## Thomas McKay

Print publication date: 2006

Print ISBN-13: 9780199278145

Published to Oxford Scholarship Online: January 2007

DOI: 10.1093/acprof:oso/9780199278145.001.0001

ContentsFRONT MATTER

# Set Theoretic Semantics

Chapter:
(p. 103 ) 5 Set Theoretic Semantics
Source:
Plural Predication
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199278145.003.0006

# Abstract and Keywords

This chapter presents the development of the semantics within a set-theoretic context. It also provides a base for reconsidering limitations of the set-theoretic approach.

The semantics of Chapter 3 provides a systematic interpretation for the sentences of a language with non‐distributive predication and plural reference and quantification. The present chapter is not really needed. However, set theory has become so much the semantic standard, that some people will feel more satisfied about the interpretability of plurals when they see a set‐theoretic semantics. In addition, a semantics in this form may facilitate some comparisons with related work of others. So I will develop a set‐theoretic semantics here; in any case, that will enable us to consider again the limitations of the set‐theoretic approach in the case of plurals.

# Set Theoretic Representation of Non‐Distributive Predication

Since we will now allow ourselves reference to sets, there is an obvious semantics for a non‐distributive predication like:

1. (1) Alicia, Betty and Carla are meeting together.

2. (2) F⌊a, b, c⌉

The sentence (2) is true iff FX is true relative to an assignment of {Alicia, Betty, Carla} to X. A model will assign a set of sets of individuals to a monadic predicate, its (plural) extension in the model, and we can use ‘FX M ’ to refer to the set of sets that is the plural extension of the predicate FX in a model M. If a M , b M , c M are the individuals that a model M assigns to terms ‘a’, ‘b’ and ‘c’, then

‘F⌊a, b, c⌉’ will be true in a model M iff {a M , b M , c M } ∈ FX M .

If Alice, Betty, Carla, Doreen and Ellen surrounded the Chancellor, then {Alice, Betty, Carla, Doreen, Ellen} will be one of the sets whose (p. 104 ) members jointly satisfy ‘X surrounded the chancellor’. 1 The fact that this is a non‐distributive predicate means that there is no guarantee that each of the subsets (of a set of surrounders) will be a set of things that jointly satisfy that predicate or that any individual person satisfies that predicate. Even if Alice, Betty, Carla, Doreen and Ellen surrounded the Chancellor, it does not follow that Alice and Betty did. They may have been standing together to the Chancellor's left, for example. And of course it does not follow that Alice did.

This leaves us with a choice to make concerning predicates that are distributive. Since a plural term can occur in a single sentence in both distributive and non‐distributive argument positions, and since plural variables will do so as well, we need to be able to give a semantics for distributive and non‐distributive predication that allows that. Since the plural extension of a non‐distributive predicate is a set of sets, our semantics will work most uniformly if we also make the plural extension of a distributive predicate a set of sets. Consider the predicate Sx, x is a student (and the plural SX, X are students). In standard (singularist) treatments of first‐order logic, a semantics would assign a set of individuals in a model's domain to Sx. 2 If K is such a set of individuals (the set of students) in an ordinary model M, then we have two evident possibilities for the plural extension in our semantics. Using SX M to stand for the plural extension of the predicate in a model M:

“Plural‐set” interpretation of ‘SX’: SX M would be ℘(K)—{Ø}

“Unit‐set” interpretation of ‘SX’: If a, b, c, . . . are the members of K, then SX M would be {{a},{b},{c}, . . . }

Either choice provides us with a set of sets as the extension of ‘X are students’.

The plural‐set option might seem natural, since every subset of a set of students is also a set of students. However, there is good reason to prefer the unit‐sets (singletons) choice for the interpretation of sentences with names and without quantifiers.

