1.1. Good Runners

You are observing today’s 5k. First come the leading runners; they are very fast and in excellent shape. The speed and athletic excellence of the runners gradually decreases as time goes by. The runners towards the middle are not quite as fast and athletic. After some time you observe the last participants. They are slow and out of shape. A friend approaches you and asks: “Did you have a chance to see good runners?” To which you reply: “Yes, all and only the fast ones were good runners.”

Here we can find a sorites series. The first members of the series are clearly good runners, the last members are clearly not good runners, and the running ability of the members of this series gradually decreases as they go by. Crucially, the running abilities of adjacent members of the series are indistinguishable for any practical purpose.

Now, based on your assertion, we can certainly classify some members of the series as good runners. The leading runner is clearly a good runner, given that she is very fast, and others close to her count as good runners as well. It is also clear that based on your assertion you did not classify some members as good runners; the last ones were not classified in that way. Thus, you have used “good runner” -and “fast”- to classify some members of the series in a certain way and not others. This much is uncontroversial, or at least it should be.

It is also clear that based on your assertion there is no piece of information available to us that could help us point at the last member of the series that has been classified as a good runner. The dominant position is that this is so because vague predicates -like “x is a good runner” and “x is fast”- do not draw sharp boundaries between cases where the predicate applies and all the rest.⁷ Relative to the sorites series in the previous example, “x is fast” is not the kind of predicate that applies to, say, Julia but not Eli, who has been a centimeter behind Julia during the whole race.⁸ This is why this phenomenon is so puzzling. If there is no sharp boundary between the positive cases of application and all the rest, how can it be that there are both positive and negative cases of application of the predicate? How can it be that some of the 5k runners are fast whereas others are not? This is the kind of consideration that fuels the sorites paradox.

So here we are with our standard mathematical tool-box and a classical metalanguage trying to model this phenomenon. Let me present you with an old worry directed at this kind of model-theoretic approach to the phenomenon of vagueness.⁹ The objective of the presentation is not to prove that the worry is correct -although I suspect that it is. For my purposes it is enough to show that it is reasonable and that we should take it seriously. In the rest of the paper I will argue that in the event that this worry is correct, we should adopt something like a Chrysippian attitude when modeling vague languages.

Here is the worry. Among the things we want to model is the range of application of vague predicates relative to suitable soritical domains. Much of what we say about the phenomenon of vagueness depends on this.¹⁰ Now, as we have seen, what is distinctive about vague predicates is that they do not draw sharp boundaries. What is distinctive about sets -our core mathematical tools- is that they draw sharp boundaries. A set clearly divides the objects that are members of it from all the others -a fuzzy set clearly divides the objects that are members of that set to a certain degree from all the rest. That is precisely what the range of application of a vague predicate does not do. As such, it seems odd to try to model the range of application of vague predicates by using sets -it may seem that sets are simply not the right kind of tool for this task.

There are some complications -the first one has to do with the notion of borderline case, and the second one with higher-order vagueness. A serious attempt to model vagueness as a semantic phenomenon has to be more ingenious. One may think that in between the positive and negative cases of application of the predicate there are borderline cases.¹¹ If one thinks this can be modeled with three disjoint and mutually exhaustive sets -or with two disjoint but not mutually exhaustive sets- one would face a well known objection. Vague predicates do not draw sharp boundaries between positive and borderline cases just as they do not draw those boundaries between positive and negative cases. However, sets used in this way do draw those boundaries. Thus, the reasoning goes, if we use sets in this way we can only misrepresent the semantic values of vague predicates. It is worth pointing out that most theorists of vagueness would agree with this. However, most of them will argue that if we take into account the phenomenon of higher-order vagueness, the worry can be dismissed.

It is tempting to think that the transition from the positive to the borderline cases is vague, just as the transition from the positive to the negative cases is vague. Perhaps we can make sense of this while representing the range of application of vague predicates using sets. This line of defense typically uses a determinacy operator (D) to argue that there is no sharp division between the clear cases and the borderline cases, and the negative cases and the borderline cases. The thought is that just as there are borderline cases between the positive and the negative cases, there are second-order borderline cases between the determinate cases and the borderline cases.¹² The hope is that we can iterate this idea for each order of vagueness.

