{X}\).[32] Table 1 lists some of the better studied principles that may be appropriate for actuality with their frame conditions.[33]

Table 1. Familiar axioms and their frame conditions

There are some additional principles one might want to adopt whose frame conditions for RMS-frames are not well studied. To make negation fully commute with the actuality operator, we need \(\sf{(Dc)} \sim\! \! \mathbb{A} \mathrm{A}\rightarrow \mathbb{A} \sim\! \! \mathrm{A}\), whose frame condition is

\( \sf{(FDc)}\; \forall \mathrm{x}\forall \mathrm{y}\forall \mathrm{z}(\mathrm{S}{\mathrm{x}^*}\mathrm{y}\land \mathrm{Sxz}\Rightarrow {\mathrm{y}^*}\leq \mathrm{z}). \) [34]

The adoption of \(\sf{(D)}\) and \(\sf{(Dc)}\) is sufficient to ensure both that actuality commutes with disjunction and that negations flanking an actuality operator cancel out, as recorded in lemma 4 in the Appendix.

Next, I will turn to the axiom \(\sf{(U)} \mathbb{A} (\mathbb{A} \mathrm{A}\rightarrow \mathrm{A})\).[35] This corresponds to the frame condition

$$ \sf{(FU)}\; \forall \mathrm{x}\in \mathrm{N}, \forall \mathrm{y}(Sxy\Rightarrow \forall \mathrm{z}\forall \mathrm{u}(Ryzu\Rightarrow Szu)). $$

In the context of the classical modal logic of Crossley and Humberstone (1977), the embedded converse, \(\sf{(Uc)}\; \mathbb{A}(\mathrm{A}\rightarrow \mathbb{A} \mathrm{A})\), is a consequence of the other axioms and rules, replacing the relevant implication with the classical material conditional. Provided the contraposition axiom, \(\sf{(B7)}\), \(\sf{(U)}\), and \(\sf{(Dc)}\) are in the logic, then the embedded converse follows from the other principles, for which proof see the Appendix. If the logic is lacking these principles, then one can validate (Uc) manually, with the following frame condition.

$$ \sf{(FUc)}\; \forall \mathrm{x}\in \mathrm{N}, \forall \mathrm{y}(Sxy\Rightarrow \forall \mathrm{u}\forall \mathrm{v}(Ryuv\Rightarrow\forall \mathrm{z}(Svz\Rightarrow \mathrm{u}\leq \mathrm{z}))) $$

There are further principles that one could add, following Crossley and Humberstone (1977). Although many of these principles follow from the other axioms in the classical modal context, they may be independent in the relevant context. Rather than attempt an exhaustive catalog of axioms, I will turn to discussion of some philosophical issues. In the Appendix, I address some metatheoretic issues.

6. Discussion

In this section, I will discuss some of the philosophical issues arising from the preceding work. After a brief summary, which will diagnose the problem with ubiquity, I will begin by discussing the interpretation of the frames (§6.1) and then highlight some natural candidates for a preferred logic of actuality (§6.2). Following this, I will look at how the view presented here fits into two debates in metaphysics, namely the debate between indexical and absolute theories of actuality and the debate between actualism and possibilism in ontology (§6.3). I will then discuss issues arising in the interplay between actuality and necessity, in particular concerns about the fixedly operator of Crossley and Humberstone (§6.4), before closing by briefly mentioning directions for future work (§6.5).

One might have thought that it would be no problem to add an actuality operator to relevant logics. Natural ways of extending the Routley-Meyer frames to accommodate the actuality operator lead to direct conflicts with relevance intuitions that motivate work in the area. As indicated in §4, the problems stem from the interpretation of actuality. In an RMg-model, the interpretation of actuality is constant across the model. This leads to true actuality formulas being ubiquitous in the models.

The problems with the RMg-frames were solved by moving to RMS-frames. The cost of RMS-frames is that one adopts a somewhat removed view of the actual world in the sense that the points need not all agree on which world, or proposition, is actual. The Ubiquity Lemma suggests why global agreement on what is actual will be difficult to maintain together with the relevance intuitions. Maintaining the relevance intuitions seems to require that the models allow for the possibility that somewhere, things can, in a sense, “go awry”. We can see this playing out in the basic Routley-Meyer models. In Routley-Meyer models, there can be points at which logical truths fail. Putting the point in terms of ubiquity, there must, in general, be points witnessing a failure of ubiquity for logical truths. Were there not such points, then \(\mathrm{A}\rightarrow (\mathrm{B}\rightarrow \mathrm{B})\) and its kin exemplifying irrelevance would be valid. Similarly, there have to be points that, in a sense, get things wrong with respect to what actually is the case. That is the diagnosis of the problem with ubiquity. I will now turn to the issue of interpreting the RMS-frames.

7. Appendix: Derivability, Soundness, and Completeness

In earlier sections, several axioms and frame conditions were considered. In this section, I will prove some of the results claimed, including derivability, soundness, and completeness. I will begin with some basic facts about derivability.

7.3. Completeness

With soundness in hand, I will now turn to completeness. The approach of this section is to use the Henkin-style canonical model construction familiar from work in the area, leaning on the particular developments of Fuhrmann (1990) and Restall (2000). For a given logic, one defines a frame and then shows that it satisfies all the conditions needed to be a frame for the logic. Then, one shows that the evaluation clauses work as expected, and so one can define a model on the frame to refute the target non-theorem. At the end of this subsection, I will sketch a proof demonstrating that the logics of actuality from RMS-frames respect a necessary condition on relevance, Belnap’s variable sharing constraint.

It follows from the work of Fuhrmann (1990) that, for a logic \(\sf{L}\) extending \(\sf{B}\) with axioms from \(\sf{(B1)}\)–\(\sf{(B8)}\), \(\sf{L.MX}\) is sound and complete with respect to \(\mathcal{F}(\sf{L.MX})\), where \(\sf{X}\subseteq\{\sf{(Nec)}, \sf{(4)}, \sf{(D)}, \sf{(K)}\}\). I will, then, focus on axioms not already covered. In the interest of making this paper more self-contained, I will reproduce some of the standard definitions.