As we have seen, some predicates allow for satisfaction either by a single individual or by several individuals. ‘X lifted the table’ can (p. 105 ) be true of a single (strong) individual or of many (perhaps weaker) individuals (in a joint effort). We need to distinguish distributive and non‐distributive plural claims involving such predicates. For example:

1. (3) Alice, Bob and Carla lifted the table.

can be true in virtue of either a non‐distributive or a distributive satisfaction of the predicate. We can distinguish these by saying, for example:
1. (4) They lifted the table collectively (together) and did not lift the table distributively (individually).

The plural‐set choice for the interpretation of non‐distributive predicates would have the unfortunate result of leading us to have two ways for the set {a, b, c} to be a member of the extension of ‘X lifted the table’. Both their lifting it individually and their lifting it together would be represented by that set's being in the extension of the predicate, and our semantics would provide no way to distinguish importantly different cases. In particular, these cannot be distinguished:
1. (5) They lifted the table distributively (individually) and lifted the table collectively (together).

2. (6) They lifted the table distributively (individually) and did not lift the table collectively (together).

In both these cases, {a}, {b}, {c} and {a, b, c} would all be members of the plural extension of the predicate, if we chose the plural‐set interpretation for distributive predicates. The unit‐set interpretation of distributive satisfaction, though, will distinguish these cases semantically, with {a, b, c} being an element of the extension in models that verify (5) but not so for (6). Thus we will choose the unit‐set approach to the interpretation of distributive satisfaction of predicates, and purely distributive predicates will always have a set of singletons as the plural extension.

We also need a way to syntactically mark the distinction between distributive and non‐distributive predication in the formal language itself. As we have seen, marking each argument place with ‘D’ or ‘C’ to indicate whether the argument place is to be taken distributively or non‐distributively with respect to that use of the predicate is not totally clear, unless we introduce ways to indicate the scope of the distributivity. So we continue to use such sentences only as a convenient shorthand, when scope is clear enough.

1. (7) L⌊a, b, c⌉ C∧¬L⌊a, b, c⌉ D

(p. 106 ) These subscripts are no part of our formal language. Our better notation is to use the predicate ‘x is among Y’ (or ‘x is one of Y’) in a distributing universal quantifier marking the distributive case, thus providing a way to mark the ambiguous scope of the distributing universal:
1. (8) They lifted the table, but no one person among them lifted it.

2. (9) L⌊a, b, c⌉ ∧ [∀x:x is a person among ⌊a, b, c⌉ ]¬Lx

3. (10) They lifted the table, but it is not the case that each person among them lifted it.

4. (11) L⌊a, b, c⌉ ∧¬ [∀x:x is a person among ⌊a, b, c⌉ ] Lx

With C and D subscripts, we would distinguish two types of relationship of a plural term to a predicate.

‘F⌊a, b, c⌉ C’ will be true in a model M iff {a M , b M , c M } ∈ FX M

‘F⌊a, b, c⌉ D’ will be true in a model M iff {{a M }, {b M }, {c M }} ⊆ FX M

These two relationships are also simple consequences of marking distribution by a distributing universal quantifier.

‘F⌊a, b, c⌉’ will be true in a model M iff {a M , b M , c M } ∈ FX M

‘[∀x:xA⌊a, b, c⌉ ] Fx’ will be true in a model M iff {{a M }, {b M }, {c M }} ⊆ FX M (i.e., iff each of {a M }, {b M }, {c M } ∈ FX M ).

In general, a model will interpret an n‐place predicate as a set of n‐tuples of sets of items in the domain. So if the only liftings represented in a model J are these

Seven students (together) lifted Bernie.

Carlos lifted Bernie.

Alice lifted three tables (at once)

then the interpretation in J would be a set including these three pairs:

〈{s1, s2, . . . ,s7}, {Bernie}〉

〈{Carlos}, {Bernie}〉

〈{Alice}, {t1, t2, t3}〉

(Here ‘sn’ stands in for a name of a particular student in the model, and ‘tn’ for the name of a particular table.)

# (p. 107 ) Quantifiers

The simplest kind of quantified sentence for us is a sentence involving only non‐distributive predicates.

