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BE Is Not the Unique Homomorphism That Makes the Partee Triangle Commute

Partee (1986) claimed without proof that the function BE is the only homomorphism that makes the Partee triangle commute. This paper shows that this claim is incorrect unless “homomorphism” is understood as “complete homomorphism.” It also shows that BE and A are the inverses of each other on certain natural assumptions.


Introduction
In a famous and influential paper, Partee (1986)  She pointed out that these operators satisfy the equality BE • lift = ident, so the following diagram, now often referred to as the Partee triangle, commutes. Partee declared that BE is "natural" because of the following two "facts." Fact 1. BE is a homomorphism from e, t , t to e, t viewed as Boolean structures, i.e., BE(P 1 P 2 ) = BE(P 1 ) BE(P 2 ), BE(P 1 P 2 ) = BE(P 1 ) BE(P 2 ), BE(¬P 1 ) = ¬BE(P 1 ).
Fact 2. BE is the unique homomorphism that makes the diagram commute.
While Fact 1 is immediate, Fact 2 is not obvious. Partee (1986) nevertheless did not give a proof of Fact 2, but only a note saying, "Thanks to Johan van Benthem for the fact, which he knows how to prove but I don't." Meanwhile, van Benthem (1986) referred to Partee's work and stated Fact 2 on p. 68, but gave no proof either. Despite this quite obscure exposition, because of the classic status of Partee's and van Benthem's work, I suspect that many linguists take Fact 2 for granted while unable to explain it. Not only is this unfortunate, but it is actually as expected, because Fact 2 turns out to be not quite correct unless "homomorphism" is read as "complete homomorphism." The main purpose of this paper is to rectify this detrimental situation.
Van Benthem (1986) took the domain of entities to be finite, writing, "Our general feeling is that natural language requires the use of finite models only" (p. 7). Fact 2 is indeed correct on this assumption. However, natural language has predicates like natural number whose extensions are obviously infinite. Also, if we take the domain of portions of matter in the sense of Link (1983) to be a nonatomic join-semilattice, then the domain of entities will surely be infinite, whether countable or uncountable. It is a fact that a single sentence of natural language, albeit only finitely long, can talk about an infinite number of entities, as exemplified in (1).
(1) a. Every natural number is odd or even. b. All water is wet. (Link, 1983) Given this, it is linguistically unjustified to assume the domain of entities to be finite. Since Partee (1986) herself discussed Link (1983), she was certainly aware that the domain of entities might very well be infinite, so it is unlikely that Partee followed van Benthem about the size of the domain of entities.
What difference does it make if the domain D e of entities is infinite, then? Fact 2 would be correct if "homomorphism" were read as "complete homomorphism." A complete Boolean homomorphism is a Boolean homomorphism that in addition preserves infinite joins and meets. It is clear from the equalities given in Partee's Fact 1 that she did not mean complete homomorphism by the word "homomorphism." When D e is finite, this does not matter since in that case, D e,t ,t is also finite, and consequently, every Boolean homomorphism from D e,t ,t is necessarily complete. However, when D e is infinite, so is D e,t ,t , and in that case, a Boolean homomorphism from D e,t ,t can be incomplete, and "Fact" 2 turns out to be false. This paper essentially consists of extended notes on Partee (1986). Section 2 shows that BE is the unique complete homomorphism that makes the Partee triangle commute and also that it is not the unique homomorphism that does so if D e is infinite. Section 3 discusses why it is important that BE is complete by examining its interaction with A. Finally, Section 4 shows that A is special among the many inverses of BE. The paper assumes the reader's basic familiarity with Boolean algebras and does not provide definitions or explanations of the technical terms that are used. I would suggest Givant and Halmos (2009) as a good general reference.

