Ted Sider on Vagueness, Logic and Reality

For my Philosophy of Time course, my students and I read the second chapter of Ted Sider’s Four-Dimensionalism (2001). It’s called “Against Presentism” and serves very well as an introduction to attacks on that particular position. Perhaps I’ll blog about this chapter later on: as a defender of presentism, I certainly have some critical thoughts. But today I want to look at something else. I want to look at Sider’s ideas about logic and vagueness; and I want to use the opportunity to reflect on his general vision of philosophy.

First, some background. Ted Sider is a big name in what we can call analytic metaphysics, the part of analytic philosophy that discusses questions about time, space, identity, parthood, modality, causation, and so on. When I read his work — apart from Four-Dimensionalism I’ve also read his 2011 book Writing the Book of the World, though it’s been a while — I read it in part because Sider is good at what he’s doing. He is clear, knowledgable, and sharp. But I also read him because I’m curious, in an almost anthropological way, about these ‘analytic metaphysicians’ and how they conceive of philosophy.

But Victor, I hear you say, aren’t you an analytic metaphysician yourself? I guess I am. But Sider is typical of a particular subculture within analytic philosophy, a subculture that I’m definitely not part of. I can’t precisely define or delineate this subculture, but some of its characteristics are that (1) it is strongly influenced by David Lewis; (2) it cares nothing for the history of philosophy; (3) it takes natural science, and especially physics, to be much more in touch with reality than any humanistic discipline; and (4) it lives in self-chosen isolation from any kind of philosophy that might be called continental, pragmatist, postmodern, or even Kantian or Wittgensteinian or logical-positivist — any kind of philosophy, that is, which does not uncritically embrace the independence of reality from human concerns and ways of thinking.

Sider is quite explicit about most of this, stating, for instance, in Writing the Book of the World that a “knee-jerk” realism is his most fundamental convictions and that he has “never questioned it”. In his very good review of that work, Adrian Moore says that he doesn’t believe that Sider has never questioned his realism, and then adds: “If I could be persuaded of the truth of that claim, then my incredulity would give way to despair for the current state of philosophy.” Let me add that there is hope for the more credulous; after all, Adrian Moore himself is also part of the current state of philosophy, and his books are fantastic.

Anyway. Sider’s aim in Four-Dimensionalism is to argue that perdurantism is true, that is, that objects exist across times rather than at times; or, as he himself ends up phrasing it, that objects have temporal parts. The idea is that a book, say, is not a three-dimensional object that is spread out only across space, but rather a four-dimensional object that is spread out across space and time. Books, humans, and molecules are all four-dimensional entities. (There are some subtleties here — Sider might want to argue that our common word ‘book’ actually refers to a three-dimensional temporal part of a perduring object rather than to the four-dimensional object itself — but these subtleties need not concern us.)

One of Sider’s favourite arguments for perdurantism is his argument from vagueness. The outline of the argument is as follows. First, Sider wishes to establish the thesis of unrestricted composition: if A is an object and B is an object, then the sum of A and B is also an object. (So if I’m an object and you are an object, than there is also an object that consists of the two of us, strange as this may sound.) If unrestricted composition is true, and if the past and future are real (that is, if presentism is false), then it immediately follows that there are four-dimensional objects with temporal parts. Take me right now and me five minutes ago; together, we form an object that is spread out across time; and this object clearly has temporal parts.

But how to establish the thesis of unrestricted composition? Through a reductio: if unrestricted composition is false, then there must be cases where composition does take place and cases where it does not take place. But if there are such cases, Sider argues, then there must also be other cases in which it is unclear, in which it is vague, whether or not composition takes place. But composition cannot be vague. Hence, unrestricted composition is true.

That there are vague cases is something the defender of restricted composition can hardly deny. Taking my own intuitions as an example, I would be fine in saying that I am an object, all the hairs on my head included. If one of those hairs is detached from my head and lands on my desk, I would like to say that it is no longer a part of me, and that I and the hair do not compose an object. But at which point did the hair and the rest of me stop composing an object? How far did it have to move from my body? How loose did it have to be? Surely, one cannot draw a sharp line between the hair and me still being one object and the hair and me no longer being one object. Or rather, one can draw this line, but one can only draw it arbitrarily. So I will grant Sider that if we are committed to restricted composition, we are also committed to vague cases of composition.