Definition 2. A set \(\mathrm{X}\) of formulas is an \(\sf{L}\)-theory iff

• if \(\mathrm{A}\in \mathrm{X}\) and \(\vdash_{\sf{L}}{\mathrm{A}\rightarrow \mathrm{B}}\), then \(\mathrm{B}\in \mathrm{X}\), and
• if \(\mathrm{A}\in \mathrm{X}\) and \(\mathrm{B}\in \mathrm{X}\), then \(\mathrm{A}\land \mathrm{B}\in \mathrm{X}\).

• A theory \(\mathrm{X}\) is prime iff \(\mathrm{A}\lor \mathrm{B}\in \mathrm{X}\) implies \(\mathrm{A}\in \mathrm{X}\) or \(\mathrm{B}\in \mathrm{X}\).
• A theory \(\mathrm{X}\) is \(\sf{L}\)-regular iff \(\sf{L}\subseteq \mathrm{X}\).

For all the defined concepts, when \(\sf{L}\) is clear from context, it will be dropped.

The canonical model \(\mathrm{M}\) for the logic \(\sf{L.MX}\) will be defined as follows, where for sets of formulas \(\mathrm{X}\), \(\mathrm{X}^{-\mathbb{A}}=\{\mathrm{A}: \mathbb{A} \mathrm{A}\in \mathrm{X}\}\).

• \(\mathrm{K}_{\mathrm{M}}\) is the set of prime \(\sf{L.MX}\)-theories.
• \(\mathrm{N}_{\mathrm{M}}\) is the set of \(\sf{L.MX}\)-regular, prime \(\sf{L.MX}\)-theories.
• \(\mathrm{R_{M}abc}\) iff \(\forall \mathrm{A}\forall \mathrm{B}(\mathrm{A}\rightarrow \mathrm{B}\in \mathrm{a}\land \mathrm{A}\in \mathrm{b}\Rightarrow \mathrm{B}\in \mathrm{c})\).
• \(\mathrm{a}^{*_{\mathrm{M}}}=\{\mathrm{A}: \sim\! \! \mathrm{A}\not\in \mathrm{a}\}\).
• \(\mathrm{S_{M}ab}\) iff \(\mathrm{a}^{-\mathbb{A}}\subseteq \mathrm{b}\).
• \(\mathrm{V}_{\mathrm{M}}(\mathrm{p})=\{\mathrm{a}\in \mathrm{K}_{\mathrm{M}}: \mathrm{p}\in \mathrm{a}\}\).

In what follows, the subscript ‘\(\mathrm{M}\)’ will be omitted to avoid clutter.

The canonical frame obeys the required frame condition for RMS-frames,

• \(\mathrm{a\leq b\Rightarrow Sb\subseteq Sa}\).[57]

Further, by standard arguments one can show that it satisfies the conditions required by the base logic \(\sf{L}\).[58] We need to show that the frame underlying the canonical model for the logic \(\sf{L.MX}\) satisfies the frame conditions \(\sf{(FA)}\), for each axiom \(\sf{(A)}\in \mathrm{X}\). I will provide proofs for the axioms \(\sf{(Dc)}\), \(\sf{(U)}\), and \(\sf{(Uc)}\).

Lemma 9. If the logic \(\sf{L.MX}\) contains \(\sf{(Dc)}\), then the canonical frame obeys the condition \(\sf{(FDc)}\).

Proof. Suppose \(\sf{L.MX}\) contains \(\sf{(Dc)}\). Further, suppose \(\mathrm{S}{\mathrm{a}^*}\mathrm{b}\), \(\mathrm{Sac}\), but not \({\mathrm{b}^*}\leq \mathrm{c}\). Then there is some \(\mathrm{B}\in{\mathrm{b}^*}\) such that \(\mathrm{B}\not\in \mathrm{c}\). Therefore, \(\mathbb{A} \mathrm{B}\not\in \mathrm{a}\), so \(\sim\! \! \mathbb{A} \mathrm{B}\in {\mathrm{a}^*}\). Since \({\mathrm{a}^*}\) is a theory and \(\vdash_{\sf{L.MX}}{\sim\! \! \mathbb{A} \mathrm{B}\rightarrow \mathbb{A} \sim\! \! \mathrm{B}}\), \(\mathbb{A} \sim\! \! \mathrm{B}\in{\mathrm{a}^*}\). Thus, \(\sim\! \! \mathrm{B}\in \mathrm{b}\), so \(\mathrm{B}\not\in {\mathrm{b}^*}\), contradicting the assumption.

\(\Box\)

Lemma 10. If the logic \(\sf{L.MX}\) contains \(\sf{(U)}\), then the canonical frame obeys the condition \(\sf{(FU)}\).

Proof. Suppose \(\sf{L.MX}\) contains \(\sf{(U)}\). Suppose \(\mathrm{a}\in \mathrm{N}\), \(\mathrm{Sab}\), \(\mathrm{Rbcd}\), but not \(\mathrm{Scd}\). Then there is some \(\mathrm{B}\) such that \(\mathbb{A} \mathrm{B}\in \mathrm{c}\) and \(\mathrm{B}\not\in \mathrm{d}\). By definition, \(\mathbb{A} \mathrm{B}\rightarrow \mathrm{B}\not\in \mathrm{b}\), so \(\mathbb{A}(\mathbb{A} \mathrm{B}\rightarrow \mathrm{B})\not\in \mathrm{a}\). But, as \(\mathrm{a}\in \mathrm{N}\), \(\mathbb{A}(\mathbb{A} \mathrm{B}\rightarrow \mathrm{B})\in \mathrm{a}\), which is a contradiction.

\(\Box\)

Lemma 11. If the logic \(\sf{L.MX}\) contains \(\sf{(Uc)}\), then the canonical frame obeys the condition \(\sf{(FUc)}\).

