Abstract
Epistemic modal predicate logic raises conceptual problems not faced in the case of alethic modal predicate logic: Frege’s “HesperusPhosphorus” problem—how to make sense of ascribing to agents ignorance of necessarily true identity statements—and the related “HintikkaKripke” problem—how to set up a logical system combining epistemic and alethic modalities, as well as others problems, such as Quine’s “Double Vision” problem and problems of selfknowledge. In this paper, we lay out a philosophical approach to epistemic predicate logic, implemented formally in Melvin Fitting’s FirstOrder Intensional Logic, that we argue solves these and other conceptual problems. Topics covered include: Quine on the “collapse” of modal distinctions; the rigidity of names; belief reports and unarticulated constituents; epistemic roles; counterfactual attitudes; representational versus interpretational semantics; ignorance of coreference versus ignorance of identity; twodimensional epistemic models; quantification into epistemic contexts; and an approach to multiagent epistemic logic based on centered worlds and hybrid logic.
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Notes
 1.
We do not mean to suggest that there is a consensus on the proper semantics for alethic modal predicate logic. What we have in mind here is the standard development of Kripkestyle semantics for modal predicate logic (see, e.g., [12]). To the extent that we support the Conservative Approach so understood, one might expect that epistemic operators could be smoothly introduced into alethic modal predicate logic developed in other ways as well. As another point of qualification, we are not arguing for conservativeness with respect to the question of relational versus neighborhood semantics for epistemic logic (see [3]).
 2.
Wettstein [65] uses ‘cognitive fix’ in a more demanding sense, requiring not merely a way of thinking about an object, but also accurate beliefs about what distinguishes the object from others.
 3.
It seems that this leaves Russell without a solution to the problem of the cognitive fix, relative to logically proper names. For a discussion of this issue, see [66].
 4.
See [53] for a fuller account of the slingshot and Føllesdal’s treatment of it.
 5.
What if we amend (B) and (C) so that they begin with sequestered terms? Extending ‘Dthat’ to class abstracts, we have:
 (A):

van Benthem is a human.
 (B\({^\prime }\)):

\( \text {Dthat}(\left\{ x \mid x = \emptyset \, \& \right. \) van Benthem is a human\(\left. \right\} ) = \{\emptyset \}\).
 (C\({^\prime }\)):

\( \text {Dthat}(\left\{ x \mid x = \emptyset \, \& \right. \) van Benthem is a logician\(\left. \right\} ) = \{\emptyset \}\).
 (D):

van Benthem is a logician.
Could Quine still argue that (A) and (B\({^\prime }\)), and (C\({^\prime }\)) and (D), have the same modal status? (D) is clearly contingent; the multitalented van Benthem could have been a computer scientist. Is (C\({^\prime }\)) contingent? The result of applying Dthat to a description or class abstract is supposed to be a rigid designator, or more generally, a modally loyal term in the sense defined above. Thus, evaluating

