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The Logical Pluralist

Interview by Richard Marshall.

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It’s not true that everything in the world makes it true that there’s either beer in my fridge or not—there are some parts of the world that are completely silent about my fridge and its contents. You can maintain the classical insights as being correct, but still say that they don’t tell you all there is to know about consequences and logical connections.

Greg Restall is Professor of Philosophy at the University of Melbourne. He received his Ph.D. from the University of Queensland in 1994, and has held positions at the Australian National University and Macquarie University, before moving to Melbourne in 2002. His research focuses on formal logic, philosophy of logic, metaphysics, and philosophy of language, and even some philosophy of religion. He is a Fellow of the Australian Academy of the Humanities. Here he discusses logical pluralism and why in his view an argument can be valid in one sense but invalid in another, the difference between manipulating the formal machinery and doing philosophy of logic, classical logic’s relationship with alternatives, why pluralism doesn’t mean abandoning universal, true and valid classical intuitions or rejecting classical logic at all but rather adds to the universe of logical connections, substructural logic, the liar paradox, Bradwardine’s theory, why philosophy of logic is a humanities subject and new vistas opening up in contemporary philosophy of logic.

3:AM: What made you become a philosopher?

Greg Restall: I was the first of my immediate family to go to university—I grew up in a working class neighbourhood of Brisbane, and having showed some skill in mathematics at school, I enrolled in a B.Sc. to study computer science and mathematics at the University of Queensland. This was the mid 1980s, and if I was thinking of anything beyond University, I suppose thought I might head into a career as a computer programmer. In the first year of that degree, I enjoyed the mathematics more than the computer science, so I concentrated there, mostly pure mathematics, probability theory and statistics. Meanwhile, I was attempting to sort out what I believed about things for myself, having grown up as a member of a conservative Baptist church. My world opened up at University as I read many of different books in theology and politics and philosophy, and got to know a broader circle of people. In all of that, it was the philosophy that I enjoyed the most. By the end of my degree, I decided to try to move from mathematics over into philosophy. At that time, Graham Priest had arrived in Brisbane to take up the Chair in Philosophy at the University of Queensland, and to my great good fortune, he agreed to my proposal to supervise me in a one-on-one reading course in logic as a part of the final honours year of my B.Sc. I learned more in that course than in any other, and my final paper for grew into my first publication). Once I completed my Honours degree, the Philosophy Department accepted me into their Ph.D. program under Graham’s supervision. One thing followed on from another, and I found myself becoming a philosopher, working in formal logic, philosophical logic, philosophy of language and related issues.

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3:AM: What is logical pluralism and are you a logical pluralist? Can a pluralist logician be a classicist some of the time and if she can, isn’t that cheating? And is pluralism an Australian thing?

GR: Put simply, logical pluralists take there to be more than one deductive logic, and yes, I’m a logical pluralist.

Maybe an analogy might help explain the position: this is the same sort of attitude to deductive logic that many people have to logic in general. Most of us are pluralists about ways that arguments can be good. Many people think that an argument can be inductively strong, or defeasibly good but not deductively valid. (Take the argument from “Tweety is a bird” to “Tweety flies”, to pick a well worn example.) One way for an argument to be good is that the premises give you good reason to believe the conclusion, or if the premises significantly raise the probability of the conclusion. Another, different way for an argument to be good is for it to be deductively valid—for the conclusion to indubitably logically follow from the premises. Those two evaluations of arguments measure different things. The pluralist about deductive validity takes the same thing to hold for deductive validity itself—there’s more than one way that an argument can be deductively valid.

To answer the question about classical logic, I’ll need to explain some more, so I’ll get back to that in a minute. But I can tell you it’s not just an Australian thing. While many Australian logicians lean toward pluralism, my collaborator Jc Beall is very much American (though many count him as an honorary Australian since he’s spent so much time here), and colleagues like Patrick Allo, Ole Hjortland and Francesco Paoli fly the pluralist flag in Europe, and Gillian Russell and Stewart Shapiro are working on logical pluralism in North America. The view is spreading.

