This paper explores an unfamiliar and potentially revealing interaction between natural language quantification and semantic anomaly. Semantically anomalous utterances are utterances of grammatical sentences with meaningful constituents that seem to resist conventional interpretation, as would most uses of (1)–(3).1

  • (1)
    * Quadruplicity drinks procrastination.2
  • (2)
    * This stone is thinking about Vienna.
  • (3)
    * He put an event in the hole.

I'll begin by arguing that semantic anomaly triggers a form of quantifier domain restriction with several unique features, setting it apart from other known forms of domain restriction. The domain restriction is important because it requires a wholly different kind of explanation from domain restrictions influenced by contextual salience or relevance. But it is also important because of its potential to give us indirect insight into the semantic properties of anomalous utterances. This ‘indirect' information about anomaly is valuable because it allows us to bypass more controversial, and perhaps unreliable, truth-value judgments about anomaly. To show all this, I propose an explanation of the source of the domain restriction that essentially appeals to the truth-valuelessness of anomalous material and then argue that the other most natural explanations of the phenomenon are extensionally inadequate, or ad hoc.

The argument has two interesting downstream consequences. First, it motivates a compositional semantics which unconventionally permits truth-valueless material to make positive contributions to the interpretability of truth-evaluable utterances. Second, the logic of the resulting semantics may provide us with the clearest available violations of classical inference schemes, and interesting motivations for characterizing logical consequence not as truth-preservation, but as truth-preservation among truth-evaluables.

1. Anomaly and Quantifiers

  1. Top of page
  2. 1. Anomaly and Quantifiers
  3. 2. The Semantics of Anomaly and Quantification
  4. 3. Logical Consequence
  5. Appendix
  6. References

1.1 Domain Restriction

Consider the following narrative.

Trees and Planks. Bob owns a house with a large yard. In the yard there are six trees and six beautiful hand-carved Scandinavian planks, but nothing else—no bushes, brush, grass or anything of the sort: just dirt. Bob wants to build a fire to keep warm in the winter but is loathe to use those wooden planks. Consequently Bob uproots the six trees and uses them as firewood.

In response to such a story, speakers are typically willing to classify (4) as true and (5) as false.3

  • (4)
    Bob uprooted everything in his yard and burned it.
  • (5)
    Bob burned everything in his yard.

The assessments present us with a puzzle: (4) should entail (5) as is witnessed in what look to be plausible renditions of their logical form below.

  • (4′)
    [inline image: InYard(x, Bob)][Uproot(Bob, x) ∧ Burn(Bob, x)]
  • (5′)
    [inline image: InYard(x, Bob)][Burn(Bob, x)]

In a sense there is a ready explanation, transparent to English speakers, as to why the inference is blocked. In (4) the planks of wood aren't being considered part of “everything” whereas in (5) they are. That is, the domain of quantification in (4) is restricted so as to exclude the planks, while in (5) the domain is broader.

I want to focus on a question about the source of this domain restriction: what brings it about that the planks are removed from the quantifier domain in (4)? This question about the broad sources of the domain restriction is one that we can and should settle before settling whether the domain restriction is implemented at the semantic or pragmatic level—though I'll return to the question of implementation shortly. We want a general explanation of why (4) doesn't communicate information about planks, whether those reasons ultimately influence the semantics of quantifiers or pragmatic inferences that speakers draw in response to a typical utterance of (4).

It's clear that part of our explanation has something to do with the anomalous status of (6).

  • (6)
    * Bob uprooted the planks.

The reason is that we can construct similar instances of domain restriction systematically correlated with the presence of anomaly among substitution instances like (6). Some further examples supporting this claim follow later in the paper.

My question about the source of the domain restriction in (4) is non-trivial in part because some of its features distinguish it from better known forms of quantifier domain restriction—like those in (7)—which have been extensively studied by linguists and philosophers and whose sources seem relatively clear.

  • (7)
    All the beer is in the fridge.

You won't hear an utterance of (7) used to communicate that a given fridge contains all the beer in the world. Rather, some suitably salient instances of the beverage will be up for discussion—for example all the beer the speaker bought on a given day, or all such beer except the two bottles that the speaker dropped on the way home.

Sentences like (7) are of interest because their domains of quantification are highly sensitive to features of conversational context, and so form central cases for investigating general theories about how context interacts with language use. For example, whether (7) is usable to communicate the first or the second of my two readings above might depend essentially on whether it is used in a context in which two bottles broke on the trip home, and this fact is apparent to all parties in the conversation.

The domain restriction in (4), however, exhibits at least two features which distinguish it from cases like (7). First, the domain of quantification shifts from (4) to (5), whereas the context need not alter significantly between their assessments. That is, evaluators of those sentences who read them one right after the other nonetheless make the truth-value attributions indicative of a domain shift. The shift even occurs when the sentences are evaluated in reverse order.4

For example, contrast the pairs (4)/(5) and (8)/(7). Getting readings of (8) and (7), read or heard in immediate succession, on which (8) sounds true and (7) sounds false, will require a noteworthy intervening change in context.

  • (8)
    All the beer is in the fridge and next to some wine.

For example, there should be two different salient groups of beer, and context should make clear somehow that one group is discussed in (8) and a different group is discussed in (7). With (4) and (5), by contrast, the domain shift responsible for the change in truth-value judgments, given the setup, seems to occur naturally in the reading of the sentences themselves.

A second way in which (4) differs from (7) is in terms of its responsiveness to considerations of salience and relevance. One way of explaining why the domain of quantification in (7) includes only the beer bought at the store on the day of the utterance (say) even though there is also some beer stashed in the basement, is that the former beer is more salient than the latter for the purposes of the conversation at hand. Were the other beer made salient enough, and relevant to the topic of conversation, it would fall into the domain of quantification as well, as is witnessed in (9).

  • (9)
    You remember that beer that we bought at the store? Well, it turns out there was even more in the basement. And guess what: all the beer is in the fridge.

Typically, rendering an object suitably salient or relevant ensures it will be included in the domains of subsequently used quantifiers.

The domain restriction in (4) is not responsive to salience and relevance in the same way.

  • (10)
    Bob was cold the other day and looking for kindling to keep warm. The type of trees that grow on Bob's property were not really any good for making fires, but the Scandinavian planks in his yard were spectacularly flammable. Bob didn't really value those planks at all. Anyway, at the end of the day he uprooted everything in his yard and burned it.

It is hard to make the planks more salient than in (10). They are not only clearly up for discussion, but are stressed as directly relevant to the topic of the last sentence of the monologue: what Bob burned. Nonetheless they are kept out of the quantifier domain of that final sentence—a variation of (4).

In cases like (7), we can claim that the lack of salience of certain objects, or the fact that those objects are not obviously ‘up for discussion' in the conversational context explains why those objects fall outside the relevant quantifier domains. This has a good deal speaking in its favor, since it not only conforms to the data, but makes sense of why contextual domain restriction arises: it is part of a strategy to gain efficiency in communication by letting conversational context dictate the bounds of quantifier domains instead of having speakers explicitly delimit them. But since the domain restriction in (4) is not responsive to salience or relevance in the right ways, this explanation for (7) cannot be transposed to (4).

If we can't explain the restriction in (4) in the same way as for (7), what are our other choices? A second option involves claiming that the domain restriction occurs because, were it not performed, the quantified statement would be false as it would have a false instance correlated with (6). On this view, the quantifier domain restriction is in place in (4) to avoid having it express a falsehood (or an obvious falsehood). Perhaps there is a semantic mechanism which restricts the quantifier domain to have this effect, or a kind of charity of interpretation leads interlocutors to search for the relevant reading.

While the sort of mechanism this option posits perhaps arises in other contexts, it is unpromising as an explanation of the quantifier restriction in (4). This time, although the explanation has the potential to capture the datum given by (4), the general principle it invokes simply does not apply in the majority of cases involving quantification over false, or even trivially false, instances. That is, the principle over-generates.

Suppose, for example, that the story of Trees and Planks were modified so that Bob didn't burn one of the trees in the yard. Then language users immediately, and unproblematically, take (4) to be false. Language users make no attempt to rectify the utterance by removing the relevant tree from the domain of quantification. Moreover, such behavior would be quite bizarre—surely we should allow that people can make mistakes.

Consider another, more forceful example: if one mathematician utters (11) to another mathematician, it is implausible that any domain restriction would be accommodated.

  • (11)
    Every number between 2 and 5 is prime.

The interlocutor would almost certainly conclude there had been an oversight on the part of the speaker. This is so even if, as we have supposed, the relevant parties are experts and highly unlikely to make the relevant mistakes. It is also the case even if the falsehood in question is both an obvious and a necessary, rather than a contingent, falsehood.

There are perhaps cases where something's being obviously false might create a domain restriction via something like normal contextual forms of quantifier domain restriction. Consider the following case: Clyde runs into a room where Al is standing, grabs a small pile of books on the table, leaving only a pen on it, and runs out. Bill enters the room, sees only the pen on the table, and asks Al what happened. Al might successfully communicate the facts by uttering (12).

