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Paradoxes and Their Logic

Interview by Richard Marshall.


Bruno Whittle is an assistant professor of philosophy at Yale. His work is centred around paradoxes and logical results related to them. He has written about infinite size, the extent to which theories can prove things about themselves, paradoxes for truth and propositions, a puzzle affecting belief, and the nature of necessary truth. Here he broods on whether there’s a connection between metaphysics and logic, the liar paradox, whether true contradictions exist, why dialetheism doesn’t live up to its most important promise of doing away with Tarski’s hierarchies, Curry’s paradox, epistemically possible worlds and propositions, brute necessities and whether they exist, the connection between formal semantics of natural languages and logic, whether non-metaphysicians can learn from metaphysicians and why science can’t answer everything, even questions motivating scientific research. So now, riddle me this, riddle me that…


[Pic: Mehdi Ghadyanloo]

3:AM: What made you become a philosopher?

Bruno Whittle: A combination of British educational restrictions and Wittgenstein’s Tractatus.

When I was deciding what to do at university I knew I wanted to do mathematics—but also that I didn’t want to do just that. I would have done maths and English, but that wasn’t possible. So I looked around for something I could combine maths with. My school organized a session to introduce students to university subjects they might not be familiar with, one of which was philosophy. The ‘sample’ philosophical question was something like: are people motivated exclusively by self-interest? I was convinced that the teacher who was giving a presentation on this had missed (what I then saw as) the most important point—and so it began!

I certainly didn’t though think that I would continue with philosophy beyond being an undergraduate. But in my second year at university I ended up taking a course on Wittgenstein (again, as I recall, because I couldn’t take the class on ethics that I really wanted to). We begun with the Tractatus, and I was completely hooked straight away. The idea of this simple—but far-reaching and revisionary—account of language, thought and the (rest of the) world just seemed incredibly exciting. I realized at that point that I wanted to continue with the subject.

3:AM: Is there a metaphysical commitment embedded in the sort of logic a person adopts? Logic used to get advertised as neutral towards commitments to anything metaphysical. Was this false advertising?

BW: Actually, I don’t think that there is any obvious or straightforward connection between logic and metaphysics. That is, I think that pretty much any logic is compatible with pretty much any metaphysics. Just as I think that pretty much any mathematical theory is compatible with pretty much any metaphysics. Here by a ‘metaphysics’ I mean an account of what the world is ‘really’ or ‘fundamentally’ like.

Different logics are correct for different languages. There are perfectly coherent languages for which classical logic is correct, but there are also such languages for which various non-classical logics are. And to choose a language is not to choose a metaphysics—any more than to choose an operating system is.

It is true that one’s metaphysical commitments might affect one’s choice of language—but so might many other things.

To illustrate, consider the issues arising from paradoxes such as the Liar, i.e. the paradox involving sentences such as ‘this very sentence is not true’. These show that one’s language can’t have all of the properties that one might initially have expected it to have. Specifically, it can’t have the property that every sentence of the following form is true: ‘‘A’ is true if and only if A’ (for a sentence A); and also the property that whenever sentences A1, … An are true, and B follows from these in standard (i.e. classical) logic, B is also true. (Because if it did have both of these properties, then every sentence of the language would be true—which is clearly unacceptable!) Thus, any choice of a language is going to involve giving something up. And we now have a lot of work exploring the various possibilities, i.e. various possible languages.

Many of the most promising give up classical logic in some way. But the factors that are going to lead one to choose one of these rather than another are unlikely to be metaphysical—they are far more likely to concern what these languages allow us to do. For example, one of the things that we would like to able to do is to talk about the language that we in fact speak. So we want a language that can talk about itself in a range of ways. It seems that the languages that are best able to do this are those that revise classical logic in some way. However, considerations like this are very far from metaphysical.

So I would argue that logic and metaphysics are in fact relatively independent.

3:AM: You’re interested in logic and one of your earliest papers looked at the position made well known by Graham Priest and possibly embedded in eastern philosophical beliefs and perhaps Fichte/Hegel that true contradictions exist. Could you first outline dialetheism? It seems strange at first blush to think that this seemingly paradoxical belief can sit comfortably with orthodox views about rationality doesn’t it?

BW: Dialetheism is the view that there are true contradictions, i.e. true claims of the form ‘P and not-P’ (or ‘P and it is not the case that P’). For example, I mentioned ‘this sentence is not true’ earlier. According to standard versions of dialetheism, this sentence is in fact both true and not. So the claim to this effect would be an example of a true contradiction.