A view of this kind is certainly ingenious. It is not unreasonable to hope that this treatment of higher-order vagueness allows us to model the phenomenon of vagueness using precise mathematical tool without worrying about it. However, a worry does persist. Sainsbury (1996) makes this point in a colorful way: “This hope, however, is groundless. Indeed, its very structure should be unappealing: you do not improve a bad idea by iterating it” (p. 255). The bad idea is to use sets in order to model the range of application of vague predicates in terms of positive, negative, and borderline cases. The iteration consists in using the same idea to understand each of the orders of vagueness. If this three-fold set-theoretic distinction is too precise to model vagueness, why think that iterating it will make it better? If at all, it gives us more things to worry about. There is much room for debate here. However, even if one is optimistic about this way of understanding higher-order vagueness, it is hard not to feel the worry.

There is a further source of concern. It may be that higher-order vagueness, understood in the standard way, is paradoxical. This is a technical debate with an abundant literature. The general form of the debate is this. It is argued that given a small set of very reasonable assumptions one can prove that this way of understanding higher-order vagueness leads to inconsistencies, and therefore that it is deeply flawed.¹³ One can complain that some of the assumptions in the proofs should be rejected.¹⁴ To this one could reply that there are other proofs with similar results that do not rely on those problematic assumptions, but only on other very reasonable ones.¹⁵ And probably there can be plausible replies to those replies, and so on.¹⁶ The jury is still out. As such, one should take both sides of the debate seriously. Perhaps the standard way of thinking about higher-order vagueness is incoherent -for what it is worth, I suspect that it is.¹⁷ If it is, then we cannot appeal to it in trying to advocate for the use of standard mathematical tools in our semantic models of vague languages. If there is a justification for using those tools, it has to be quite different.¹⁸

So let me suppose that these worries are well founded, and that an attempt to use set theoretic machinery to model the complete range of application of vague predicates is bound to misrepresent the phenomenon one attempts to model -it is important to keep in mind this dialectical point throughout the rest of the paper. What to do then? One can declare the project futile, and abandon it altogether. I think it would be a bit soon to take this stance. Perhaps one may think that we can only aspire to offer a homophonic semantic theory with semantic clauses of this kind: “tall” is true of an object if and only if that object is tall.¹⁹ Semantic clauses of this kind are hard to reject. Also, they are not very substantial -they are trivial and not very informative.²⁰ Or perhaps one can deny that vague predicates have a range of application relative to soritical domains. The idea is that only relative to those domains are vague predicates truly paradoxical. If so, we can easily use sets to model the semantic values of vague predicates relative to non-soritical domains -these are domains where, say, there is a considerable gap between those who are tall and those who are not (for instance, everyone is either taller than 1.90m or shorter than 1.50m).²¹ This view is attractive, however it seems wrong to deny that “good runner” has a range of application relative to our example above -notice that in that example the domain is clearly soritical. To my mind it is part of the data to be explained that we use vague predicates to classify objects relative to soritical domains. If so, then it is not very appealing to deny that these predicates have a range of application in those cases.

In section 2, I explore a different approach. I call it semantic quietism. According to it, vague predicates do have a range of application relative to soritical domains, and, with a touch of Chrysippian silence, we can use sets to model that phenomenon. The worries above are up to something important, but if we are ready to sacrifice a bit, we can get around them. In section 2.3, I explain the important ways in which semantic quietism is different from other similar views in the market. In section 3, I respond to some potential worries. In section 4, I develop a version of the view that preserves classical logic for a significant fragment of the language. This is not to say that semantic quietism is committed to classical logic -only some ways of implementing it are.