  • (12)
    Clyde grabbed everything on the table and ran with it.

He might do this, despite the pen being on the table in plain view, counting on Bill to recognize from the context that the pen is not among the things talked about. Note, for example, that if instead Al had said that Clyde grabbed almost everything on the table, that too could be interpreted as a truth. This suggests that false substitution instances can influence quantifier domains contextually, through some process of charitable interpretation. However, the fact that the restriction proceeds via the normal contextual avenues means that considerations of salience and relevance may defeat it. Suppose Bill enters the room, sees the pen, and utters (13).

  • (13)
    I can't believe I left my precious antique pen on the table where Clyde could just take it. I see that Clyde grabbed everything on the table and ran with it. I'm so happy he didn't grab that pen on the table.

Though it is possible to figure out what Bill means, his utterance sounds contradictory. The tension between the fact that Bill's utterance has rendered the pen pertinent to the taking, and the falsity of Bill's quantified statement on its most general reading is very readily felt.

Thus the falsity of certain substitution instances—even their obvious, egregious, and necessary falsity in the face of mutually aware interlocutors—does not in general produce a domain restriction. False instances may restrict quantifier domains indirectly via normal modes of contextual quantifier domain restriction, but in this case the domain restriction will ultimately be sensitive to considerations of salience and relevance. We have already seen that the kind of domain restriction we are out to explain isn't like this.

Both of the first two attempted explanations of the domain restriction occurring in (4) suffer from an obvious defect: they fail to take into account that the domain restriction looks to be connected with a property specially bound up with anomalous status of the substitution instance, (6), which engenders the restriction in (4). That anomalous character almost certainly constitutes some kind of defect or infelicity. The first account I examined, which transposes the account of the domain restriction in (7), does not do any justice to the idea that the domain restriction in (4) is responsive to the special status of that sentence, rather than simply to features of the conversational setting. The second account, which took the alleged falsity—perhaps the egregious, obvious, and necessary falsity—of (6) as the grounds for the restriction, at least looked to some properties of (6) to track the relevant restriction. The problem is that it didn't look to anything that was specially connected with anomaly. To that extent, it ended up positing a general principle of domain restriction which tended to over-generate.

A third possible explanation takes its cue from these failures, and acknowledges that there is some feature of anomaly which helps enforce the domain restriction, but insists that it is a non-semantic feature of some kind. On this account, there are just things which it is ‘odd' or ‘awkward' to talk about using certain predicates. Planks, for example, exhibit this oddness as concerns the predicate “uproot”. Our judgments of anomalous status tend to track this oddness. Moreover, it is this oddness of predications, and not necessarily their falsity, which results in a domain restriction.

To test this third explanation we of course need to make it a little more specific. What exactly makes a particular ascription of a predicate odd in the relevant sense? The more specific one is about what the peculiarity consists in, the more implausible a domain restriction over odd substitution instances becomes. Here, for example, are three possible ways of spelling out the oddness in more detail.

  • (a)
    Predicating F of a is odd if people tend not to make such predications, or tend not to be moved to make them.
  • (b)
    Predicating F of a is odd if it describes a highly fantastical or wondrous situation.
  • (c)
    Predicating F of a is odd if it is particularly confusing or difficult to understand.

I won't dwell on these elaborations, since it should be fairly clear that there is no general domain restriction over the members of any of the classes described by (a)–(c). There are of course other ways of spelling out the ‘oddness' alluded to in the third response. But the attempts given by (a)–(c) point to two interconnected problems with the general strategy here. First, non-semantic specifications of the oddness (if they don't simply appeal to judgments of anomaly themselves) again tend to over-generate. Second, even if there were a quantifier domain restriction over the members of (a)–(c), it would remain a bit of a mystery why the restriction occurred. As I said, these two problems are connected. Sometimes speakers believe, and want to communicate, things that are unusual, fantastical, confusing, or obviously false. A restriction that precluded this would lead to unnecessary expressive limitations, or an increased risk of misinterpretation. We won't find restrictions over any sets like (a)–(c), and with good reason.

The real problem here stems from ignoring the prima facie case that anomaly exhibits a distinct form of semantic aberration. A fourth explanation might concede this and recognize anomalous defect itself as relevant to the domain restriction, but end the explanation there. On this view, anomaly is a sui generis semantic property of an utterance (perhaps further explained as a form of categorial mismatch), and anomaly just happens to generate the relevant domain restriction.

The broad form of explanation here isn't completely unprecedented. For example, bound readings of pronouns in languages with gender marking may be absent if gender agreement isn't maintained. The absence of the readings owes to a more or less arbitrary convention on which gender agreement constrains interpretation, and gender itself is a sui generis grammatical feature. The explanation for the lack of bound readings doesn't seem to go any deeper than this.

But applying this kind of explanation to the domain restriction from anomaly is extremely unsatisfying. Anomalous defect doesn't seem to be an arbitrary feature attaching to expressions, like gender in Romance languages. Genders of synonymous nouns vary from language to language, for example, but it is hard to see how anomaly could be arbitrarily detached and reattached to otherwise synonymous utterances in the same way. This to state the obvious: anomaly is simply not sui generis, but warrants further characterization in terms of its semantic or communicative effects. Additionally, we should at least hope that the nature of the explanations of anomalous defect will somehow be connected with the domain restriction, perhaps naturally giving rise to it. The current proposal is really no explanation at all, amounting to a form of defeatism: unlike other forms of quantifier domain restriction, like that in (7), there is no interesting explanatory source of our restricting quantification in sentences like (4). We should not be content with such a shallow, ad hoc explanation if we can find a natural one that goes just a little deeper.

I believe there is such an explanation—indeed, one that is a natural conclusion to our failed list of alternatives. To give this explanation, we must help ourselves to one version of a common but controversial claim: that anomalous utterances are truth-valueless, where failure of truth-evaluability marks some semantic obstacle to conventional interpretability. This claim is typically motivated by contentious and potentially unreliable truth-value intuitions. But these are not my motivations here. Instead I think we should adopt the claim because it furnishes us with a semantic feature specially borne by anomaly, and one specific enough to form part of an informative explanation for the existence of the quantifier domain restriction to which anomaly gives rise. Of course, in saying failures of truth-evaluability are ‘specially' borne by anomaly, I don't mean to presume that anomaly alone is truth-valueless. I just mean if anomaly was truth-valueless, this would distinguish it from obvious falsehoods, and ‘odd', ‘fantastical', or ‘confusing' claims.

Taking anomalous utterances to be truth-valueless is, of course, insufficient to explain the domain restriction on its own. We also need to explain how this truth-valueless status interacts with quantifier domains. The basic idea is that the domain restriction is the product of a general interpretive strategy on the part of language users to maximize truth-evaluable (i.e. conventionally interpretable) content. This strategy has implications for the semantics of quantified statements with anomalous instances on the following plausible claim: that some uttered quantified statements would be truth-valueless were their domains of quantification to include all of their truth-valueless substitution instances. Some evidence for this claim, on the assumption that anomalous utterances are truth-valueless, comes from basic projection data for anomaly. For example, sentences with quantifiers uninterpretable unless forced to range over anomalous instances, such as “the number eight is red” or “all towels are prime”, tend to be anomalous. The claim also gains motivations from many standard trivalent semantics. For example, the claim will be made true by some uses of universal quantification on both the Strong and Weak Kleene schemes.

If some quantified statements inherit truth-valuelessness from some of their truth-valueless substitution instances, and anomalous utterances are truth-valueless, then a policy of restricting the domains of quantification to exclude relevant anomalous substitution instances preserves the truth-evaluability of many whole quantified statements, leading to a straightforward increase in conventional expressive power, with no obvious costs.

To see what I mean by this, consider maximally general assertions like (14).

  • (14)
    Everything has a moral if only you look for it.

What would a speaker uttering (14) be talking about? Possibly books, fables, tall-tales, but also perhaps the lives of great men and women and incidental events in one's own daily life. Note, however, that there are of course things which the speaker clearly is not talking about: tea doilies, bowling alleys, and socks among them. To say of these things that they have a moral would be anomalous. If these instances are truth-valueless and would render a fully general interpretation of (14) truth-valueless as well, then we have an explanation for why it might be advantageous to restrict the domain of quantification in (14) over non-anomalous objectual substitution instances. This would enable (14) to serve as a truth-evaluable, and quite possibly true, statement.

If, by contrast, we allow quantification to range over all objectual substitution instances, we would be in danger of condemning many generalities like (14) to fail to express truth-evaluable propositions relative to a context—a strict loss of conventional expressive power with no corresponding gain. Moreover, as the example hopefully makes clear, there is value to being able to express the content of the corresponding generalities with restricted domains.5

If the hypothesized policy of restricting some quantifier domains to exclude certain truth-valueless substitution instances were in place, it would mostly only be apparent in cases like (14) which can, on their face, be accounted for equally well by standard explanations of contextual quantifier domain restriction. Cases like (4), where salience plays no role, signal that a different phenomenon is at work.