Seven classmates are meeting (together).

‘[7X: CX] FX’ is true in a model M iff some set S is such that S ∈ F M and S ∈ 7 M (C M ). 3 (The members of S are meeting together and the members of S are seven classmates.)

Here ‘7 M ’ stands for a function that selects the seven‐membered sets that are members of the set it applies to. (For any set k, 7 M (k) ⊆ k, every member of 7 M (k) has seven members, and every seven‐membered member of k is a member of 7 M (k).)

We must allow for multiple quantification, and that requires that we introduce a function R that makes assignments to free variables. For plural variables, R assigns some set of things to each variable of the language. For a singular variable, R will assign a unit set to the variable. To give the semantics for multiple quantification, we must consider alternative ways of assigning sets of individuals to variables, so we need to define:

Rν/s is the assignment to variables that is just like R except that it assigns s to ν (where ν is a variable and s is some non‐empty set).

Then we can give the conditions in which a clause is true, relative to a model M and an assignment R to the variables. (For example, an open sentence ‘X are classmates’ will be satisfied by an assignment of {Alice, Betty, Carlos} to X iff Alice, Betty and Carlos are classmates, i.e., {Alice, Betty, Carlos} ∈ C M .)

To interpret quantifiers, we can define the extension of an open sentence in a model, relative to an assignment of a variable, and relative to a particular variable. For example, consider the model J (indicated above) and an assignment R to variables that assigns {Alice} to X and {Bernie} to Y. The extension of LXY relative to X according to J and R, LXY J, R, X , would be {{s1, s2, . . . ,s7}, {Carlos}}, since s1, s2, . . . , s7 lifted Bernie (together) and Carlos lifted Bernie (and that is all), according to the model. LXY J, R, Y would be {{t1, t2, t3}}, since those are the things that Alice lifted (at once). In general, an extension is a set (p. 108 ) of sets. For a plural variable ν and a sentence G, we define the extension of G (in model M relative to variable‐assignment R) as follows:

E ∈ G M, R iff M, Rν/E | = G. (G M, R ⊆ ℘(D)–{Ø}) 4

The quantificationally relevant extension must be defined differently for singular variables, and this is the principal place where the difference between the two kinds of variables will show up. Roughly put (and with the help of brackets), in the case of a singular variable ν, we want G M, R to include all unit sets of [things that satisfy G] whereas with plural ν, we want all [sets of things] that satisfy G.

The set representing the property of being a student will be a set of (one‐tuples of) unit (singleton) sets. Nothing in that set is seven‐membered. When we consider a quantificational sentence like ‘Seven students are meeting’, however, we need semantic access to a seven‐membered set of individuals (students). So when ν is a singular variable, we define G M, R in the following way:

E ∈ G M, R iff E ≠Ø, and ∀dE, M, Rν/{d}| = G. (G M, R ⊆ ℘(D)–{Ø})

Note that we have two things that might be called the extension of a distributive predicate F. One is the unit‐set interpretation of Fx, a set of singletons that M associates with F, F M . The other is a set of sets of individuals Fx M, R,x , where Fx M, R,x = ℘({i:{i} ∈ F M })–{Ø}. Although we saw good reason to take the unit‐set interpretation (F M ) of the extension of a distributive predicate for the purposes of interpreting unquantified sentences, we must also have access to the plural‐set interpretation (Fx M, R,x ) of the extension for the purpose of interpreting quantified sentences.

The easiest way to see why we must give different treatment to the singular and plural variables here, is to think about the difference between these two sentences:

Seven classmates are meeting (together).

Seven students are meeting (together).

Seven classmates are classmates together; it is the seven people together who satisfy the predicate ‘X are classmates’. The sentence about classmates is true if some seven‐membered set is both a set of classmates (p. 109 ) and a set of people meeting together. But each one is a student. Each satisfies ‘x is a student’. A quantifier must select a subset of G M, R as its interpretation (the seven‐membered sets that are classmates (together) or the seven‐membered subsets of the set of students, for example). A quantifier Q in a model M (we will write Q M ) selects a subset of ℘(D)–{Ø}. In fact, for any set k ⊆ ℘(D)–{Ø}, Q M k ⊆ k.