Proof. Suppose \(\sf{L.MX}\) contains \(\sf{(Uc)}\). Suppose \(\mathrm{a}\in \mathrm{N}\), \(\mathrm{Sab}\), \(\mathrm{Rbcd}\), \(\mathrm{Sde}\), but not \(\mathrm{c}\leq \mathrm{e}\). Then there is some \(\mathrm{B}\), \(\mathrm{B}\in \mathrm{c}\) but \(\mathrm{B}\not\in \mathrm{e}\). By definition, \(\mathbb{A} \mathrm{B}\not\in \mathrm{d}\), so \(\mathrm{B}\rightarrow\mathbb{A} \mathrm{B}\not\in \mathrm{b}\). Therefore, \(\mathbb{A}(\mathrm{B}\rightarrow \mathbb{A} \mathrm{B})\not\in \mathrm{a}\). But, as \(\vdash_{\sf{L.MX}}{\mathbb{A}( \mathrm{B}\rightarrow \mathbb{A} \mathrm{B})}\) and \(\mathrm{a}\in \mathrm{N}\), \(\mathbb{A}(\mathrm{B}\rightarrow \mathbb{A} \mathrm{B})\in \mathrm{a}\), which is a contradiction.

\(\Box\)

The frame underlying the canonical model for a logic containing any of the preceding actuality axioms obeys the frame condition for that axiom. It is fairly routine to establish the following.

Lemma 12 (Truth Lemma). In the canonical model for the logic \(\sf{L.MX}\), for all formulas \(\mathrm{A}\), \(\mathrm{A}\in \mathrm{a}\) iff \({\mathrm{a}}\Vdash{\mathrm{A}}\).

Proof. See Fuhrmann (1990) or Restall (2000: Ch. 11).

\(\Box\)

When the preceding is combined with the results from Fuhrmann (1990), we have the following completeness result.

Theorem 3 (Completeness). Let the logic \(\sf{L}\) extend \(\sf{B}\) with any of the axioms from \(\sf{(B1)}\)–\(\sf{(B8)}\), and suppose \(\sf{X}\subseteq\{\sf{(Nec), (U), (Uc), (D), (Dc), (K), (4)}\}\). Then, \(\models_{\mathcal{F}(\sf{L.MX})}{\mathrm{A}}\) only if \(\vdash_{\sf{L.MX}}{\mathrm{A}}\).

With the completeness results in hand, I can now argue that the RMS-frames preserve relevance intuitions in the sense that they have the variable sharing property, just as the underlying relevant logics do.

Theorem 4 (Variable sharing). For any of the logics \(\sf{L.MX}\) proved complete above, for all formulas \(\mathrm{A}\) and \(\mathrm{B}\) not containing \(\sf{t}\), if \(\vdash_{\sf{L.MX}}{\mathrm{A}\rightarrow \mathrm{B}}\), then \(\mathrm{A}\) and \(\mathrm{B}\) share a propositional variable.

Proof. (Sketch) The proof is adapted from proofs that \(\sf{R}\) has the variable sharing property, as described by Dunn and Restall (2002: §1.6), using the algebraic method of Anderson and Belnap (1975: 252–254), whose Hasse diagram and matrices are in Table 2.[59] The designated values are marked \(+\) and \(\sf{t}\) is always assigned the value \(+0\). For the present context, interpret \(\mathbb{A}\) as the identity operator. All actuality axioms considered in this paper are designated on all valuations on these matrices, \(\sf{(Nec)}\) preserves the property of being designated, and similarly for all axioms and rules of \(\sf{R}\).

Table 2. Hasse diagram and matrices for the variable sharing property

Suppose that \(\vdash_{\sf{L.MX}}{\mathrm{A}\rightarrow \mathrm{B}}\) for some \(\sf{t}\)-free formulas \(\mathrm{A}\) and \(\mathrm{B}\) but \(\mathrm{A}\) and \(\mathrm{B}\) do not share a propositional variable. Let \(\mathrm{A}'\) and \(\mathrm{B}'\) be \(\mathrm{A}\) and \(\mathrm{B}\), respectively, with all instances of the actuality operator removed. Then \(\mathrm{A}'\) and \(\mathrm{B}'\) take the same values as \(\mathrm{A}\) and \(\mathrm{B}\), respectively. Since \(\mathrm{A}\) and \(\mathrm{B}\) do not share a propositional variable, neither do \(\mathrm{A}'\) and \(\mathrm{B}'\), so there is a valuation on the matrix such that \(\mathrm{A}'\rightarrow \mathrm{B}'\) is not designated, which implies that \(\mathrm{A}\to \mathrm{B}\) is undesignated. Therefore, \(\mathrm{A}\to \mathrm{B}\) is not provable, contradicting the assumption. As all the relevant logics considered in this paper are sublogics of \(\sf{R}\), the argument works for all of them.

\(\Box\)

Thus, we see that the logic of the actuality operator, as delivered by the RMS-frames, maintains the necessary condition on relevance, an indication that they at least modestly respect the relevance intuitions with which we started.

Acknowledgments

I would like to thank Greg Restall, Rohan French, Dave Ripley, Lloyd Humberstone, Melissa Fusco, Fabio Lampert, John Bigelow, and the audience at the Melbourne Logic Seminar for comments and discussion that have greatly improved this paper. I would also like to thank the anonymous referees whose feedback significantly improved this paper. This research was supported by the Australian Research Council, Discovery Grant DP150103801.