As already noted, anomalous expressions might not be the only ones that are truth-valueless. Accordingly, appealing to truth-valuelessness as I have to explain the anomalous domain restriction raises an over-generation worry, much like those I raised previously for accounts that tried to explain the domain restriction by appeal to falsity or awkwardness. Do all truth-valueless expressions generate a form of domain restriction, and if not, why not? Here the over-generation challenge can be met, treating key classes of possibly truth-valueless expressions case by case.

Some of the best candidates besides anomaly for failure of truth-evaluability are non-referring expressions, such as non-referring names. But non-referring names can't even in principle generate a form of domain restriction unless they have proper connections to recursive characterizations of a satisfaction relation. These connections will be absent unless we accommodate substitutional quantifiers that do not seem to be present in natural languages.6

Presupposition failures may also involve failures of truth-evaluability. But we shouldn't expect presuppositions to be exploited in communication in quite the same way as anomaly. If presupposition failures exhibit a form of semantic defect, that defect is, by definition, one that is contingent on a particular presupposition failing to be satisfied. Unlike with anomaly, a default policy of domain restriction for these cases would come with regular communicative costs. If a speaker and hearer unwittingly disagree about whether a presupposition of an instance of an unrestricted quantifier is satisfied, they will interpret one and the same quantified utterance in different ways. We would thus have an undetected communication failure whenever a speaker and hearer weren't in agreement about relevant presuppositions.

Finally there are special philosophical cases as, for example, liar sentences. But if liar sentences are truth-valueless, the story about why will be much more complex than that for simple instances of anomaly. The philosophical intractability of the liar strongly indicates that we shouldn't expect speakers to be immediately aware of its particular aberrations, the way that normal instances of anomalous defect simply ‘jump out' at speakers. If so, we shouldn't expect a domain restriction to occur. The domain restriction is supposed to be a means for speakers to exploit each others' clear awareness of uncontroversial defect. Liar sentences can't be exploited as such, even if in the end philosophical theorizing drives us to treat them as merely truth-valueless.

So we can make use of my explanation of the domain restriction from anomaly that appeals to truth-valuelessness without fear of over-generation. This is in part because my account doesn't propose to explain why we restrict our quantifier domains simply by appeal to truth-valuelessness, but rather by appeal to truth-valuelessness of a particular kind: that which speakers can generally count on each other to recognize merely in virtue of knowing the meanings of the words they speak. This feature is arguably unique to the class of anomalous utterances, even if there are several other classes of expression that constitute serious forms of semantic defect.

Let me get clearer about just how strong a thesis about semantic defect we need to afford ourselves the explanation I'm proposing. We would need to endorse the following.

  • (C)
    When an uttered sentence exhibits the resistance to interpretation characteristic of anomaly in a given context, that utterance is not truth-valued.

Though (C) is a highly controversial claim, as I've already noted, there are several noteworthy respects in which (C) is weak. For example, (C) is non-committal as to whether anomalous status precludes an utterance from expressing a proposition in its context. Whether this is so might turn, for example, on the question of whether trivalence is a property propositions could coherently bear. (C) is also non-committal as to whether anomalous character arises in a context-independent way, and so leaves open that a sentence figuring in an anomalous utterance could be used in other circumstances, or in embedded contexts, truth-evaluably.

(C) affords us a plausible and extensionally adequate explanation of why we witness the special kind of domain restriction that anomaly produces—something which none of the other four responses were even able to achieve. But this is not all. Adopting (C) has three additional virtues. First, as we have seen, it gives an intuitive explanation of why the domain restriction occurs which shows it to be a communicatively beneficial linguistic mechanism, just as standard explanations of contextual quantifier domain restriction do for sentences like (7). Second, the explanation validates what seems to be an obvious fact about the domain restriction: it is a kind of response to a semantic feature which anomaly in particular bears. Finally, it also helpfully explains the features of the domain restriction which distinguish it from other kinds of quantifier domain restriction—for example its characteristic robustness in the face of considerations of salience and relevance. Regardless of how salient a given object is for the purposes of the discussion at hand, this won't change the fact that the relevant anomalous instance fails to express a truth-evaluable proposition. Accordingly, a speaker who knows the meanings of her words, and who focuses attention on an object, which figures as an anomalous substitution instance of her quantified statement, cannot simply be making a blatant mistake about what the facts are, the way the mathematician who uttered (11) could be. On these hypotheses, there is no definitive mistake about how the world is for the speaker to make. Thus there is no immediate cause to reinterpret her statement as having a quantifier which ranges more broadly.

If this is right, we have strong support for (C). It figures as an indispensable part of the best account of the source of the unique form of quantifier domain restriction witnessed in (4) and similar cases of domain restriction from anomaly. For this reason, I propose that we provisionally accept (C) and appeal to it in exploring what changes the domain restriction that motivates it might require of us. Before doing this, however, I need to address the question as to whether or not the domain restriction from anomaly is semantically enforced, and how prevalent it is.

1.2 Syntax, Semantics, Pragmatics

The argument of the previous subsection for the claim that anomalous utterances are not truth-valued proceeded in abstraction from the question of whether or not the domain restriction from anomaly was enforced syntactically, semantically, or pragmatically. One could, for example, accept (C) on the basis of the arguments I gave and still maintain any of these three options: the domain restriction could be the product of phonetically null syntactic material, it could be the result of a systematic semantic mechanism, or perhaps a statement like (4) should be counted as straightforwardly truth-valueless with pragmatic principles leading speakers to reinterpret the utterance in the appropriate way. In this section, I'll argue that the restriction is best construed as operating at the semantic level. This claim will play an important role in §'The Semantics of Anomaly and Quantification' and §'Logical Consequence'.

First, just as in cases of salience-sensitive quantifier domain restriction, maintaining that the domain restriction in (4) is syntactic is difficult due to the problem of underdetermination.7 Very few, if any, theorists think that the domain restriction in (7) is the result of added, unarticulated syntactic material.

  • (7)
    All the beer is in the fridge.

The reason is that there seems to be no principled way to pick out one of many extensionally equivalent expressions allegedly present in a given utterance of (7) to restrict the quantifier domain appropriately. For example, in one particular context “all the beer which we just bought” does just as well as “all the beer which we just bought today”, and “all the beer which we carried in together” and so forth. In the same way there are extensionally equivalent ways of bringing out the domain restriction in (4): “which has roots”, “which is rooted in the ground”, “which is planted in the yard”, and so on. In addition to the problem of selecting one from among several possible candidates for syntactic ellipsis, sometimes it is difficult to even find one. Consider again, in this regard, (14).

  • (14)
    Everything has a moral if only you look for it.

On its maximally general interpretation, which only excludes anomalous instances, (14) is restricted to a highly diverse array of things. Even if one could find the right syntactic material to perform the restriction, it would have to be implausibly long, and completely unavailable to the speakers supposedly generating the relevant syntactic structure.8

The real question is whether the phenomenon is semantic or pragmatic in nature. Three things point to a semantic treatment. First, the domain restriction, where it occurs, is fairly robust: it is salience-insensitive and gives evidence of being quite systematic. Second, unlike with more familiar forms of contextual quantifier domain restriction, the domain restriction due to anomaly is triggered by the presence of a semantic feature—anomalous status—rather than by special features of the context of use. As I've allowed above, whether or not an utterance is anomalous may turn out to be sensitive to conversational context. But within a fixed context, judgments of anomalous status are clearly tracking some kind of semantic aberration. It seems reasonable to suppose that interpretive shifts clearly responsive to the presence of a semantic feature are themselves semantic. Both of these first two reasons for treating anomaly semantically are connected with the fact, which I'll explore in §'The Semantics of Anomaly and Quantification', that we can systematize the information relevant to the domain restriction so as to track when it occurs.

A third reason for treating the domain restriction from anomaly semantically is that there are noteworthy arguments for considering even the more seemingly pragmatic phenomenon of domain restriction from salience or relevance as semantic in nature. These can be extended in an indirect way to apply to the domain restriction from anomaly. The arguments, owing to Stanley and Szabó (2000a), capitalize on a binding phenomenon arising from the interaction of multiple quantifiers. For example, on its most natural interpretation, the domain of quantification of “every bottle” in (15) varies with the different rooms in the domain of the first quantifier.

  • (15)
    In every room in John's house, every bottle is in the corner.

Stanley and Szabó have argued that pragmatic accounts of how quantifier domains are restricted can have a hard time explaining how the natural reading is arrived at in (15) since there appears to be a kind of binding.9 (15) reads roughly as “every x such that x is a room in John's house is such that every y in x such that y is a bottle is in the corner of x.” Semantic accounts which posit a variable that interacts with the discourse context to produce contextual domain restrictions can account for this phenomenon very easily, since the binding in question can occur over the relevant, syntactically realized variable.