3:AM: Before getting into details, we’d better step back and ask what you think logic is. You argue that logic is about consequences but what exactly do you mean by logical consequence because it’s not a clear cut notion is it?

GR: The kind of pluralism that Jc Beall and I have written about fleshes this pluralism out in the following way. One part of logic is the notion of logical consequence or validity. At a general level of description, we can say that an argument is valid if the conclusion follows from the premises. It’s invalid if there’s a counterexample, some way of making the premises true and the conclusion untrue. A pluralist in the Beall/Restall sense takes the notion of validity to be plural. There’s no single one best or most correct notion of validity.

I (and here I’m talking for me, not for Jc, who differs from me on this) think that classical propositional logic, for example, gets something very right about deductive validity. According to classical logic, the argument from the contradictory premises p and not-p to the conclusion q is a valid one, because there’s no way for the premises to be true and the conclusion false—because there’s no way for the premises to be true. They contradict each other.

I also think that relevant logics get something very right about validity. According to many relevant logics, the argument from those contradictory premises to an unrelated conclusion is invalid because there is a circumstance (an impossible circumstance—a way that things can’t be, if you like) according to which p and not-p are both true and q is false. Although the premises are contradictory, that inconsistency is not enough to bring with it every unrelated conclusion. I think that both classical logic and relevant logic are equally good (correct, useful) ways to evaluate arguments, and that’s how I’m a logical pluralist.

To get back to your earlier question, a classical logician can, I think, be a pluralist, if she thinks that classical logic doesn’t tell us everything about logic that there is to know. (Another example I like to think about is the mathematician who uses classical logic to prove theorems, but also explores theories in weaker logics like intuitionistic logic, to explore what is constructively derivable.) But I think there’s something unhelpful in thinking of this as being “classical some of the time” and “non-classical on other times”. You might think of it that way when you’re producing arguments, and you’re trying to produce arguments that are valid in some sense or other. But I like to think of it in another way: the arguments we produce can be evaluated using all of the logics we endorse, and that will always tell us something about the strengths and weaknesses of those arguments. The same goes for using inductive strength and deductive validity. If an argument is not deductively valid (in any sense) but is inductively strong, then this tells us something useful.

As to whether this is cheating: I don’t think it is, but to understand why depends on what you take logic to be for. Do you think that for any argument to be any good, it must be formally and deductively valid? Nobody should think that—arguments are have many different uses, and can succeed in many different ways. Many formally invalid arguments are still good arguments.

3:AM: is there a difference between being able to do logic in the sense of manipulating the formal machinery and doing philosophy of logic? Is an interesting example of this division Kit Fine’s work on substructural logic which you say ‘ is formally astounding but philosophically opaque’ and does it follow from your view that logic has a place in the humanities?

GR: Yes, there’s a big difference between logic as the elaboration of a technical device and the core ideas of logic in the deeper and broader sense. There are many different things that a formal logical system might be used for, and you can study logical machinery as a purely technical device or a mathematical structure, and that is definitely logic, but studying a range of different logical structures doesn’t make you a logical pluralist in our sense. For that, the different formal systems have to be equally applicable in evaluating arguments.

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Think of what goes on when you spot what you think is an error in an argument. It’s one thing to notice that you’ve slipped in some equivocation or what you thought was a perfectly good step of modus ponens was actually affirming the consequent. That tells you something (namely, that the argument doesn’t have the shape you thought it had), but it doesn’t tell you all you might want to know about the argument. To show that an argument is invalid in some sense or other, you look for a counterexample, some way of rendering the premises true which doesn’t make the conclusion true too. If you’ve come up with that, you have genuinely useful information—you’ve learned that the conclusion doesn’t arise out of the premises. Logic moves beyond some technicalities used for calculation when the structures and interpretations that we study can give us genuine understanding of how our claims hang together or fall apart.