Uniqueness and Nonuniqueness Proofs
Since it is cumbersome to work with functions, let's adopt set talk. The operators lift, ident and BE and the Partee triangle can be rendered as follows, where D = D e is a nonempty set of entities and ℘ denotes power set. 1 1 Here and below, λ's are used merely to describe functions; they are not meant to be symbols in a logical language that are to be interpreted.
Proof. It suffices to show that BE preserves arbitrary unions and complements (denoted by c ). If For all P ∈ ℘ ( ℘ (D)), Lemma 2. Let h be a homomorphism from ℘ ( ℘ (D)) to ℘ (D). The following conditions are equivalent.
(ii) ⇒ (i). We show the contrapositive. Suppose Then, either there is some a ∈ D such that a ∈ h(P) and {a} / ∈ P or there is some a ∈ D such that a / ∈ h(P) and {a} ∈ P. In the latter case, we have a ∈ h(P) c = h(P c ) and {a} / ∈ P c . Thus, in either case, (ii) does not hold. Proof. We first show that h({∅}) = ∅. Assume that there exists some a ∈ h({∅}). Since ∅ / ∈ lift(a), we have {∅} ⊆ lift(a) c . Since h is a homomorphism and hence preserves order, Since a ∈ h({P }), we obtain b = a, a contradiction.
Theorem 4. BE is the unique complete homomorphism that makes the Partee triangle commute.
Proof. To see that BE makes the Partee triangle commute, observe that for any a ∈ D, Now, let h be a complete homomorphism from ℘ ( ℘ (D)) to ℘ (D) such that h • lift = ident. We show that h = BE. Let a ∈ D and P ∈ ℘ ( ℘ (D)) satisfy a ∈ h(P). By Lemma 2, it is sufficient to show that {a} ∈ P. Since h is a complete homomorphism, It follows that for some P ∈ P ∩ lift(a), By Lemma 3, P must be a singleton set. Since the only singleton set contained in lift(a) is {a}, we have {a} = P ∈ P ∩ lift(a), so {a} ∈ P.
Note that Theorem 4 immediately follows from Keenan and Faltz's (1985) Justification Theorem (p. 92) as well. Individuals in Keenan and Faltz's theory can be identified with the elements of the is necessarily a complete homomorphism, so by Theorem 4, BE is automatically the unique homomorphism that makes the Partee triangle commute. This is not the case, however, when D is infinite. To consider such cases, the following lemma plays an important role of giving (unique) representations of homomorphisms that make the Partee triangle commute.
Lemma 5. Let h be a function from ℘ ( ℘ (D)) into ℘ (D). The following conditions are equivalent.
To begin with, we show that U x is an ultrafilter in the Boolean algebra ℘ (lift(x)). First, since (ii) ⇒ (i). Assume (ii). To show that h is a homomorphism, it suffices to check that it preserves finite union and complement. Being an ultrafilter, U x is a prime filter. Therefore, for all are complements of each other in ℘ (lift(x)) and since U x is an ultrafilter in ℘ (lift(x)), Lemma 6. Let U x be a principal ultrafilter in ℘ (lift(x)). The following conditions are equivalent.
We show that Q = {{x}}. For every y = x, we have lift(x) ∩ lift(y) / ∈ U x , and because U x is an ultrafilter, this implies that its complement Lemma 6 then implies that U x is not a principal filter.
Proof. Let D be infinite. By Lemma 5, a ho- where U x is an ultrafilter in ℘ (lift(x)) such that lift(x) ∩ lift(y) / ∈ U x for all y = x. By Lemmata 6 and 7, there are at least two such ultrafilters U x for each x ∈ D: a principal one and a nonprincipal one. For each x, different choices for U x clearly give rise to different homomorphisms. It follows that the cardinality of the set of homomorphisms h such that h • lift = ident is at least 2 |D| .
Observe that since we can write Thus, in Lemma 5's representation of BE, each U x is a principal filter in ℘ (lift (x)). This also explains why BE has to be the unique homomorphism that makes the Partee triangle commute when D is finite, because in that case, each ℘ (lift(x)) is finite, and every filter in a finite Boolean algebra is necessarily principal. Now suppose D is infinite. By Theorem 8, there is a homomorphism h = BE that makes the Partee triangle commute. By Theorem 4, we know that h is not a complete homomorphism. It may be illuminating to confirm this fact directly. The observation in the previous paragraph implies that in Lemma 5's representation of h, there is some a ∈ D such that U a is a nonprincipal ultrafilter in ℘ (lift(a)). We have {{a}}∩lift(a) = {{a}} / ∈ U a because {{a}} ∈ U a would imply U a = ↑{{a}} but U a is nonprincipal. Also, for all x = a, we have {{a}} ∩ lift(x) = ∅ / ∈ U x because an ultrafilter does not contain the bottom element. Thus Since {{a}} is the only singleton set in lift(a), for every P ∈ lift(a) such that P = {{a}}, we have h({P }) = ∅ by Lemma 3. It follows that  So h does not generally preserve an infinite union.
3 Why do we need a complete homomorphism? Partee (1986) proposes that BE is a type-shifting operator naturally employed in natural language semantics on the grounds that it is a Boolean homomorphism and it makes the Partee triangle commute. As we have seen, however, when D is infinite, there are infinitely many such homomorphisms. Couldn't they then perhaps be employed as type-shifting operators in place of BE? What distinguishes BE from all the rest is the fact that it is the only complete one. So the question boils down to this: how should being a complete homomorphism matter?
To answer this question, let's recall Partee's (1986) discussion of the functions THE and A from D e,t into D e,t ,t , which in set talk can be rendered as the following functions from ℘ (D) into ℘ ( ℘ (D)).
Partee argues that THE and A are "natural" since they are inverses of BE in the sense that for all P ∈ ℘ (D),