Now Sider wants to claim that there cannot be vague cases when it comes to composition. Of course there can be vague cases about other things. His own example is baldness: our use of the word ‘bald’ is simply not precise enough to determine exactly how many hairs someone is allowed to have and still count as bald. So there are cases where it seems we can only arbitrarily declare that someone is or is not bald. Vagueness as such is not a problem. But vagueness about composition is a different matter, according to Sider.

To understand why, we must understand Sider’s central thesis about vagueness, the linguistic theory of vagueness. According to this theory, vagueness is always a semantical phenomenon:

There is no vagueness ‘in the world’; all vagueness is due to semantic indecision. (Sider 2001, p. 125)

Let’s return to the example of baldness. Our term ‘bald’ can be precisified in many ways: we could, for instance, stipulate that only someone with less than one hundred hairs is bald. We don’t in fact do that, and therefore our term is vague. But this is vagueness in our language, not in the world.

What makes it possible for us to say that the term bald is vague? Clearly, it is the fact that we can describe the state of someone’s head in two different ways. On the one hand, we can describe the exact number of hairs, their position, length and thickness. On the other hand, we can use the terms bald and not bald. We find out that bald is a vague term by realising that when asked to move from the first to the second way of describing a case, we do not always know what to say. For certain numbers of hairs of certain lengths and thicknesses, we’re unsure whether or not to apply the term bald. The first, more precise, vocabulary allows us to map out a space of possibilities; and then we find out that while some regions of that space can be clearly labelled ‘bald’ and some can be clearly labelled ‘not bald’, there are also regions where we don’t know what to say.

The same is true for other examples, for instance the famous heap. Here the more precise language is that in which we specify the number of grains of sand that have been brought together. We then find out that, clearly, two grains of sand don’t form a heap of sand; clearly, twenty thousand grains of sand do form a heap of sand; but there’s also a region where we’re unsure what to say.

The kind of vagueness we are talking about, then, is a relation between two parts of our language. (There may be other forms of vagueness, but I am not concerned with them now.) That this is so, that non-linguistic reality has no role to play, is easily seen from the following. Suppose someone were to empirically discover that in our world there are only two kinds of people: those who have no hair at all, and those with luxuriously flowing manes. Would it follow from this that baldness is not, after all, vague? No. That baldness is vague can be established by describing, using one part of our language, merely conceivable heads of which it is unclear (to competent speakers) whether or not they should be described as ‘bald’.

So Sider is clearly correct to say that all vagueness is linguistic. But he is so clearly correct that it is hard to understand how anyone could think something different; indeed, it is hard to understand why Sider would even want to make the claim that there is no vagueness in the world. The world is also not prime, and not happy — but there seems no point in making this explicit. What adds to the mystery is that it is unclear how any of this could help Sider in arguing that composition cannot be vague. After all, composition is a concept, a word in our language; showing that the-world-in-itself cannot be vague seems of absolutely no help in showing that the concept of composition cannot be vague.

I should stress at this point that Sider is not confused about any of this. He knows precisely what he’s doing. And to understand what it is that he’s doing, we now need to look at his ideas about reality and language. Sider, of course, accepts that whenever we make (possibly true) claims about things, we use language. And he also accepts that there are many different ways of speaking; that there are many ways in which our language can ‘cut up’ the world, so to speak. But at this point Sider will want to assert his basic realist point of view, which is roughly the following:

  1. The world is what it is independently of the concepts we use to describe it.
  2. The world (the world-in-itself) has structure.
  3. It is possible, and desirable, for our concepts to precisely fit this structure of the world. (In this case Sider says that our concepts ‘cut reality at its joints’.)

I want to say a little more about these theses, starting with the first one. On page 130 of Four-Dimensionalism, Sider says:

A claim so wild that I will not consider it is that the world is the way it is because we talk in a certain way.

