Too much reference: Semantics for multiply signifying terms

Abstract: The logic of singular terms that refer to nothing, such as ‘Santa Claus,’ has been studied extensively under the heading of free logic.  The present essay examines expressions whose reference is defective in a different way: they signify more than one entity.  The bulk of the effort is directed at developing an acceptable formal semantics based upon an informal and intuitive idea introduced by Field (1973) and discussed by Camp (2001); the basic strategy is to use supervaluations.  The idea, as it stands, encounters serious difficulties, but with suitable refinements it can be salvaged.  Two other options for a formal semantics of multiply-signifying terms are also presented, and the relative merits of the three semantics are briefly discussed.  Finally, possible modifications to the standard logical regimentation of the notion of existence are considered.

Greg Frost-Arnold

1. Introduction: failures of univocality—two kinds
	In Meaning and Necessity, Carnap writes: “Every expression used as a name…is a name of exactly one entity.”  Carnap dubs this necessary condition for a singular term to qualify as a name “the principle of univocality.”  Now, there are two ways in which a singular term t can fail to satisfy the principle of univocality, and thereby fail to be a genuine name.  First, t might refer to no object in the universe of discourse—or at least, the object t refers to might not be actual.  The singular term ‘Santa Claus’ is an example of the first species of failure.  And if one holds 
(contra Russell) that definite descriptions should be treated as singular terms, then ‘the greatest natural number’ is a second example of this kind of failure to meet the principle of univocality.  This type has been extensively explored, both proof-theoretically and semantically, under the heading of free logic.  The second way a singular term can fail to satisfy the principle of univocality is, in a sense, the inverse of the first: t could be associated with more than one object in the universe of discourse.  In this case, one might describe t as referring multiply, ambiguously, equivocally, or, following Hartry Field’s terminology, “partially.”  One could also characterize t as referring, following Joe Camp, ‘confusedly,’ for to be confused is to conflate things that are actually distinct, i.e., to take two or more things to be one.  In what follows, I will primarily use the term ‘multiple,’ for it seems more neutral than ‘confused’ or ‘ambiguous’; that is, I do not want to pre-judge whether multiple reference, as just described, is the best explication of the usual notions of confusion and ambiguity.
Let us characterize multiple reference using the apparatus of model theory.  We begin with the familiar notion of an interpreted language L = <L, M, >, where L carries the syntactic information for the language, M is an interpretation or model that fixes signification of the nonlogical constants of L, and  determines the truth-value of a compound expression formed using logical connectives, given the truth-values of its components.  Thus far, the treatment is classical.  The models of languages containing multiply-referring terms, however, will diverge from the standard case.  Classically, M = <D, f>, where D is a nonempty set, and f is an interpretation function which assigns a member of D to each individual term of L, a set of ordered n-tuples Dn to each n-ary predicate of L, and a member of DnD to each n-ary function symbol of L.  To accommodate languages with multiply signifying terms, I propose we generalize our notion of model by replacing the interpretation function with a binary interpretation relation, RI(x,y).  Furthermore, RI is allowed to be one-many.  For example: suppose t multiply denotes exactly two objects, a, bD.  The only ordered pairs in the extension of RI that have t as their first member are (t, a) and (t, b); for all other xD, RI(t, x) does not hold.  For any ordered pair in the extension of RI, if the first entry is a singular term from L, the second term will be an element of D.  This idea can easily be extended to predicates.  Suppose P is an n-ary predicate of L, and S and T are sets of ordered n-tuples such that S  T.  Then P multiply signifies (‘refers’ is probably not the correct term for predicates) if RI(P, S) and RI(P, T) both hold.  Functions, obviously, can be treated analogously; I will ignore them for the remainder of the paper.  We can now define a multiply-signifying model MMS as an ordered pair <D, RI>, in which RI is one-many, for if RI is not one-many, then it can be converted into a function (though only a partial function, if we allow terms with empty signification).  A multiply-signifying language is an interpreted language with a multiply-signifying model. 