2.1. The Sound of Silence

So we need to model the extension of a predicate in an incomplete way -that is what it is for the model to be silent about certain cases. Consider a simple example. Let our domain be the natural numbers from 1 to 10. The predicates “is an even number” and “is not an even number” have an extension relative to this domain. To a first approximation, here is how we can represent those extensions in an incomplete way (for convenience I will call an incomplete representation of an extension, an incomplete extension):

This is not the official notation, but it is colorful enough to get across the main ideas -see Section 4 for a rigorous alternative. The commas with nothing in between represent the incomplete aspect of the representation. They leave open the possibility that there is something else in the extension, without committing to it. Now, here is the sense in which these are incomplete extensions. They say something about how these predicates classify, but they do not say everything there is to say. For instance, this representation takes 2 and 4 into the positive extension of the predicate, and 1 and 3 into its negative extension, but nothing has being said about whether 5, 6, 7, 8, 9, and 10 are in the extension or counter-extension of this predicate. It is in this sense that we are dealing with an incomplete extensions. It is a fact that 6, 8, and 10 are in the range of application of “is even", however this incomplete extension is entirely silent about this fact. To clarify, the representation does not say that 2 and 4 are the only members of the extension, and it does not say that 5, 6, 7, 8, 9, and 10 are negative or borderline cases. This incomplete extension is silent about the status of these numbers with respect of the range of application of “is even”. It is in this sense that the extension (or rather, this representation of it) is incomplete.²³

Here is an analogy that provides an intuitive way of understanding the notion of the incomplete extension of a predicate. Imagine an artist made a sculpture of you. This sculpture is very accurate in all respects, except that it is missing some parts; there is nothing corresponding to your lower torso and upper legs (see Figure on page 13). This is an incomplete sculpture. The sculpture on the right represents correctly the upper and lower parts of your body, but it says nothing about the middle part of your body. Now, that sculpture does not represent you as not having middle body parts, nor does it represents you as having an indefinite middle body (let us stipulate this was not the intention of the artist). Rather, that sculpture is completely silent about your middle body, and it is so in virtue of being incomplete.

FIGURE Fragments, 2016, by Bruno Catalano. Bronze Patiné, 3 pieces, 100 × 115 × 42 cm. (Photograph courtesy of De Medicis Gallery, Paris.) 

Incomplete extensions, like incomplete sculptures, accurately capture some aspects of what they try to model. However, both incomplete extensions and sculptures remain silent about some aspects of what they represent. Incomplete extensions are silent about whether the relevant predicates apply to some members of the domain, even if there is a fact of the matter about whether the predicate applies to some of them.

2.2. Incomplete Extensions and Vagueness

Hold on. Isn’t that a sharp boundary between positive cases and the silence region? Yes, but not all boundaries are harmful; it all depends on what they divide. Quietism draws harmless boundaries. A bad boundary is one that divides two distinct semantic categories, placing two objects that are indistinguishable for practical purposes on different sides of the division. This kind of boundary is bad, because what is distinctive about vague predicates is precisely that they do not draw this kind of division. A theory of vagueness that postulates that kind of cut-off fails to do justice to the phenomenon of vagueness. Here is an example of bad cut-offs (this time consider the vague predicate “x is blond”):

Now, we need to draw boundaries of some kind; that is the nature of our formal tools. We cannot draw them between all the positive cases and the first borderline case, because there is no such boundary. Where else could we draw them? According to quietism, for the boundaries to be harmless the first one should be drawn between two positive cases and the second one between two negative cases. Thus, this is the picture I recommend:

Notice that there is at least one member of the series who is blond, and our incomplete extension does not classify him as such -nor is he classified as not blond, or borderline. This little model is simply silent about the status of that member of the series with respect to the range of application of the predicate. Similarly, there is at least one member of the series that is not blond, but the model is completely silent as to whether he is blond.

Why are these boundaries not of the bad kind? The crucial thing to notice is that they are perfectly compatible with the existence of vague classifications. This incomplete way of modeling the range of application of “blond” does not misrepresent the relevant vague classification; it is just that it does not represent it completely. What makes this possible is, of course, that this incomplete extension becomes silent before it is forced to misrepresent the vague classification -this is the semantic analog of a Chrysippian attitude towards a forced march. What a quietist boundary represents is this: it represents the point at which the semantic model becomes silent, not the transition between two distinct semantic categories.²⁴ As such, this boundary -and the one between negative cases and the silence region- is harmless.

