Abstract
Should objects count as necessarily having certain properties, despite their not having those properties when they do not exist? For example, should a cat that passes out of existence, and so no longer is a cat, nonetheless count as necessarily being a cat? In this essay I examine different ways of adapting Aldo Bressan’s ML^{ν} so that it can accommodate an affirmative answer to these questions. Anil Gupta, in The Logic of Common Nouns, creates a number of languages that have a kinship with Bressan’s ML^{ν}, three of which are also tailored to affirmatively answering these questions. After comparing their languages, I argue that metaphysicians and philosophers of language should prefer ML^{ν} to Gupta’s languages in most applications because it can accommodate essential properties, like being a cat, while being more uniform and less cumbersome.
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Notes
For an introduction to ML^{ν} that is more accessible than Bressan’s book, see [1]. Since submitting this essay for review, Belnap and Müller have published two essays selfconsciously developing the first order fragment of ML^{ν} [3] & [2]. In the first of these, they do an especially nice job of explaining the virtues of ML^{ν} and of their own CaseIntensional First Order Logic (CIFOL), in comparison to other quantified modal logics. I refer the reader to this essay for a more comprehensive discussion of the related languages than I will provide.
Bressan first suggests how his account can be modified to deal with objects that may not exist in all cases in [4, p. 89]. He amends this suggestion in [5, p. 372]. In both of these discussions, he passes over our problem for the semantics of necessity in silence. Gupta, however, struggles mightily with the issue in developing his L_{3}. It is worth noting that Belnap and Müller end up treating the semantics of necessity in CIFOL in a very similar manner to the way that I suggest we should treat it in ML^{ν} [3, esp. 419].
Following Bressan, I will be using the term ‘case’ in order to stay neutral between interpreting modal indices either as worlds or times. And although, of course, there are quite important differences that arise when interpreting modal indices in different ways, as much as possible I will be attempting to work at a level of abstraction that is above these.
Gupta presents an argument for the distinct logical treatment of common nouns that bring with them such criteria of identity [10, esp. ch. 1, §5]. Bressan offers some assessment of this argument [5, §N6]. And McCawley has a nice brief discussion [11].
Although it will be predicates that provide principles of transcase identity in these languages, we need not take this to be an implicit endorsement of ‘contingent’ or ‘relative identity .’ Rather, we just need to think common nouns like ‘cat,’ ‘horse,’ or ‘person’ have distinctive semantic properties that distinguish them from predicates like ‘red’ or ‘smooth’ that do not, and that this difference is worth modeling in our language.
cf. e.g., [6, ch. 15 & 16]
Specifically, as a practicing physicist, Bressan wanted to capture Mach’s definition of “mass” in terms of possible experiments.
[12]. Another major motivation for this extensionalism seems to have come from reading intension as on the side of the mental. Although there was plenty of historical warrant for this, it is clear that Bressan’s intensions are patterns of extensionsatacase, as the case varies. There is nothing mental about them.
Bressan puts the point that common nouns, extensional predicates, and intensional predicates will be treated in a syntactically uniform way in ML^{ν} rather strongly by claiming that “no a priori distinction is made in ML^{ν} between common nouns and 1ary predicates” [5, p. 351]. Bressan treats all predication as intensional. There are not different semantic rules for assessing extensional and intensional predicates. Nonetheless, extensional predicates are distinctive, since their truth in a case only depends on the extension in that case of the individual concepts falling under it, not the extensions of these concepts in other cases. This allows Bressan to preserve a uniform treatment of predication while still capturing significant semantic differences for extensional predication. (The relevant technical details for understanding how this works will be presented in the next section.)
I give a formal statement of these differences in the Appendix on Gupta’s languages.
Bressan, Gupta, and Belnap all designate nonexistence by having a single nonexistent entity of each type. A few other ways of representing nonexistence seem available to us. First, we could leave individual concepts undefined in the cases in which they don’t exist. Second, we could have lots of nonexistent entities—most intuitively one for every possible object. Using the second of these would have the advantage of trivializing the problems that arise for Bressan with the introduction of nonexistence. Since that would make for an uninteresting essay, and mean accepting a vast menagerie of nonexisting things into our ontology, I will leave it aside. The first option, however, will come up again below. (Gupta considers these two options briefly [10, p. 68].)