Let us couch Carnap’s principle of univocality in our new terminology.  Let c be an individual constant of L.  Term c qualifies as a (genuine) name in L only if RI assigns to c exactly one member of D.   If RI assigns no elements to c, or assigns two or more elements to c, then c is no longer a genuine name. The two types of failure can be precisely contrasted thus: the first species (‘zero reference’) modifies the standard interpretation function to a partial function, whereas the second species (‘multiple reference’) uses a one-many interpretation relation instead of a function.  The second, multiply-referring type of failure to satisfy the principle of univocality has not been explored as extensively as the first, though Field and Camp are notable exceptions.  This essay aims to fill that lacuna, by proposing formal semantics for multiply-signifying languages.  The bulk of the effort is directed at spelling out more precisely an informal but intuitive idea introduced by Field and discussed by Camp for assigning semantic values to sentences with multiply-signifying terms; two other semantic options will be presented, however.  
Philosophical impetus for this project can be drawn from the central problem addressed in Joe Camp’s Confusion, which Camp illustrates concretely with the following example.  Suppose a person purchases an ant colony.  There are many small ants in the colony, and two big ants, Ant A and Ant B.  However, the owner of the ant colony does not realize that there are actually two big ants, because he only sees one big ant at a time.  (We assume the ants’ behavior is coordinated to satisfy this condition.)  The owner believes that there is one big ant in the colony, and the owner decides to name this ant ‘Charley.’  The philosophical issue such a story raises is this: what is the semantic status of sentences that include the term ‘Charley’?  Consider the following examples: ‘Charley is eating now,’ ‘Charley is an ant,’ ‘Charley is eating now or Charley is not eating now.’  The difficult semantic issues are not restricted to sentences alone: how do we appraise inferences concerning Charley?  I will use the example of Charley throughout the paper to illustrate various proposals for handling multiply-signifying languages.
Camp’s story of “Charley the ant” is, of course, a toy example.  There are, fortunately, examples that are more ‘realistic’ and/or independently interesting.  If one accepts that the above formalism captures the notion of ambiguity in ordinary language, then the present analysis will enjoy a wide range of application.  The primary example Field offers is the concept of Newtonian mass, viewed from the standpoint of the special theory of relativity.  In the special theory of relativity, there are two distinct concepts of mass: relativistic mass and proper mass (relativistic mass is total energy/c2, proper mass is non-kinetic energy/c2).  Thus, Field proposes (in my terminology) that, for a physicist committed to special relativity, the term ‘mass’ in Newtonian theory multiply signifies relativistic and proper mass.  Other examples can be drawn from scientific revolutions; for example, from the point of view of classical Newtonian physics, the concept of mass that occurs in general relativity multiply signifies gravitational mass and inertial mass, which are distinct in classical physics.  We do not need fully-blown scientific revolutions to encounter these kinds of cases, however.  In general, any situation in which one language draws a distinction that another does not, the more ‘complex’ language will view the ‘simpler’ one as multiply signifying.  And the untranslatability that results from such situations is (one type of) what (the later) Kuhn called ‘incommensurability.’  So here we have another motivation for developing a formal semantics for multiply signifying languages: hopefully, it could help clarify a perenially sticky philosophical problem.  In addition to ordinary language and scientific cases, other applications for multiply signifying languages are possible.  Intuitionists hold that classical logicians conflate p with p; i.e., from the intuitionist viewpoint, the classical logician’s p multiply signifies p and p.  Perhaps philosophers of Christian religion might characterize the term ‘God’ as multiply referring to the Father, the Son, and the Holy Ghost.  Russell introduces the idea of ‘typical ambiguity’ in order to eliminate extravagances in his theory of types.  So, though this essay will use ‘Charley’ as its primary example, the value and interest of a semantics for multiply-signifying languages will apply to cases that are more interesting and philosophically relevant, but more complex.