Why adopt such a crazy view? The apparently most promising argument, which had been stressed by Graham Priest, was essentially this: one should adopt this view because any more orthodox solution of paradoxes such as the Liar will involve severe expressive restrictions. Specifically, any such solution must eventually appeal to some sort of hierarchy—and that, Priest argued, is too high a price to pay.

Thus, the ‘classic’ orthodox solution was that proposed by Alfred Tarski in the 1930’s. According to this, truth is stratified along the following lines. Suppose that we start with some language L0 that doesn’t contain the resources for talking about truth. If we want to talk about the truth of the sentences of L0, then the way in which Tarski proposes that we do this is by introducing a term ‘true-0’ that applies precisely to the true sentences of L0. This will result in an extension of the language, L1. But since ‘true-0’ applies only to sentences of L0—and not to the new sentences of L1—if we want to talk about the truth of the sentences of L1 more generally, we are going to need to introduce another new term, ‘true-1’, giving a further extended language L2. And so on. Let L* be the language that results from adding all of these new terms. The problem with L*—with this approach—is that L* does not contain a general term for talking about the truth of its own sentences, i.e. a term that applies precisely to the true sentences of this language. It contains a term (‘true-0’) that applies precisely to those sentences that belong to (the ‘sublanguage’) L0, another for L1, etc. But none that works generally. What this means is that one cannot, in L*, make straightforward claims about L* itself, such as ‘no sentence (of L*) is both true and false’. One can make claims of the form ‘no sentence is both true-n and false-n’, but these will not express what one wants to, since they won’t contain a notion of truth that applies to sentences of L* generally. But then, if L* is the language that we in fact speak, we are going to be unable to make such claims about our own language. This is a serious problem! Other, more recent, orthodox approaches aim to be less restrictive. But, Priest argued, they always have to appeal to some sort of hierarchy along these lines, and with similarly unacceptable consequences. In contrast, Priest claimed that dialetheism allows one to avoid such hierarchies altogether: since, essentially, it is only to avoid contradiction that we are forced to appeal to these.

To answer the last part of your question: this is not a position that sits comfortably with orthodox views of rationality (any more than it sits comfortably with such views of logic or truth). For example, one standard principle is that one should never believe both P and not-P. But according to dialetheism there are cases in which one should do exactly that.

3:AM: So you think there are problems with the view.

BW: I don’t think that it lives up to its most important promise. That is, I don’t think that it really does allow us to get by without hierarchies, any more than familiar orthodox solutions do. In the paper that I wrote about this—‘Dialetheism, Logical Consequence and Hierarchy’—I consider a paradox involving the notion of logical consequence. I explain how this forces the dialetheist to appeal to a hierarchy in just the way that more orthodox solutions are forced to do that—and, further, that this leads to just the same sort of problems, i.e. just the same sort of expressive restrictions.

The basic strategy of the paper is as follows. Orthodox solutions are forced to appeal to hierarchies by ‘strengthened’ paradoxes, that is, paradoxes that the machinery of these solutions cannot handle. Does dialetheism face such paradoxes? All dialetheism does, essentially, is ‘defuse’ contradictions. That is, it provides a framework that allows one to make sense of the idea that some of these are true, the central component of which is a logic in which contradictions do not entail everything (as they do in classical, and many non-classical, logics). But there are paradoxes that lead to unpalatable conclusions without going via contradictions. For example, Curry’s paradox, which involves sentences such as ‘if this sentence is true, then 0 = 1’.

(To see the paradox: call this sentence C, and suppose that C is true. Then what C says must be the case: i.e. if C is true, then 0 = 1. But we are supposing that C is true, so it follows that 0 = 1. That is, if C is true, then 0 = 1 follows. But that’s what C says! So C is true—from which it follows that 0 = 1.)

Thus, the main idea behind dialetheism doesn’t help at all with such paradoxes. It is then natural to expect that these will lead to problems similar to those that strengthened paradoxes cause for orthodox solutions. What I show in the paper is that this is indeed the case. Specifically, I consider a version of this paradox involving a logical consequence connective *: where A*B means that if A, then it follows logically that B. Of course, dialetheists were aware of Curry’s paradox. However, they had not realized that it could form the basis of objections to their position very similar to those that they were busy wielding against more orthodox ones.


3:AM: Is this a problem for all paraconsistent logics? Does this mean classical logic is actually the only game in town, and deviant logics all fail?

BW: It is a serious problem for what had seemed to be the best motivation for using a paraconsistent logic. But it is not an argument to the effect that there will never be any use for such logics. And it is certainly not an argument for the claim that we should stick to classical logic—at least not as it is normally understood. Indeed, I think that there are good reasons for revising this: I have argued that the best account of truth has it that there are classically valid arguments ‘P1, …, Pn, therefore Q’ such that P1, …, Pn are all true, but Q is not. I make this case in my paper ‘Truth, Hierarchy and Incoherence’.