\( \Box \,\text {Dthat}(\left\{ x \mid x = \emptyset \, \& \right. \) van Benthem is a logician\(\left. \right\} ) = \{\emptyset \}\)
amounts to checking for every possible world whether ‘\(y= \{\emptyset \}\)’ is true there, where \(y\) is assigned the object that is designated by ‘\( \text {Dthat}(\left\{ x \mid x = \emptyset \, \& \right. \) van Benthem is a logician\(\left. \right\} )\)’ in the actual world (or the world of the context of utterance). Since the object designated by ‘\( \text {Dthat}(\left\{ x \mid x = \emptyset \, \& \right. \) van Benthem is a logician\(\left. \right\} )\)’ in the actual world is \(\{\emptyset \}\), the check succeeds, so (C\({^\prime }\)) is necessary. Hence (C\({^\prime }\)) and (D) do not have the same modal status; so the modalities do not collapse.
 6.
 7.
 8.
Cf. Lewis [44, p. 543] on watching as a “relation of acquiantance”.
 9.
While our strategy is based on the approach of Crimmins and Perry [19], those authors took cognitive fixes to be notions and took notions to be unarticulated constituents of beliefreports. Subsequently Perry has developed an account that is basically similar, but takes cognitive fixes to be notionnetworks, basically intersubjective routes through notions. Of course, traditionally cognitive fixes have been taken to be individual concepts. We believe that the concept of a role provides a general framework into which all of these candidates can be fit.
 10.
Cf. Lewis [44, p. 542]: “If I have a belief that I might express by saying “Hume was noble”, I probably ascribe nobility to Hume under the description “the one I have heard of under the name of ‘Hume’ ”. That description is a relation of acquaintance that I bear to Hume. This is the real reason why I believe de re of Hume that he was noble”.
 11.
Cf. Lewis [45, 10f] on “relations of epistemic rapport” or “relations of acquaitance”.
 12.
There are a few small differences. First, Fitting allows relation symbols \(Q\) (but not \(=\)) to apply to what we call role variables—his intensional variables—whereas to simplify the definition of the language (to avoid typing relations), we do not; this is why we do not count role variables among the terms \(t\) in Definition 22.1. Second, we include constant symbols in the language, whereas for convenience Fitting does not. Third, we have a bimodal language with and , whereas Fitting has a monomodal language with \(\Box \). Finally, where we write \(\mathsf {P}(t,r_i)\), Fitting would write \(\mathsf {D}(r_i,t)\).
 13.
These models are almost the same as those for “contingent identity systems” in [48] (cf. [37]) and [54], but for a few differences: we follow Fitting in allowing \(F\) to contain partial functions; Parks does not deal with constants; and while Priest does deal with constants, he treats them as nonrigid. The differences between our models and Fitting’s [24, 25] are that we deal with constants, and Fitting defines \(V\) so that predicates can apply not only to elements of \(D\), but also to elements of \(F\) (cf. [37]).
 14.
We are not suggesting that all there is to a role is a partial frunction from \(W\) to \(D\); but such a function captures an important aspect of a role, namely the players of the role across worlds.
 15.
Some minor changes must be made, e.g., since we include constants in the language (recall note 12), but we will not go into the details here.
 16.
One may then wish to add the assumption that for all \(f\in F\), if \(f(w)=d\), then \(d\in V(\mathsf {E},w)\), i.e., if an object \(d\) plays a role for the agent in \(w\), then \(d\) exists in \(w\), validating \(\mathsf {P}(t,r_i)\rightarrow \exists _ax \,t=x\).
 17.
 18.
We did not define the interpretation of role variables in Definition 22.4, since we do not officially count them as terms (recall note 12), and they only appear in the \(\mathsf {P}(t,r_i)\) clause in Definition 22.5, where the assignment \(\mu \) takes care of them directly; but the definition would be \(\left[ r_i\right] _{\fancyscript{M},\,w,\,\mu } = \mu (r_i)\).
 19.
The exception among the authors referenced in note 17 is Carlson, who allows \(F\) to contain partial functions and uses a threevalued semantics to deal with undefined terms.
 20.
Remember that in (27) and (28), we read as “it is doxastically necessary that \(\varphi \)” or “in all worlds compatible with the agent’s beliefs, \(\varphi \)”. The whole of (27) gives the condition that the truth of the belief report imposes on the actual world and the space of the agent’s doxastic alternatives, so we would not read the second conjunct as “the agent believes that there exists ...”.
 21.
Belardinelli and Lomuscio [6] include both \(x\) and \(z\) variables in their multiagent quantified epistemic logic. Instead of distinguishing two types of variables, we could instead distinguish two types of quantifiers, in the tradition of Hintikka’s [33] distinction between \(\exists y\) and \(\mathrm {E}y\). By understanding \(\exists z\) quantification in terms of agentrelative roles, we are following Perry [51].
 22.
This point is inspired by Crimmins and Perry [19] on notion provision vs. notion contraint.
 23.
We could have two sorts of function variables, \(r_1,r_2,\dots \) for roles and \(s_1,s_2,\dots \) for nonrole functions. Or we could indicate the difference between role functions and nonrole functions by a oneplace predicate \(\mathsf {Role}\) whose extension contains only functions to be thought of as roles.
 24.
According to this intuitive understanding, the extension of \(\mathsf {Stip}\) should be a functional relation: if \(\mathsf {Stip}(r,s)\) holds, then \(\mathsf {Stip}(r,s{^\prime })\) should not hold for any \(s{^\prime }\not = s\).
 25.
The last of the quoted sentences occurs in footnote 7 of [1].
 26.
One may try to apply a similar strategy to predicate symbols in order to model agents who do not believe/know that two (necessarily) coextensive predicates are coextensive.
 27.
Compare this to the “fixedly actually” operator of Davis and Humberstone [21].
 28.
To deal in the 1D framework with an agent who does not believe, e.g., that something contains water iff it contains H20, we would need to generalize the notion of role so that properties (understood extensionally, intensionally, or hyperintensionally) can play roles for an agent.
 29.
Note that (55)/(56) does not require the existence of anyone who actually plays \(r_1\). We can express a reading that requires the existence of a roleplayer with: .
 30.
Aloni considers it an advantage of this more general semantics that we can have

(69)
\(\fancyscript{M},w\vDash _\pi \exists z^i\varphi (z^i)\wedge \lnot \exists z^j\varphi (z^j)\),
as if there is a shift in context midformula. Instead of doing this with one of Aloni’s models, we could consider two regular models \(\fancyscript{M}=\langle W,R_a,R_d,D,\pi (i),V \rangle \) and \(\fancyscript{M}{^\prime }=\langle W,R_a,R_d,D,\pi (j),V \rangle \), each associated with a different context, such that

(70)
\(\fancyscript{M},w\vDash _\mu \exists z\varphi (z)\) and \(\fancyscript{M}{^\prime },w\vDash _{\mu {^\prime }}\lnot \exists z\varphi (z)\).
Aloni’s motivation for considering (69) is the following kind of reasoning:

(I)
Ralph believes that the man with the brown hat is a spy.