Montague treated terms like John as denoting not an individual, John, but a set of John’s properties, where properties are intensional, mapping indices to sets of extensions. This Russalian treatment made a corresponding treatment of nonexistent objects natural. Accordingly, when translating a sentence like ‘John seeks a unicorn’ into IL, ‘a unicorn’ will be treated as a property of properties . (For a nice explanation see [8, ch. 7, §V].) This kind of approach means that one can avoid countenancing nonexistent objects in one’s ontology, give a decent rendering of the sentence, and get close to the specificity that one could have by allowing a different nonexistent entity for every merely possible object. This kind of strategy is very different from the ones we will be pursuing, in part because we will treat singular terms as designating individuals in a Fregean manner.
This means that what A ranges over will shift according to application—variables and constants will be atomic expressions of one type, truth values will be another, predicates taking constants as arguments another, etc.
With these definitions in place, for the most part, in the rest of the essay I will suppress types because they are largely irrelevant to the issues under consideration.
Belnap defines this notion as a counterpart to Bressan’s quasiintension function.“ Q I _{ I }(A)”signifies “the quasiintension on interpretation I of A”. This gives a function from the set of cases Γto expressions of the appropriate type τ,that is, a member of I n t _{ τ }.
This makes formal issues over contingent vs. strict identity quite clear. For a nice defense of why treating equality this way is preferable in ML^{ν} see [1, p. 3637].
Furthermore, parallel to the doubling of option (3), in order to preserve unrestricted quantification on option (2) one would need two distinct domains, one which includes merely possible objects, and one that doesn’t. Such a complication seems worth avoiding, if possible.
Gupta tinkers with Bressan’s notions. Although every common noun does provide a principle of identity for tracing the objects it is true of across cases, not every common noun designates a kind of substance. For example, ‘man born in Jerusalem’ does not, since being born in Jerusalem is not an essential property of the man. Unlike Bressan, who is only concerned with modeling substance kinds through his absolute concepts, Gupta marks this difference. He does this by distinguishing between sorts and substance sorts, which are the intensions assigned to the two kinds of common nouns. Roughly, sorts provide principles of identity that allow one to trace an object from case to case because they are separated intensional predicates, while substance sorts also indicate essential properties, and so are constant. Gupta maintains that for every sort, there is a substance sort that underlies it, which accounts for why it is separated (for discussion, see [11]). Although Gupta’s substance sorts correspond to Bressan’s absolute concepts, and Gupta does not need to substantively alter Bressan’s modal constancy, because Bressan’s modal separation is case relative it will not do for modeling a principle of transcase identity. For some of the technical details on how Gupta modifies modal separation so that it can effectively model the principles of identity of common nouns see the Appendix.
Gupta goes so far as to suggest an added condition on sorts: that they never apply to nonexistent objects [10, p. 69n].
In defining QuasiModal Separation [5, p. 372],Bressan accidentally omits the diamond. (The diamond is not forgotten in[4, p. 94].)The ramifications of this omission illuminate the relative importance of QMC and QMS.Leaving it out weakens the requirement because without the diamond a predicate can still bequasimodally separated in a case even if two individual concepts falling under it overlap, aslong as they don’t overlap in the case under consideration. Keeping the diamond means theindividual concepts can’t overlap in any case if the concept they are falling under is separatedin that case. Since quasiabsolute concepts are also quasimodally constant, whether thediamond is included or not makes no difference to them. Still, including the diamond ispreferable because quasimodal separation is intended to let us trace the same object fromcase to case, and if two objects overlap in some case, then from that case it is impossible toknow which object to trace back through the other cases. These considerations help showthat separation is more important than constancy for tracing objects. In Chapter 4 of hisbook, Gupta elaborates an elegant solution to Chisholm’s transworld identity problemfor inanimate objects (like bikes or Theseus’s ship) that admits such objects aren’t evenquasimodally constant but which shows that as long as the corresponding sortal predicate(e.g. ‘x is a bike’) is quasiseparated in every world, this is enough to trace them across worlds[10, p. 86107,esp. p. 104106].
