# Philosophical Applications of Modal Logic

### Schedule:

Wednesday June 13:

- 09:30-11:00 Jessica Leech (KCL),
*Relative Necessity Extended* - 11:15-12:45 Sven Rosenkranz (Barcelona),
*Towards a logic for being in a position to know* - 13:30-15:00 Brian Rabern (Edinburgh),
*Toward a solution to the nesting problem for two-dimensionalism* - 15:15-16:45 Susanne Bobzien (Oxford),
*Intuitionism and the Modal Logic of Vagueness*(joint work with Ian Rumfitt)

Thursday June 14:

- 09:30-11:00 Peter Fritz (Oslo),
*Possible Worlds in Higher-Order Logic* - 11:15-12:45 Wesley Holliday (Berkeley),
*From Worlds to Possibilities…and Back?* - 13:30-15:00 Øystein Linnebo (Oslo),
*Predicativism and potential infinity*(joint work with Stewart Shapiro) - 15:15-16:45 Stephan Leuenberger (Glasgow),
*Fragmentation and introspection in epistemic logic*(joint work with Martin Smith)

### Abstracts:

#### Jessica Leech (KCL), *Relative Necessity Extended*

One can fruitfully define various kinds of alethic necessity as relative, that is, as the logical consequences of a particular set of true propositions. For example, one might take the natural necessities to be what is logically necessary relative to the laws of nature. One merit of such an approach is that we can explain the commonality between different necessities; they are all relativizations of one and the same fundamental necessity. One might then ask: what about non-alethic (and epistemic) necessities? There is a sense in which epistemic, doxastic, legal, deontic, and other necessities are also necessities. But the relativist approach, in taking these to be the logical consequences of, say, known propositions, believed propositions, the laws of morality, etc., runs in to serious and familiar problems (that arise from the logic). In my talk, I assess various options for solving these problems, with a view to exploring whether a larger-scale unification of necessity – alethic and non-alethic – is possible (or desirable) for the relativist.

#### Sven Rosenkranz (Barcelona), *Towards a logic for being in a position to know*

My concern is with an epistemic logic governing the notion of being in a position to know. Such a logic is of independent interest. My own interest stems more specifically from my work on justification. According to the view I’ve defended elsewhere, *p* is propositionally justified iff one is in no position to know that one is in no position to know *p*. More recently, I have argued that *p* is doxastically justified iff one is in no position to know that one doesn’t know *p* – where a belief is justified iff it is held under circumstances under which *p* is doxastically justified in this sense. I will not here argue for these claims. Instead, I wish to explore the features that the underlying epistemic logic and semantics ought to have. After introducing principles of knowledge, and of being in a position to know, that should be acceptable to almost everyone, I suggest two non-standard principles governing these notions and provide a rationale for them. After highlighting a number of interesting theorems, I then proceed to argue that, just like knowledge-operators, operators for being in a position to know behave non-normally and create hyperintensional contexts, with well-known consequences for formal semantics. I make some suggestions of what shape a suitable semantic treatment should take, and dwell on some unresolved issues to which this treatment gives rise, and whose proper resolution bears on the larger project of characterising justification in the ways proposed.

#### Brian Rabern (Edinburgh), *Toward a solution to the nesting problem for two-dimensionalism*

Post-Kripkean theorising concerning modal epistemology accepts a misalignment between apriority and necessity. Two-dimensional semantics provides a framework in which to analyse the Kripkean phenomena of the contingent a priori and the necessary a posteriori. And, as the name suggests, it does so in terms of two “dimensions” of meaning or two levels of semantic value associated with any linguistic expression. But there is a problem concerning how these two levels compose and interact. In particular the problem concerns nested environments: environments where sentences are nested under both modal and epistemic operators (see Soames 2005, Dever 2007, and Forbes 2011). Soames, in fact, insists that nested environments pose the “chief technical problem” for two-dimensionalism. There is a general problem here for a multi-modal logic with operators for metaphysical necessity and epistemic necessity. In light of the contingent priori we face a dilemma: Either (i) what is a priori is a contingent matter or (ii) it is possible that something is a priori but false. In this talk, I will survey some options for addressing this problem from the perspective of a two-dimensional propositional modal logic with operators both for the modalities of “necessity” and “apriority”. I will demonstrate the undesirable consequences of Chalmers and Rabern’s (2014) semantics for apriority, which appeals to a liveness constraint, and then argue that the alternative proposal of Johannesson and Packalén’s (2016) suffers from a Gettier-style objection. I conclude by exploring a solution which adds a truth predicate to the two-dimensional system.