2. The Field semantics—and its problems
What is Field’s suggestion for sentences involving multiply signifying terms?  The basic strategy is supervaluational.  Put roughly and informally: suppose the term t multiply refers to objects (or extensions) a1, a2, …an.  A sentence S containing the term t will be supervaluated true if S is true on all n disambiguations of t, S will be supervaluated false if S is false on all n disambiguations of t, and S lacks a supervaluation otherwise.  For example, ‘Charley is an ant’ will be supervaluated true, because there are only two disambiguations of this sentence, namely ‘Ant A is an ant’ and ‘Ant B is an ant,’ and both are true.  Some people’s linguistic intuitions reject assigning the sentence ‘Charley is an ant’ the value true; two formal semantics for such skeptics will be provided later.  Nonetheless, Field’s idea does appeal to others.  However, I will argue that even for those who are willing to call sentences like ‘Charley is an ant’ true, the Field semantics yields certain undesirable consequences. 
But first, I provide a formal explication of the Field strategy.  To do this, a precise characterization of a complete disambiguation is needed.  A complete disambiguation of a multiply signifying language L assigns, for each multiply-signifying term ti of L, exactly one of ti’s designata.  (A ‘term’ is either a singular term, a predicate, or a function symbol.)  A complete disambiguation is similar, though not identical, to the notion of a completion model found in supervaluational semantics in free logic.  
The model MC (=<D, fC>) is a complete disambiguation of the multiply-signifying model MMS (=<D, RI>) if the following conditions are met:

(MC 1)	RI(a, fC(a))	(a any individual constant that is not zero-denoting)

(MC 2)	RI(Pn, fC(Pn))	(Pn any n-ary relation symbol)

(MC 3)  fC is a function: no individual constant is assigned more than one object in D by fC, and no n-ary relation symbol is assigned more than one set of ordered n-tuples Dn by fC.  (If we wish to allow ‘zero-reference’ terms in L, say t, then we say that fC(t) is undefined, so fC is a partial function.)

Working through Camp’s example may clarify the notion of a complete disambiguation.  Suppose ‘Charley’ is the only multiply-signifying term in L; ‘Charley’ denotes two objects: Ant A, Ant B D.  Then there will be a total of two complete disambiguations of MMS (call them MC1 and MC2): in MC1, the interpretation function fC1 assigns Ant A to ‘Charley,’ while the other model, MC2, fC2 assigns Ant B to ‘Charley’ instead.  There are no other complete disambiguations.  In general, if in L there are n multiply-signifying terms, and the first denotes m1 objects (or ordered n-tuples), the second m2 objects, and so on, then there are exactly m1  m2  …  mn complete disambiguations of MMS.  Finally, note that in each complete disambiguation, bivalence holds for sentences that do not contain any zero-signifying terms (or other semantic ‘funny business’): every such sentence is either true or false.  
Now we can assign truth-values to sentences about Charley in accordance with the Field suggestion as follows.  A sentence is supervaluated true in MMS if it is true in all complete disambiguations of MMS, supervaluated false in MMS if it is false in all complete disambiguations, and is undefined otherwise.  So, ‘Charley is an ant’ is supervaluated true, ‘Charley can speak English’ is supervaluated false, and ‘Charley is eating now’ will lack a definite value whenever one ant is eating, and the other is not.  The law of excluded middle (‘pp’ is always true) holds for interpreted languages containing no semantic deviance other than multiple reference, since it will be true in every complete disambiguation; but the law of bivalence (p is true or p is false) does not hold, since some sentences will not be assigned any truth-value by the supervaluational scheme.  The machinery of supervaluation was originally developed as a semantics for terms that do not denote any existing objects, i.e., zero-reference terms such as ‘Pegasus.’  The supervaluational technique has also been extended to handle certain problems surrounding the issue of vagueness, but such proposals have not gone unchallenged.  In what follows, supervaluations will be adapted to serve as the basis for a model-theoretic account of multiply-signifying terms.  Intuitively, the basic idea of supervaluation seems to fit cases of multiple reference even better than it fits cases of zero-reference, for in the multiple-reference case there is an independent motivation (namely, the very fact that a term multiply refers) for assigning more than one member of D (or Dn) to a single term.
There is an additional constraint we may wish to place on the definition of a complete disambiguation, involving the treatment of identity claims between two multiply-referring terms.  Suppose, in more casual moments, Fred sometimes calls his big ant by the name ‘Chuck.’  It would be natural to formalize this as: the extension of {x | RI(Charley, x)} is identical to the extension of {x | RI(Chuck, x)}.  So it seems plausible that ‘Charley=Chuck’ should be true (at least in the Field semantics).  But on the supervaluational scheme as just outlined, it lacks a value: there is a complete disambiguation in which Ant A is assigned to ‘Charley’ and Ant B to ‘Chuck.’  If we consider this a problem, then it can be remedied by adding a condition to our definition of complete disambiguation:
(MC 4) Given any two constants (individual or predicate) k1, k2 of L, if {x | RI(k1, x)} and {x | RI(k2, x)} are coextensive, then fC(k1) = fC(k2).  