This is the gist of the footnote on p. 70 of [10].
Again, I give some of the details of Gupta’s account of these notions in the Appendix.
For Gupta’s discussion of the problem, see chapter 3, §2 [10, p. 7178].
There is a pardonable abuse of notation here that I will continue in what follows. \(QE_{\gamma _{3}, I}C(eh^{*})=F\) abbreviates: \(QE_{\gamma _{3}, I}C(x)=F\) where I(x) = e h ^{∗}, and similarly for \(QE_{\gamma _{3}, I}C(f^{**})=F\).
How much we agree with Gupta here will depend on what we are using our logic for and how we are interpreting our cases. At first at least, it will seem we won’t want to count merely possible cats as cats, if we interpret cases as worlds, since we don’t want to have to consider all of the merely possible men in the room when we talk about men. On the other hand, if cases are interpreted as times or moments in possible histories, and we want to model, “Mama could have had two more kittens than she in fact had” it seems odd to insist that what we are referring to are not cats.
This rule of thumb will certainly not be hard and fast, and I do not take deciding between these options to be a matter for logic. Still, perhaps it is worth mentioning that my own view is that merely possible cats, men, or unicorns, should usually count as of their kind, in line with Fig. 3 and against Gupta. Possible, dead, or imaginary men seem to be no less men than do living ones, and their nonexistence is marked by their having the nonexistent object as their extension. (Kant’s remark about the hundred Thalers comes to mind (CpR, A599/B627).)
L _{4}, however, is an exception. It gives up on the thought that possibly nonexistent cats are necessarily cats and is much closer to ML^{ν} than Gupta’s other languages because its variables range over individual concepts rather than extensions.
This is closely related to Gupta’s “initial intuition”[10, p. 71].
Perhaps the semantics can, somewhat controversially, be extended to two (or more) place relations by the following maneuver:
$$QE_{\gamma, I} \psi(i, j)=T^{*} \text{ iff } QE_{\gamma, I} \psi(i, j)\neq T \text{ and either } i(\gamma)=* \text{ or } j(\gamma)=* $$Here, on a temporal reading, “I am the great grandchild of my great grandfather” would be T* (just as with “Socrates is a man”), since my great grandfather has passed away, if we treat the predicates as ‘is a cat’ in Fig. 2. If we treated them as in Fig. 3, however, their value would be T, and we need not treat all predicates one way or the other.
N.B. if formulas only depend on objects whose extension is nonexistent and are false of those, they will still come out T*. For example, “Socrates is sitting” is T*.
Arguably, an example might be, ‘there is something that is a flying horse,’ where the intensional object that makes this true is Pegasus.
The full tables for the two place logical operators are:
$$\begin{array}{lccr} \begin{array}{lcccc} \wedge & T & T^{*} & F^{*} & F\\ T & T & T^{*} & F^{*} & F\\ T^{*} & T^{*} & T^{*} & F^{*} & F\\ F^{*} & F^{*} & F^{*} & F^{*} & F\\ F & F & F & F & F\\ \end{array} & & &\begin{array}{lcccc} \vee & T & T^{*} & F^{*} & F \\ T & T & T & T & T\\ T^{*} & T & T^{*} & T^{*}(T) & T^{*}\\ F^{*} & T & T^{*}(T) & F^{*} & F^{*}\\ F & T & T^{*} & F^{*} & F\\ \end{array} \end{array} $$$$\begin{array}{lccr} \begin{array}{lcccc} \rightarrow & T & T^{*} & F^{*} & F \\ T & T & T^{*} & F^{*} & F\\ T^{*} & T & T^{*}(T) & F^{*} & F^{*}\\ F^{*} & T & T^{*}(T) & T^{*}(T) & T^{*}\\ F & T & T & T & T\\ \end{array} & & &\begin{array}{lcccc} \leftrightarrow & T & T^{*} & F^{*} & F \\ T & T & T^{*} & F^{*} & F\\ T^{*} & T^{*} & T^{*}(T) & F^{*}(F) & F^{*}\\ F^{*} & F^{*} & F^{*}(F) & T^{*}(T) & T^{*}\\ F & F & F^{*} & T^{*} & T\\ \end{array} \end{array} $$I have listed two truth values for some of the operations because while they will usually have the first value, if the reasons that the values of the component formulas was T* or F* in the first place line up, then it seems they should have the second value. For example, if two formulas involving universal quantification are T*, then, for each, there will be individual concepts, d _{1}…d _{ n }, whose extensions are nonexistent and which falsify them. Taking one of these formulas as the antecedent and the other as the consequent, if those individual concepts that falsify the antecedent are a superset of those on which the consequent is false, then it should be: T ^{∗}→ T ^{∗} = T. Or if the antecedent is F* and the consequent T*, then the way the conditional will come out T is if none of the values on which F* is true are also those on which the consequent is false. So that the operations are truth functional, it makes sense to assign the value that is not in parentheses, despite the fact that in specific cases the assignment of the other value can be justified.