Now we are in a position to give the semantics for our basic non‐proportional quantifiers, for example, ∃ (some), κ (many), σ (several) and simple numericals. 5

For variable ν, and quantifier Q (from among ∃,κ,σ, and the simple numericals (1, 2, etc.) ):

M, R| = [QN: G] H iff some set s is such that s ∈ H M, R and s ∈ Q M (G M, R, ν).

The universal quantifier ∀ will be defined:

[∀ν: G] H = df¬[∃ν: G] ¬H

Look briefly at how our semantics works for sentences mixing distributive and non‐distributive predication.

Seven students are meeting (together).

[7X: [∀y:yAX] Sy] MX

M, R| = [7X: [∀y:yAX] Sy] MX iff some set s is such that s ∈ MX M, R,X and s∈ 7 M ([∀y:yAX] Sy M, R,X ).

Consider the semantics for [∀y:yAX] Sy.

M, R| = [∀y:yAX] Sy iff M, R | = ¬[∃y:yAX] ¬Sy iff it is not the case that some set k is such that k ∈ ¬Sy M, R,y and k ∈ ∃ M (yAX M, R,y ).

k ∈ ¬Sy M, R,y iff ∀d∈ k, M, R y/{d}| = ¬Sy.

k ∈ yAX M, R,y iff ∀d ∈ k, M, R y/{d}| = yAX.

k ∈ ∃ M (yAX M, R,y ) iff k∈ yAX M, R,y . (The existential quantifier function on a non‐empty set of sets is the identity function.)

M, R| = ¬[∃y:yAX] ¬Sy iff it is not the case that some set k is such that ∀d ∈ k, M, R y/{d}| = ¬Sy and ∀d ∈ k, M, Rν/{d}| = yAX; (p. 110 ) i.e., iff no set k is a set of non‐students and a set of things among X; i.e., iff no non‐student is among X; i.e., iff every individual among X is a student (according M and R).

[∀y:yAX] Sy M, R,X is the set of all such sets; i.e., the set of all (non‐empty) sets of students. So 7 M ([∀y:yAX] Sy M, R,X ) is the set of all seven‐membered sets of students. Thus M, R| = [7X: [∀y:yAX] Sy] MX iff some set s is such that s ∈ MX M, R,X and s ∈ 7 M ([∀y:yAX] Sy M, R,X ); i.e., some set s is a member of the set of sets of individuals meeting together and a member of the set of seven‐membered sets of students.

# Proportional Quantifiers

As we saw in Chapter 3, we must give a different interpretation to proportional quantifiers. “Most (all) jazz musicians” cannot mean “jazz musicians who are most (all) in number.” The quantifier indicates a portion (most or all) of some specified reference class of individuals, and so we need to build in the reference to that class. And since there may be no appropriate reference class in the case of some non‐cumulative predications, we need to recognize the possibility of semantic anomaly.

When Q is a proportional quantifier, [QX: AX] has an interpretation only when some things are identifiable as “the As”. As we noted in Chapter 3, this always happens when A is cumulative, but it may also happen for purely contingent reasons. For example, the sentence

All students meeting together in the next room are wearing hats.

is acceptable when there is only one meeting in the next room, even though it would not be acceptable if there were several separate meetings. As in Chapter 3, we parallel natural language, allow for the syntactic acceptability of such sentences, and provide for cases in which they are semantically anomalous.

Let's identify a condition K that must be fulfilled if semantic anomaly is to be avoided with proportional quantifiers. If G is any clause with plural variable ν and extension G M, R:

K: ∃S, S ∈ G M, R such that ∀S′, S′ ∈ G M, R, S′S; i.e., ∪(G M, R) ∈ G M, R.