Later on, on page 150, there’s a passage where Sider expresses the independence of world and language as follows. He discusses the claim that ‘we decide the persistence conditions of clubs’:

This is tempting to say, but what exactly does it amount to? On one hand it threatens to amount to nothing more than the banal point that if we had meant something different by ‘club’, different sentences would have been true. All facts, even facts about electrons, are conventional in this sense, for any bit of language could have meant something different. On the other hand, it threatens to imply that if you and I were to talk differently then clubs would vanish. But surely we have no such mystical powers.

But we do have such powers. If we all start talking differently, clubs will vanish; and depending on how we talk differently, so might democracy, or unemployment, or the Eurovision Song Festival. Clearly, many things, including physics and philosophy, only exist because of the way people talk. There’s nothing wild about this; of course the world is the way it is because we talk a certain way!

But, Sider would no doubt interject, this is not true for all things. Some things exist independently of human beings and the ways that they talk: stars, for instance, and electrons, and bacteria. It’s only those parts of the world there were meant by the thesis that the world exists independently of our concepts.

Not everyone might agree, and I think one of the most interesting sources of disagreement would lie in a puzzle raised by Sider’s second and third thesis. The world, independently of human beings, has structure; and this structure can be captured by our concepts. If this is true, it means that the world has a conceptual structure, a structure of precisely the kind that can be grasped by the human power of understanding. Really? How can we explain such a marvellously convenient arrangement, except through some theological narrative about humans being created in the image of God? And how could we know this convenient arrangement to be true? How could we discern the difference between a world which fits our cognitive powers and a world that merely appears to us as fitting our cognitive powers? Is the idea of a world not fitting our cognitive powers even thinkable? Asking these questions would set us on the road towards a confrontation with all the philosophical traditions — Kantianism, phenomenology, positivism, and so on — that Sider wishes to avoid talking to and talking about.

(I want to note in passing that none of these questions can be answered by appealing to Darwinism. Any Darwinian narrative about the emergence of human cognitive powers presupposes rather than explains that the world is structured in such a way that it can be captured by human cognitive powers. It presupposes this precisely in the act of describing the world using our Darwinian concepts. And of course this point generalises to all forms of naturalism.)

Here lies my deepest disagreement with Sider. He should confront these questions; they are among the most essential questions of philosophy, almost all of the greatest philosophers since Kant have discussed them, and not confronting them seems to me an act of almost wilful blindness; an act that risks turning the brilliant works of Sider (and some of his colleagues) into parochial tracts that will be forgotten by history. If this sounds overly dramatic, consider that even Carnap is already beyond the limits of Sider’s philosophical circle. Carnap figures in the introduction to Four-Dimensionalism as someone whose views on language-relativity would undermine the kind of metaphysics that Sider wants to do. Sider spells out the difference of opinion, but pointedly refuses to engage in a real confrontation. (“I do not pretend to refute Carnap” he says on page xxi; and “my purpose has not been to convince Carnap” we read on page xxiii.) Of course, this form of deliberate distancing is still vastly to be preferred over Sider’s remark in Writing the Book of the World that if we refuse to believe in an epistemically accessible objective structure of the world, “then the postmodern forces of darkness have won.”

Nevertheless, I want to investigate Sider’s argument further, because more can be learned from it when we look at the details. So let us grant Sider his three realist theses. There is a world independent of us; this world has the kind of structure that can, in principle, be grasped by our understanding; and our language can and should capture this structure, even though it does not always manage to do so. Now we can understand how Sider’s argument from vagueness is supposed to work. Sider wants to show that composition is a term that captures the structure of the world; he has already claimed that the world itself is not vague; and therefore, composition cannot be vague either. But anyone who denies unrestricted composition is committed to vague composition. Therefore, unrestricted composition must be embraced.

Clearly, the crucial new idea here is that composition captures the structure of the world. Actually, that’s not quite what Sider says. Instead, Sider argues that composition cannot be vague because standard first-order logic captures the structure of the world. Suppose that composition is vague. It’s unclear whether or not the sum of me and my loose hair forms an object. But if we say yes, there will be one more object in the world than if we say no. So if composition is vague, so is the number of objects in the world. Now statements of the from “There are exactly N objects in the world” can be phrased using nothing but the central terms of first-order logic. And so, if composition is vague, at least some of the terms of first-order logic must be vague as well. But first-order logic cuts nature at its joints; so it cannot be vague; contradiction; QED.