Again, for technical details see the Appendix.
There are, however, serious issues with the semantics of quantification and necessity for this strategy. Specifically, as Gupta notes [10, p.7275], although intuitively the assignments of variables that are not free in a formula should be ignored when figuring out their semantic value, implementing this is difficult, and for both L_{2} and L_{3}, the schema ‘ A → (∀K ,x)A’ is invalid.
Gupta comments on this [10, p. 7778].
This means the problem with Q A b s that Gupta points to in his footnote (which was discussed on page 13 above) is not one for Q A b s ^{MC} [10, p. 70].
Belnap and Müller hit upon the same conception of an essential property in developing CIFOL [3, § 5.3]. They do a nice job of showing how absolute concepts will not be the only essential properties. Properties like the sex of a horse, which we commonly take to be essential to it qua horse, will also come out as essential in this sense.
Fine is after this distinction with his example of Socrates and his singleton [9, p. 241].
For more details and discussion see [10, ch. 1, §12].
References
Belnap, N. (2006). Bressan’s typetheoretical combination of quantification and modality. In Lagerlund, H., Lindström, S., & Sliwinski, R. (Eds.), Modality Matters: TwentyFive Essays in Honour of Krister Segerberg, volume 53, pages 31–53. Dept. of Philosophy, Uppsala University, Sweden.
Belnap, N., & Müller, T. (2014a). BHCIFOL: Caseintensional first order logic. Journal of Philosophical Logic, 43(5), 835–866.
Belnap, N., & Müller, T. (2014b). CIFOL: Caseintensional first order logic. Journal of Philosophical Logic, 43(2), 393–437.
Bressan, A. (1972). A General Interpreted Modal Calculus: Yale University Press.
Bressan, A. (1993). On Gupta’s book The Logic of Common Nouns. Journal of Philosophical Logic, 22(4), 335–383.
Cresswell, J., & Hughes, G. (2004). A New Introduction to Modal Logic: Taylor & Francis.
Cresswell, M. (1975). Hyperintensional logic. Studia Logica, 34(1), 25–38.
Dowty, D., Wall, R., & Peters, S. (1981). Introduction to Montague Semantics, volume 11 of Synthese language library. D. Reidel Publishing Company.
Fine, K. (1995). The logic of essence. Journal of Philosophical Logic, 24(3), 241–273.
Gupta, A. (1980). The Logic of Common Nouns: An Investigation in Quantified Modal Logic: Yale University Press.
McCawley, J. (1982). Review of The Logic of Common Nouns. Journal of Philosophy, 79, 512–517.
Quine, W.V.O. (1953, 1980). Reference and modality. In From a Logical Point of View: 9 Logicophilosophical Essays, Logicophilosophical essays, 2nd edition: Harvard University Press.
Acknowledgments
My research into Bressan’s and Gupta’s languages began with a semester of funding as the Allan Ross Anderson fellow in the spring of 2008. During this time the essay began to take shape under the patient guidance of Nuel Belnap, to whom I am very grateful. I should also note that during the revision process, an anonymous reviewer at JPL offered truly exceptional feedback that substantially improved the final version of the essay. In addition, Shawn Standefer and Anil Gupta gave me helpful comments through out the process.
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Appendix: Gupta’s Languages
Appendix: Gupta’s Languages
In this appendix I will give some of the technical details concerning two aspects of Gupta’s Languages. First, I will present Gupta’s various versions of the notions of a substance sort, constancy, and separation, which correspond to Bressan’s absolute concepts, modal constancy, and modal separation. Second, I will give some of the details of Gupta’s treatment of the semantics of quantification and necessity. When he introduces nonexistent objects, Gupta struggles mightily to adapt the semantics of necessity so that possibly nonexistent cats can still be necessarily cats, but as I mentioned in the conclusion, he does not ultimately arrive at a satisfying formulation. (Gupta recognizes this, to a degree [10, p. 75].) I will try to give some sense of why, without going through the full story [10, ch. 3]. All page numbers included with the definitions are to the corresponding definitions in [10].
Before doing either of these, some preliminaries are in order. In addition to the standard logical categories, Gupta’s L_{1} includes a category for common nouns. Although the syntactic rules of L_{1} are fairly straightforward and I will not rehearse all of them, quantification is always restricted to quantification over a certain sort of thing by a common noun, so the syntactic rules governing these are worth presenting:
Definition 12 (Some of Gupta’s syntax; cf. p. 7)