If Q is a proportional quantifier, then [QX: GX] has a well‐defined interpretation relative to a model M only if G M, R,X fulfills condition (p. 111 ) K. Then we can give the semantics for proportional quantifiers like ‘all’ (Λ) and ‘most’ (μ) as follows:

If G M, R fulfills condition K, then

S ∈ Λ M (G M, R) iff S ∈ G M, R and all members of ∪ G M, R are in S;

S ∈ μ M (G M, R) iff S ∈ G M, R and most members of ∪ G M, R are in S.

The general semantic clause governing quantifiers (and, ultimately, other clauses of the semantics) needs to recognize the possibility of semantic anomaly.

For plural variable ν, and proportional quantifier Q (e.g., Λ and μ), if G M, R fulfills condition K, then M, R| = [Qν: G] H iff some set s is such that M, Rν/s | = H and s ∈ Q M (G M, R).

If ν is plural and G M, R does not fulfill condition K, then [Qν: G] H is semantically anomalous with respect to M and R (when Q is a proportional quantifier).

# The Formal Language and Semantics

The syntax is the same as for the formal language of Chapter 3.

A model M on a domain D will represent some sentences as true. If a sentence has free variables, then it may be represented as true relative to an assignment of values to the variables.

If c is an individual constant, then c M D.

If Pn is an n‐place relation, then Pn M ⊆ (℘(D)–{Ø})n.

If Q is a quantifier, then Q M is a partial function such that for any E ⊆ ℘(D)–{Ø}, Q M (E) ⊆ E (or Q M is undefined for E). 6

To treat quantifiers as part of the logical vocabulary, we must constrain the functions assigned to quantifiers so that for each quantifier word, every M assigns the same function to it; e.g., if E ⊆ ℘(D)–{Ø}:

S ∈ ∃ M (E) iff SE. (∃ M is the identity function.)

S ∈ κ M (E) iff SE and many things are in S.

S ∈ σ M (E) iff SE and several things are in S.

(p. 112 )

S ∈ 7 M (E) iff SE and seven things are in S.

etc.

For a proportional quantifier, the condition is more complicated.

S ∈ μ M (E) iff SE, and ∃S*, S*E such that ∀S′, S′E, S′S*, and most of S* are in S; i.e., iff SE, ∪(E) ∈ E, and most of ∪(E) are in S.

Terms

An interpretation of a term with respect to M and a function R (that assigns values, non‐empty subsets of D, to variables) is some non‐empty set S such that S ⊆ D.

For an individual constant c, if c M = d, then c M, R = {d}. 7

For a variable ν, ν M, R D. If ν is a singular variable, then ν M, R is a singleton set.

For a term ⌊T1, . . . ,Tn⌉, ⌊T1, . . . ,Tn M, R = T1M, R ∪ . . . ∪ TnM, R .

Clauses

Atomic: M, R| = (Bn T1 . . . Tn) iff 〈T1M, R , . . . , TnM, R 〉 ∈ Bn M .

M, R | = T1 A T2 iff T1M, R ⊆ T2M, R .

¬: If G is semantically anomalous with respect to M and R, then ¬G is semantically anomalous. Otherwise, M, R | = ¬G iff it is not the case that M, R | = G.

∧: If G is semantically anomalous or H is semantically anomalous with respect to M and R, then (G ∧ H) is semantically anomalous. Otherwise, M, R | = (G ∧ H) iff M, R | = G and M, R | = H.

Singular quantification: For singular variable ν and quantifier Q, if G is semantically anomalous or H (p. 113 ) is semantically anomalous, then [Qν: G] H is semantically anomalous. Otherwise, M, R | = [Qν:G] H iff some set s is such that s ∈ H M, R and s ∈ Q M (G M, R).