This condition, if imposed, has the effect of ‘throwing out’ those complete disambiguations in which ‘Charley=Chuck’ is false.  (Note: an analogous problem arises in positive free logics, but it cannot be dealt with there as it can here.  Positive free logics evaluate ‘Santa=Santa’ as true; however, ‘Santa=Saint Nick’ will not be.  For even though ‘Saint Nick’ is another name for Santa Claus, the supervaluational apparatus cannot distinguish between ‘Saint Nick’ and ‘Zeus’—and we certainly do not want ‘Santa=Zeus’ to be supervaluated true.)
The Field semantics for multiply-signifying languages, unfortunately, encounters serious difficulties with certain sentences that contain both identity and multiply-signifying terms.  Consider the open formula ‘x=Charley.’  No member of D satisfies this formula, for neither Ant A nor Ant B is identical to the reference of ‘Charley,’ and there are no other possible candidates in D.  The only other intuitive possibility is that the reference of ‘Charley’ is somehow identical to the set {Ant A, Ant B}, but that obviously cannot be a value taken by the variable x, since it only ranges over the individuals of D.  Then, if we assume the usual definition of satisfaction for existential quantification, x(x=Charley) should be false in MMS.  However, the Field semantics does not yield this answer.  In each complete disambiguation of MMS, there is an element of D that satisfies ‘x=Charley’: in MC1, the singular term ‘Charley’ is assigned to Ant A, so obviously Ant A satisfies the formula ‘x=Charley’ in MC1.  Similarly, Ant B will satisfy it in MC2.  Thus, in each complete disambiguation, ‘x=Charley’ is satisfiable, so ‘x(x=Charley)’ is evaluated as true on the Field proposal.  The Field policy has led us into a problem: if x(x=Charley) is true, as the Field policy would have it, then the usual semantic rules for = are violated in MMS.  To my mind, the point is not merely that Field’s semantics conflict with the standard rules of formal semantics (that is not surprising, given that we are working under deviant semantic assumptions).  The point is primarily an intuitive one: there is nothing that is identical with the reference of ‘Charley,’ but the Field semantics says that there is.
One could conceivably resolve this problem by substantially revising the definition of satisfaction for identity (or the existential quantifier, or the individual variable), so that x(x=Charley) would come out true in MMS.  However, many might prefer to avoid such a departure from the usual notions of existence and identity, if possible.  Instead, I will abandon the notion that truth in a multiply-signifying language is truth in all complete disambiguations.  A slightly more nuanced approach can salvage their fundamental idea without simultaneously committing ourselves to the truth of ‘There exists something identical to Charley’ and the like.  Along the way, two other options for a formal semantics of multiply-signifying terms will also be presented; the first treats ‘Charley is an ant’ as meaningless, the second treats it as false.

3. Semantics for multiply signifying languages
	In free logic, there are three kinds of semantics: positive, negative, and neutral.  They are distinguished by how they treat sentences containing nondenoting terms.  For example, consider ‘Pegasus=Pegasus.’  This (along with other logical truths involving non-denoting terms)is true in a positive semantics, false under negative semantics, and it lacks a truth-value in neutral semantics.  In what follows, an analogous trichotomy of semantics for multiply-signifying languages will be presented, and I will follow the usage of free logicians and call them positive, negative, and neutral respectively.  (The Field proposal is a positive semantics.)  The construction of these three semantics proceeds in several steps.

A. First step: Define restricted model (MR)

First, from the base model MMS (=<D, RI>) we construct a restricted model MR = <D, fR>, where fR is the (partial) function based on RI satisfying the following conditions.
  
(MR 1) For each singular term s: If RI assigns to s exactly one object b D, fR(s) = b; otherwise fR(s) is undefined.

(MR 2) For each n-ary predicate letter Pn: If RI assigns to Pn exactly one set of ordered n-tuples SnDn, then fR(Pn)=Sn; otherwise, fR(Pn) is undefined. 

Intuitively, the difference between the original model MMS and its restricted counterpart MR is that the restricted model treats multiply-signifying terms as if they did not signify at all. That is, the restricted model is ‘blind’ to the information in RI about which objects multiply-denoting singular terms denote, and which extensions multiply-signifying predicates signify. 