Quietism tackles the problem of compatibility with the phenomenon of vagueness at the first-order, rather than appealing to the higher-order vagueness to get the job done. As we shall see, this puts fewer constraints on the quietist treatment of higher-order vagueness -this, in turn, opens the door to some ways of avoiding the paradoxes other theories face.²⁵

2.3. Borderline Cases

Before we move on it is important to further clarify the quietist perspective on borderline cases. As I have already pointed out, if a quietist model is silent as to whether Marco is tall, it does not thereby take Marco to be borderline tall. The silence region is not the borderline region.²⁶ As it turns out, quietism is quite flexible with respect to which conception of borderline cases one wishes to adopt. This view is perfectly compatible with a semantic conception according to which borderline cases of predicate “F” are those objects such that it is neither true nor false that they are F (^{Fine 1975}, ^{McGee and McLaughlin 1995}, ^{Tappenden 1993}). If we go this way, we can model the range of application of a vague predicate as having positive, borderline, and negative cases, with a silence region between the positive and the borderline, and the negative and the borderline. As expected, these silence regions, if selected properly, contain some positive and negative cases or some borderline cases. In section 4, I develop a superquietist framework, showing how this may be rigorously done.

As a quietist one could also characterize borderline cases in epistemic terms (^{Williamson 1994}). One could say, for instance, that borderline cases of predicate “F” are those objects such that its not knowable whether they are F and it is not knowable whether they are not F. Moreover, a quietist could think of borderline cases in terms of permissibility (^{Shapiro 2006}). For instance, she could say that an object is borderline F just in case it is permissible to assert that it is F and it is also permissible to assert that it is not F. Finally, a quietist could claim that something is borderline “F” just in case there is no fact of the matter about whether it is F (^{Field 2003}). All this flexibility is possible because quietism is, first and foremost, a theory designed to ensure compatibility with the phenomenon of vagueness, not a theory designed to capture a specific conception of borderline cases.

Now, let’s contrast quietism and Crispin Wright’s agnosticism (^{Wright 2001} and ^{Wright 2003}). At a certain level of abstraction these two views resemble each other.²⁷ According to Wright’s view, borderline cases should be understood in terms of a particular mental state. ^{Wright (2001)} calls this mental state a quandary.²⁸ As such, this view recommends agnosticism regarding borderline cases and the law of excluded middle -as ^{Wright (2003)} points out, intuitionism is a natural ally. Now, and this is the point of resemblance, agnosticism and silence about p are compatible with p being true (or false). However, this similarity can only be found at the surface. Agnosticism is a propositional attitude, whereas silence is, primarily, a property of formal models. It is because of this that a quietist can know that Maria is tall, and, yet, let her model be silent about it.²⁹ However, one cannot know that Maria is tall and be agnostic about it -these two attitudes are flat-out inconsistent. A further difference between these two views is that agnosticism recommends that one be agnostic with respect to the law of excluded middle whereas some varieties of quietism can accept such a law (see Section 4 for more details about this). I suspect that quietism and agnosticism can play well with each other. However, it is worth keeping in mind that they exhibit sharp differences at a basic level.

It is instructive to understand the way in which quietism is different from the vagueness theory in ^{Soames 1999}. According to Soames, vague predicates are only partially defined.

The first step in the construction of the model is to treat vague predicates as partially defined. As we saw […], when F is partially defined, there will be objects to which it applies, objects to which it does not apply, and objects for which it is undefined. The latter are objects for which the linguistic rules governing the predicated and giving it meaning have nothing to say about whether or not F applies to them. (^{Soames 1999, pp. 206-207})

There are similarities between this view and quietism. Soames even talks about the linguistic rules governing a predicate being silent as to whether the predicate applies to an object. These similarities do not run deep. A quietist may agree with Soames in thinking that the borderline cases are those objects such that the linguistic rules are silent about them. However, the quietist insists that a semantic model compatible with the phenomenon of vagueness also has to be silent in a different way. There have to be objects such that the model is silent as to whether the linguistic rules governing a predicate are silent about them. That is to say, there have to be some objects that are in fact borderline cases and where the model is silent about whether or not this is the case. The quietist’s silence is the one that avoids the worries stated at the beginning of this paper. In contrast, Soames’ view is precisely the kind of view those worries target directly.