For me, this is one reason why logic—including the most technical aspects of the discipline—has a place in the humanities. It provides the tools for a kind of cognitive or mental discipline, in helping us delineate possibilities and connections between our claims. Constructing proofs and models for different parts of our vocabulary helps us understand them better.

3:AM: So what are you claiming when you say there is a plurality of logics? Is it more than simply saying that there are different languages and so we have to adapt logics to fit each of them? Is there a historical dimension – do we have pluralism back in medieaval times before Frege and Russell et al?

GR: The idea that different languages have different logics is one way to be a pluralist, but that’s not my kind of pluralism. While I do think that different languages can bring with them concepts that have different behaviours and ‘logics,’ that’s only a part of the pluralism I endorse. On my view, even in the one and the same language, more than one logic might be appropriate. To be a bit technical for a moment, I think that classical, intuitionistic and relevant logic all have their place in the analysis of arguments using the very simple logical vocabulary of ‘and’, ‘or’ and ‘not’. It’s the one and the same argument from p and not-p to q that is valid classically but invalid relevantly. It’s the very same argument from not-not-p to p that is classically valid but intuitionistically invalid. On my view, the pluralist’s judgement that the argument is valid in one sense but invalid in another in each of those cases tells us more information about the arguments than the single judgement delivered by one favoured logic.

I’m no historian of logic, but I think that we can find kinds of pluralism in pre-modern times when we look at different ways of evaluating reasoning. Aristotle’s syllogistic logic, the theories of obligations, suppositions, and consequence all seem to be different logical theories marking out different sorts of logical consequence relations important in making sense of reasoning, discourse and dialogue. To the extent that you think that these theories are not rivals and are marking out different . I’m not an expert in the history here, but I’ve learned a lot from Catarina Dutilh Novaes and her work in the history of logic. Start with her Formalizing Medieval Logical Theories for more on these topics.

3:AM: Classical logic seems to work with consequences that strike many as being universal such as that a proposition is either true or false, or that a proposition can’t be both true and false and so on. Does the pluralist case depend on this intuition being wrong?

GR: No, you can think that those deliverances of classical logic are true and universal and valid, and be a pluralist at the same time. It’s one thing to think that every proposition is either true or false (and that, as a result, the disjunction either *p* or not-*p* is true) it’s another thing to reject any logic which rejects the law of the excluded middle as a theorem. You can think that it’s true without thinking that every logic must tell you that it’s true. An example comes out of the literature on truthmaking. Let’s agree with with the classical logician that for every statement p, either p or not-p is true. It doesn’t follow from this fact that anything makes that disjunction true. We can use the notion of truthmaking to motivate two different sorts of consequence. The “classical” kind of consequence is straightforward—an argument is good if no matter how things go, if something makes the premises true, then something makes the conclusion true, too. But there is now a more discriminating “relevant” consequence relation. Any particular thing that makes the premises true also makes the conclusion true too. You could think that the claim “either p or not-p” is a tautology in the first sense, since no matter how things go, something makes it true that either there is beer in my fridge or there isn’t (the state of my fridge, for example), but it’s not a tautology in the second sense. It’s not true that everything in the world makes it true that there’s either beer in my fridge or not—there are some parts of the world that are completely silent about my fridge and its contents. You can maintain the classical insights as being correct, but still say that they don’t tell you all there is to know about consequences and logical connections.

3:AM: So where would pluralism be more useful than classical logic?

GR: Pluralism is useful wherever there is need to keep track of more than one way that things can be logically connected. The notions of truthmaking or dependence are one place which points to finer distinctions than classical logic can draw, but it seems to me that there are others, too, like constructive reasoning in mathematics and theoretical computer science. There’s a well developed but minority body of work on constructive formal reasoning, where every proof corresponds to some kind of construction of the conclusion in terms of the premises. A pluralist can have the benefits of constructive reasoning as well as classical reasoning, provided she is happy to understand them as modelling different ways that information hangs together.