(Sider’s argument assumes that the total number of objects in the world is finite; since if it’s not, adding one new object makes no difference to the total number. I doubt whether this assumption is innocuous, but I won’t pursue this line of thought any further.)

That first-order logic captures the structure of reality is a claim Sider had already put forward in the introduction to his book:

A related but more contentious assumption is that modern logic’s quantificational apparatus mirrors the structure of reality: I assume an ontology of things. Moreover, I assume that there is a single, objective, correct account of what things there are. (p. xvi)

As a partial justification, he tells us:

a thing-ontology begins to look more attractive because of the power of the modern logic that presupposes it. Quantificational logic since Frege has proved to be a powerful tool inside and outside of philosophy. Think of contemporary semantic research in linguistics, philosophical logic, or even mathematics. For that matter, think of the conceptual scheme of ordinary thought, which appears to model the world as a world of things. If humans have been so successful with thing-thinking, it takes a strong reason to make us give it up. (p. xix)

Few people would deny Sider the right to speak in terms of an ontology of things, or to use first-order logic as much as he wants. As he correctly points out, logic can be a powerful tool. We would not easily give it up. But the thesis that Sider has to establish goes far beyond this. What he has to convince us of is that the world-in-itself has the same structure as first-order logic. Note that this goes beyond the three realist theses indicated earlier. It’s one thing to believe that nature has joints and that our language can sometimes cut it at those joints; it is something else entirely to claim that one is able to identify the parts of language that succeed in doing this; and yet something else to claim that first-order logic is one of those parts.

How does Sider know that first-order logic cuts nature at its joints? How can he step out of our ways of thinking to compare those ways of thinking with the world-in-itself? Less metaphorically, how can he justify such a claim about the world-in-itself? Given that Sider embarks on precisely the kind of project that a critical philosophy would deem to be impossible, we would really like to hear more about his epistemology of metaphysics. Sider writes:

Unfortunately, I also share with my fellow practitioners the lack of a good answer to a very hard follow-up question: why think that a priori reasoning about synthetic matters of fact is justified? I have no good epistemology of metaphysics to offer. It should not be thought, though, that this uncertainty makes metaphysics a worthless enterprise. It would be foolish to require generally that epistemological foundations be established before substantive inquiry can begin. Mathematics did not proceed foundations-first. Nor did physics. Nor has ethics, traditionally. (p. xv)

I find it amusing that Sider presents as an argument for non-critical philosophy precisely what Kant presents as the mistake of non-critical philosophy: the analogy with mathematics. Kant writes the following in his late, unfinished article “What real progress has metaphysics made in Germany since the time of Leibniz and Wolff?”

The first and oldest steps in metaphysics were not ventured merely as risky attempts, say, but were made, rather, with complete confidence, though without having first initiated any careful inquiries as to the possibility of a priori cognitions. What was the cause of this trust that reason had in itself? Its imagined success. For in mathematics reason succeeded in knowing a priori the constitution of things, well beyond all expectations of the philosophers; why should there not be just as much success in philosophy? (20: 262)

Kant goes on to say that a fundamental difference between philosophy and mathematics was overlooked, since mathematics constructs its concepts in intuition and metaphysics cannot do this. We might say that mathematics investigates man-made axiomatic systems, and thus ventures no guesses as to the world-in-itself. Its success can hardly justify Sider’s project. Physics is an empirical discipline, and as such also does not dream of claiming to capture the world-in-itself. As for ethics — it is hardly a paradigm case of epistemic success, certainly no more so than metaphysics itself.

But let us leave such generalities behind and look more closely to the case at hand: first-order logic and its application to vagueness. The essential elements of first-order logic for Sider’s purposes are the universal and existential quantifiers; for they require for their use (on a common though not perhaps inevitable interpretation of the formalism) a domain of objects over which quantification takes place. If the world-in-itself is structured like first-order logic, then there has to be a well-defined domain of quantification, and hence a well-defined class containing all and only the objects. That is what Sider needs for his argument against the vagueness of composition: that the world-in-itself comes with a well-defined class of objects. So what Sider ought to convince us of, is that the quantificational structure of first-order logic mirrors the structure of the world-in-itself. And we have seen that the one thing that Sider says to justify the special status of logic, is that it is such a powerful tool.