References

• Adams, Robert Merrihew (1974). Theories of Actuality. Noûs, 8(3), 211–231. https://doi.org/10.2307/2214751
• Anderson, Alan Ross and Nuel D. Belnap (1975). Entailment: The Logic of Relevance and Necessity, Vol. I. Princeton University Press.
• Avron, Arnon (1992). Whither Relevance Logic? Journal of Philosophical Logic, 21(3), 243–281. https://doi.org/10.1007/bf00260930
• Beall, Jc, Ross Brady, J. Michael Dunn, A. P. Hazen, Edwin Mares, Robert K. Meyer, . . . , Richard Sylvan (2012). On the Ternary Relation and Conditionality. Journal of Philosophical Logic, 41(3), 595–612. https://doi.org/10.1007/s10992-011-9191-5
• van Benthem, Johan, Nick Bezhanishvili, and Wesley H. Holliday (2017). A Bimodal Perspective on Possibility Semantics. Journal of Logic and Computation, 27(5), 1353–1389. https://doi.org/10.1093/logcom/exw024
• Bimbó, Katalin (2006). Relevance Logics. In Dale Jacquette (Ed.), Philosophy of Logic (723–789), Vol. 5 of Handbook of the Philosophy of Science. Elsevier.
• Blackburn, Patrick and Maarten Marx (2002). Remarks on Gregory’s “Actually” Operator. Journal of Philosophical Logic, 31(3), 281–288. https://doi.org/10.1023/a:1015726824270
• Brady, Ross T. (1984). Natural Deduction Systems for Some Quantified Relevant Logics. Logique Et Analyse, 27(8), 355–377.
• Braüner, Torben (2006). Axioms for Classical, Intuitionistic, and Paraconsistent Hybrid Logic. Journal of Logic, Language and Information, 15(3), 179–194. https://doi.org/10.1007/s10849-006-9013-2
• Braüner, Torben (2007). Why Does the Proof-Theory of Hybrid Logic Work So Well? Journal of Applied Non-Classical Logics, 17(4), 521–543. https://doi.org/10.3166/jancl.17.521-543
• Braüner, Torben (2011). Intuitionistic Hybrid Logic: Introduction and Survey. Information and Computation, 209(12), 1437–1446. https://doi.org/10.1016/j.ic.2011.10.001
• Braüner, Torben (2017). Hybrid Logic. In Edward N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2017 ed.). Retrieved from https://plato.stanford.edu/archives/sum2017/entries/logic-hybrid/
• Braüner, Torben and Valeria de Paiva (2006). Intuitionistic hybrid logic. Journal of Applied Logic, 4(3), 231–255. https://doi.org/10.1016/j.jal.2005.06.009
• Chalmers, David J. (2004). Epistemic Two-Dimensional Semantics. Philosophical Studies, 118(1/2), 153–226. https://doi.org/10.1023/b:phil.0000019546.17135.e0
• Costa, Diana and Manuel A. Martins (2017). Paraconsistency in Hybrid Logic. Journal of Logic and Computation, 27(6), 1825–1852. https://doi.org/10.1093/logcom/exw027
• Crossley, John N. and Lloyd Humberstone (1977). The Logic of “Actually”. Reports on Mathematical Logic, 8, 11–29.
• Davies, Martin (2004). Reference, Contingency, and the Two-Dimensional Framework. Philosophical Studies, 118(1/2), 83–131. https://doi.org/10.1023/b:phil.0000019544.39800.50
• Davies, Martin and Lloyd Humberstone (1980). Two Notions of Necessity. Philosophical Studies, 38(1), 1–31. https://doi.org/10.1007/BF00354523
• Davis, Wayne A. (2013). The Semantics of Actuality Terms: Indexical vs. Descriptive Theories. Noûs, 49(3), 470–503. https://doi.org/10.1111/nous.12046
• Dunn, J. Michael and Greg Restall (2002). Relevance Logic. In Dov M. Gabbay and Franz Guenthner (Eds.), Handbook of Philosophical Logic (2nd ed., Vol. 6, 1–136). Kluwer.
• Evans, Gareth (1979). Reference and Contingency. The Monist, 62(2), 161–189. https://doi.org/10.5840/monist197962220
• Fine, Kit (1988). Semantics for Quantified Relevance Logic. Journal of Philosophical Logic, 17(1), 27–59. https://doi.org/10.1007/BF00249674
• French, Rohan (2012). An Argument against General Validity? Thought: A Journal of Philosophy, 1(1), 4–9. https://doi.org/10.1002/tht3.1
• French, Rohan (2015). In the Mood for S4: The Expressive Power of the Subjunctive Modal Language in Weak Background Logics. Studia Logica, 103(2), 239–263. https://doi.org/10.1007/s11225-014-9557-0
• Fritz, Peter (2013). A Logic for Epistemic Two-Dimensional Semantics. Synthese, 190(10), 1753–1770. https://doi.org/10.1007/s11229-013-0260-x
• Fritz, Peter (2014). What Is The Correct Logic of Necessity, Actuality and Apriority? The Review of Symbolic Logic, 7(3), 385–414. https://doi.org/10.1017/s1755020314000136
• Fuhrmann, André (1990). Models for Relevant Modal Logics. Studia Logica, 49(4), 501–514. https://doi.org/10.1007/bf00370161
• Fusco, Melissa (2015). Deontic Modality and the Semantics of Choice. Philosophers’ Imprint, 15(28), 1–27. Retrieved from http://hdl.handle.net/2027/spo.3521354.0015.028
• Gale, Richard M. (1989). Lewis’ Indexical Argument for World-Relative Actuality. Dialogue, 28(2), 289–304. https://doi.org/10.1017/s001221730001578x
• Giambrone, Steve (1992). Real Reduced Models for Relevant Logics without WI. Notre Dame Journal of Formal Logic, 33(3), 442–449. https://doi.org/10.1305/ndjfl/1093634408
• Gilbert, David (2016). Two-Dimensional Tableaux. Australasian Journal of Logic, 13(7), 143–170. https://doi.org/10.26686/ajl.v13i7.3894
• Gilbert, David R. and Edwin D. Mares (2012). Completeness Results for Some Two-Dimensional Logics of Actuality. Review of Symbolic Logic, 5(2), 239–258. https://doi.org/10.1017/S1755020311000384
• Girard, Patrick and Zach Weber (2015). Bad Worlds. Thought: A Journal of Philosophy, 4(2), 93–101. https://doi.org/10.1002/tht3.162
• Gregory, Dominic (2001). Completeness and Decidability Results for Some Propositional Modal Logics Containing ‘Actually’ Operators. Journal of Philosophical Logic, 30(1), 57–78. https://doi.org/10.1023/A:1017579410231
• Hanson, William H. (2006). Actuality, Necessity, and Logical Truth. Philosophical Studies, 130(3), 437–459. https://doi.org/10.1007/s11098-004-5750-8
• Hazen, Allen P., Benjamin G. Rin, and Kai F. Wehmeier (2013). Actuality in Propositional Modal Logic. Studia Logica, 101(3), 487–503. https://doi.org/10.1007/s11225-012-9395-x
• Holliday, Wesley H. (2014). Partiality and Adjointness in Modal Logic. In Rajeev Goré, Barteld Kooi, and Agi Kurucz (Eds.), Advances in Modal Logic (Vol. 10, 313–332). College Publications.
• Humberstone, Lloyd (1981). From Worlds to Possibilities. Journal of Philosophical Logic, 10(3), 313–339. https://doi.org/10.1007/bf00293423
• Humberstone, Lloyd (2004). Two-Dimensional Adventures. Philosophical Studies, 118(1–2), 17–65. https://doi.org/10.1023/B:PHIL.0000019542.43440.d1
• Humberstone, Lloyd (2011). The Connectives. MIT Press.
• Humberstone, Lloyd (2016). Philosophical Applications of Modal Logic. College Publications.