B. Define true and false in MR

	The next step is to define truth in the restricted model, keeping in mind that we (eventually) want ‘there exists exactly one big ant in the colony’ to be evaluated false.  To do this, we avail ourselves of the usual definition of the satisfaction of formulae by sequences as far as possible.  However, modifications are necessary in order to handle sentences that are neither true nor false in MR.  Why?  In the classical case, a formula is true if every sequence satisfies it, and is false if no sequence satisfies it.  This creates problems for sentences that we intend to have an undefined truth-value, because no sequence will satisfy a formula with undefined terms or predicate letters, so that formulae without truth-values would collapse into false ones.  That is, in classical contexts, satisfaction is bivalent: a sequence either satisfies a formula or it does not.  However, in the cases of interest here, we need to introduce ‘satisfaction’/ ‘anti-satisfaction’ (analogous to ‘extension’/ ‘anti-extension’ for predicates), in order to leave open the possibility that a sequence could do neither. 
We first need to slightly modify the notion of a valuation (or evaluation).  Let t be an individual constant or a variable.  Given MR, let  be the set of all denumerable sequences of D.  For each sequence s=(s1, s2, s3, …) we define a function s* (often called a valuation or evaluation) that takes terms in L to elements of D:  
If t is a variable xi , then s*(t) is si.
If t is an individual constant ai, then s*(t) = fR(ai); if fR(ai) is undefined, so is s*(t).  (This final clause is the only change to the usual notion of a valuation.)

A wff  is defined on M (=<D, f>) iff f assigns values to all (individual, predicate, and function) constants in .

For atomic formulae defined on MR, the conditions for satisfaction are just the standard ones familiar from Tarski onwards.

A sequence s anti-satisfies an atomic formula A in M if either:
 A is defined on M, and s does not satisfy A in M, 
or 
 A is an atomic sentence of the form t1=t2, and exactly one of s*(t1) and s*(t2) is undefined.

As a result, the open formula ‘x=Charley’ is anti-satisfied by every sequence.

For all other atomic formulas not defined on MR, no sequence s satisfies or anti-satisfies A.

We can carry over the usual definition of satisfaction for compound expressions, so long as the sub-definitions do not rely on the notion of ‘s does not satisfy A’; for those occurrences, we substitute ‘anti-satisfies.’  Thus, we should modify the satisfaction-condition for equivalence and for negation, as follows: 
s satisfies A  B iff s satisfies both A and B or s anti-satisfies both A and B.
s satisfies A iff s anti-satisfies A.

The conditions for anti-satisfaction of compound expressions can be formulated in an obvious way: 
s anti-satisfies A iff s satisfies A.
s anti-satisfies A  B iff s anti-satisfies A and s anti-satisfies B.
s anti-satisfies A  B iff s anti-satisfies A or B or both. [[Strong Kleene]] 
s anti-satisfies xi A iff every  sequence that differs from s in at most the ith place anti-satisfies A. 

We can now define true and false in MR:
A formula  is true (resp. false) in MR if every sequence satisfies (resp. anti-satisfies) ;  lacks a truth-value in MR otherwise.

Note that x(x=Charley) will be false in MR.

C. Neutral, negative, and positive semantics

Using the restricted model as a basis, we can formulate three species of multiply-signifying semantics: neutral, negative, and positive.
(a) Neutral semantics.  Simply adopt the truth-value assignment of MR.  If desired, (an analogue of) the Weak Kleene valuation scheme could be adopted—the above truth-definition follows the Strong Kleene scheme for dealing with sentences without truth-value. 
(b) Negative semantics.  Evaluate the atomic closed formulae that MR leaves undefined as false.  Then the classical rules for satisfaction and truth can be applied.  Why are only the atomic undefined expressions evaluated false?  Suppose that, in MR, A is true and B undefined; then ‘A or B’ will be undefined in MR.  If we do not make the restriction to atomic sentences, then ‘A or B’ will be false under the negative semantic theory—an obviously undesirable consequence.
(c) Positive semantics.  Use a supervaluational strategy.

First, we need the notion of true in MC with respect to MR:

(i.)  If the formula  has a truth-value in MR, then the truth-value of  in MC w.r.t. MR is the same as the truth-value of  in MR.