(i)
If K is a common noun, x is a variable, and A is a formula, then (∀K ,x)A is a formula.

(ii)
If K is a common noun, x is a variable, and A is a formula, then (K,x)A is a common noun.
The second clause allows for complex common nouns built from simpler ones, such as ‘Man who likes Margret.’^{Footnote 40}
The semantics of Gupta’s languages begins standardly enough with:
Definition 13 (Model Structure for L_{1}; p. 18)
A model structure for L_{1}is an ordered triple < W,D,i ^{∗} > ,where:

(i)
W is a nonempty set,

(ii)
D is a function that assigns to each member of W a nonempty set,

(iii)
i ^{∗}is a function that assigns to each member w of W a member of D(w).
Think of W as the set of possible worlds (or cases), D(w) as the set of objects that exist in w, and i ^{∗} as the individual concept whose extension in all worlds is the nonexistent object (i.e. i(w) = ∗ in all w).
As with Bressan’s absolute concepts, Gupta models substance sorts through an intensional property that is constant and separated:
Definition 14 (Gupta’s Substance Sort in L_{1}; p. 35)
A substance sort in a model structure is a modally constant and separated intensional property.
Modal constancy in L_{1} is not substantively different from Bressan’s, although Gupta states it slightly differently. Where is a model structure:
Definition 15 (Gupta’s Modal Constancy; p. 27)
An intensional property \(\mathcal {S}\) in is modally constant iff \(\mathcal {S}(w)=\mathcal {S}(w^{\prime })\)at all worlds w,w ^{′}∈ W.
That is, \(\mathcal {S}\) will be constant if the individual concepts in the extension of \(\mathcal {S}\) for any world are the same.
I mentioned in footnote 18 that Gupta adjusts Bressan’s notion of modal separation because it is caserelative, and so it will not do for modeling a principle of transcase identity. Gupta’s preferred modification is his own ‘separation:’
Definition 16 (Gupta’s Separation; p. 29)
An intensional property \(\mathcal {S}\) in is separated iff all individual concepts i,i ^{′}that belong to \(\mathcal {S}\)at any worlds w,w ^{′}are such that if i(w _{1}) = i ^{′}(w _{1})at a world w _{1},then i = i ^{′}.
He uses this to define his notion of a sort:
Definition 17 (Gupta’s Sort in L_{1}; p. 33)
A sort in a model structure is an intensional property in which is separated.
In general Gupta uses variables ‘\(\mathcal {S}\)’, ‘\(\mathcal {S^{\prime }}\)’, ‘\(\mathcal {S}_{1}\)’, etc. to range over sorts in a fixed model structure .
In addition to Gupta’s separation, there is also a weaker notion that is like Bressan’s worldrelative separation, except that it holds at every case.
Definition 18 (Gupta’s WorldRelative Separation; p. 29n)
An intensional property \(\mathcal {S}\) in is separated in the world w iff all individual concepts i,i ^{′}that belong to \(\mathcal {S}\)at w are such that if i(w _{1}) = i ^{′}(w _{1})at a world w _{1},then i = i ^{′}.
Definition 19 (Gupta’s Weak Separation; p. 29n)
An intensional property\(\mathcal {S}\) in is weakly separated iff\(\mathcal {S}\)is separated in every world.
Intuitively, the difference is that while separation says that the extension of two individual concepts that are \(\mathcal {S}\) in any world (even if these worlds are different) will never overlap at a world, weak separation just says that at each world the individual concepts that are \(\mathcal {S}\) in that world will not have the same extension in any world. Weak separation can provide a principle of identity, and Gupta develops a closely related notion that also incorporates a treatment of nonexistence in L_{5} of chapter 4.