(II)
The man with the brown hat is Ortcutt.

(III)
So Ralph believes of Ortcutt that he is a spy.

(IV)
Ralph believes that the man seen on the beach is not a spy.

(V)
The man seen on the beach is Ortcutt.

(VI)
So Ralph does not believe of Ortcutt that he is a spy.
Aloni concludes that
 (71)
should be satisfiable, which it is in her semantics. However, it seems to us to be a mistake to conclude (VI) on the basis of (IV) and (V). Instead, by analogy with (I)(III), one should conclude
 (VI\({^\prime }\)):

So Ralph believes of Ortcutt that he is not a spy.
Then we can express the compatibility of (III) and (VI\({^\prime }\)) by the satisfiable sentence

(72)
.
This is not to say, however, that there are not other good motivations for the more general semantics.

(69)
 31.
Thus, we have a termmodal logic in the sense of [23].
 32.
In English, to say “from the point of view of \(t\), \(\varphi \)”, might suggest that \(t\) believes \(\varphi \), but this it not the intended reading of \(\mathsf {pov}_t\varphi \), as its formal truth definition below makes clear.
 33.
While we take a centered world to be any element of \(W\times D\), one may wish to only admit pairs \(\langle w,c\rangle \) such that \(c\) is an agent (in some distinguished set \(Agt\subseteq D\)) and \(c\) exists in \(w\) (using the existence predicate of Sect. 22.3, \(c\in V(\mathsf {E},w)\)), but for simplicity we do not make these assumptions here. Also for simplicity, we are not putting explicit times into the centered worlds. Adding a temporal dimension to our framework would expand its range of application to further interesting issues.
 34.
By the truth definition, we have iff for all \(w{^\prime }\in W\), \(c{^\prime }\in D\), if \(\langle w,[t]_{\fancyscript{M},\,w,\,\mu }\rangle R_d\langle w{^\prime },c{^\prime }\rangle \), then \(\fancyscript{M},w{^\prime },c{^\prime }\vDash _\mu \varphi \). The problem with taking the operators as primitive instead of is that we would then lose important results of modal correspondence theory. For example, requiring that \(R_d\) be reflexive (thinking of it now as an epistemic accessibility relation) would not guarantee the validity of , since the reflexivity of \(R_d\) would not guarantee that \(\langle w,[t]_{\fancyscript{M},\,w,\,\mu }\rangle R_d\langle w,c\rangle \). But reflexivity would guarantee the validity of , as desired.
 35.
One may want to define the selfrole such that for all worlds \(w\) and agents \(c\), \(f_{self}(w,c)=c\).
 36.
A similar analysis applies to other wellknown problems in the theory of reference, such as Castañeda’s [16] puzzle about the first person. Surely through most of his life after 1884, Samuel Clemens believed that he wrote Huckelberry Finn. But one can imagine that in his dotage, Clemens held a copy of the book in his hand, saw that it was written by Mark Twain, but couldn’t remember that ‘Mark Twain’ had been his pseudonym and had no inclination to say “I wrote this”. Castañeda made the point, with many similar examples, that even in the latter case, we could say

(74)
Samuel Clemens believes that he wrote Huckelberry Finn.
since he is Mark Twain, and he believes that Mark Twain wrote Huckelberry Finn. However, in the sense in which it was true through much of his life that he believed he wrote Huckelberry Finn, at this moment late in his life, it is not. There is a reading of (74) on which it is false.
In the case we are imagining, Samuel Clemens plays (at least) two roles in Samuel Clemens’ life, the selfrole \(\mathbf {r}_{self}\) and the role \(\mathbf {r}_{MT}\) of being the source of the ‘Mark Twain’ namenetwork that is exploited by the use of that name on the book he holds in his hands. Given this, we can distinguish between the two readings of (74), the first false and the second true, as follows:

(75)
;

(76)
.

(74)
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Acknowledgments
For helpful discussion or comments on this paper, we thank Johan van Benthem, Russell Buehler, Thomas Icard, David Israel, Ethan Jerzak, Alex Kocurek, Daniel Lassiter, John MacFarlane, Michael Rieppel, Shane SteinertThrelkeld, Justin Vlasits, and Seth Yalcin.
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Holliday, W.H., Perry, J. (2014). Roles, Rigidity, and Quantification in Epistemic Logic. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/9783319060255_22
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