3:AM: A challenge to classical logic might be vagueness but Timothy Williamson for one doesn’t concede that classical logic can’t handle it. If classical logic can handle something as tricky as vagueness, is there any case that needs an alternative?

GR: Vagueness is certainly a delicate matter—and I don’t have anything like a settled view on how best to understand it—but I think that this is the wrong way to understand the relationship between classical logic and pluralism. Adopting a pluralist perspective doesn’t necessarily involve rejecting classical logic. It’s being open to the possibility that there are also logical distinctions in addition to those drawn by classical logic. If there is a satisfactory purely classical logic approach to vagueness, that’s no bad thing. It’s another thing to think that classical logic is all the logic you will ever need or want. I think that the evidence here paints a richer picture of logic. The universe of logical connections includes classical logic, but it also includes much more than that.

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3:AM: Substructural logic is another area of interest for you and you’re a leading expert in this field. How would you summarise what substructural logics are for a non-specialist, and why are they important?

GR: Right. For this we need to step back a little. The term comes from proof theory—that part of logic that concentrates on the structure and behaviour of proofs. You can think of different logical rules in terms of proof. For example, think of how you might derive a claim of the form “if A then B”. One way to do that would be to suppose A (together with everything else you’ve already supposed) and then derive B from those suppositions. Once you’ve done that, you’ve derived the conditional if A then B from your other suppositions, and A is no longer supposed. One way to concisely state this is as a rule of proof. If you have a proof from X (a collection of your other suppositions) together with A to the conclusion B, then you can extend this into a proof from X to the conclusion if A then B. That’s a rule of proof that features the logical concept of the conditional. There are rules governing other connectives too, like conjunction (if you have a proof from X to A and a proof from Y to B then you can combine these into a proof from the premises X, Y (combine the two collections of premises together) to the single conjunctive conclusion A and B. There are other rules of proof which seem to govern all kinds of judgements of any shape (not just conjunctions, conditionals, disjunctions, etc), and these are called structural rules. One example is the rule of weakening. It says that if you have a proof from X to B, then you also have a proof from the weaker collection of premises X, A (where you’ve assumed another claim) to B. The rule of contraction says that if you have a proof from X, A, A to B then you have a proof from X, A to B too—the number of times you repeat an assumption doesn’t matter.

Substructural logics are logical systems where you leave out some of the standard package of structural rules. The idea behind the name seems to be the image of digging below the surface of a logical system (like classical logic) to see how the structural rules do their work in the foundations of a system of proof or of logical consequence.

3:AM: How does this link up with notions of relevance, resource consciousness, and ordering logics (and what are these in a nutshell?)?

GR: If you think that a proof from premises X to a conclusion B indicates that each of the premises in X are used on the way to prove B—that B is true ‘in virtue of’ X, then you can see why the rule of weakening might be a problem. Relevant logics leave out the weakening rule in order to find space for a genuine connection between premises and conclusions. If you want to keep track of how the information in the premises gives rise to the conclusion, and in particular, how many times a piece of information has been used, you might also want to limit the rule of contraction, too. This seems to be useful in some computer science applications where proofs correspond to processes of construction, and can think of premises as resources in memory. The number of repetitions of a premise correspond to the number of times it needs to be accessed before the memory location can be cleared. Very weak substructural logics even restrict the ordering in which premises can appear (if you have a proof from A,B to C, it doesn’t necessarily follow that you have a proof from B,A to C) and these have applications in syntactic theory in what is now known as the Lambek Calculus.

The field of substructural logics is filled with a rich body of techniques, arising out of very different traditions (philosophical logic, computer science, linguistics), and branching out into many different applications.

3:AM: Is substructural logic restricted to propositional logics?