I don't want to take a stand on whether Stanley and Szabó are right. I only want to note that if they are right, this provides additional support for treating anomalous domain restriction semantically. This is because the domain restriction from anomaly can interact with this binding phenomenon.

Jose's Cocktails. At a gastronomical competition Jose served two courses of tuna and veal accompanied by different rum cocktails. Jose added three kinds of cocktail bitters to the first cocktail, and two distinct kinds of cocktail bitters to the second. The judges enjoyed each meal. But after sipping from the cocktails, they declared that they would like to try them with more of the bitters that Jose had added. Jose accordingly added more of his first three bitters to his first cocktail, and more of his subsequent two bitters to the second cocktail, and presented them again for judging.

  • (16)
    The judges sipped from everything Jose served before Jose added more of each cocktail bitter to it.

There is a true reading of (16), used to describe the above story. To get this true reading two forms of quantifier domain restriction must occur. First, “everything” must be restricted to speak only about drinks. This is the restriction created by the anomalous character of the claim that the judges sipped non-liquids. Second, “each cocktail bitter” should quantify not over every cocktail bitter but, for each drink, over every cocktail bitter added to that given drink. That is, (16) should have a reading with a form roughly like “For every x such that Jose served x, and for every y added to x such that y is a cocktail bitter, the judges sipped from x before Jose added more of y to x.” What is important about this example is that if we follow Stanley and Szabó in taking the second restriction to be semantic, we have good reason to maintain that the first, anomalous restriction is semantic as well. If instead we take the anomalous restriction to be pragmatically enforced, and other restrictions to be semantic, we will still end up needing a pragmatic explanation of the kind Stanley and Szabó claim is challenging to provide: one which requires pragmatic inferences from a claim involving a semantically unrestricted quantifier (in this case, ranging over food and drinks, say), to influence not only the values of that quantifier (so as to involve drinks), but also to influence the bound reading of a distinct quantifier (that is, to influence exactly which bitters lie in the domain of quantification of the second quantifier). In other words, we can't be selective about which forms of quantifier domain restriction are encoded semantically, since the two kinds of restriction sometimes must operate in tandem.

What I've been arguing so far is that the case for taking anomalous domain restrictions to be semantic in nature is strong—significantly stronger, for example, than the case for taking contextual quantifier domain restriction to proceed via a semantic influence of context. We've just seen that the best reasons in favor of the latter case, including the binding phenomenon, also favor construing anomalous restriction semantically. Moreover, unlike with contextual quantifier domain restriction, the insensitivity of anomalous domain restriction to shifts in contextual salience, and the fact that it arises from a responsiveness to a semantic feature both further motivate giving it a systematic semantic treatment. For these reasons, I'll proceed now on the working assumption that the domain restriction from anomaly is semantically enforced to explore what consequences this has for semantic theorizing and logic.

But before I continue, I need to mention a brief caveat. When I claim that the domain restriction is semantic, I do not mean to claim the phenomenon exhibits no sensitivity to context, or is without exception. It is not necessarily insensitive to context because anomalous status itself may be context-sensitive.10 It need not be without exception because, if the story of §'Domain Restriction' is on track, the restriction is merely a default interpretive mechanism, which may be overcome by other factors. This is important because there is evidence the domain restriction is not entirely uniform. I suspect, however, the explanations for the lack of uniformity are diverse. In addition to the possible context-sensitivity of anomalous status, processing costs may be relevant. Speakers might be less likely to enforce the domain restriction if the anomalous triggering material occurs late in a sentence, past the point where the ordinary material contributing to the quantifier restrictor occurs as in (17).

  • (17)
    Bob burned everything in his yard with due precaution, not long after having uprooted it.

Also, there are degrees of anomalousness, and there is an open question of ‘how anomalous' something must be to count as truth-valueless, and generate the domain restriction. Unfortunately I don't have the space to explore the interactions between these phenomena and the domain restriction here. What's important is that the view I mean to be defending leaves open that that the domain restriction could have exceptions for these reasons and perhaps others.

2. The Semantics of Anomaly and Quantification

  1. Top of page
  2. 1. Anomaly and Quantifiers
  3. 2. The Semantics of Anomaly and Quantification
  4. 3. Logical Consequence
  5. Appendix
  6. References

If, as I have argued, the quantifier domain restriction of §1 is a semantically enforced phenomenon responsive to the truth-valuelessness of anomaly, what can we learn from this about the shape of our compositional semantic theories? Here, I'll make some general remarks about what I take to be a plausible answer to this question. The appendix contains a formalism which encapsulates the main ideas.

If we accept (C) owing to the domain restriction from anomaly, we are committed to representing the defects of anomaly in our semantic theories. We cannot, for example, treat anomaly as malformed in the same way as ungrammatical sentences, thereby excluding it from the purview of our semantic apparatus. This is because the semantics will ultimately need to retrieve information about which utterances are anomalous in order to adequately track the quantifier domain restriction due to anomaly. Syntax must admit the anomalous sentences if the semantic apparatus is to systematize the relevant information.

Once anomalous sentences are treated by the semantic theory, the latter will naturally have to make special provisions on pain of misrepresenting the semantic values of anomalous and non-anomalous sentences alike. Purely bivalent theories (which have only two possible extension assignments for sentences) will, for obvious reasons have to go by the board. We need a third semantic status over and above truth and falsity and use this value to recursively track instances of anomaly and their effect on quantifier domains.

The idea of accommodating a third semantic status is wholly familiar—even in application to anomaly.11 But there is little agreement on how this third value should behave, and what exactly its significance is. My proposal here is going to have some striking consequences for these issues, at odds with most trivalent theories: we'll soon see that a third semantic value may play a systematic, positive role in truth-evaluable interpretation. But before I can say more clearly what I mean by a ‘positive' role in interpretation, and why this is unique to the phenomena of §'Anomaly and Quantifiers', I need to say a little more about the compositional behavior of anomaly and the domain restriction it creates.

Let me begin by quickly sketching some familiar recursive tools we need to track anomaly. At the base level of our recursion the semantics will clearly need, for each predicate in the language, information about which objects that predicate can truth-evaluably be used to talk about. I'll do this by associating with each predicate a set of objects comprising what I will call, as a nod to Russell, its domain of significance.12 Since I won't have time to address issues concerning the structure of those domains here, I will presently take the most non-committal formalization of them possible: one allowing for arbitrary sets of objects, or n-tuples of objects for predicates of higher adicity than one.13

Such domains constitute the information needed to classify which ‘atomic' ascriptions of a predicate to an n-tuple of objects are anomalous. Using these domains, perhaps along with the extensions of predicates, to recursively track how anomaly projects into coordinative constructions is simple to achieve, though controversial in precisely how to implement. Anomaly seems to have an ‘infectious' character: wholes with anomalous parts tend to be anomalous as seen in (18)–(20).

  • (18)
    * Relapses demote the undertow.
  • (19)
    * Relapses demote the undertow and ice cream tastes great.
  • (20)
    * If relapses demote the undertow, I'm going on vacation.

However, anomaly may not always project in these ways. Possible exceptions include embeddings of anomaly under some uses of negation, and into disjunctions and counterfactual conditionals. My goal here is not to take a stand on how anomaly projects in any of these cases. It suffices to say that many different ways of tracking truth-valueless projection behavior across simple connectives are already well-explored (for example, in Kleene's Strong and Weak schemes). Moreover, these can all be implemented in a framework drawing only on domains of significance (and extensions) for atomic predicates, and so don't bear directly on the novel techniques I want to introduce to cope with the case that has preoccupied me here, namely that of quantification.

So let me turn to the question: when is a quantified statement anomalous? The answer to this question will turn out to be helpful in deciding how to represent quantifier domain restriction in non-anomalous cases. Consider the following two simple instances of quantified anomaly.

  • (21)
    * Some primes are red.
  • (22)
    * Every tomato is polarized.

From both a logical and a semantic perspective (22), for example, doesn't seem very problematic. A standard rendition of the logical form of (22) might be as

  • display math

Certainly it should be permissible to predicate a variable with “Tomato”. Likewise for “Polarized”. So, if anything, something must go wrong at the level of appending the quantifier. Similarly (21) merely asserts a non-empty intersection between two sets—the set of primes and the set of red things. But the intersection of those sets is empty. Why isn't the claim simply false?

On reflection there seems to be a ready explanation for why both (21) and (22) are anomalous, brought out by consideration of similar instances of anomaly. There are plenty of primes and plenty of red things, but nothing non-anomalously talked of as both. That is, for any object o either “o is prime” or “o is red” is anomalous. Likewise for tomatoes and polarization. What renders quantified anomalous sentences problematic appears to be non-intersective domains of significance. Otherwise put, a quantified sentence seems to be anomalous if it has only anomalous objectual substitution instances. Following this lead, I'll be developing a semantics on which having only anomalous substitution instances is both necessary and sufficient for a quantified statement's becoming anomalous. There are special cases that apply pressure to deny both the necessity and sufficiency of this condition, and I am open to weakening the proposal.14 Nonetheless, it has an attractive simplicity that will at least be useful as a starting point.