• Kamp, Hans (1971). Formal Properties of “Now”. Theoria, 37(3), 227–273. https://doi.org/10.1111/j.1755-2567.1971.tb00071.x
• Kripke, Saul A. (1963). Semantical Analysis of Modal Logic I: Normal Propositional Calculi. Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 9(56), 67–96. https://doi.org/10.1002/malq.19630090502
• Lampert, Fabio (2018a). Actuality, Tableaux, and Two-Dimensional Modal Logics. Erkenntnis, 83(3), 403–443. https://doi.org/10.1007/s10670-017-9896-0
• Lampert, Fabio (2018b). Actuality and the A Priori. Philosophical Studies, 175(3), 809–830. https://doi.org/10.1007/s11098-017-0894-5
• Lewis, David (1970). Anselm and Actuality. Noûs, 4(2), 175–188. https://doi.org/10.2307/2214320
• Lewis, David (1986). On the Plurality of Worlds. Wiley-Blackwell.
• Logan, Shay Allen (2019). Notes on Stratified Semantics. Journal of Philosophical Logic, 48(4), 749–786. https://doi.org/10.1007/s10992-018-9493-y
• Mares, Edwin D. (1992). Semantics for Relevance Logic with Identity. Studia Logica, 51(1), 1–20. https://doi.org/10.1007/BF00370329
• Mares, Edwin D. (1996). Relevant Logic and the Theory of Information. Synthese, 109(3), 345–360. https://doi.org/10.1007/bf00413865
• Mares, Edwin D. (2000). CE is Not a Conservative Extension of E. Journal of Philosophical Logic, 29(3), 263–275. https://doi.org/10.1023/a:1004731401855
• Mares, Edwin D. (2004). Relevant Logic: A Philosophical Interpretation. Cambridge University Press.
• Mares, Edwin D. (2008).General Information in Relevant Logic. Synthese, 167(2), 343–362. https://doi.org/10.1007/s11229-008-9412-9
• Mares, Edwin D. (2017). An Informational Interpretation of Weak Relevant Logic and Relevant Property Theory. Synthese. Advance online publication. https://doi.org/10.1007/s11229-017-1524-7
• Mares, Edwin D. and Robert Goldblatt (2006). An Alternative Semantics for Quantified Relevant Logic. The Journal of Symbolic Logic, 71(1), 163–187. https://doi.org/10.2178/jsl/1140641167
• Mares, Edwin D. and Robert K. Meyer (1993). The Semantics of R4. Journal of Philosophical Logic, 22(1), 95–110. https://doi.org/10.1007/BF01049182
• Mares, Edwin D. and Koji Tanaka (2010). Boolean Conservative Extension Results for Some Modal Relevant Logics. The Australasian Journal of Logic, 8, 31–49. https://doi.org/10.26686/ajl.v8i0.1812
• Menzel, Christopher (2018). Actualism. In Edward N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2018 ed.). Retrieved from https://plato.stanford.edu/archives/sum2018/entries/actualism/
• Meyer, Robert K. and Edwin D. Mares (1993). Semantics of Entailment 0. In Peter Schroeder-Heister and Kosta Došen (Eds.), Substructural Logics (239–258). Oxford Science Publications.
• Nelson, Michael and Edward N. Zalta (2012). A Defense of Contingent Logical Truths. Philosophical Studies, 157(1), 153–162. https://doi.org/10.1007/s11098-010-9624-y
• Priest, Graham (2006). In Contradiction: A Study of the Transconsistent (2nd ed.). Oxford University Press.
• Priest, Graham (2008). An Introduction to Non-Classical Logic: From If to Is. Cambridge University Press.
• Read, Stephen (1988). Relevant Logic: A Philosophical Examination of Inference. Basil Blackwell.
• Restall, Greg (1999). Negation in Relevant Logics: How I Stopped Worrying and Learned to Love the Routley Star. In Dov Gabbay and Heinrich Wansing (Eds.), What Is Negation? (53–76), Vol. 13 of Applied Logic Series. Kluwer Academic Publishers.
• Restall, Greg (2000). An Introduction to Substructural Logics. Routledge.
• Restall, Greg (2012). A Cut-Free Sequent System for Two-Dimensional Modal Logic, and Why It Matters. Annals of Pure and Applied Logic, 163(11), 1611–1623. https://doi.org/10.1016/j.apal.2011.12.012
• Routley, Richard (1977). Ultralogic as Universal? Relevance Logic Newsletter, 2, 51–89. Reprinted as Appendix in Richard Routley (1980) Exploring Meinong’s Jungle and Beyond (893–962). Page numbers are for the reprinted version.
• Routley, Richard (1984). The American Plan Completed: Alternative Classical-Style Semantics, without Stars, for Relevant and Paraconsistent Logics. Studia Logica, 43(1–2), 131–158. https://doi.org/10.1007/BF00935746
• Routley, Richard and Robert K. Meyer (1972a). The Semantics of Entailment—II. Journal of Philosophical Logic, 1(1), 53–73. https://doi.org/10.1007/BF00649991
• Routley, Richard and Robert K. Meyer (1972b). The Semantics of Entailment—III. Journal of Philosophical Logic, 1(2), 192–208. https://doi.org/10.1007/bf00650498
• Routley, Richard and Robert K. Meyer (1973). The Semantics of Entailment. In Hughes Leblanc (Ed.), Truth, Syntax, and Modality: Proceedings Of The Temple University Conference On Alternative Semantics (199–243). North-Holland.
• Routley, Richard, Val Plumwood, Robert K. Meyer, and Ross T. Brady (1982). Relevant Logics and Their Rivals (Vol. 1). Ridgeview.
• Schroeter, Laura (2017). Two-Dimensional Semantics. In Edward N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Summer 2017 ed.). Retrieved from https://plato.stanford.edu/archives/sum2017/entries/two-dimensional-semantics/
• Sedlár, Igor (2015). Substructural Epistemic Logics. Journal of Applied Non-Classical Logics, 25(3), 256–285. https://doi.org/10.1080/11663081.2015.1094313
• Slaney, John K. (1987). Reduced Models for Relevant Logics without WI. Notre Dame Journal of Formal Logic, 28(3), 395–407. https://doi.org/10.1305/ndjfl/1093637560
• Stalnaker, Robert (1999). Assertion. In Context and Content (78–95). Oxford University Press.
• Stephanou, Yannis (2010). The Meaning of “Actually”. Dialectica, 64(2), 153–185. https://doi.org/10.1111/j.1746-8361.2010.01222.x
• Sylvan, Richard (1988). Radical Pluralism: An Alternative to Realism, Antirealism and Relativism. In Robert Nola (Ed.), Relativism and Realism in Science (253– 291). Kluwer.
• Tennant, Neil (2017). Core Logic. Oxford University Press.
• Urquhart, Alasdair (1972). Semantics for Relevant Logics. Journal of Symbolic Logic, 37(1), 159–169. https://doi.org/10.2307/2272559
• Weber, Zach, Guillermo Badia, and Patrick Girard (2016). What Is an Inconsistent Truth Table? Australasian Journal of Philosophy, 94(3), 533–548. https://doi.org/10.1080/00048402.2015.1093010
• Wehmeier, Kai Frederick (2004). In the Mood. Journal of Philosophical Logic, 33(6), 607–630. https://doi.org/10.1023/b:logi.0000046193.69391.18
• Zalta, Edward N. (1988). Logical and Analytic Truths that Are Not Necessary. The Journal of Philosophy, 85(2), 57–74. https://doi.org/10.2307/2026992