GR: They’re certainly the focus of most of the work, any of the major substructural logics can be equipped with the standard universal and existential quantifiers without too much difficulty. (However, finding an appropriate model theoretic `worlds’ semantics for the quantifiers in relevant logics did prove to be hard work. Thankfully, some recent results of Rob Godlblatt and Ed Mares seem to have answered many of the important questions here, and we have tools to understand the behaviour of the quantifiers from the perspective of truth and of models.) However, I think that many questions remain open, particularly concerning identity and its interaction with relevance and resource sensitivity. This is a major open question in substructural logics, just as quantified modal logics are filled with difficult questions in the formal theory and in its interpretation.

3:AM: How do you solve the liar paradox? Are you a spandrel guy like the legendary Jc Beall? Why is Bradwardine’s Theory of Truth relevant?

GR: Ha! If only I could solve the liar paradox! The liar paradox is much too difficult for me to offer anything like a definitive solution—and although I have a lot of time for Jc’s spandrel account of truth, I don’t think it’s right. I do have rather a lot of fondness for Stephen Read’s recovery of the medieval logician Thomas Bradwardine’s theory of truth, and I think that it’s worth exploring more for insights into the behaviour of the truth predicate. I’ll try to explain why.

Many raditional formal theories of truth postulate the biconditional T if and only if A for each sentence A, or maybe as many of these as you can accept, without falling into paradox. We say that “snow is white” is true if and only if snow is white, and so on. This seems fair enough, but it can’t be the whole story about truth, since this biconditional is clearly false for many sentences of English (the sentence “I am sitting” uttered by you is not true if and only if I am sitting—changes in context between the sentence being evaluated for truth and the sentence used as the criterion can break the connection) and of course there are true sentences expressed in languages I do not speak. We need more than the raw biconditional in that crude form. (And I must emphasise, the crude biconditional as I have stated it does is not due to Tarski, despite often being lumbered with the name the ‘Tarski biconditional’). Bradwardine’s insight is an account of truth that can apply much more generally. To put it simply, what can we say about x (for anything x whatsoever) if x is true, in the relevant sense? Well, we can conclude that if x says that p, then (since x is true), p. So, if you make some utterance, if that utterance is true, and you said that snow is white, then indeed, snow is white. This is true even if you’re speaking German or some other language: if your utterance said that snow is white, and your utterance is true, then indeed, snow is white. If you said “I am sitting” and this utterance said that you were sitting, then indeed, you were sitting. Bradwardine’s theory can be summarised like this. For any x, x is true if and only if for every p, if x says that p, then p.

Bradwardine’s theory brings with it a creative and interesting analysis of liar paradoxical sentences. A sentence l is liar paradoxical if it says that it’s not true. That is, l is a liar if and only if l says that ~Tl. What can we say about liars? Well, the theory tells us that if l says that ~Tl, then we can’t have Tl (that would be a contradiction), so indeed, the sentence l is not true. But it doesn’t necessarily follow that the liar is also true, because for that we would need to be sure that if l said p, then p. But if l says something else (something other than ~Tl) as well, then it is safe to say that the liar sentence is not true. And that’s Bradwardine’s analysis. What else might l say? A plausible candidate is that l says of itself that it is true (that l also says that Tl) and that’s certainly something that is not true.

There are many open questions about how to best understand this sort of theory of truth, how to relate it to other theories of truth, and to analyse its costs and benefits, and I’ve written about these topics in a series of four papers. I’m not convinced that the theory can be made to work, I think that it’s a breath of fresh air and fruitful in what has become a very ‘worked over’ problem domain, and that it will reward further attention.

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3:AM: If you were to summarise what are the big new vistas being opened up by philosophical logic at the moment? Are we living through a good and important time for this work and using your crystal ball what do you think will be discovered in the near future that we’ll all be noticing?

GR: This is a good question, but a difficult one. It can be really hard to see the wood for the trees when you focus on detailed research in your own specialisation. But when I try to take the broad view, I’m challenged an excited by a number of different developments in logic and its relationship to philosophy.