In order to accommodate nonexistence, in chapter 3 Gupta modifies L_{1} in three different ways, treating the semantics of quantification and necessity slightly differently in each of L_{2}, L_{3}, and L_{4}. Still, the way that he treats modal constancy and separation in each of these is the same:
Definition 20 (Gupta’s Near Constancy; p. 6970)
An intensional property \(\mathcal {S}\)is nearly constant in iff, if an individual concept i belongs to \(\mathcal {S}\)at any world w, then i belongs to \(\mathcal {S}\)at all worlds w ^{′}such that i(w ^{′})≠i ^{∗}(w ^{′}).
Definition 21 (Gupta’s Near Separation; p. 69)
An intensional property \(\mathcal {S}\)is nearly separated in iff all individual concepts i,i ^{′}that belong to \(\mathcal {S}\)at some worlds (i.e., \(i\in \mathcal {S}(w_{1})\)and \(i^{\prime }\in \mathcal {S}(w_{2})\),for some w _{1},w _{2} ∈ W)are such that if i(w) = i ^{′}(w)≠i ^{∗}(w)at a world w, then i = i ^{′}.
As I just noted here and in footnote 18, Gupta is concerned with distinguishing two forms of principles of identity associated with common nouns, sorts and substance sorts, where only the latter apply to essential properties.
Definition 22 (Gupta’s Sort in L_{2}L_{4}; p. 69)
\(\mathcal {S}\)is a sort in a model structure iff \(\mathcal {S}\)is an intensional property and \(\mathcal {S}\)is nearly separated in .
Definition 23 (Gupta’s Substance Sort in L_{2}L_{4}; p. 70)
\(\mathcal {S}\)is a substance sort in iff \(\mathcal {S}\)is a sort in and \(\mathcal {S}\)is nearly constant in .
As with Gupta’s separation and Bressan’s modal separation, Gupta’s near separation differs from Bressan’s quasimodal separation in that it is not caserelative and so can serve as a principle of transcase identity. In chapter 4, Gupta modifies separation and constancy again, this time endorsing for his ‘quasiseparation’ something like Bressan’s ‘quasimodal separation’ but in every case. For the details, as well as the corresponding required adjustment of constancy for L_{5}, see [10, p. 104, & p. 107].
There are a few background notions that we need to have in place before we can look at Gupta’s treatment of the semantics for quantification and necessity. First, he defines two sets of brackets. Given a sort \(\mathcal {S}\) in , he designates by \(\mathcal {S}[w]\) the set of objects that fall under \(\mathcal {S}\) in w, and by \(\mathcal {S}[{\kern 2.3pt}[ w ]{\kern 2.3pt}]\) the set of objects that are possibly \(\mathcal {S}\).
Definition 24 (p. 35)
Definition 25 (p. 36)
Next he defines what it means to be ‘the same \(\mathcal {S}\)’ and ‘an \(\mathcal {S}\) counterpart:’
Definition 26 (‘the same \(\mathcal {S}\)’ in L_{1}; p. 36)
d in w is the same \(\mathcal {S}\)as d ^{′}in w ^{′}iff there is an individual concept, i, that belongs to\(\mathcal {S}\)at some world, and i(w) = d and i(w ^{′}) = d ^{′}.
Definition 27 (‘an \(\mathcal {S}\) counterpart’; p. 36)
The \(\mathcal {S}\)counterpart in w ^{′}of the individual d in w (abbreviated \(\mathcal {S}(w^{\prime }, d, w)\))is the unique individual d ^{′}such that d ^{′}in w ^{′}is the same \(\mathcal {S}\)as d in w.
As Gupta points out, “\(\mathcal {S}(w^{\prime }, d, w)\) is well defined if \(d\in \mathcal {S}[{\kern 2.3pt}[ w ]{\kern 2.3pt}]\). For if \(d\in \mathcal {S}[{\kern 2.3pt}[ w ]{\kern 2.3pt}]\), then there is an individual concept i belonging to \(\mathcal {S}\) at some world such that i(w) = d. The separation of \(\mathcal {S}\) implies that i is unique. Hence there is a unique d ^{′}, namely i(w ^{′}), which is the same \(\mathcal {S}\) as d in w” [10, p. 36].
Now to define the assignment function, we first need the notion of a model:
Definition 28 (Model for L_{1}; p. 37)
A model for L_{1}is anordered quintuple < W,D,i ^{∗},m,ρ > ,where:

(i)
< W,D,i ^{∗} > is a model structure,

(ii)
m is a function that assigns (a) to each individual constant ofL_{1}an individual concept, (b) to each nary predicate an nary relation, and (c) to eachatomic common noun a sort,

(iii)
ρ ∈ W.
Through the function m a model in L_{1} assigns an intension to each atomic expression and ρ specifies the real world.
With the notion of a model, an assignment is:
Definition 29 (Assignments for L_{1}; p. 38)
An assignment for L_{1}relative to a model M =< W,D,i ^{∗},m,ρ > is a function that assigns to each variable ofL_{1}an ordered pair \(< \mathcal {S}, d >\),where \(\mathcal {S}\)is a sort relative to the model structure and d ∈ U(= ∪_{ w∈W } D(w)).
Here if a is an assignment, a _{ o }(x) is the object assigned to x by a and a _{ s }(x) is the sort assigned to x by a. Using this, Gupta defines a few notions that he then deploys in defining the semantic value of formulas involving quantification and necessity:
Definition 30 (Normal assignments for L_{1}; p. 38)
An assignment a (for L_{1}relative to a model M) is normal in w iff a _{ o }(x) ∈ a _{ s }(x)[[w]]for all variables x.
Definition 31 (\(\mathcal {S}\) variants for L_{1}; p. 3839)
An assignment a^{′} is an \(\mathcal {S}\) variant of a at x in w iff

(i)
a ^{′}is just like a except perhaps at x (abbreviated to\(a^{\prime } \underset {x}{\bumpeq } a\)),

(ii)
\(a^{\prime }_{s}(x) = \mathcal {S}\),

(iii)
\(a^{\prime }_{o}(x)\in \mathcal {S}[w]\).
Definition 32 (World variants for L_{1}; p. 39)
The w^{′} variant of a relative to w (abbreviated to f(w ^{′},a,w)) isthe unique assignment a ^{′}that meets the following conditions:

(i)
\(a^{\prime }_{s}(x)=a_{s}(x)\)at all variables x,

(ii)
\(a^{\prime }_{o}(x)\) in w ^{′}is the same \(a^{\prime }_{s}(x)\)as a _{ o }(x)in w, at all variables x.
If these conditions are not met by any assignment then f(w ^{′},a,w) is undefined.
Now, having defined M, w, and a, Gupta then defines through induction on the length of expression α the concept: “the semantic value of α at a world w in a model M relative to the assignment a normal in w” [10, p. 40]. He abbreviates this to \(V^{w}_{M, a}(\alpha )\). Before giving this definition, however, it will help with quantification to have defined one more function that gives the intension of an expression α in a model M, for an assignment a, and a world w:
Definition 33 (Intension function for L_{1}; p. 40)
Let M,w,a,and α be as above.Then \(I^{w}_{M, a}(\alpha )\)is a function with domain W that satisfies the followingcondition:
Gupta uses m to define the valuation function V as expected for individual constants, variables, and common nouns, and the value of an equality or truth function are found in the standard ways [10, cf. p. 4041]. I include the definition of V for nary relations to give a better sense of how things run:
Definition 34 (Part of \(V^{w^{\prime }}_{M, a}(\alpha )\); p. 4041)
Let M,w,a,and α be as above. Then V isdefined by induction on α:

(i)
If α is the atomic formula F(t _{1},…,t _{ n }),then \(V^{w}_{M, a}(\alpha )=T\) if \(\langle V^{w}_{M, a}(t_{1})\),…,\(V^{w}_{M, a}(t_{n})\rangle \in m(F)(w)\).Otherwise \(V^{w}_{M, a}(\alpha )=F\).

(ii)
If α is the formula □A,then \(V^{w}_{M, a}(\alpha )=T\) if \(V^{w^{\prime }}_{M, f(w^{\prime }, a, w)}(A)=T\)at all worlds w ^{′}∈ W.Otherwise \(V^{w}_{M, a}(\alpha )=F\).

(iii)
If α is the formula (∀K ,x)A,then \(V^{w}_{M, a}(\alpha )=T\) if \(V^{w}_{M, a^{\prime }}(A)=T\)for all assignments a ^{′}that are \(I^{w}_{M, a}(K)\)variants of a at x in w. Otherwise \(V^{w}_{M, a}(\alpha )=F\).