Plural quantification with non‐proportional quantifiers: For plural variable ν and non‐proportional quantifier Q (for example, ∃, κ, σ, and the simple numericals (1, 2, etc.) ), if G is semantically anomalous or H is semantically anomalous, then [Qν: G] H is semantically anomalous. Otherwise, M, R |= [Qν:G] H iff some set s is such that s ∈ H M, R and s ∈ Q M (G M, R).

Plural quantification with proportional quantifiers: For plural variable ν, and proportional quantifier Q (Λ or μ, for example), if G is semantically anomalous or H is semantically anomalous, then [Qν: G] H is semantically anomalous. Also, if G M, R does not fulfill condition K, 8 then [Qν: G] H is semantically anomalous. Otherwise, M, R | = [Qν:G] H iff some set s is such that s ∈ H M, R and s ∈ Q M (G M, R).

Closed sentence S: M | = S (S is true in model M) iff for all R, M, R | = S.

Logical truth: S is a logical truth iff for all models M, M | = S.

# Relationship Between Singular and Plural Quantification

In Chapter 4 we indicated that it is important to prove that there is a relationship between the two different representations of a simple distributive quantification:

Many dogs bark.

x: Dx] Bx

X: [∀z: zAX] Dz] [∀z: zAX] Bz

The sentences are semantically equivalent. To show that the two versions of ‘Many dogs bark’ are equivalent, it suffices to show that these are the same set, for any distributive predicate F:

Fx M, R,x

[∀y:yAX] Fy M, R,X

Ignoring the complication of semantic anomaly, which is not relevant in this kind of case, our semantics for quantified sentences is this:

M, R | = [Qν:G] H iff some set s is such that s ∈ H M, R and s ∈ Q M (G M, R).

(p. 114 ) The full proof here becomes a bit more cumbersome than the corresponding proof in Chapter 4, but the following outline should suffice.

If F is distributive, then Fx M, R,x = [∀y:yAX] Fy M, R,X .

Proof: S ∈ Fx M, R,x iff ∀d ∈ S,M, R y/{d} | = Fx

iff ∀d ∈ S, 〈{d}〉 ∈ F M

iff S ∈ [∀y:yAX] Fy M, R,X .

# The Set‐Theoretic Approach and the Goal of Semantics

The set‐theoretic approach just outlined will lead to paradox if we take it as a very general approach to semantics. Since every quantification requires a set, a sentence such as:

1. (17) Some things are the non‐self‐membered sets.

will create a problem. That sentence is true, but the semantics requires that there be a set of non‐self‐membered sets in order for it to be true. Of course, the formal correlate of it will be true in some models:
1. (18)Xy (yAX ↔ (Syy∉y) )

It can be true in a model as long as the set that the semantics requires for the truth of the sentence is not itself a thing in the model (or at least is not something that ‘S’ applies to in the model). But then the semantics requires that there be a set that ‘Sx’ does not apply to, and so ‘Sx’ does not mean ‘x is a set’, 9 and so we have not really verified the original sentence. 10 Put another way: if we insist that every set that the semantics requires is a set in our model, we will get a contradiction if we try to make (17) true in a model.

If a semantics is just a technical device for showing that some language is interpretable in some model or other (to show consistency of some sentences, for example), then it is not necessary for predicates to have assignments in the model that give their true meaning. But (p. 115 ) when we are indicating how meanings of complex linguistic items are constructed from the meanings of simpler ones, and we hope to be able to include the very language that we speak in that, then we have a problem if the semantics requires that certain sets exist and also requires that the predicate that means ‘x is a set’ cannot apply to them. The set‐theoretic semantics developed in this chapter crashes into the set‐theoretic paradoxes, if we envision that semantics as something that at least leads the way towards a semantics for natural language.

This is a difficult issue, though. First, there are attitudes towards sets themselves that might cushion the collision. Sometimes sets are said to be inexhaustible, meaning that there can never be a fixed domain of all sets. Our semantics for (17) must reach out to a set that cannot be an individual in the domain of the interpreting model, but that fact might be seen as simply a symptom of a more general feature (bug?) of set theory, that “whenever we have formed a conception of quantification over some range of sets, we can define a set which isn't in this range.” 11 Williamson 2003 argues very effectively against this kind of response when he considers views that would prevent us from talking about everything.