If we were only concerned about these kinds of instances of quantified anomaly, how should we track their occurrence? The answer is simple: use domains of significance to recursively keep tabs on which assignments to unbound variables result in a truth-evaluable whole. We can call this set of variable assignments the domain of significance of the open formula. If the domain of significance of an open formula is empty, this means any way of binding all its variables results in an anomalous sentence.

We've codified our commitments about when quantified statements are anomalous, or not. But to complete our semantics we need to say more about what the interpretation of non-anomalous quantified sentences is. This is a more delicate issue since, according to the view I defended in §'Anomaly and Quantifiers', we interpret quantified statements by restricting their domains to avoid having to interpret them in ways that would prevent them from being truth-evaluable. For some quantifiers, there are relatively uncontroversial ways of construing how failures of truth-evaluability sometimes project given unrestricted quantification. For example, most trivalent semantics for universal quantification (including both Strong and Weak Kleene, for example) have that quantifier inherit truth-valuelessness from any truth-valueless substitution instances within its quantifier domain, provided the other instances are true. Quantifiers like “some”, and “most” present us with more controversial options. Will “some Fs are G” always exhibit truth-valuelessness just so long as it has at least one truth-valueless substitution instance involving an object from its quantifier domain? Some semantics—like a Strong Kleene semantics—deny this, while others—like a Weak Kleene semantics—affirm it. Similar considerations apply to “most”.

Different proposals for the projection behavior of truth-valueless instances in unrestricted quantifier domains will interact with my proposed views on quantifier domain restriction due to anomaly to generate different empirical predictions. This leads to two virtues. First, my proposal is flexible: it can accommodate any view about truth-valueless projection not only for connectives, but also for unrestricted quantifiers. Second, my proposal opens up the possibility of working backwards from facts about quantifier domain restriction to claims about anomalous status and projection in unrestricted quantifiers. This methodology is quite useful since, as already noted, intuitions about domain restriction are often much more stable than intuitions about anomalous status, or its projection.

To see this second virtue in operation, here's an application of the ‘working backwards' methodology to “most”.

Vera's Patient. Vera has one of her patients, Marla, begin their therapy session by producing drawings and text on a single sheet of paper. Marla scrawls a dozen or so images and writes out the first ten words that come to her mind. Vera picks up the paper and, after reading the first two words in her head, reads the next eight, which seem more significant, out loud to Marla.


  • (23)
    Vera read most things Marla scrawled on the page out loud.

Speakers tend to read (23) as true, even when the images on the page are explicitly described as scrawled on it (and the words are not). This would only be predicted, given the views I've been articulating, on the assumption of two facts: it is anomalous to say of Marla's drawings that they are read, and the semantics for “most As B” should treat it as truth-valueless when there are objects o in its quantifier domain such that inline image is truth-valueless. This shows how we can arrive empirically at facts about the semantics of unrestricted quantifiers using data involving restricted quantifiers, via (C) and the domain restriction hypothesis concerning anomaly.

So to reiterate, any view about projection of truth-valuelessness in unrestricted quantifier domains can be integrated with my proposed views on domain restriction, so there is no need to make general commitments as to what the original projection behavior is, and it is my preference here not to do so. Nevertheless, the theory of domain restriction can serve an instrumental purpose in helping to empirically settle facts about projection in unrestricted contexts. The theory interacts with proposals for unrestricted quantification to yield predictions about the conventional truth-values of truth-evaluable quantified claims. This allows us to test views about projection from quantifier to quantifier—perhaps with results more helpful than tests that appeal to intuitions about truth-valueless claims.

Now, since a recursive characterization of domains of significance for open formulas of the kind I mentioned earlier would implicitly contain information about the truth-evaluability of various substitution instances, that same recursive characterization has all the information needed in our definitions of the truth-conditions of quantifiers to adequately capture the quantifier domain restriction due to anomaly. Truth or falsity of a quantified statement is ascertained by evaluating objectual substitution instances of the open formulas over which it quantifies that would not lead to failure of truth-evaluability were the quantifier domain to include them. So however we negotiate the details of the quantifier domain restriction, recursively characterized domains of significance for open formulas will be sufficient not only to track the presence of anomaly in quantified statements, but to interpret non-anomalous, truth-evaluable quantified statements.

This concludes my sketch of a semantics for anomaly: add domains of significance for predicates, use this information to recursively track the presence of quantified anomaly, and also to characterize the interpretation of non-anomalous quantification. The novel, and most important, part of my proposal comes of course in the third step. For the more formally inclined, a more detailed implementation of these ideas can be found in the appendix.

For now, we're in a position to appreciate the importance of a claim I made earlier: that truth-valuelessness can play a positive role in truth-evaluable interpretation. To understand what I mean by this, consider what happens in various trivalent semantics under an expansion of a predicate's domain of significance—the addition of some objects to a predicate's domain of significance which antecedently lay outside it. In standard trivalent semantics, producing such an expansion never produces a shift between truth-evaluables. That is, such an expansion never changes a true claim to a false claim, or a false claim to a true one. This is a property of interpretation functions that belongs to all interpretation schemes of familiar trivalent semantics.

For example, consider the Weak Kleene scheme, on which truth-valuelessness always projects through connectives and quantifiers. If any sentence S changes its truth-value after the expansion of a domain of significance of a predicate P with an object o, it must be because the truth-value arising from predicating P of o matters to the truth-value of S. In the Weak Kleene scheme the only way predicating P of o could have originally influenced the truth-value of a sentence S is by rendering it truth-valueless. So no expansion of a domain of significance on this semantics moves us from a true claim to a false claim, or a false claim to a true claim. The same is true of both the Strong Kleene scheme and a supervaluation scheme.

This commonality isn't incidental. On a prevalent, usually tacit, assumption reflected in trivalent semantics of many kinds—especially those applied to anomaly—truth-valuelessness merely interferes with truth-evaluable interpretation. Only the extent to which it so interferes is contested. On the Weak Kleene scheme it interferes as much as possible. On the Strong Kleene, less so. And on a supervaluation scheme, less still.

This changes once we move to a semantics which integrates an involvement of truth-valuelessness in producing a quantifier domain restriction, for the obvious reason. Sometimes a statement in which the domain restriction takes place, like (4), has the conventional truth-value it does (true, in this case) because a particular predication is truth-valueless. If we could expand the domain of significance of “uproot” to include planks, we might well change the value of (4) to false.

What this means is that on this semantics we abandon a core, shared feature of most standard trivalent semantics. We must do this because truth-valuelessness isn't merely interfering with conventional, truth-evaluable interpretation. It's contributing to it. Language users attend to anomaly as they try to figure out what they are conventionally, and successfully saying to each other. Aside from its intrinsic interest, this result has the potential to help us answer foundational questions about the character of our third semantic status. A key question about semantics that appeal to a third value is whether the third value is being used to model competences that speakers have (the positive ability to recognize aberration), or competences that they lack (mere inability to interpret). Our result obviously favors the former. But perhaps more importantly, unusual positive contributions to truth-evaluable interpretation from semantic defect may have equally unusual effects on characterizations of logical consequence. Let's turn to these effects now.

3. Logical Consequence

  1. Top of page
  2. 1. Anomaly and Quantifiers
  3. 2. The Semantics of Anomaly and Quantification
  4. 3. Logical Consequence
  5. Appendix
  6. References

The domain restriction from anomaly has some interesting implications for issues in philosophical logic. There are two ways the domain restriction may interact with logical form. Regardless of which option one takes, the presence of anomaly generates failures of classical logic in characterizing important classes of natural language inference—failures of a kind that no other known phenomenon generates. Additionally, on very weak assumptions about the projection of anomaly in quantified contexts, the interaction of anomaly with quantifiers may supply new motivations for thinking that capturing a useful set of inferences owing to logical form requires the importation of at least some semantic information. Let me take each idea in turn.

In §1, I argued that a speaker's truth-value attributions to (4) and (5) owed to a semantically enforced domain restriction responsive to anomaly.

  • (4)
    Bob uprooted everything in his yard and burned it.
  • (5)
    Bob burned everything in his yard.

How the truth of (4) and falsity of (5) affect the logical consequence relation depends on how the quantifier domain restriction from anomaly interacts with the logical forms of these sentences.

A first construal takes the logical forms of (4) and (5) to be something like (4′) and (5′).

  • (4′)
    [inline image: InYard(x, Bob)][Uproot(Bob, x) ∧ Burn(Bob, x)]
  • (5′)
    [inline image: InYard(x, Bob)][Burn(Bob, x)]

If this is the case, a straightforward classical inference is violated among truth-evaluable sentences relative to the same context.15 Such a failure is arguably uniquely generated by the presence of anomaly. For example, this kind of inference failure is not as clearly manifested by forms of salience-sensitive quantifier domain restriction, context-sensitivity in general, or even by phenomena which otherwise have the potential to motivate a shift to trivalent semantics, such as presupposition failure. Let me say a little more by way of defending this claim.