4.1. Superquietism

The core superquietist idea is that it is problematic to understand (super)truth in terms of quantification over all admissible (classical) interpretations of the language -to do so is a good way of getting into higher-order vagueness problems. Instead, what the theory does is to understand (super)truth in terms of quantification over some admissible interpretations and a region of silence comprising some admissible and some inadmissible interpretations. As technical details will soon reveal, by going this way superquietism can think about vague classifications as follows: relative to a sorites series for “tall”, there is a silence region between the positive and the borderline cases, and the negative and the borderline cases. These regions of silence are just what you would expect; they contain some positive cases, some negative cases, some borderline cases, and a paradoxical region we had better be silent about.

The superquietist approach to the sorites paradox is, to some extent, supervaluationist. In both theories (super)truth is a matter of being true in a certain collection of classical interpretations -the difference lies in what they take this collection to be. Given that in both cases the interpretations are classical, the inductive premise of the sorites paradox is always false, and its negation is always true. As is explained in the next section, superquietism also validates all penumbral connections formulated in the object language.

4.2. Admissible Interpretations

Our starting point is the notion of an admissible interpretation. In a standard supervaluationist framework (^{Fine 1975}, ^{McGee and McLaughlin 1995}, ^{Keefe 2000}) the range of admissible interpretations is what determines, in good part, the division between positive, borderline, and negative cases. Thus, if we want to include a region of silence between the positive and the borderline, and between the negative and the borderline, we have to play around with the range of admissible interpretations. Given that “x is an admissible interpretation” is vague by any reasonable standards, the quietist should use her tools to interpret this predicate. Here is one way in which this can be done. We can represent the positive extension of “x is an admissible interpretation” as a pair 〈P _A ,S _A 〉 and the negative extension as 〈N _A ,S _A 〉, where P _A contains some admissible interpretations, N _A contains some inadmissible interpretations, and S _A comprises the silence range -where the subscript “A” stands for “admissible”.³¹ Needless to say, S _A should contain a few admissible and a few in-admissible interpretations. This, again, is how quietism gets around higher-order vagueness worries.

How should we decide which interpretations go into P _A and S _A ? The answer to this question heavily relies on our particular theoretical goals. Normally we would be attending to a particular conversation and we would have a set of goals we want our model to meet. For the sake of simplicity, let’s abstract away from these complications and focus on a language whose only predicate is “tall” and whose domain of discourse only contains a sorites series for this predicate. Let us call this series “T” and let’s assume that it has 100 members, where 1 is the shortest and 100 is the tallest. (Clearly, a sorites for this predicate would have more members. I am just simplifying the example in a harmless way). An admissible interpretation of this language -classical as it is- draws a sharp boundary somewhere in T. If an interpretation draws a boundary between, say, 41 and 42, then, on this interpretation all the members of the series between 1 and 41 (inclusive) are not tall, and the rest are tall. If two interpretations are different, then they classify the members of T differently -they draw the cut-off at different locations in T.³² A good way of selecting the members of P _A is by taking a cluster of interpretations that draw cut-offs in a sub-region of the borderline region -these are uncontroversial admissible interpretations. Let’s assume, for simplicity’s sake, that this sub-region contains the members 40-60. Notice that this is only a proper sub-region of the borderline region, meaning that, at the very least, 39 and 61 are also borderline cases. Thus, the interpretations in P _A are the ones that draw a cut-off at any place between 40 and 60.

Now, there are interpretations that draw the cut-off to the left (towards 1) and others to the right (towards 100) of this borderline sub-region. Some of them go into S _A . How many? For sure the one that draws the cut-off between 39 and 40, and the one that does it between 60 and 61; these are some of the admissible interpretations that the model is silent about. Which others? Well, we have to go far enough as to include some inadmissible interpretations that draw the cut-off among clearly tall/not-tall members of T. Recall that a good region of silence must include some of the clear cases (clearly admissible interpretations in this case) and some clear negative cases (clearly inadmissible interpretations). Let’s say that 20 and 19 are clearly not tall. Let’s also say that 79 and 80 are clearly tall. An interpretation that draws a cut-off between any of these pairs is inadmissible. We can say, then, that the members of S _A are the interpretations that draw a cut-off somewhere between 40 and 20, or somewhere between 60 and 80.³³ All the interpretations that are not in P _A or S _A , are in N _A .