Second, though, is the very real question about what semantics achieves. We can focus on that by first considering some cases other than the case of set‐theoretic semantics.

Event semantics. If Donald Davidson is right about the semantics of adverbs, then the semantics for ‘John is walking slowly’ involves quantification with respect to an event. (∃e (e is a walking ∧ John is an agent of e ∧ slowly(e) ). It seems, if we accept this, that we should conclude that the following principle is true in virtue of the meaning of ‘John is walking slowly’:

E If ‘John is walking slowly’ is true, then at least one event exists.

Possible worlds semantics. If possible worlds semantics is right about the meaning of modality, then it would seem that the following principle is true in virtue of the meaning of ‘I could have stayed home’:

P If ‘ I could have stayed home’ is true, then at least one possible world exists.

(p. 116 ) Tarskian semantics. If the ordinary Tarskian semantics gives the meaning of predicative sentences, then it seems that the following principle is true in virtue of the meaning of ‘ Alice is happy’:

T If ‘Alice is happy’ is true, then at least one set exists.

The example based on the Tarskian semantics (T) is most relevant to our concerns (and we discussed the possible worlds example in Chapter 2). Few would say that the meaning of predicative sentences alone would verify T. Yet, according to the Tarskian semantics, the truth of the antecedent requires that there is a set of happy things. Let's call the principle “semantic realism”.

Semantic realism: If the semantics for S requires the existence of an entity e for the truth of S, then if S is true, e exists.

If the semantics actually tells us the meaning of the sentence, then presumably whatever entities are required in verifying models that correctly interpret the sentence must exist. So ‘happy’ must be associated with the set of happy things, if the Tarskian semantics provides the real meaning of the predicate. 12 I take this as a reason to think that Tarski's semantics does not give the meaning of predicative sentences but rather provides a formal model that illustrates some semantic properties. In particular, ‘non‐self‐membered set’ can't be associated with the set of such things, so semantic realism does not allow the possibility that the Tarskian semantics in terms of sets can give the actual meaning of ‘non‐self‐membered set’. 13

The semantics of Chapter 3 provides an approach to the semantics of plurals that may more plausibly be said to give the meaning of plural sentences. It will endorse the following principle:

If ‘Alice and Betty (together) lifted two tables (together)’ is true, then at least one relation exists.

In Chapter 3 we associated each seriously plural (non‐distributive or non‐cumulative) predication with a property or relation. There we say that some things may have a property (like being neighbors) that is (p. 117 ) associated with a predicate. Chapter 3 avoids set‐theoretic paradoxes, but doesn't the assertion of the existence of properties involve us in perfectly analogous paradoxes? Consider the property of not applying to (not being true of) itself.

The answer on behalf of the semantics of Chapter 3 has several parts. First, one might give an answer like the inexhaustibility answer for sets. If we can make a firm distinction between the object language and the metalanguage, we identify the things D that are the individuals of the object language, and it is only in the metalanguage that we treat properties of things as additional individuals. Of course, when we consider the metalanguage itself as a plural language of the type we are considering, we then must consider the status of these properties. We must then introduce properties of those properties in the appropriate meta‐metalanguage, etc., or we must have a theory of properties that is adequate for the semantics and that tells us that they cannot be treated as individuals.

Alternatively, we might interpret predicates by translation rather than by assigning properties (that are treated as individuals). One might explain translation as interpreting object‐language predicates with predications, activities rather than objects. But what is predicated? A property? The answer can be that to talk of predicating something is misleading. In the limiting case of monadic plural predication, we say that some things are (or were or will be) in a certain way: they are neighbors, they are sitting in a circle, etc. Translation interprets by associating some such activity (the activity of predication, of saying that some things are somehow) with each predicate. Then, however, in defining logical truth we must consider all possible ways of translating or all ways of predicating, and (as we mentioned in Chapter 3) that specification is also troublesome.