Notes

1. See Dunn and Restall (2002) or Bimbó (2006) for reader-friendly overviews of relevant logics.

2. How to understand the connection at issue is a delicate matter. Anderson and Belnap (1975) suggest a kind of connection based on use. An alternative conception is the sufficiency view of Routley (1977: 895ff.). A different approach to relevance is offered by Tennant (2017).

3. This holds at least when sentential constants aren’t in the language. When sentential constants, such as \(\sf{t}\) below, are around then caveats are needed, but I will leave those on the sidelines for this paper.

4. Some work in the area draws a distinction between ‘implication’ and ‘conditional’, but I will use the terms interchangeably.

5. I will leave somewhat open the extent of the kindred axioms and rules. The particular demarcation adopted here excludes, e.g., \(\sf{R-Mingle}\), which is sometimes classed as a relevant logic. \(\sf{R-Mingle}\) is obtained by adding the mingle axiom \(\mathrm{A}\rightarrow (\mathrm{A}\rightarrow \mathrm{A})\) to \(\sf{R}\). Its status as a relevant logic is disputed on the grounds that it has theorems violating the variable sharing condition, such as \(\sim\! \! (\mathrm{A}\rightarrow \mathrm{A})\rightarrow (\mathrm{B}\rightarrow \mathrm{B})\). See Avron (1992) for a discussion of \(\sf{R-Mingle}\) and a defense of its intensional fragment.

6. Routley-Meyer frames were introduced by Routley and Meyer (1972a; 1972b; 1972). See Restall (2000: 253–274 for more recent developments in frame semantics. Routley changed his name to Sylvan later in life. I will cite his work by the published name.

7. This is notwithstanding the quotations to follow, which refer to some of the points as worlds. I will note that Mares and Meyer (1993) and Sedlár (2015) provide ways to define worlds as distinguished points.

8. See Beall et al. (2012) for a discussion of the interpretation of the ternary relation. See Restall (1999) for a discussion of the interpretation of the Routley star.

9. Disjunction is defined, so this clause is, technically, a lemma rather than a stipulation.

10. The logical framework adopted here is FMLA, rather than one of the more general consequence relation frameworks, e.g., SET-FMLA. See Humberstone (2011: 103ff.).

11. The symbol ‘\(\Rightarrow\)’ does double duty as the metatheoretic material conditional and the separator between premises and conclusion of the rules of the object logic.

12. The rule \(\mathrm{A}\to\sim\! \! \mathrm{B}\Rightarrow \mathrm{B}\to\sim\! \! \mathrm{A}\) is commonly used instead of \(\sf{(R5)}\). This rule can be obtained from \(\sf{(R5)}\), and vice versa, given the other rules and axioms.

13. For the correspondence results, which are standard, see Routley, Plumwood, Meyer, and Brady (1982: Ch. 4).

14. These are somewhat redundant axiomatizations. For example, \(\sf{(R5)}\) is redundant once \(\sf{(B7)}\) is added, and \(\sf{(B6)}\) follows from \(\sf{(B4)}\). See Brady (1984), on which this presentation is based, for non-redundant axiomatization.