It is important to notice that our selection of P _A , S _A , and N _A was semi-arbitrary. We could have chosen a slightly bigger or smaller S _A , and, as a consequence, a bigger or smaller P _A and N _A . Whether our choice was a good one is determined by how well we can capture the flow of the particular conversation we are concerned with. Our choice, however, was not completely arbitrary; no inadmissible interpretation could have been admitted in P _A and no admissible one could have been admitted in N _A .

4.3. Supertruth

We have taken our first step -we can understand the range of admissible interpretations in a way acceptable by quietist standards. The second step is to find a good way of defining supertruth and the determinacy operator. Given that this operator will be defined in terms of supertruth, we shall focus on this notion first.

Textbook supervaluationism defines supertruth as truth in all admissible interpretations. It may seem, then, that superquietism should define supertruth as true in all the interpretations in P _A . Doing so, however, leads to disaster. Consider 60 in our previous example. She is borderline tall, but the model is silent about it. However, “60 is tall” is true in all the interpretations in P _A . This is so because all the interpretations in this set draw the cut-off somewhere between 40 and 60, so there is no interpretation in P _A that classifies 60 as not tall. Therefore, if supertruth is true in all the interpretations in P _A , then “60 is tall” is supertrue. But this is wrong, since 60 is borderline tall. A moment of reflection shows that a version of this result obtains in every proper superquietist model.

The way around this problem is to let supertruth quantify over the union of P _A and S _A . This is perfectly in line with quietism’s ideology; given that “admissible interpretation” is vague, it is not clear that we can quantify over all and only the admissible interpretations without drawing nasty boundaries. Perhaps the best thing we can do is to quantify over some admissible interpretations and a region of silence. This may sound risky, because in S _A there are some interpretations that are not admissible. However, if we are careful enough in how we assign supertruth-conditions, no unpleasant result will follow. In the next section I will show how to do that.

4.4. Semantic Clauses and the Silence Region

Before we cash out these ideas with more precision, let’s settle on some terminology. By an admissible interpretation of the language we shall understand -following ^{McGee and McLaughlin 1995}- a classical model that satisfies both penumbral connections and classificatory constraints. Each of these models is a complete sharpening of the base model.³⁴ The base model is a non-classical model of vague language with a partial interpretation function -it is easy to interpret a partial function in a quietist way. Following quietism, we represent the positive extension of “x is an admissible interpretation of L” as the pair 〈P _A ,S _A 〉 -recall that if we choose this pair wisely, there are some admissible and some inadmissible interpretations in S _A .

Given this, we partially define supertruth in a model and the determinacy operator (D) as follows (where Φ is a formula, M is a model, and σ is a variable assignment):³⁵

There are two very important things to notice right away. The first one is that in clauses 1-4 we don’t offer conditions that are both necessary and sufficient -necessary and sufficient conditions typically repel vagueness. The second one is that, in 1 and 3, both P _A and S _A are built into the sufficient condition, but in clauses 2 and 4 only P _A appears in the sufficient condition. Notice, however, that these sufficient conditions are quite uncontroversial and, as it will become clear soon, this asymmetry in the clauses is exactly what we need.

These clauses give us precisely what we wanted -a region of silence between the positive, borderline, and negative cases. This is, of course, only provided that we choose P _A , S _A , and N _A wisely. For an object to be borderline tall, it has to satisfy ¬DTall(x)&¬D¬Tall(x). For this to be the case it is enough if a person is tall relative to a model in P _A and not tall relative to a different model in P _A . For a superquietist model to place someone in the silence region between the positive and borderline cases, that person has to be tall in all the models in P _A and not tall in some models in S _A . A model like this is silent as to whether it is supertrue that that person is tall, since the sufficient conditions in clauses (1) and (2) are not satisfied. Therefore, the model is also silent as to whether it is determinate that she is tall. Hence, it is also silent as to whether she is borderline tall. This can be so, of course, even if it is in fact determinate that that person is tall -for this to be the case, she only has to be not tall in those models in S _A that are inadmissible. A good superquietist model has cases like that. It is trivial to specify what it would take for a superquietist model to place someone in the silence region between the borderline and negative cases.