(ii.)  If the formula  lacks a truth-value in MR, then the truth-value of  in MC w.r.t. MR is the same as the truth-value (or lack thereof) of  in MC.

Now (finally) we can define the notion of truth on a supervaluation S:

(S 1)  Formula  is true (resp. false) on S iff  is true (resp. false) in MC w.r.t. MR for every complete disambiguation MC of MMS.

(S 2)   lacks truth-value on S iff  is true in MC w.r.t. MR for some complete disambiguations MC, but false in other complete disambiguations.

D. Which semantics?

Which (if any) of these three semantics is the ‘right’ one?  Most logicians seem to have gotten over this kind of question, so I will only remark upon the relative interest, value, and applicability of these systems.  First, for many people, the negative and/or neutral semantics perhaps match up better with everyday linguistic intuitions.  However, these semantics share two closely related features that make them less interesting as objects of study.  First, they basically assimilate the case of multiple designation to the case of ‘zero’ designation: the negative and neutral semantics treat ‘Charley the ant’ in the same way free logics treat the term ‘Santa Claus.’  The phenomenon of multiple reference in its own right disappears—multiply-referring terms cannot be distinguished from zero-referring ones.  Secondly, the negative and neutral frameworks, unlike the positive approach, effectively ‘throw away’ a fair amount of the information contained in a multiply-signifying model: the purported name ‘Charley’ conflates only two things in the universe.  Even if ordinary language may discard that information (namely, that Charley does not refer to the number 5, or to my cat, or…), philosophers can study methods for retaining and extracting that information from a multiply-signifying language.
An analogy from mathematics may be instructive.  Functions are one of the most studied objects in mathematics.  But curves which are not functions can still be investigated rigorously and fruitfully.  A simple example is the curve algebraically described by x=y2.  One cannot properly speak of ‘the unique value of y at x=3,’ since there will be two such values.  Nonetheless, one could say that, if x=9, then y satisfying the relation will be odd, or any such y will have absolute value 3, etc.  Contrast this case with the relation defined by y=1/x, evaluated at x=0.  The ‘value’ 1/0 is simply not defined, and so we cannot say that it is odd or even, or that it has a particular absolute value.  These two parallel the cases, respectively, of multiple- and zero-reference in semantics.  We cannot properly speak of the unique ant named ‘Charley,’ but it seems that we could say that Charley is less than one inch long, is not a dog, etc., for that information is contained in the multiply-signifying model MMS.  But for a singular term that refers to nothing, no such information is contained in its language.

4. Does Charley exist?
In Section 3, we constructed all three of our semantics such that ‘x(x=Charley)’ is evaluated false.  Of course, this formula is just the canonical logical paraphrase of ‘Charley exists.’  So, if one accepts that the usual symbolic regimentation of ‘t exists’ is adequate, as well as any of the three semantics above, then one accepts that Charley does not exist.  For some, this is likely to be an unproblematic conclusion.  However, for others—and especially for those who favor a positive semantics for multiply-signifying languages—this may be an unwelcome consequence.  For though neither Charley nor the Easter Bunny exist in the same way you or I do, these two cases feel intuitively different—especially to someone who is sympathetic to the positive semantics.  Such a sympathizer might therefore worry that the cases of multiple-reference and zero-reference were being collapsed, since ‘x(x=Charley)’ and ‘x(x=Easter Bunny)’ would have the same semantic status.  
However, the difference in existential status between Charley and the Easter Bunny could perhaps be captured in other ways in a positive semantics, so that the two cases would not collapse.  Here is one.  Take all the sentences containing the term ‘Charley’ that are evaluated true (under positive semantics) that are not theorems of logic and do not contain ‘=’: 
1[Charley], 2[Charley], 3[Charley],… .  
We can then form a true existential sentence: 
x(1[x/Charley]  2[x/Charley]  3[x/Charley] …).  
However, we cannot do likewise in the case of zero-denoting singular terms: in the positive free logic case, a sentence  containing a zero-denoting term, but not containing ‘=’, will be true only if  is a logical theorem.  Thus, there are no true sentences, not containing identity, about the Easter Bunny that are not theorems of logic (according to supervaluational semantics).  So we cannot infer a substantive (i.e. not logically true) existential statement from truths about the Easter Bunny, while we can do so for Charley.  This is one way of making clear the asymmetry between the case of Charley and the Easter Bunny, without having to evaluate ‘x(x=Charley)’ as true.  
The above method of making out the existential difference between Charley and the Easter Bunny took for granted the adequacy of the canonical logical paraphrase of ‘t exists’. However, the adequacy of that paraphrase is not forced upon us, and by resisting it, we can give another account of the existential difference between multiply-referring and zero-referring cases.  Upon reflection, ‘x(x=t)’ appears to assert something perhaps more specific than the ordinary language assertion ‘t exists’: namely, ‘t exists uniquely,’ or ‘There exists exactly one t.’  This is simply a consequence of adopting the usual semantic rules of satisfaction.  And even the advocate of positive semantics for multiply-signifying languages should hold that the sentence ‘There exists exactly one Charley’—which is what our canonical regimentation really says—should be false.  Thus one might hold that the question ‘Does Charley exist?’ is not equivalent to asking whether ‘x(x=Charley)’ is true, for the latter involves a uniqueness claim which is not necessarily present—at least prima facie—in the former.  Questions of existence and of uniqueness are standardly kept separate in mathematical proofs; perhaps we should do the same for Charley and his ilk.  
Free logics often introduce a one-place predicate of existence, E!x (read ‘x exists’), which, in languages containing identity, is defined as E!tx(x=t).  Analogously, a primitive one-place predicate of existence (E) could be introduced into multiply-signifying languages, but here the predicate would not prejudge the issue of uniqueness.
(E: Liberal policy)	Et (‘t exists’) is true in M (=<D, RI>) iff there exists at least one element in the domain D such that RI(t, a) holds; Et is false otherwise.