15. There has been comparatively little written on actuality in non-classical logics. A near relative is the work on intuitionistic hybrid logic of Braüner and de Paiva (2006) and Braüner (2006; 2007; 2011), as well as the work on paraconsistent hybrid logic of Costa and Martins (2017).

16. In the early work, such as Routley and Meyer (1973), there was no set \(\mathrm{N}\). Instead, \(\mathrm{g}\) did the work of defining the heredity ordering and being used to define truth in a model. The distinguished world, \(\mathrm{g}\), was sometimes denoted by ‘0’.

17. The “factual world” terminology is also used by Routley (1984: 149).

18. Sylvan, Meyer, and Slaney were working at ANU, and Slaney, Giambrone, and Fuhrmann were all doctoral students there. In addition, there were regular visits by other logicians from around Australasia working in the relevant logic tradition.

19. While these quotations call the points ‘worlds’, they are often not regarded as such in the literature on relevant logics, as worlds settle too much. I will come back to the interpretation of the points in §6.

20. Kripke (1963: 69) calls the distinguished world of the models the “real world”. This usage is not restricted to classical modal logic or Routley-Meyer frames. See, e.g., Priest (2006: 85–87), where the distinguished world of the model is described as “the ‘real world’ or assignment which is in accord with the actual” and this idea is used to argue in favor of certain properties of the distinguished world.

21. See, e.g., Crossley and Humberstone (1977), Hazen, Rin, and Wehmeier (2013), Gilbert and Mares (2012), or Fritz (2013; 2014). For proof-theoretic approaches to actuality, see Restall (2012), Gilbert (2016), and Lampert (2018a).

22. Davies and Humberstone (1980) note that this axiom is redundant, given the rest of the axioms and rules. It does not, however, follow from the other actuality axioms with the addition of a \(\sf{(Nec)}\) rule for actuality. As it is somewhat distinctive for actuality, I include it here. I thank Fabio Lampert for drawing my attention to the redundancy.

23. See, e.g., Davies and Humberstone (1980), Humberstone (2004), and Lampert (2018b).

24. See Schroeter (2017) for an overview.

25. There are further related topics in the area that would be worth exploring. Wehmeier (2004) shows an equivalence between classical \(\sf{S5}\) with actuality and a modal language with subjunctive predicates in addition to regular predicates. Would such an equivalence hold in the relevant logical context or would there be some expressive difference between languages with subjunctive atoms and those without subjunctive atoms but with an actuality operator? French (2015) shows that a difference emerges in between the languages when a classical modal logic weaker than \(\sf{S5}\) is used, so it seems likely that differences will emerge against a relevant logical background. I thank Fabio Lampert for this suggestion and Rohan French for discussion.

26. See Davies and Humberstone (1980), Zalta (1988), Humberstone (2004), Hanson (2006), Nelson and Zalta (2012), and French (2012) for discussion.

27. There is a third option for defining actuality that was not included above, since it involves a connective that is not primitive in the language, namely fusion, \(\circ\). If we add fusion to the language, then we also need to add the rules \(\mathrm{A}\rightarrow(\mathrm{B}\rightarrow \mathrm{C})\Rightarrow ({\mathrm{A}}\circ{\mathrm{B}})\rightarrow \mathrm{C}\) and \(({\mathrm{A}}\circ{\mathrm{B}})\rightarrow \mathrm{C} \Rightarrow \mathrm{A}\rightarrow(\mathrm{B}\rightarrow \mathrm{C})\). A natural way to attempt to define an actuality operator using fusion is as \({\sf{t}}\circ{\underline{\phantom{\mathrm{A}}}}\). The problem with this is that it makes \(\mathrm{A}\leftrightarrow({\sf{t}}\circ{\mathrm{A}})\) valid, even in \(\sf{B}\). To see this, note that \(\sf{t}\rightarrow(\mathrm{A}\rightarrow \mathrm{A})\) is a theorem of \(\sf{B}\), which by one fusion rule yields \(({\sf{t}}\circ{\mathrm{A}})\rightarrow \mathrm{A}\). For the other direction, we have \(\sf{t}\rightarrow(\mathrm{A}\rightarrow({\sf{t}}\circ{\mathrm{A}}))\) by \(\sf{(A1)}\) and a fusion rule. From \(\sf{(A8)}\) and \(\sf{(R1)}\), we then have \(\mathrm{A}\rightarrow({\sf{t}}\circ{\mathrm{A}})\). As with the implicational definition of actuality, whether \(\mathbb{A} \mathrm{A} \leftrightarrow \mathrm{A}\) is valid is not something that the underlying logic should settle.

    In weaker logics, \({\sf{t}}\circ{\mathrm{A}}\) is not equivalent to \({\mathrm{A}}\circ{\sf{t}}\), suggesting a potential option for defining actuality as \({\underline{\phantom{\mathrm{A}}}}\circ{\sf{t}}\). This option is a non-starter, as it would result in \(\mathbb{A} \mathrm{A}\land\mathbb{A} \mathrm{B}\rightarrow\mathbb{A}(\mathrm{A}\land \mathrm{B})\) being invalid in \(\sf{T}\) and weaker logics. I would like to thank an anonymous referee for asking about these potential definitions of the actuality operator.

28. Mares (1992) provides a way to model identity that does not generate ubiquitous truth and irrelevancy. The extension of the identity predicate is required to be reflexive over \(\mathrm{N}\), but not elsewhere. Hence, \(\mathrm{i}=\mathrm{i}\) will turn out valid while \(\mathrm{p}\rightarrow \mathrm{i}=\mathrm{i}\) can be invalidated.

29. In the Kripke frames for classical modal logic, arbitrary sets of points can be propositions. In the present setting, arbitrary sets of points will not do.