3:AM: What’s the issue with epistemically possible worlds and propositions?

BW: Propositions are the things that attitudes like belief, desire, hope, fear etc. are attitudes to. So if you hope that Chelsea are going to win, whereas I fear it (!), then the idea is that we have different attitudes to the same object. The issue that I raise in the paper of that title—‘Epistemically Possible Worlds and Propositions’—is related to the things that we have just been talking about. For just as the notion of truth for sentences gives rise to paradoxes, so does that of a proposition. Again, a natural (and common) way of solving these involves the appeal to some sort of hierarchy. For example, one according to which any proposition belongs to some ‘level’, and can only talk about propositions at lower levels. These block the paradoxes very effectively, but unfortunately they block a lot of other things too. They prevent us from talking about propositions generally—so we can never talk about knowledge or belief generally, for example.

On the traditional, perhaps most intuitive, approach propositions are ‘structured’ in a way that (more or less) mirrors the structures of the sentences that express them. Thus, the proposition expressed by ‘Hillary knows Lloyd’ is similarly made up of three constituents, which might be: Hillary, the relation of knowing, and Lloyd.

But it turns out that one way of blocking the paradoxes without the need for any sort of hierarchy is to give up the idea that propositions are structured in this way. In particular, to move from such a view to that on which propositions are sets of possible worlds (i.e. sets of ‘ways the world might have been’). Intuitively, the idea is to identify a proposition with the set of worlds at which it is true: so the proposition that Hillary knows Lloyd would be identified with the set of worlds at which she does in fact know him.

There is at least something natural about such accounts. After all, one of the things that a proposition ‘does’ is divide the space of possibilities in two: into those possibilities in which the proposition is true, and those in which it isn’t. But the simplest thing that does that is a set of worlds (or, equivalently, a function that sends possible worlds to either true or false). So why not just identify propositions with such sets?

The drawback is that such accounts face rather serious problems. The most obvious is the fact that on such an account there is only one necessarily true proposition (i.e. the set of all possible worlds). This means that every necessarily true sentence—e.g. ‘1+1 = 2’ or ‘water is H2O’—says the very same thing.

A natural question is thus: can one keep the benefits of this account (i.e. the ability to solve the paradoxes non-hierarchically) while lessening the costs? That is the question I consider in the paper. In particular, I consider the apparently promising strategy of moving from possible worlds—as standardly conceived—to ‘epistemically’ possible worlds: something like ways the world might be, for all we know; or ways the world might be, for all we can know a priori. Thus, such an account would allow us to distinguish the proposition that 1+1 = 2 from that to the effect that water is H2O: since there are epistemically possible worlds in which 1+1 = 2, but water isn’t H2O. Further, the notion of an epistemically had recently been put to a whole range of philosophical uses, most notably by David Chalmers.

What I show in the paper, however, is that moving to epistemically possible worlds is not in fact a promising strategy: because the most natural accounts of such worlds are in fact inconsistent. It may be possible to give consistent accounts of such worlds, but any such account will fundamentally compromise the basic idea behind epistemic possibility. Of course, this is a problem not just for accounts of propositions in terms of epistemically possible worlds, but for accounts of anything in terms of these.

3:AM: A necessarily true sentence is ‘brute’ if it does not rigidly refer to anything and if it cannot be reduced to a logical truth. Is the philosophical issue about brute necessities an example of where modal logic matters to the metaphysical picture we hold? Can you expand on what brute necessities are and give an example or two to allow us to get a grip on what this issue is about.

BW: Here are some apparent brute necessities: ‘there are infinitely many prime numbers’ or ‘for any things x and y, x is taller than y if and only if y is shorter than x’. Neither of these are logical truths, and neither can be reduced to one by replacing terms with their definitions or analyses: because no logical truth asserts the existence of a specific sort of thing (and so the claim about prime numbers is neither a logical truth nor reducible to one), and it would be arbitrary to define tallness is terms of shortness, or vice versa (so the claim about tallness is also neither a logical truth nor reducible to one).

Necessities of this sort can seem puzzling, even problematic, in part because they seem to resist the sort of explanations we can give of other necessary truths, such as ‘it is either raining or it’s not’ (which can be explained in terms of the meaning of distinctive logical terms like ‘or’ and ‘not’) or ‘Socrates is human’ (which can be explained in terms what it is for something to be Socrates, i.e. Socrates’s essence).