4.5. Superquietist Logic

It is time to say something about the notion of validity. Traditionally there are at least two options open to Supervaluationism; global and local validity.³⁶

Thus, a globally valid argument is one that preserves supertruth, and a locally valid argument is one that preserves truth in all admissible interpretations.

Whether supervaluationism should accept global or local validity has been the subject of debate. ^{Williamson (1994)} and ^{Fara (2010)} argue that supervaluationism must hold to global validity and that doing so leads to horrible problems. ^{Keefe (2000)} argues that even though supervaluationism should accept global validity, there are no problems that emerge from doing so. ^{McGee and McLaughlin (1995)} and ^{Varzi (2007)} argue that the right notion of validity is the local one.

In this paper I shall only be concerned with what superquietism looks like if we accept local validity. Now, it is clear that we need to modify the definition of local validity if it is to be acceptable by quietist standards. In particular, in our definition we have to allow for quantification over all the interpretations in P _A ∪ S _A . The resulting clause is this:

The superquietist version of local validity delivers some nice results. Given that all the interpretations in P _A ∪ S _A -for whatever choice of P _A and S _A - are classical, all classical valid inferences hold in them. Therefore, all classically valid inferences hold according to this version of superquietism. Also notice that all instances of Φ∨¬Φ and any other logically valid sentence are valid (and supertrue) given that they are true in all the (classical) interpretations in P _A ∪ S _A .³⁷Please keep in mind that this is not to say that quietism assumes classical logic -it is only this particular way of developing the view that has this assumption. There can be non-classical ways of being a quietist.

Another nice feature is that all D-free penumbral connections are valid. The reason for this is that even though not all interpretations in P _A ∪ S _A are admissible -because not all interpretations in S _A are- all of them satisfy penumbral connections. Some interpretations in S _A are not admissible because they violate some classificatory constraints -they classify clearly tall/not tall people as not tall/tall- however, that is not to say that they violate penumbral connections. Given that all penumbral connections are true in all interpretations in P _A ∪ S _A , all penumbral connections are supertrue.

Let’s consider a specific example. Julia is slightly taller than Carla. Thus, the following sentence is an instance of a penumbral connection:

As such, (a) is supertrue according to superquietism. This can be shown simply by pointing out that all interpretations in P _A ∪ S _A respect all penumbral connections. Hence, (a) is true in all of them. The very same thing can be said about every other penumbral connection -so long as the sentence in question does not contain D.

As it turns out, superquietism can tackle the sorites paradox in the same way supervaluationism does.³⁸ According to superquietism, relative to a sorites series for F, (b) and (c) are both supertrue (where x ^′ is the successor of x):

This is so because, regardless of our choice of P _A and S _A , all the interpretations in these two sets are classical and satisfy all the penumbral connections. Thus (b) and (c) are true in all the interpretations in these sets.

As expected, relative to a soritical domain for predicate F, the following sentence is supertrue:

Relative to this kind of domain, this sentence is always supertrue according to superquietism. Notice that a good superquietist model won’t be silent as to whether that sentence is supertrue.³⁹ Consider an arbitrary member of the domain, a. Suppose the model is silent as to whether DF(a). Then, a is in the silence region between the positive and borderline cases.⁴⁰ Thus a ^′ is in that same silence region or it is a borderline case -this, of course, is an inclusive disjunction.

If the semantic model declares it a borderline case, then it follows right away that ¬D¬F(a'). If a ^′ is in the silence region between the positive and the borderline cases, then F(a') is true in all the members of P _A and false in at least one member of S _A . But then it is not supertrue that ¬F(a') since that sentence is false in at least one model in P _A (in fact it is false in all of them). Either way, it is not supertrue that ¬F(a ^′) and, therefore, it is the case that ¬D¬F(a'). Now suppose that the model is silent as to whether D¬F(a ^′). Then, a ^′ is in the silence region between the borderline and negative cases. Given that a is the predecessor of a ^′, it is in that same silence region or it is a borderline case. Either way, it is not supertrue that F(a). Therefore, it is the case that ¬DF(a). Hence, if the model is silent about one conjunct, the other one is false. Thus, we can conclude that ¬(DF(a) ∧ D¬F(a')). Since a is arbitrary we can conclude that (d) is the case.