Context sensitivity may seem to provide a wealth of potential failures of classical inferences schemes. For example even the inference from “It's precisely 5 o'clock” to “It's precisely 5 o'clock” may be suspect, since the first utterance can be true and the second false owing to minuscule changes in their contexts of utterance. Similarly, consider the ‘inference' from the potentially true utterance of “All the beer is in the fridge” as used before the beer in the basement is made salient to the potentially false utterance of “All the beer is in the fridge” after that beer has been made salient.

Crucially, however, such examples only pose a threat to classical inference schemes on the assumption that context does not impact parameters in logical form. If the time of the context of utterance, or the context of utterance itself, forms part of the logical form of any utterance of “It's precisely 5 o'clock”, these won't provide counterexamples to classical schemes. They will simply motivate (relatively minor) complications in our conception of logical form. Relative to fixed contexts, classical schemes keeping track of identical contributions to logical form from context can be preserved. Similar remarks hold for familiar forms of contextual quantifier domain restriction. In fact, Stanley and Szabó have used the binding phenomenon to argue precisely that the influence of context on quantifier domains is mediated by variables present in logical form.

Other phenomena that might motivate a shift to trivalence, such as certain strong forms of presupposition failure, could have more of an effect on classical inference schemes. But the effect of shifting to trivalence alone is not necessarily as damaging to classical logic as one might expect. The complications arising from trivalence can lead to re-defining consequence not as truth-preservation, but as truth-preservation among truth-evaluable sentences. This restricted relation largely factors out the influence of a third truth-value, again enabling us to pick out a quite substantial body of ‘inferences' (now reconstrued) which are truth-preserving-among-truth-evaluables in virtue of logical form. The resulting relation, for obvious reasons, tends to vindicate classical inference schemes, fostering the view that classical logic is the logic of truth-evaluables.

By contrast, the truth of (4) and falsity of (5), given the proposed logical forms (4′) and (5′), in some sense constitute as real and substantial a violation of classical logic as one could get. If we adopt (4′) and (5′) and leave the task of effecting domain restriction to the clauses of the semantics for the universal quantifier, we have kept logical form too simple to allow a move like that typically used to safeguard classical schemes in the face of context-sensitivity. Moreover, unlike with other engagements with trivalence, redefining inference to track truth-preservation-among-truth-evaluables won't help in this case, as (4) and (5) are both truth-evaluable. And of course this particular failure isn't the only one—many other failures of standard quantified inferences will have to go by the board, increasing the sense that classical inference schemes are inadequate for capturing basic quantified inferences in natural language, even just for “all” and “some”.

All this is true if we stick to (4′) and (5′) as the proper logical forms for (4) and (5). But there is a second way of construing their logical forms. Rather than enforcing the domain restriction in the recursive semantic clauses of quantifiers, we can take it to be effected by an element realized in the logical form of quantifier restrictors. This would most likely be done by systematically accommodating special variables or functions in quantifier restrictors. The logical form of (4) and (5) might then look something like:

  • (4″)
    [inline image: InYard(x, Bob) ∧ inline image][Uproot(Bob, x) ∧ Burn(Bob, x)]
  • (5″)
    [inline image: InYard(x, Bob) ∧ inline image][Burn(Bob, x)]

Given the arguments of §'Syntax, Semantics, Pragmatics', the values of the variables or functions are not here supplied by features of the context of linguistic use, but by the semantics of the sentences themselves. Even so, what is important about this construal is that the apparent classical violation in the blocked inference from (4) to (5) is merely apparent: the logical forms of (4) and (5) are more complex than their surface grammar reveals. So, in a way, classical logic is safeguarded.

But we don't merely care about what the logic of our language is; we also care about how often it applies to inferences we actually make. Though the strategy adopted on the second construal avoids violating classical inference schemes, it does so without safeguarding its applicability to some of the most common natural language inferences. The reason is that on this construal a vast range of quantified natural language inferences have logical forms making them classically invalid. To take just one example, consider the inference from (24) to (25).

  • (24)
    Every man left.
  • (25)
    Every short man left.

On the current proposal, this inference will not be valid, even holding the contributions of context of use fixed. This is because our current proposal has two commitments: First, the mechanism by which domain restrictions due to anomaly are assessed owe to values attached to tacit elements in quantifier restrictors. Even if the domains of significance of predicates are perfectly broad, or are all supersets of the denotations of the overt elements in the quantifier restrictor, we need these tacit elements in place if only to represent the semantically inert outcome of the process by which anomalous domain restriction is assessed. Second, the process by which the element associated with a given quantifier is assigned a value in logical form involves operations over the domains of significance of every predicate attaching to a variable bound by that quantifier. This means that the values of the elements associated with the quantifiers of (24) and (25) must be distinct, since they involve operations over distinct sets of domains of significance. We accordingly have the following logical forms, with distinct assignments to inline image and inline image.

  • (24′)
    [inline image: Man(x) ∧ inline image][Left(x)]
  • (25′)
    [inline image: Man(x) ∧ Short(x) ∧ inline image][Left(x)]

Note that even if the product of the operations over domains of significance in (24) and (25) is the same, this product is settled in our semantics, and is unavailable to syntax. Accordingly the inference from (24′) to (25′), and hence from (24) to (25), isn't safeguarded merely in virtue of logical form.

Also note that this problem doesn't afflict inferences involving sentences with standard forms of context sensitivity, since the inferences that are good logical inferences are ones where it's quite plausible that contributions from context to logical form can be held fixed. For example, the same problem won't arise for analogous treatments of contextual quantifier domain restriction, modeled with tacit variables or functions. In those cases, whatever shifts the values of the tacit material (when unbound) is clearly exhausted by facts about context of linguistic use. If I safely infer “All the beer is in the fridge or on the counter” from “All the beer is in the fridge”, despite a domain restriction in both, we can continue to construe the inference as a logical one since contextual contributions to the restrictions are plausibly the same.16

Thus, on this second way of treating the domain restriction, though traditional logic gets the semantics of the quantifiers right, it no longer on its own captures anywhere near as substantial and interesting a body of natural language inferences as traditionally conceived. Since the problem stems from the semantic features of the sentences used in inference, we can't sidestep this issue in the way we might for corresponding ‘failures' owing to context-sensitivity: it is fruitless to try to preserve a substantial body of inferences by restricting attention to a fixed context. What this means is that on this second proposal, logic as applied to natural language inference would, in effect, be reduced to an awkward and meager extension of propositional logic.

So, regardless of whether the domain restriction from anomaly is built into the recursive clauses for quantifier interpretation, or whether it is mediated by an element in logical form, this domain restriction threatens the applicability of classical logic to natural language inference in ways that no other known phenomenon does.

Appreciating this point should reinforce the idea that anomaly interestingly transforms fairly standard conceptions of logical consequence. What then should a revised consequence relation look like? Our first move in recapturing a set of valid quantified inferences should be relatively straightforward given our work in §2. Letting Γ be a set of sentences and ϕ a sentence, a typical consequence relation ⊧ is given by the following definition:

  • display math

where inline image ranges over bivalent models. To get a better consequence relation, we need only allow inline image to range more broadly over the trivalent models, such as those sketched in §'The Semantics of Anomaly and Quantification' (and detailed more rigorously in the appendix), that track the influence of anomaly on the domains of quantifiers.

Such a move, however, brings other problems to light which concern the potential projective behavior of truth-valuelessness. The stronger its projective behavior, the more inference schemes are lost. To take one example, if truth-valuelessness projects across disjunction, the above definition will ensure inline image. As I've tried to stress, I mostly want to stay neutral on the question of how truth-valuelessness projects in non-quantified contexts, since this is a controversial and delicate matter I don't have the time to discuss. What we can say here is that very minimal assumptions about the projective behavior of anomaly in quantified contexts on their own ensure that anomaly unduly perturbs even a consequence relation redefined over trivalent models as above. Consider:

  • (26)
    Few men ate.
  • (27)
    * Few men refrained around the discipline and ate.
  • (28)
    No buildings are grey.
  • (29)
    * No pubescent buildings are grey inversions.

If (27) and (29) (or suitable variants of them) are anomalous, and my arguments from §'Anomaly and Quantifiers' are sound, these examples give violations of quantified inferences which are importantly different from those I've examined so far. The domain restriction from anomaly on its own provides no reason to expect a violation of the inference, for example, from “No Fs are Hs” to “No F-and-Gs are H-and-Is”. After all, further winnowing the objects satisfying the restrictor and matrix of “No” typically only increases the likelihood its use will come out true. The problem seems to arise from the fact, noted in §'The Semantics of Anomaly and Quantification', that quantified claims with only anomalous substitution instances tend to come out anomalous.

As these examples should help reveal, this phenomenon has the potential to greatly perturb the inferential schemes licensed by our definition of consequence above. In general, inferences which move between quantifiers while adding more material to the quantifier scope or restrictor will be threatened. This results in a set of valid quantified inferential schemes that looks erratic, and somewhat uninteresting.