#### Susanne Bobzien (Oxford), *Intuitionism and the Modal Logic of Vagueness *(joint work with Ian Rumfitt)

Intuitionistic logic provides an neat solution to the *Sorites* Paradox which avoids the implausible sharp cut-offs in classically based theories. Its acceptance as the correct logic for vagueness has been hampered by two factors. First, the lack of an agreed semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of *why* intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic does not appear to help with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic status alongside *true *and *false* is best elaborated by taking the normal modal system **S4M** to be the sentential logic of the operator ‘it is clearly the case that’. **S4M** is one of the modal counterparts of the intuitionistic sentential calculus (**IPC**) and we use this fact to explain why **IPC** is a correct sentential logic to use when reasoning with vague statements. Our explanation assumes nothing about the form of a semantic theory for a language with vague terms. To start with, the sentential logic that underpins **S4M** is assumed to be classical, but we also show that our key results go through in an intuitionistic version of **S4M**. In both its classical and intuitionistic versions, **S4M** opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion.

#### Peter Fritz (Oslo), *Possible Worlds in Higher-Order Logic*

Let intensionalism be the view that necessarily equivalent propositions are identical. Assuming intensionalism, there is a promising account of possible worlds according to which they are special propositions, namely those which are possible although maximally strong. For such propositions to behave as worlds are widely expected to behave, it is necessary that each possible proposition can be strengthened to a maximally strong one; call this claim "atomicity". Using higher-order logic as a framework in which to regiment our talk of propositions, properties and relations, this talk will explain why atomicity does not follow straightforwardly from intensionalism, but also show that it follows with plausible additional assumptions once the framework is expanded to include higher-order analogues of plural quantifiers. These considerations will presuppose necessitism, the claim that it is necessary what there is. The talk will conclude by sketching some of the ways in which the situation becomes much more complicated when necessitism is rejected.

#### Wesley Holliday (Berkeley), *From Worlds to Possibilities…and Back?*

Possibility semantics for modal logic, originating in Lloyd Humberstone’s 1981 paper “From Worlds to Possibilities”, is a generalization of possible world semantics based on partially ordered sets of region-like “possibilities” instead of only point-like “worlds". A key aspect of the generalization is in not imposing the atomicity requirement that every possibility is refined by a maximally specific possibility or “world". This is the source of the mathematical and logic interest of possibility semantics for basic modal languages (see https://escholarship.org/uc/item/0tm6b30q). Imposing the atomicity requirement would render possibility semantics for these languages no more general than possible world semantics, though there would still be advantages for extended modal languages. In this talk, I will briefly outline the mathematical and logical interest of possibility semantics, but my main goal is to discuss whether possibility semantics—with the rejection of the atomicity requirement—is of philosophical interest. Some philosophers, including Bob Hale and Ian Rumfitt, have explicitly rejected the atomicity requirement for possibilities. Other philosophers, including Kit Fine, have argued that every proposition is entailed by a world proposition, which in the context of possibility semantics implies the atomicity requirement. I will aim to refute these arguments, as formalized in modal logic with propositional quantifiers and an actuality operator. In the course of my response, I will discuss a distinctive kind of propositional contingentism that can be incorporated in possibility semantics.

#### Øystein Linnebo (Oslo), *Predicativism and potential infinity* (joint work with Stewart Shapiro)

We develop some predicativist approaches within the modal framework for potentiality that was developed in Linnebo (2010) and Linnebo and Shapiro (2018). The result is illuminating, as it puts predicativism into a more general framework and helps to sharpen some of the key theses.

#### Stephan Leuenberger (Glasgow), *Fragmentation and introspection in epistemic logic* (joint work with Martin Smith)

All standard epistemic and doxastic logics legitimate something akin to the principle of closure. And yet the principle of closure, particularly in its multiple premise guise, has a somewhat ambivalent status within epistemology. In this paper we describe a family of weak logics in which closure fails, and describe two alternative semantic frameworks in which these logics can be modelled. One of these – which we term plurality semantics – is relatively unfamilar and unexplored. What makes this framework significant is that it can be interpreted in a very natural way in light of one motivation for rejecting closure: that epistemic agents may be fragmented. Fragmentation is one way of falling short of an epistemic or doxastic ideal. Another one, which has taken central stage in traditional epistemic and doxastic logic, is lack of introspection. The paper then investigates the relationship between these two dimensions of non-ideality.