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⌉

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

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}, . . . }

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.

1. (4) They lifted the table collectively (together) and did not lift the table distributively (individually).

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).

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

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

‘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

‘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)

〈{s1, s2, . . . ,s7}, {Bernie}〉

〈{Carlos}, {Bernie}〉

〈{Alice}, {t1, t2, t3}〉

(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.)

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).

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,ν} iff M, Rν/E | = G. (G _{M, R,ν} ⊆ ℘(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,ν} 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,ν} in the following way:

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

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).

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,ν} and s ∈ Q _M (G _{M, R, ν}).

[∀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} ).

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).

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.

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,ν}:

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

If G _{M, R,ν} fulfills condition K, then

S ∈ Λ^∃ _M (G _{M, R,ν}) iff S ∈ G _{M, R,ν} and all members of ∪ G _{M, R,ν} are in S;

S ∈ μ _M (G _{M, R,ν}) iff S ∈ G _{M, R,ν} and most members of ∪ G _{M, R,ν} are in S.

For plural variable ν, and proportional quantifier Q (e.g., Λ^∃ and μ), if G _{M, R,ν} 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,ν}).

If ν is plural and G _{M, R,ν} 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

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

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

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

(p. 112 )

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

etc.

S ∈ μ _M (E) iff S ∈ E, and ∃S*, S* ∈ E such that ∀S′, S′ ∈ E, S′ ⊆ S*, and most of S* are in S; i.e., iff S ∈ E, ∪(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, . . . ,T_n⌉, ⌊T₁, . . . ,T_n⌉ _{M, R} = T_{1M, R} ∪ . . . ∪ T_{nM, R} .

Clauses

Atomic: M, R| = (Bⁿ T₁ . . . T_n) iff 〈T_{1M, R} , . . . , T_{nM, R} 〉 ∈ Bⁿ _M .

M, R | = T₁ A T₂ iff T_{1M, R} ⊆ T_{2M, R} .

∧: 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,ν} and s ∈ Q _M (G _{M, R,ν}).

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,ν} and s ∈ Q _M (G _{M, R,ν}).

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,ν} 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,ν} and s ∈ Q _M (G _{M, R,ν}).

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.

[κx: Dx] Bx

[κX: [∀z: zAX] Dz] [∀z: zAX] Bz

Fx _{M, R,x}

[∀y:yAX] Fy _{M, R,X}

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

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.

1. (18) ∃ X∀y (yAX ↔ (Sy∧y∉y) )

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.

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.

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.

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.

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.