I’m encouraged by the development of ‘formal epistemology’ as an established part of the discipline. Seeing formal methods applied to a range of philosophical problems in a sustained way over a period of over 25 years is interesting, because we can begin to see what things bring genuine insight and what things don’t. Simply sprinkling symbols over a problem and using formulas instead of natural language is mere obfuscation rather than insight. But when it comes to proving striking results and relating disparate concepts through creative modelling and better understanding the behaviour of our core concepts—that’s where formal methods can have their power. It seems to me that we have enough of a track record to be able to look back at our successes and our failures and to begin to learn from our own history, in just the same way that the late 20th Century flowering of history of analytic philosophy has helped us understand how analytic philosophy has come to be and the strengths and weaknesses of that tradition.

In that vein, I think that the recent interest in normative pragmatics and the connections between norms of language use and inferential practice are really very exciting. Once you look at language and logic through that series of questions and problems, you can see a rich body of work connecting people as diverse as Nuel Belnap, Robert Brandom, Michael Dummett, Rebecca Kukla, Mark Lance, Rae Langton and Per Martin-Löf. These connections tie together considerations of norms, social practice and expression on the one hand, and formal considerations deep in the heart of proof theory on the other. I think that there will be more exciting work in the foundations of logic in this tradition in the coming years, which will involve the development of new formal techniques

Another development that I find very exciting is the clash between statistical and structural models of meaning in natural language. The rise of ‘big data’ has shown us that we can learn a lot about the meanings of words with the ‘dumb’ technique of measuring their contexts of occurrence in a large corpus. Two words with what we take to be similar meanings tend to appear in similar contexts, and so-called ‘distributional models’ perform really well in representing word meaning, without imposing anything like a predetermined grammar. On the other hand, it seems clear that meanings compose, in the sense that the meaning of a complex sentence should depend on the meanings of its parts. It’s no easy task to combine the quantitative results of distributional models with the great strides we have taken in formal and structural theories of meaning, inspired and informed by the semantics of formal logics. I expect there to be interesting insights coming ahead as people begin to connect the worlds of search engines, data mining, text analysis with tools in syntax, semantics, logic. See the book Quantum Physics and Linguistics below to get a hint of how we might start to connect those worlds.

3:AM: And for the readers here at 3:AM, are there five books other than yours that you could recommend that will take them further into your philosophical world?

GR: I read more papers than books in my own line of research, and my book reading is much more wide than logic.

Robert C. Roberts, Spirituality and Human Emotion (Eerdmans, 1982). This is not a logic book, but it’s the book, more than any other, that got me excited about philosophy. The example of philosophical analysis applied matters of human concern showed me that philosophy could be clear, engaging, warm and insightful all at once.

J. Alberto Coffa, The Semantic Tradition from Kant to Carnap (Cambridge University Press, 1993) and I saw that analytic philosophy had a rich history, and that the development of new techniques in logic by Bolzano, Frege, Russell, Tarski and Gödel played a crucial role in the changes in understanding of the analytic and the a priori.

Mark Lance and Rebecca Kukla, ‘Yo!’ and ‘Lo!’: the pragmatic topography of the space of reasons (Harvard University Press, 2009). This book is a rich treatment of the norms of discourse which take us beyond the ‘declarative fallacy’—the assumption that the heart of language is what we can express by means of assertions. Any account of logic and of meaning has to come to terms with the rich texture of everything we can do in language beyond assertion and denial.

Chris Heunen, Mehrnoosh Sadrzadeh and Edward Grefenstette: Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse (Oxford University Press, 2013). If you want to see the kinds of techniques we can use to develop richer theories of meaning that can combine vector models of meaning in terms of word use and the structural grammar of a compositional language.

Katalin Bimbó: Proof Theory: Sequent Calculus and Related Formalisms (Chapman and Hall, 2014). This is an exciting book on two topics close to my heart: substructrural logics, and proof theory. Kata’s book is a thorough and comprehensive introduction to both fields.

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Richard Marshall is still biding his time.

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First published in 3:AM Magazine: Sunday, June 5th, 2016.