To cope with this problem, it is natural to take a strategy I alluded to earlier: since truth-valueless whole sentences tend to render the consequence relation uninteresting, we can get a better grip on the class of inferences mediated by logical form by simply ‘factoring out' the influence of these problematic sentences on the consequence relation. That is to say, we can recharacterize the consequence relation as one which relates truth-evaluables, as follows.

  • display math

I'll call inline image the relation of formal logical consequence, for reasons that will be clear soon.

Formal consequence succeeds in picking out a significant class of schemes conducive to inference including many involving quantified sentences, while blocking those from the domain restriction due to anomaly. For example, as long as anomaly projects over conjunction, and unrestricted universal quantifiers inherit truth-valuelessness from any truth-valueless substitution instances in their matrix,

  • display math

in line with the evaluations of (4) and (5) in §1. On the other hand, other quantified inferences such as

  • display math

are safeguarded. Note, for example, that added anomalous restriction owing to the presence of inline image can only further narrow the quantifier domain of the rightmost formula, which will never interfere with the inference.

What the final relation looks like, of course, depends on our choice of projection schemes. To take one example, if we adopt a Weak Kleene scheme for propositional connectives the formal consequence relation only eliminates quantified validities from the bivalent setting that need to be jettisoned due to the quantifier domain restriction from anomaly.

Proposition 3.1 Let BIVALENT denote the set of valid bivalent inferences, PROP denote the set of valid inferences of bivalent propositional logic, and FORMAL denote the set of formally valid inferences (for trivalent models using a Weak Kleene scheme (as in the appendix). Then the following relations hold:

  • display math

Proof. Suppose Γ propositionally entails ϕ in the bivalent setting, and a trivalent model inline image (of the kind given in the appendix) is such that inline image, and inline image. Then, since we are working in a Weak Kleene scheme, each inline image which is a truth functional component of either ϕ or a formula in Γ is such that inline image. But then we are essentially in the bivalent case, so we have inline image. This shows the first containment. The second containment follows from the fact that bivalent models are just trivalent models with degenerately broad domains of significance. That the containments are proper is witnessed by the two examples of which quantified inferences are, and are not, formally valid given just above.□

Other schemes may of course result in a very different formal consequence relation.

Adopting the formal logical consequence relation, however, comes with an important philosophical cost. inline image does not model sound inference, but only sound inference among truth-evaluables. So there will be very many sentences ϕ and ψ, and models inline image, such that inline image while inline image and inline image. This is significant because of a standard construal of what a logical consequence relation should be.

Formal logic, in the sense I'm alluding to, is conceived as in the business of tracking which inferences are truth-preserving in virtue of logical form. It does this through attention to how the truth-conditions of complex sentences systematically correlate with aspects of their logical form. Logic, though sometimes employed in conjunction with semantics to swell the relevant body of inferences tracked, is thought to have an autonomous domain. The idea, from the ancients down to the early analytics, is that there is a substantial and interesting body of inferences which are entirely content or subject-matter independent. The idea that there is such a body of inferences is threatened by the projection behavior of anomaly.17 The influence of anomaly in generating truth-valueless quantified claims threatens to make a class of quantified inferences which appeals solely to logical form look impoverished and erratic, as already noted.

What makes the set uninteresting is a problematic interaction between two desiderata: aiming to track the contributions that pure logical form makes to truth-preserving inference; and aiming to track truth-preserving inferences themselves. To adopt the formal consequence relation is to concede the force of this tension, and jettison the second of these desiderata in favor of the first. The formal consequence relation ceases to pick out only truth-preserving inferences. But it succeeds in picking out inferences that are recognizable merely on the basis of their logical forms—forms that are conducive to, but insufficient for, truth-preserving inference. We can recapture genuine truth-preserving inference with a second consequence relation, which I'll call the semantic logical consequence relation, that imports a minimal amount of semantic information to capture genuine truth-preserving inference due to logical form.

  • display math

By appealing to a model parameter, we can capture information about truth-evaluability needed to ensure that a transition from Γ to ϕ is one which is guaranteed to preserve truth, and by appealing to inline image we ensure the transition is indeed mediated by logical form, to the extent logical form can make contributions to inference. Just as with inline image though, inline image sacrifices one intuitive hallmark of a logical consequence relation for another: It captures all and only sentence transitions which are genuinely truth-preserving due to their logical form. However, it does this at the expense of incorporating semantic information via the model parameter, and thus can no longer be construed as tracking inferences which are entirely subject matter independent.

The fact that the semantic consequence relation imports information from a model has important philosophical consequences. For it shows that assessing whether a legitimate inference has been drawn between sentences may require basic information about the semantics of those sentences. Put another way: we cannot simply look at the syntactic features of sentences to discover information about truth-preserving inference.

So anomaly's interaction with quantification has two potentially interesting implications for philosophical logic. On the one hand, the ways in which anomaly does not project in quantification afford us our most substantial threat to the utility of classical inference schemes. On the other hand, the ways in which anomaly does project in quantification may provide us with reasons to doubt that there are substantial and regular bodies of sentence to sentence transitions which preserve truth solely in virtue of their logical form.

There is a tendency in the literature on semantic defect to appeal to controversial truth-value intuitions to motivate its existence, and to guide the creation of semantic and logical theories modeling its behavior. By contrast, I've aimed to show that in the special case of anomaly we can pinpoint, and diagnose, interesting forms of semantic defect indirectly, through its influence on uncontroversially truth-evaluable utterances. The possibility of such influence is typically dismissed or simply overlooked, as is reflected in the range of better-known logical theories that treat defect as merely interfering with truth-evaluable interpretation. As just argued, it is in fact the indirect influence of semantic defect which actually seems to create some of the more interesting and revealing effects on our logical theories.


  1. Top of page
  2. 1. Anomaly and Quantifiers
  3. 2. The Semantics of Anomaly and Quantification
  4. 3. Logical Consequence
  5. Appendix
  6. References

What follows is an idealized trivalent model-theoretic semantics for anomaly, assuming the simplest projection scheme for connectives and quantifiers: the Weak Kleene scheme. I'll work in a language incorporating two binary quantifiers ∀ and ∃, and connectives ¬, ∧, ∨ with a familiar syntax.18 To the standard characterization of a model, we need to add only information about which predicates truth-evaluably apply to which objects. I use a double-bracket notation inline image inline image for such domains. Otherwise the definition of a model is the familiar one.

A model inline image is a tuple inline image consisting of a non-empty universe of discourse M, and an interpretation function inline image which maps:

  • each constant c to an element inline image of M;
  • each n-ary R to a pair inline image containing
  • (i)
    an extensioninline image; and
  • (ii)
    a domain of significanceinline image with inline image.

Denotations are also computed in standard fashion: constants acquire their denotation from the model and variables from a variable assignment. Using such denotations we can specify the domains of significance for more complex expressions. Recall: the domain of significance of an open formula is the set assignments of objects to free-variables in that expression which would result in truth-evaluability.

In making this definition, as I said, I'll use a Weak Kleene characterization of the behavior of ¬, ∧, and ∨. As noted earlier, nothing important hangs on this choice for present purposes (aside from simplicity of exposition), and we can let empirical considerations guide our selection of projection schemes. Quantified formulas, however, merit special commentary. When a formula is appended with a quantifier binding a variable v we need two effects on the domain of significance. First, if every assignment to the unbound variables in the formulas comprising the quantifier restrictor and matrix renders either the restrictor or matrix non-truth-evaluable, the quantified formula should inherit this defect. Otherwise the quantified formula itself will be truth-evaluable according to the criteria I've given, and so should itself have a non-empty domain of significance. Now that v is bound, though, the assignments in the formula's domain of significance should be ‘indifferent' to the value on v. Both effects are achieved by appropriately importing the domain of significance of the restrictor and matrix and allowing its assignments to be arbitrarily permuted on v.19

The domain of significance of an expression e in a model inline image, written inline image, is a subset of inline image, the set of variable assignment functions, given as follows:

  • inline image for variable v
  • inline image for constant c
  • inline image | inline image for atomic inline image.
  • inline image
  • inline image
  • inline image | inline image for all quantifiers Q.

We can say an open formula ϕ is rendered truth-evaluable by an assignment g if inline image. A sentence ϕ (without free variables) is truth-evaluable just in case inline image (or equivalently inline image). Note that the definition of the domain of significance of e only appealed to facts about domains of significance, and not about extensions or anti-extensions. It's worth mentioning this is a unique feature of the Weak Kleene scheme, in which facts about truth-valuelessness are ‘separable' in this way. In other projection schemes, we need to incorporate facts about extensions and anti-extensions in recursively tracking truth-valuelessness, integrating the recursions for domains of significance and satisfaction.