BE(THE(P )) =
P if P is a singleton, ∅ otherwise, BE(A(P )) = P.
One should then wonder whether analogous equalities hold with other homomorphisms that make the Partee triangle commute. It is immediate that an analogous equality holds with THE.
With A, by contrast, an analogous equality does not generally hold, and this is where (in)completeness becomes crucial.
Theorem 10. Let h be a homomorphism from In particular, if P is finite, h(A(P )) = P.
Proof. Let P ∈ ℘ (D). We have Thus, for every x ∈ P , lift(x) ⊆ A(P ) and hence If P is finite, then the homomorphism properties of h ensure that Theorem 10 suggests that homomorphisms other than BE are undesirable as a type-shifting operator to replace BE, even if they make the Partee triangle commute. To see this point, imagine that some such homomorphism h = BE were actually employed as a type-shifter.
First, consider the following example.
Following Partee (1986), let's assume that the verb be is semantically vacuous and a type-shifter is inserted to convert a quantifier into a predicate.
(2) would then be analyzed as (6) a. π is a natural number. b. This is some water.
These would be analyzed as in (7), assuming that this denotes an entity and that some = a = A.
In contrast to the previous case, natural number and water ought to be infinite sets. According to Theorem 10, what we can know is then only that What these inequalities imply is that even though π / ∈ natural number , (7-a) might hold and hence (6-a) come out true, and similarly, even if this / ∈ water , (7-b) might hold and so (6-b) come out true. Such states of affairs would be clearly undesirable. This suggests that h should not be used as a type-shifter in natural language semantics.
The above argument does not show, however, that undesirable states of affairs necessarily ensue, as the inequality h(A(P )) ⊇ P in Theorem 10 is not necessarily a proper inclusion. Then, even in a case where D is infinite, might there perhaps be a homomorphism h = BE such that h(A(P )) = P for all P ∈ ℘ (D)? The following theorem tells us that this possibility never obtains. Note that it also characterizes BE without directly mentioning completeness or the property of making the Partee triangle commute.
Theorem 11. BE is the unique homomorphism h from ℘ ( ℘ (D)) to ℘ (D) such that h • A is the identity map on ℘ (D).
Proof. Since BE is a complete homomorphism, by substituting BE for h in the last set of equalities in the proof of Theorem 10, we can see that BE(A(P )) = P for all P ∈ ℘ (D).
To show the uniqueness, assume to the contrary that there is a homomorphism h = BE such that h • A is the identity map. By Lemma 2, for some a ∈ D and some P ∈ ℘ ( ℘ (D)), we have a ∈ h(P) and {a} / ∈ P. Since and since {a} / ∈ P, we have Since h • A is the identity map, it follows that This contradicts a ∈ h(P).
It follows from Theorems 10 and 11 that if h = BE is a homomorphism from ℘ ( ℘ (D)) to ℘ (D) and h • lift = ident, then there exists some infinite set P ∈ ℘ (D) such that h(A(P )) P . Indeed, we can find a concrete example. Since h = BE, in Lemma 5's representation, there is some a ∈ D such that U a is a nonprincipal ultrafilter in ℘ (lift(a)). We have {{a}} / ∈ U a since U a is nonprincipal. Since is the complement of {{a}} in ℘ (lift(a)) and since U a is an ultrafilter in ℘ (lift(a)), we have According to Lemma 5, this implies that a ∈ h(A(D\{a})).
The discussion in this section is just another example demonstrating the significance of the notion of completeness of the Boolean structures and homomorphisms between them that are employed in natural language semantics, which was extensively argued for by Keenan and Faltz (1985).