The semantics of Chapter 3 provides a real advantage over the set‐theoretic semantics in this chapter when we consider the semantics of natural language. We can develop a more homophonic semantics; we don't have to change the subject (to a set) and change the predicate (to a predicate of sets) when we interpret non‐distributive plurals. So we have the right things filling theta‐role positions. When we say that some students are surrounding Adams Hall, the students are the subject of the predication, not some other individual (like the set of students) that the students bear a relationship (like membership) to. If some students surround a building, then the students are the agents of the surrounding; our semantics should not force us to identify a property that is somehow (p. 118 ) related to surrounding but that has a set as its “agent”. The semantics of Chapter 3 avoids the distorting change of subject and predicate that accompanies the development of a set‐theoretic semantics, and so it is able to associate the actual agents (patients, etc.) with theta‐roles rather than associating their set‐theoretic surrogates. 14

In addition, if a language for set theory itself is desired, then the semantics of Chapter 3 provides something with some potential. The paradoxes guarantee that relying on a set‐theoretic semantics will produce difficulties of paradox or inexhaustibility if we try to use plurals as a language for set theory. We can avoid those problems with sets if we shift to a plural language with a semantics based on properties or on translation. This may not seem like much of a gain, but there is an asymmetry to consider here. It seems that we cannot give an adequate set‐theoretic semantics for a language that would serve for a full theory of properties; but it seems that we can give an adequate property‐theoretic semantics for a plural language that could serve for a full theory of sets.

Plural language has some advantages in metaphysics, in the semantics of ordinary language, and in the language for set theory itself. It is difficult to see any advantage in the set‐theoretic approach to the semantics of plurals.

## Notes:

(1) Although availing ourselves of set‐theoretic resources, we should not go so far as to say that the set surrounds the Chancellor or even that a set satisfies the predicate ‘X surrounded the Chancellor’, though the latter may be too tempting to avoid.

(2) Actually, a semantics often assigns a set of one‐tuples of individuals to a monadic predicate.

(3) Strictly speaking, 〈S〉 ∈ F M .

(4) Note that we allow that G M, R may be empty, but it is not possible that Ø is a member of G M, R.

(5) The quantifiers that are defined here are the basic (existence‐entailing) non‐proportional quantifiers. Basic (existence‐entailing) proportional quantifiers (most, 1/3, some interpretations of all, for example) will require a separate definition, and we will define additional quantifiers in terms of the basic quantifiers, as in Ch. 3.

(6) Q M is undefined if Q is a proportional quantifier and E does not fulfill condition K.

(7) Although we can think of an individual constant as referring to an individual, when we evaluate the constant's role in sentences, we must always consider the singleton set containing that individual. This is a peculiarity of this set‐theoretic semantics that does not exist in the semantics without set theory.

(8) K: ∃S, S ∈ G M, R such that ∀S′, S′ ∈ G M, R, S′S. (∪(G M, R) ∈ G M, R.)

(9) Cf. Boolos 1998, 30ff., for related discussion.

(10) Having classes in addition to sets merely postpones the problem (cf. Lewis 1990, 65ff), if classes are individual things. If classes are not individual things, then introducing classes may just be a misleading way of adopting the plural solution. (See Ch. 2, 24, and Ch. 6, 146–7, for further discussion of classes “as many”.)

(11) Linnebo 2004. This is his way of putting the view, which he attributes to Michael Dummett, Charles Parsons and Michel Glanzberg.

(12) Presumably no one thinks of it in that way.

(13) Tarski identified other sources of paradox in semantics, and so he advocated for a strict metalanguage hierarchy; the semantics for a language is always expressed in some other language. That can't be very satisfying for any of us, but especially not for those of us who would envision a systematic semantics for natural language or who think that we can say that everything is self‐identical.

(14) There is some additional discussion of this issue in Ch. 2.