This is not, though, a case where considerations from modal logic matter to metaphysics: modal logic is concerned principally with claims that are about modality (i.e. claims containing necessity and possibility operators), whereas the issue here is rather about the modal status of claims that are not in and of themselves about that. It is however certainly a case where metaphysical claims (i.e. claims about what the correct metaphysics of the actual world is) have been argued for on the basis of considerations about modality.

3:AM: Cian Dorr says brute necessities don’t exist. Can you say why he says this and what that does to his metaphysics?

BW: Right. He claimed that such necessities would be problematic for the sort of reasons just mentioned. He was sceptical that there are such ‘laws of metaphysics’ (as he called them).

He went on to argue for a whole range of claims on the basis of the premise that there are no such necessities. For example, he argued in this way that there are no abstract objects (such as numbers, sets, propositions etc.). The gist of the argument, of course, is that if there were such objects, then they would give rise to brute necessities (such as the claim that there are infinitely many primes).

3:AM: You argue that they do exist don’t you? And Godel’s incompleteness theorem is important for your argument isn’t it?

BW: In ‘There are Brute Necessities’ I argued that this sort of blanket denial of such necessities in untenable. I did this by exhibiting a range of such necessities that it seems that anyone must accept—even those who deny that there are such things as abstract objects.

Yes, the argument was an application of Gödel’s first incompleteness theorem. What that result shows, to put it very roughly, is that mathematics outruns logic. Or, to put it slightly more carefully, that there are more mathematical truths than can be logically deduced from any given axiom system. I showed how this fact can be used to produce a range of brute necessities. The basic idea is this. I introduced a predicate that applies to concrete objects just in case they are ‘structured’ in a certain way—structured, in fact, in a way that essentially mirrors the structure of the natural numbers (assuming that those exist). This predicate gives rise to a whole range of necessary truths (corresponding to theorems about natural numbers). But it follows from Gödel’s result that, however we analyze or define the predicate in question, many of these necessary truths will be neither logical truths nor reducible to such—that is, they will be brute necessities.

Attempting to establish metaphysical claims on the basis of the principle that there are no such necessities would thus seem hopeless.

3:AM: Can you say something about the formal semantics of natural languages and whether there are current issues in logic that are impacting on formal semantics – or vice versa?

BW: I think that one of the most interesting things of this sort is that which we touched on earlier, i.e. the issues that arise from paradoxes such as the Liar. As I (in effect) said, what these show is that no language can have all of the properties that one might initially have believed that English has. That raises the question: which properties is it possible for a language to have? Or, to put it another way: are there limits on the extent to which a language can talk about itself—on the extent to which we can talk about our own language—and, if so, what are these? And these are questions on which great progress has been made by using the techniques of logic. Specifically, by developing formal models of languages that can talk about themselves in a variety of ways. For example, the work of Kripke, Priest, Gupta and Belnap, McGee, Gaifman, Field and others.

Thus, what much of this work does is show the ranges of things that are possible once one moves away from standard, classical models of languages. That is, the range of ways in which it is possible for languages to talk about themselves, once we do that. One way of so moving away is as follows. In standard models, a predicate (e.g. ‘is true’) has a set as its semantic value (i.e. its meaning). That is, the set of things that the predicate applies to, or is true of. One way of departing from such models is then to allow the truth predicate to have some different sort of thing as its semantic value. For example, Kripke considered the possibility of replacing such sets with pairs of sets: on the one hand, the set of things that the predicate is true of, and, on the other, the set of things that it is false of, but leaving open the possibility that there might be things that the predicate is neither true nor false of (i.e. that there might be things in neither set). He showed that once one moves away from standard models in this way, one can give natural models of languages that contain their own truth predicates (i.e. terms that apply to precisely the true sentences of the language).

In fact, however, although this is one way of departing from standard models, it also has significant drawbacks. As Kripke himself noted, the ‘ghost’ of Tarski’s hierarchy remains: essentially because, although these languages can contain their own truth predicates, there are very natural notions of untruth that they cannot express about themselves; that one must, that is, ascend to a ‘higher’ level of a hierarchy to express. (This is the sort of problem that I alluded to earlier in the discussion of dialetheism.) An ultimately more successful approach, I have argued, requires something fundamentally different: rather than jettisoning standard semantic values in this way, we should stick with these, but instead allow the standard ‘compositional’ rules that determine the truth value of a sentence in terms of the semantic values of its words to have exceptions. Doing this avoids the need to appeal to a hierarchy in anything like the way that other approaches have to. I discuss models of such languages, and justify these claims about them, in ‘Truth, Hierarchy and Incoherence’.

As I said, these models are developed using the techniques of logic. So this is one way in which discoveries from logic affect the theory of natural language—specifically, the theory of which natural languages are possible.