(iv)
If α is the common noun (K,x)A,then \(V^{w}_{M, a}(\alpha )\)is the set of individual concepts i such that\(i \in V^{w}_{M, a}(K)\)and \(V^{w}_{M, a^{\prime }}(A)=T\),where \(a^{\prime } \underset {x}{\bumpeq } a\)and \(a^{\prime }_{s}(x)=I^{w}_{M, a}(K)\)and a _{ o }(x) = i(w).
Working back through the definitions, with this semantics one can see how assignment functions contribute to fixing principles of transworld identity in the way described in Section 7.2.
In accommodating nonexistent objects Gupta takes over many of these definitions, only modifying them when necessary. The main difficulty comes with the semantics of necessity and how it interacts with quantification. To give the modifications of these he first revises the notions of being ‘the same \(\mathcal {S}\):’
Definition 35 (‘the same \(\mathcal {S}\)’ with nonexistents; p. 71)
d in w is the same \(\mathcal {S}\)as d ^{′}in w ^{′}iff d≠i ^{∗}(w)and d ^{′}≠i ^{∗}(w ^{′})and there is an individual concept, i, which belongs to\(\mathcal {S}\)at some world, and i(w) = d and i(w ^{′}) = d ^{′}.
Deploying this new version of ‘the same \(\mathcal {S}\)’ then has the effect of changing the sense of ‘an \(\mathcal {S}\) counterpart’ (abbreviated \(\mathcal {S}(w^{\prime }, d, w)\)), and a ‘world variant’ (abbreviated f(w ^{′},a,w)), although the wording of the definitions of these notions can stay the same.
Now the intuition that Gupta tries to capture with the semantics of necessity is “an object d of the sort \(\mathcal {S}\) satisfies □F x in w iff d satisfies F x in w, and at all worlds w ^{′} at which \(\mathcal {S}(w^{\prime }, d, w)\) is defined, \(\mathcal {S}(w^{\prime }, d, w)\) satisfies F x in w ^{′}” [10, p. 71]. He does this with the following in L_{2}, which replaces Definition 34.(ii):
Definition 36 (Box rule for L_{2}; p. 72)
Let M,w,a,and α be asabove:

(i)
If α is the formula □A,then \(V^{w}_{M, a}(\alpha )=T\) if \(V^{w^{\prime }}_{M, f(w^{\prime }, a, w)}(A)=T\) at all worlds w ^{′} at which f(w ^{′},a,w) is defined. Otherwise \(V^{w}_{M, a}(\alpha )=F\).
This definition runs into serious trouble, one aspect of which I eluded to in footnote 34, because f(w ^{′},a,w) is undefined at a world w ^{′} now whenever there is no \(\mathcal {S}\) variant at a w ^{′} for one of the variables that gets assigned an object by a. This will be true, even if the variable in question does not figure in the formula under consideration. As a result, the semantic values of the formula at worlds which should be considered end up being discounted. And even though Gupta makes progress on this issue in the semantics of L_{3}, there are still serious issues [10, cf. p. 7275].
To respond to these, in L_{4} Gupta gives up on the assignment of sorts and objects to variables and instead just assigns them individual concepts. With necessity, he now takes into account all of the worlds.
Definition 37 (Box rule for L_{4}; p. 76)
Let M,w,a,and α be asabove:

(i)
If α is the formula □A,then \(V^{w}_{M, a}(\alpha )=T\) if \(V^{w^{\prime }}_{M, a}(A)=T\)at all worlds w ^{′}∈ W.Otherwise \(V^{w}_{M, a}(\alpha )=F\).
Now that he is not doing anything to discount those cases where the objects figuring in A in various worlds have * as their extension in that world, only the cats of Fig. 3 on p. 15 will count as necessarily cats, not the ones in Fig. 2. That is, since C(e h∗) = F in γ _{3}, e h ∗ will not necessarily be a cat, even though it is one in all of the cases in which it exists. Thus, with L_{4} Gupta gives up on accommodating our ‘rough definition’ of necessity (Definition 9).
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Nunez, T. Modeling Unicorns and Dead Cats: Applying Bressan’s ML^{ν} to the Necessary Properties of Nonexistent Objects. J Philos Logic 47, 95–121 (2018). https://doi.org/10.1007/s1099201694186
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DOI: https://doi.org/10.1007/s1099201694186