Denotations of formulas relative to a model and assignment pair are mostly given as usual, again with anomalous character projected according to the Weak Kleene scheme. The main exception is in the treatment of quantifiers. In this instance, since we're assuming projection in unrestricted contexts also goes by the Weak Kleene scheme, most of the work is done by appealing to domains of significance again. Quantifiers are evaluated over domains restricted to exclude substitution instances which would generate truth-valueless status were domains to range more broadly. Given Weak Kleene projection for unrestricted quantifiers, this just means that we should restrict quantifier domains over non-anomalous substitution instances.

The denotation of a formula ϕ in a model inline image relative to an assignment g, written inline image is U if inline image, and otherwise is given as follows:

  • display math

Different construals of the projection of truth-valuelessness in unrestricted quantification will generate different, often much more complicated, clauses for enforcing the quantifier domain restriction. For example, if we adopt Strong Kleene projection for quantifiers, domains won't be restricted to elements in an open formula's domain of significance, but to some superset of that domain (that is, we will include in the domain of quantification those objects outside the domain of significance which happen not to contribute to anomalous status in unrestricted quantification). The superset in question may shift from quantifier to quantifier. In this case, domains of significance don't produce the restriction in anything like the simple way above, but the information recursively tracked in domains of significance (along with that tracked by denotations more generally) will of course need to be used in characterizing how the domain restriction occurs.

As usual, a sentence ϕ is true (simpliciter) in inline image, noted inline image, just in case for all inline image, inline image. Analogously, a sentence ϕ is truth-evaluable (simpliciter) in inline image just in case for all inline image, inline image (that is, again, if inline image).

  1. 1

    These sentences are sometimes called “category mistakes”. I prefer to avoid this terminology since it is connected with a theory about anomaly which I do not accept, but won't discuss here: that anomaly is the product of mismatches of logical types or ‘categories'.

  2. 2

    I'll use the marker “*” to mark general oddity and possible anomalous status.

  3. 3

    I won't presume the assessments are universal, and the argument I shortly give on the basis of the evaluations of (4) and (5) only requires those evaluations to occur some of the time. Along the way, and especially at the end of §'Syntax, Semantics, Pragmatics', I'll discuss some factors which would explain why judgments might waver.

  4. 4

    Of course, between any two utterances there always is some change in context, namely that produced by the passage of time and the fact that it enters into the conversational record that a new utterance has been produced. But an appeal to such changes to explain the domain restriction in (4) would be a hard sell, especially given the stability of speaker assessments in the reverse ordering.

  5. 5

    I speak of ‘conventional' expressive power since failures of truth-evaluability might still allow for the expression of ‘unconventional' trivalent propositions. I am presuming that even if there were such entities, they would be in some ways potentially detrimental to literal communication.

  6. 6

    A related case is that of non-referring demonstratives, including complex demonstratives with unsatisfied nominals. It is challenging to assess whether non-referring demonstratives do create a form of domain restriction. This is because the referents of demonstratives seem to be determined in part by the intentions of a speaker, which often must be guessed at on the basis of charitable interpretation. My sense is that a typical utterance of “Most of those [waving one's hand in the direction of a group of planks and rooted trees] will be uprooted later this afternoon” will, when possible, be interpreted as referring only to trees. But it is not clear that this is the product of a restriction on the quantifier domain, rather than because of charity in interpreting the speakers' intentions in working out the referents of the demonstrative. In any event, there is good reason to suspect that a domain restriction won't be witnessed for non-referring demonstratives: The domain restriction I've posited effectively allows speakers to gain efficiency in communication. But speakers exhibit an unwillingness or inability to capitalize on this efficiency if they take added time to explicitly demonstrate objects which are candidates for omission from quantifier domains. After all, their omission would obviate the demonstration. So it seems unlikely that a policy of domain restriction to avoid truth-valueless substitution instances would be extended to the case of non-referring demonstratives.

  7. 7

    The basic argument can be found in Stanley and Szabó (2000a) p. 236ff.

  8. 8

    These considerations against what I am calling the syntactic mode of quantifier domain restriction naturally do not speak directly against the idea that there could be, say, a demonstrative or a variable in the syntactic structures of these quantified statements which pick up their semantic values from other elements in the sentence. I'll discuss this possibility in §'Logical Consequence'.

  9. 9

    Pragmatic theories have some recourses though, at least by appealing to particular theories of structured propositions, as Stanley and Szabó (2000b) concede.

  10. 10

    What reason do we have to think anomalous status is context sensitive? Sometimes sentences whose utterances are anomalous in normal contexts can sound truth-evaluable, and sometimes even true, when interpreted against the backdrop a fantastical or wondrous story. For example “the number eight is wide” sounds anomalous, but when read in the context of a fantastical children's story in which numbers have some kind of physical reality, it may sound acceptable. One way to account for this phenomenon is to allow the interpretations of predicates to be context sensitive in ways that make anomalous status context sensitive as well. Another is to treat characteristically anomalous utterances as expressing trivalent propositions that aren't gappy at the worlds of fantastical stories. My suspicion is that we may need both kinds of sensitivity to properly account for all variations in anomalous status, but I won't explore how to do so here.

  11. 11

    See, for example, Thomason (1972) and Lappin (1981) for trivalent treatments of anomaly.

  12. 12

    See, e.g., Russell (1908), Carnap (1937), Ryle (1949), and Fodor & Katz (1964) for examples or, again, Thomason (1972) and Lappin (1981) for applications to anomaly. A note on terminology: I prefer to avoid the term “category” for the reasons alluded to in n.1. Also, one shouldn't read too much into the nomenclature. Being outside a predicate's domain of significance should not be interpreted as something to which it is not ‘meaningfully' applied in any sense that would come into conflict with the weakness of my earlier claim (C).

  13. 13

    This contrasts, for example, with first delimiting sets of logical sorts or categories, and associating an n-ary predicate with an n-tuple of such sorts. Such a theory is more restrictive and I suspect, for that reason, may lead to incorrect predictions.

  14. 14

    One reason to worry about whether having only anomalous instances is necessary to generate quantified anomaly is that, on my proposal, the domain restriction from anomaly is merely a default interpretive strategy, which might well be ‘overridden' by other factors. If so, when the domain restriction isn't operative, we may much more easily generate instances of quantified anomaly. Some reasons to worry about whether having only anomalous instances is sufficient to generate quantified anomaly come from special quantifiers like “no”, which seem like they may sometimes be exempt from requiring only non-anomalous substitution instances to be non-anomalous.

  15. 15

    Of course, the logical forms I've given above use generalized quantifiers. But since the point stands if we return to monadic quantification with a conditional, I'll stick to the notation I've used so far in the paper.

  16. 16

    Also note that my claims here don't turn on how I'm using the word ‘context', namely to apply to general features of context of use. One might grant that general features of the context of utterance needn't supply the values of inline image and inline image, when it comes to the domain restriction from anomaly, but maintain that features of linguistic context—e.g., the words used in each expression—are doing that work. I don't want to contest terminology. That's a perfectly fine way of using the word “context”. But relabeling terms doesn't avoid the problem I'm after. When one uses “context” in this sense, it's now simply the case that contributions from context must change from (24) to (25) (since any transition from (24) to (25) is ipso facto one in which contributions from linguistic context are changing). So again, one can never find an inference from (24) to (25) that is underwritten by purely logical relations.

  17. 17

    As already noted, other forms of semantic defect that are neither true nor false would also generate this threat. What is potentially special about anomaly is the strength of our arguments, given in §1, for actually taking anomaly to constitute a form of defect that is distinct from mere falsehood.

  18. 18

    I'll omit discussion of functions for the sake of brevity. I'm also a bit loose on use mention distinctions.

  19. 19

    Following convention I use inline image to denote the function differing from f at most in that inline image. Also, these definitions again predict that quantified claims are anomalous only when they have only anomalous substitution instances. See n.14 for some reasons we might eventually have to relax this requirement.


  1. Top of page
  2. 1. Anomaly and Quantifiers
  3. 2. The Semantics of Anomaly and Quantification
  4. 3. Logical Consequence
  5. Appendix
  6. References
  • R. Carnap, (1937). The Logical Syntax of Language. London: Routledge & Kegan Paul.
  • J. Fodor & J. Katz. (1964). ‘The Structure of a Semantic Theory'. In J. Fodor, & J. Katz. (eds.), The Structure of Language: Readings in the Philosophy of Language. Prentice-Hall, Englewood Cliffs, NJ.
  • S. Lappin, (1981). Sorts, Ontology, and Metaphor: the Semantics of Sortal Structure. Walter De Gruyter.
  • B. Russell (1908). ‘Mathematical logic as based on the theory of types'. American Journal of Mathematics 30(3): 222262.
  • G. Ryle, (1949). The Concept of Mind. Hutchinson, London.
  • J. Stanley & Z.G. Szabó (2000a). ‘On Quantifier Domain Restriction'. Mind and Language 15(2 and 3): 219261.
  • J. Stanley & Z.G. Szabó (2000b). ‘Reply to Bach and Neale'. Mind & Language 15(2&3): 295298.
  • R. Thomason (1972). ‘A Semantic Theory of Sortal Incorrectness'. Journal of Philosophical Logic 1(2): 209258.