3:AM: Could there be shifts in a semantic theory of natural language, for instance, or metaphysics, or epistemology, that could lead to a revision in logic?

BW: That certainly seems possible, at least in the following sense. Different languages will have different logics, i.e. different logics will be correct for such languages. Thus, empirical discoveries that tell us more about what sort of language English is, for example, could certainly have implications for the question of what the correct logic of English is.




3:AM: The relationship between philosophers of metaphysics and naturalist philosophers is often tetchy. Why can non-metaphysicians learn from metaphysics?

BW: Here is an example that I think is illustrative. Consider the question: are there different sizes of infinity? This is one of the most natural and fundamental questions that one can ask about the infinite. And it is a question that mathematicians have tried to answer.

The way in which they have done this is by considering a certain mathematically tractable relation R such that it can be proved that there are infinite sets that do not stand in R to one another. The idea is that sets are the same size if and only if they stand in relation R—in which case the mathematical results would indeed settle the question of whether there are different sizes of infinity, i.e. whether there are infinite sets of different sizes.

The relation R at issue is that which A stands in to B if there is a ‘one-to-one correspondence’ from A to B. That is, if the members of A and B can be paired off with one another. Now, it is typically taken for granted that two sets are indeed the same size if and only if they stand in R. However, there are in fact strong reasons to doubt that this is the case—or, at least, to doubt that we are currently in a position to know that it is. For when we talk about the size of a set, we are talking about a sort of property that does not in and of itself have anything to do with such correspondences. But once we realize this, and then look around for arguments that establish that size (even in the infinite case) is connected to the existence of such correspondences (in the assumed way), we discover that it is surprisingly hard to provide one.

Now, the question of whether this mathematically tractable relation R is—or is not—coextensive with the same size relation is an essentially metaphysical one. It is a question about the nature of size, i.e. these fundamental properties that we are talking about when we say that one collection is larger or smaller than another.

So the situation is this. We have a basic question—‘are there different sizes of infinity?’—that is motivating work in mathematics. But whether the mathematical results at issue really settle this in fact turns on a—far from trivial—metaphysical question. And there are, I believe, lots of other cases where metaphysics is needed in a similar way.

I talk about the question of whether there are different sizes of infinity in a paper called ‘On Infinite Size’ (and another which is in progress called ‘Size and Function’). I make the case that—contrary to what is usually assumed—the question of whether there are different such sizes is in fact wide open.

3:AM: And are there current issues in science that logic and metaphysics can usefully help with? What do you say to Stephen Hawking who says that philosophy is finished because science can answer all the questions?

BW: I would say—along the lines just sketched—that science can’t answer all the questions, at least not on its own. Indeed, there are cases (see above!) where it can’t even—on its own—answer the questions that are motivating scientific research.

3:AM: And for the curious here at 3ammagazine, are there five books you could recommend to take us further into your philosophical world?

BW: Of course. We started by talking about Wittgenstein, but the work that I would recommend as I way into the problems that he considered is Saul Kripke’s Wittgenstein on Rules and Private Language. I first read this as an undergraduate and it had an enormous impact on me. I must admit that, as time has as gone on, the basic assumptions that Kripke (and perhaps also Wittgenstein) make have come to seem increasingly implausible to me. (Essentially, despite protestations to the contrary, he seems to assume something that it is akin to behaviourism.) Nevertheless, the book is a fascinating reductio of these assumptions.

Next, I would recommend George Boolos’s collection Logic, Logic, and Logic. It is an incredibly elegant work. And it illustrates the rich seam of questions—and answers!—that can be found at the border between purely technical, or mathematical, parts of logic and more discursive parts of philosophy.

Third, I would suggest Timothy Williamson’s Knowledge and its Limits. He was my supervisor at Oxford, and I learnt an enormous amount from him. What I was most struck by—from the very first class of his I went to—was how much he seemed able to do with so little. What I mean is this. He could take some familiar set of tools, some concepts and distinctions, say, and conjure up something quite beyond what you would have thought possible. That ability is certainly on display in Knowledge and its Limits.

We’ve talked a lot about paradoxes, and there are number of excellent books about these. The one that I am most inclined to recommend is an underrated classic: A. N. Prior’s Objects of Thought. Its full of original ideas, beautifully, rigorously and economically developed.

The last three books mentioned are ones that readers will get the most from if they know at least a little logic. So I would finally recommend a book by my first logic teacher, Moshé Machover: Set Theory, Logic and their Limitations. It’s a wonderful book, presenting a wonderful body of work. A real pleasure to read and learn from.

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First published in 3:AM Magazine: Sunday, March 27th, 2016.