Modeling Unicorns and Dead Cats: Applying Bressan’s ML^ν to the Necessary Properties of Non-existent Objects

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.

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 non-existent objects, Gupta struggles mightily to adapt the semantics of necessity so that possibly non-existent 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 L₁ includes a category for common nouns. Although the syntactic rules of L₁ 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.’⁴⁰

The semantics of Gupta’s languages begins standardly enough with:

Definition 13 (Model Structure for L₁; p. 18)

A model structure for L₁is 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 non-existent 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 L₁; p. 35)

A substance sort in a model structure is a modally constant and separated intensional property.

Modal constancy in L₁ 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 case-relative, and so it will not do for modeling a principle of trans-case 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 L₁; 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 world-relative separation, except that it holds at every case.

Definition 18 (Gupta’s World-Relative 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 non-existence in L₅ of chapter 4.

In order to accommodate non-existence, in chapter 3 Gupta modifies L₁ in three different ways, treating the semantics of quantification and necessity slightly differently in each of L₂, L₃, and L₄. Still, the way that he treats modal constancy and separation in each of these is the same:

Definition 20 (Gupta’s Near Constancy; p. 69-70)

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 L₂-L₄; 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 L₂-L₄; 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 quasi-modal separation in that it is not case-relative and so can serve as a principle of trans-case identity. In chapter 4, Gupta modifies separation and constancy again, this time endorsing for his ‘quasi-separation’ something like Bressan’s ‘quasi-modal separation’ but in every case. For the details, as well as the corresponding required adjustment of constancy for L₅, 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)

$$\begin{array}{l} \mathcal{S}[w]=_{df}\{d: d\in D(w) \text{ and there is an individual concept} \\\quad\quad\quad\quad\quad i \in \mathcal{S}(w) \text{ such that}~ i(w)=d.\} \end{array} $$

Definition 25 (p. 36)

$$\begin{array}{l} \mathcal{S}[{\kern-2.3pt}[ w ]{\kern-2.3pt}]=_{df}\{d: d\in D(w) \text{ and there is an individual concept} \\ \quad\quad\quad\quad\quad i \in \mathcal{S}(w^{\prime}) \text{ for some } w' \in W \text{ such that} ~i(w)=d\}. \end{array} $$

Next he defines what it means to be ‘the same $\mathcal {S}$’ and ‘an $\mathcal {S}$ counterpart:’

Definition 26 (‘the same $\mathcal {S}$’ in L₁; 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 L₁; p. 37)

A model for L₁is 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 ofL₁an individual concept, (b) to each n-ary predicate an n-ary relation, and (c) to eachatomic common noun a sort,

• (iii) ρ ∈ W.

Through the function m a model in L₁ 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 L₁; p. 38)

An assignment for L₁relative to a model M =< W,D,i ^∗,m,ρ > is a function that assigns to each variable ofL₁an 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 L₁; p. 38)

An assignment a (for L₁relative 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 L₁; p. 38-39)

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 L₁; 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 L₁; 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:

$$(I^{w}_{M, a}(\alpha))(w^{\prime})=V^{w^{\prime}}_{M, f(w^{\prime}, a, w)}(\alpha)$$

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. 40-41]. I include the definition of V for n-ary relations to give a better sense of how things run:

Definition 34 (Part of $V^{w^{\prime }}_{M, a}(\alpha )$; p. 40-41)

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 trans-world identity in the way described in Section 7.2.

In accommodating non-existent 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 non-existents; 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 L₂, which replaces Definition 34.(ii):

Definition 36 (Box rule for L₂; 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 L₃, there are still serious issues [10, cf. p. 72-75].

To respond to these, in L₄ 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 L₄; 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 L₄ Gupta gives up on accommodating our ‘rough definition’ of necessity (Definition 9).