On this policy, ‘Pegasus exists’ will be false, but ‘Charley exists’ will be true. Of course, we are not forced to adopt this rule.  If one holds that Charley does not exist (so that to exist just is to exist uniquely), but still wants to introduce the predicate ‘E’ into the language, and then one could instead employ the following.
(E: Restrictive policy)	Et (‘t exists’) is true in M (=<D, RI>) iff there exists exactly one element in the domain D such that RI(t, a) holds; Et is false otherwise.
  
The restrictive policy is equivalent, in languages with identity, to the canonical paraphrase of ‘t exists.’  Under the liberal policy, this equivalence does not hold: ‘Charley exists’ is evaluated true, while x(x=Charley) is evaluated false.  It is in this sense that the liberal policy distinguishes between existence and uniqueness. 
Careful readers will have noticed that the two existence policies above do not fit into any of the truth definitions offered thus far; that is, the two policies do not refer to truth in MC, in MR, in MC w.r.t. MR, or truth on a supervaluation.  I suggest the two policies above not be considered as semantic rules of truth per se, but rather as conditions of adequacy.  Here’s one way to meet that condition.  At the stage of defining truth in MR (3.B), add the following rule:
Existence predicate
Let A have the form Eb (read ‘b exists’), where b is an individual constant.

(Liberal version) A is true in MR iff fR(b) is defined; if fR(b) is undefined, then A is neither true nor false. 

(Restrictive version)  A is true in MR iff fR(b) is defined; if fR(b) is undefined, then A is false.

And then, add to the definition of truth in complete disambiguations, at the level of atomic sentences: 
A is true in MC iff fC is defined; A is false otherwise.  
Such definitions will, at the level of supervaluations, meet the criteria of adequacy.  (Such definitions disallow simultaneously holding the liberal existence policy and negative or neutral semantics; but that’s not too bad, since if someone wishes to class ‘Charley=Charley’ as false (or meaningless), she will probably not want to evaluate ‘Charley exists’ as true (or meaningless).) 

5. Conclusion
	Many significant issues concerning multiply-signifying languages have been left untreated here.  The most important of these are modifications to the proof calculus needed to accommodate inferences involving multiply-signifying terms.  For example, in the positive semantics, we will need to block the inference from ‘Charley=Charley’ to ‘x(x=Charley)’; so we will have to modify the rule for existential introduction.  Another fundamental issue not yet discussed: how should definite descriptions involving multiply signifying terms be treated?  For example, should we count as true ‘The big ant in the colony=Charley’ (in the positive semantics)?  Lastly, can these semantics be applied fruitfully to philosophical issues?