But like any tool, logic has its conditions of applicability, as well as its limitations. Newtonian theory is very useful under some circumstances; but it is almost useless when one tries to predict where a dollar bill blown off the top of the Empire State Building will land. (I believe the example is due to Cartwright.) In the same way, quantificational logic is very useful under some circumstances, but useless or positively distorting under others.

Let me give one minor example before moving to my main point. We read on Wikipedia the following sentence: “The common raven is a large all-black bird.” How do we formulate this in the fomalism of quantificational logic? Something like this: “For all x, if x is a raven then x is large and x is black.” This sentence, of course, would be false if there were a single white raven in the world. And there are white (albino) ravens, so it is false.

Does this mean that Wikipedia is wrong? Not at all. The claim that ravens are black neither implies nor is implied by the claim that all ravens are black. For the latter is a claim about the overlapping extension of two predicates, whereas the former is a claim about the properties of normal ravens. Now perhaps the lesson to draw from this is that our formalisation was wrong; that there is some other, more complicated formalisation of the sentence into quantificational logic that is correct. (One reason to doubt this is Fine’s argument about modal logic and essence.) But certainly the example shows us that logic is not an unproblematic tool when one wants to understand what the ornithologist is saying. And surely much of the debate surrounding Hempel’s paradox of confirmation could have been avoided if philosophers had been more conscious of this.

Let’s leave this example behind and move to consider the idea of a domain of quantification. Sider wants us to believe that there must be a well-defined domain of objects in the world, given the success of logic. But here we should take a page from the debate on scientific realism, where the realist (using what is known as the “divide and conquer” strategy) will tell us that we should only be realists about those parts of a theory that play an essential role in its successes. Now the idea of a well-defined universal domain of objects in no way plays such a role in our successful uses of logic. Take, for instance, the formal study of arguments, such as:

  • All philosophers yearn for wisdom.
  • Ted Sider is a philosopher.
  • Therefore, Ted Sider yearns for wisdom.

Formalised, this argument would contain quantifiers that ostensibly range over all objects. But in order to use or assess the argument, to prove its validity, we don’t need to know which objects belong to this range, or even whether this range is well-defined. The idea of a universal range of quantification plays no role when we use formal logic to assess arguments. At most, this idea acts as a sort of methodological ideal which we never need to actualise. (Here we could insert a discussion about Kant’s theory of the world as an Idea of reason, but I will resist the temptation.)

When we do want to survey a range of objects to assess statements with quantifiers, as in certain computer applications, all we need to do for successful use of the formalism is to choose a finite range of objects that are contextually useful. So SHRDLU, the famous 1968-1970 computer program that could manipulate a fictional space of geometrical objects and then answer questions like “how many green objects are on top of the red cube”, needed quantifiers; but it needed them to range only over the dozen or so geometrical objects in its fictional space. Never do we need to make use of a universal and well-defined domain of objects; never, in none of our applications, do we need to so much as presuppose that such a domain exists. But surely this blocks any attempt to conclude from the success of the formalism that this well-defined universal domain must exist; and must exist in the world-in-itself, no less.

Let us look at the issue in a different way. First-order logic is a tool that requires certain conditions to be met before it can be applied. Most importantly, when we are thinking about the quantifiers, a crucial condition of the application of first-order logic is that we have a domain of well-defined objects. Now for any tool, past success can inductively justify future success; but of course, only as long as the application conditions are still met. I have good reason to believe that this hammer will be useful in the future, given its usefulness in the past; but only in situations where hammers can actually be used. I have no reason at all to believe that the hammer will help me when I need to mend a broken vase.