30. A similar idea was explored, in the classical modal setting, by Gregory (2001), with Blackburn and Marx (2002) providing further discussion and technical development. I thank Rohan French for pointing out Gregory’s paper to me and Fabio Lampert for pointing out Blackburn and Marx’s.

31. There is a second condition that is worth highlighting: \(\mathrm{Sab\land \mathrm{b}\leq \mathrm{c}\Rightarrow \mathrm{Sac}}\). This condition is optional, unlike the previous constraint, and its addition does not affect the logic. Informally, it says that if \(\mathrm{a}\) takes all actual truths to hold in \(\mathrm{b}\), which is contained in \(\mathrm{c}\), then \(\mathrm{a}\) takes all the actual truths to hold in \(\mathrm{c}\).

32. As an example, \(\sf{DW.MD}\) is the logic obtained by adding to the axioms and rules of \(\sf{DW}\), i.e., \(\sf{(A1)}\)–\(\sf{(A8)}\), \(\sf{(R1)}\)–\(\sf{(R6)}\), and \(\sf{(B7)}\), \((\mathbb{A} \land)\), \(\sf{(RM)}\), and \(\sf{(D)}\).

33. These are from Fuhrmann (1990: 507–508). I will adopt the convention that for a modal axiom or rule \(\sf{(A)}\), the corresponding frame condition is \(\sf{(FA)}\).

34. I will leave proofs of soundness results to §7.

35. The name is taken from Humberstone (2016: 33).

36. One might also take the points to be bodies of information, as is done Urquhart (1972) or by some work on channel theory, e.g., Mares (1996). Urquhart (1972) deals with operational models, rather than Routley-Meyer models, but the interpretation offered there is natural in the context of Routley-Meyer models. The discussion below carries over fairly readily to this interpretation.

37. See Mares (1996; 2004; 2008; 2017) for more on the interpretation of Routley-Meyer frames in terms of situations.

38. They can be inconsistent as well, although that will not feature in the discussion here.

39. That points can fail to be fully determinate is one of the motivations of possibility semantics. In possibility semantics, the points can be partial, but are consistent, and there is a refinement relation on them. See Humberstone (1981), Holliday (2014), and Benthem, Bezhanishvili, and Holliday (2017) for more on possibility semantics.

40. See Lewis (1970) and Lewis (1986: 92–96). For discussion, see Adams (1974), Gale (1989), Stephanou (2010), and Davis (2013).

41. This is the simple property view of Adams (1974).

42. See Weber, Badia, and Girard (2016) for the development of a related idea with respect to truth tables.

43. Sylvan (1988) denies the uniqueness of the actual world, which he views as an unstated, and false, presupposition. Thanks to an anonymous referee for drawing my attention to this article.

44. \(\sf{(SFun)}\) does not on its own require that every point have an \(\mathrm{S}\)-successor, just that there is at most one such successor. Every point having an \(\mathrm{S}\)-successor, or seriality, can be had by adopting \(\sf{(FD)}\).

45. See Restall (2000: 306–307). Conservative extension with boolean negation is subtle when there are modal operators in the language, although it can be obtained, as shown by, e.g., Meyer and Mares (1993) and Mares and Tanaka (2010). See Mares (2000) for an example of the addition of boolean negation not yielding a conservative extension.

46. For a similar sentiment regarding boolean negation, albeit in the context of quantification, see Mares and Goldblatt (2006: 183).

47. This formulation is from Menzel (2018), and for references in this active area, see the citations therein. I would like to thank an anonymous referee for suggesting this idea.

48. There are subtleties about quantification in relevant logics that I will bracket here. See Fine (1988), Mares and Goldblatt (2006), Priest (2008: Ch. 24), and Logan (2019) for more on quantification in relevant logics. As far as I am aware, there has not been any investigation of quantified modal relevant logics, at least not in the sense of quantified relevant logics with primitive alethic modal operators. One can treat implications with a fixed antecedent, \(\mathrm{A}\rightarrow\underline{\phantom{\mathrm{B}}}\), as a kind of modal operator, \([\mathrm{A}]\underline{\phantom{\mathrm{B}}}\), which is sufficient to generate many of the salient issues.

49. An example of the desired set up is a \(\sf{B.MDDc}\) frame that has 8 points, \(\mathrm{K=\{a,b,c,d, {a^*},{b^*}, {c^*},{d^*}\}}\), \(\mathrm{N}=\{\mathrm{a}\}\), \(\mathrm{d\leq c, d\leq b, {c^*}\leq {d^*}, {b^*}\leq {d^*}}\), the required reflexive pairs for \(\leq\), \(\mathrm{Sad, Sac, Sab, Sdd, Sdc, Sdb, Scd, Scc, Sbd, Sbb, S{a^*}{d^*}, S{d^*}{d^*}, S{c^*}{d^*}}\), and \(\mathrm{S{b^*}{d^*}}\).

50. Finding a countermodel is left to the reader.

51. Crossley and Humberstone (1977: 17)

52. See Davies (2004) and Chalmers (2004) for discussion.

53. For the philosophical importance of the diagonal proposition, see Stalnaker (1999), among others. For application of this idea in natural language semantics, see Fusco (2015). See Restall (2012) for development of related two-dimensional ideas in a proof-theoretic setting.

54. I would like to thank Greg Restall for the initial suggestion of using an additional ternary relation.

55. It seems plausible that quantification over \(\mathrm{N}\), or perhaps a different set of points, rather than \(\mathrm{K}\), may be more appropriate for the truth condition for \(\mathbb{F}\).

56. For more on hybrid logic, see Braüner (2017) for an excellent overview of the area, the citations therein, and the earlier citations of work on non-classical hybrid logic. I would like to thank an anonymous referee for making this suggestion.

57. The canonical frame also obeys the optional condition flagged earlier: \(\mathrm{Sab}\land \mathrm{b}\leq \mathrm{c}\Rightarrow \mathrm{Sac}\).

58. See Routley et al. (1982: Ch. 4).

59. See also theorem 8.54 of Restall (2000: 184) and Priest (2008: 205–206).