Inverses of BE
Having discussed the naturalness of BE, Partee (1986) asks what possible determiners δ are inverses of BE, i.e., BE(δ(P )) = P for all P ∈ ℘ (D). It is immediate that a necessary and sufficient condition for δ to be an inverse of BE is that (8) for all P ∈ ℘ (D), {x ∈ D | {x} ∈ δ(P )} = P , so there exist many inverses of BE. A is one but so is exactly one . Partee suggests that "nice" formal properties such as being increasing (in both arguments) and being symmetric might distinguish A from the others. Contrary to her claim, though, symmetry fails to distinguish A from exactly one , as both of these are symmetric. On the other hand, the property of being increasing certainly distinguishes A from exactly one since A is increasing and exactly one is not. Still, there are many inverses of BE other than A that are increasing, as the reader can easily check. Then, how might formal properties single A out?
At this point, we shall recall Keenan and Stavi's (1986) view that all possible determiners are expressible as Boolean combinations of "basic" determiners, which are all increasing and weakly conservative. 3 These two properties are defined as follows, where δ is an arbitrary function from ℘ (D) into ℘ ( ℘ (D)).

Keenan and Stavi
Keenan and Stavi proposed that this accounts for the apparent fact that all determiners are conservative. 4 Now, if we restrict our attention to inverses of BE that are increasing and weakly conservative, it turns out that there remain only two.
Finally, observe that by (12), . It follows that either (i) or (ii) holds.
How can we distinguish A from the other increasing, weakly conservative inverse of BE described in Case (ii) of the above lemma? One possibility might be to note that while A(D) is a sieve, δ(D) in Case (ii) is not a sieve, in the sense of Barwise and Cooper (1981): (13) P ∈ ℘ ( ℘ (D)) is a sieve ⇔ P = ℘ (D) and P = ∅.
A non-sieve is either true of every predicate or false of every predicate, and therefore would be pointless to use in normal conversation. Van Benthem (1986) suggests that a determiner δ is generally expected to be such that δ(P ) is a sieve for all P = ∅ (Variety, p. 9).
Another, presumably more appealing way is to invoke the notion of logicality, for which I refer to Westerståhl (1985). So far, we have fixed a model, whose domain of entities is D, and have not strictly distinguished linguistic expressions and their model-theoretic interpretations. We should now get rigorous about this distinction because logicality is a property of an object language symbol, and not of its interpretation in a particular model. Henceforth, let's take BE and A to be object language symbols such that for every model M = D, M , Now according to Westerståhl (1985), an object language symbol is logical if and only if it has the two properties called constancy and topicneutrality. 5 It turns out that constancy alone is sufficient to single A out. Here is the relevant definition (Westerståhl, 1985, p. 393, with slight adaptation).
A can now be characterized as in the theorem below. So long as we assume all determiners to be conservative, this theorem tells us that A is the only increasing, logical determiner that is an inverse of BE.
Theorem 13. A is the unique increasing, weakly conservative, constant inverse of BE. 6 Proof. What this theorem asserts precisely is that 5 Constancy corresponds to (invariance for) Extension (of the context) in van Benthem's (1986) terminology. Topicneutrality is a generalized notion of permutation invariance (cf. Keenan and Stavi, 1986;van Benthem, 1986). 6 Being both conservative and constant is equivalent to being conservative* in Westerståhl's (1985) terminology. So we could alternatively say that A is the unique increasing and conservative* inverse of BE. Showing (i) is straightforward. Here, let's just verify the constancy of A. Let M 1 = D 1 , M 1 and M 2 = D 2 , M 2 be models such that D 1 ⊆ D 2 . For all P, Q ⊆ D 1 , Thus A is constant.
This contradicts the constancy of δ.

Conclusion
Given that the domain of entities of a model for natural language semantics should generally be infinite, BE is characterized not as the unique homomorphism that makes the Partee triangle commute (Theorem 8), but as the unique complete homomorphism that makes it commute (Theorem 4). In light of Keenan and Faltz (1985), who have shown the importance of considering complete (rather than plain) homomorphisms in natural language semantics, this is a welcome result. BE can alternatively be characterized as the unique homomorphism that is an inverse of A (Theorem 11), while A can be characterized as the unique increasing, (weakly) conservative, constant/logical inverse of BE (Theorem 13). From this viewpoint, the naturalness of BE and that of A complement each other. On the other hand, despite Partee's (1986) conjecture that A and THE are the most "natural" determiners, it is not clear whether THE may be mathematically viewed as equally natural as A is. I hope that this paper has elucidated some finer mathematical points of the Partee triangle that have gone unnoticed and will help rid the linguistic community of any misunderstandings or confusion regarding Partee's (1986) Fact 2. Partee's statement in Fact 2 was not precise, but after all, the results of this paper reinforce her intuition that BE is nice and natural.