So where would we expect quantificational logic to founder? What are the circumstances where we should not expect it to be useful? Precisely where objects are not well-defined, that is, precisely where objecthood becomes vague. Sider is essentially arguing that the success of quantificational logic in many circumstances justifies the belief that it will always work well, and hence the belief that its conditions of application will always be met. But surely this argument is unjustified. It is like arguing that my hammer has been so useful in the past that there cannot possibly exist any problems that it cannot solve. We know in advance where hammers stop being useful, and we know in advance where the limits of logic will be.

All in all, it seems to me that Sider has provided us no reason to believe that deep ontological conclusions follow from the success — always local, always subject to limitations and preconditions — of first-order logic. Indeed, it is quite clear that no such argument will be forthcoming; that any such argument would always beg the question against those who think there are cases to which first-order logic does not unproblematically apply. (It is therefore much like the “relations must always be between things that exist; we can have relations to past objects; therefore past objects must exist” argument against presentism. Past objects are exactly where the presentist would expect the first premise to go wrong.)

But why then does Sider believe that first-order logic has a special status? That it gives us deep insight into reality? I find this genuinely perplexing, but I do have a nagging suspicion that Sider believes this simply because logic plays such an important role in the research he himself is engaged in; that we are here merely witnessing, again, the parochialism of a certain kind of philosopher. Sider’s attempt to claim, without serious justification, that the particular language that he is accustomed to speak must be the language that the world itself speaks, is curious enough that it brings to my mind the words of Sellars, who at the very end of Empiricism and the Philosophy of Mind wrote about “the philosophers who, in their efforts to break out of discourse to an arche beyond discourse, have provided the most curious dimension of all.”


  1. This passage:
    “But we do have such powers. If we all start talking differently, clubs will vanish; and depending on how we talk differently, so might democracy, or unemployment, or the Eurovision Song Festival. Clearly, many things, including physics and philosophy, only exist because of the way people talk. There’s nothing wild about this; of course the world is the way it is because we talk a certain way!

    But, Sider would no doubt interject, this is not true for all things. Some things exist independently of human beings and the ways that they talk: stars, for instance, and electrons, and bacteria. It’s only those parts of the world there were meant by the thesis that the world exists independently of our concepts.”

    put me in mind a typically wonderful and humorous essay by Michael Bérubé (once a big figure in the (left) blogosphere when there was such a thing) on the Sokal hoax. In particular he cites Searle as making precisely the kind of distinction you allude to above:

    “It was there [in a debate with Sokal] that I first unveiled my counterargument, namely, that the world really is divvied up into “brute fact” and “social fact,” just as philosopher John Searle says it is, but the distinction between brute fact and social fact is itself a social fact, not a brute fact, which is why the history of science is so interesting. Moreover, there are many things–like Down syndrome, as my second son has taught me–that reside squarely at the intersection between brute fact and social fact, such that new social facts (like policies of inclusion and early intervention) can help determine the brute facts of people’s lives (like their health and well-being).”

    This circle-squaring strongly appeals to me, as I find much to admire in the very little I understand of postmodernism (I have not read any primary literature!!), but I also cannot shake my, perhaps, knee-jerk realist intuition. It just seems off to me to put subjectivity so squarely in the center of philosophy of science when one could argue that one of the many great advances of science over the ages is to show just how non-central humans actually are, contra eons of religious tradition. But, as I say, I am almost a complete lay person on these topics, so I may be arguing (more precisely, gesturing) against a just the kind of strawman-postmodernism-as-bogeyman that Sider and Sokal were.

    The whole Bérubé article is well worth a read, by the way. As is most of his stuff.

    1. Thanks for the link! It’s indeed a good piece. In fact, Bérubé’s point is almost identical to a point made by Moore in the review of Sider’s 2011 book to which I linked. Moore points out that even if we believe, with Sider, that the concepts (plural) of physics are those that carve nature at the joints, nevertheless, the concept (singular) of physics does not. And so the claim that physics shows us the structure of the world-in-itself cannot itself be a claim about the world-in-itself. Which is just another way of saying that the distinction between a brute fact and a social fact is itself a social fact! I personally find it very hard to disagree with this.

      This is surely one of the fascinating tasks of philosophy: to square the non-centrality of humanity with the centrality of humanity. 🙂

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