We must be a bit careful here. One may argue that (d) entails a first-order gap principle: ∀x(DF(x) → ¬D¬F(x'). If higher-order gap principles are true in a superquietist model, along with suitable iterations of D in front of them, then we can run into the paradox presented in ^{Zardini 2013}. In order to avoid this problem the superquietist can remain silent as to whether some of those higherorder gap principles are true. In fact, in order to dodge the paradox it is enough to remain silent about whether some of those principles are true once we have iterated D in front of them a few times. This is easy to do by including a non-transitive accessibility relation into our semantics in the standard way. Given such an accessibility relation, different models may have a different P _A ∪ S _A assigned to them.

So let’s describe a case where DΦ is true in a model M and the same model is silent about whether DDΦ is true. Given that DΦ is true in M, Φ is true in every model in P _A ∪ S _A relative to M. Now, let’s make M be silent about DDΦ by having a M ^′ in S _A that is silent about DΦ. (If M ^′ is silent about DΦ, then we cannot say whether DΦ is supertrue relative to M and thus we have to be silent as to whether DDΦ is true relative to M.) Thus, we get that Φ is true in all models in P _A relative to M ^′ and false in at least one model in S _A relative to M'-from this it follows that S _A is not the same relative to M and M ^′. So, when Φ is a gap principle we can specify a situation where DΦ is true in a model, even though that same model is silent about whether DDΦ.

Superquietism is not as pretty as supervaluationism; it remains silent about sentences supervaluationism declares valid. This is the price superquietism pays in order to guarantee compatibility with the phenomenon of vagueness. Here is a concrete example. In a supervaluationist framework the following sentence is always supertrue:

However, sometimes, in a superquietist framework we have to be silent as to whether sentences of this form are true -which is not to say that they are false or indeterminate. Consider 61 in our example above. Our model is silent as to whether DTall(61) -61 is tall in all the interpretations in P _A , but not in all the interpretations in S _A . The model is also silent as to whether ¬DTall(61) -there is no interpretation in P _A where 61 is not tall. Therefore, the model is silent as to whether DTall(61) ∨¬DTall(61). Similarly, our model is also silent about DTall(61) → DTall(61) for the exact same reasons.

A superquietst model can be silent about pretty much any logical truth construed using D. Three things are worth noting in order to make this fact a bit more palatable. First, when a model is not silent about logical truth A, then A comes out as supertrue. Second, superquietism has no problem validating all the inference we would want a nice logic to validate. So, for example, even though a model can be silent about DF(a) ∨¬DF(a), DF(a) → F(a), and DF(a) → (F(a) ∨ G(b)), the following inferences are always valid: DF(a)⊨ _l DF(a), DF(a)⊨ _l F(a), and DF(a)⊨ _l (F(a) ∨ G(b)).⁴¹ This, of course, is a consequence of the fact that all models in P _A ∪ S _A are classical and that we are using local validity. Third, if one thinks that D is in the metalanguage, as ^{McGee and McLaughlin (1995)} do, then it is a bit less worrisome that a model can be silent about logical truths construed using that operator -this way we can rest content that our object language is well-behaved.

It is worth noting that a superquietist model can also be silent about instances of penumbral connections involving D in the silence range. For example, given that our model is silent as to whether DTall(60) and DTall(61), it is also silent as to whether DTall(60) → DTall(61). To my mind this is an acceptable price to pay. Superquietism is compatible with the phenomenon of vagueness and preserves classical logic for the fragment of the language that is D-free. These are nice features that no other theory can claim in an uncontroversial way. It is true that the fragment of the language that contains D is not as well-behaved as we may wish. However, keep in mind that if D is only a metalinguistic operator, we should not worry that much, so long as we get the penumbral connections right in the object language. That we can do.⁴²