-———Illustrasjon: Oda Aurora Norlund———-
By the late nineteenth century logic had undergone a revolution. After almost two millennia of playing around with the limited theory of Aristotelian syllogisms, inference patterns of the form
All Bs are Cs
All As are Bs
Therefore, all As are Cs,
logicians, like Gottlob Frege and Charles Sanders Peirce, greatly advanced their subject. They developed formal languages and theories for the classical propositional calculus and predicate logic of first and second order that greatly outstripped the limits of the old syllogistic approach. Alternative theories of logic were also developed in the decades to follow, such as intuitionistic logic, while some theories fell somewhat out of favor (e.g. second order logic). Underlying logicians’ disagreement about what language to state logical theories in and what principles to count as logical was the question: what is the correct logic?
Monism about logic thinks that this question has only one answer: there is only one correct logic. Most parties to the debate about what should be thought of as logic over the last century seem to have assumed some form of monism about logic. But by the mid-thirties some philosophers had started to think that maybe there is no one correct logic: maybe there are many somehow equally good logics? This pluralism about logic found one of its earliest and most radical defenders in the philosopher Rudolf Carnap. He accepted a principle of tolerance:
It is not our business to set up prohibitions, but to arrive at conventions. […] Everyone is at liberty to build up his own logic, i.e. his own form of language, as he wishes. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical arguments. (Carnap 1937:51–52)
In Carnap’s mind there was no deep philosophical question as to what counts as the correct logic. Rather, giving a logical theory was a matter of arriving at a useful system of linguistic conventions fruitful to whatever theoretical task at hand. He did not exclude the possibility that in certain cases more than one logic might be up to the task. To Carnap the pluralism afforded by the principle of tolerance was a needed liberation:
The first attempts to cast the ship of logic off from the terra firma of the classical forms were certainly bold ones, considered from the historical point of view. But they were hampered by the striving after ‘correctness’. Now, however, that impediment has been overcome, and before us lies the boundless ocean of unlimited possibilities. (Carnap 1937:xv)
Carnap’s pluralism met considerable opposition in its time, most influentially in the work of W.V.O. Quine. Yet, today pluralism about logic is a vibrant and live part of the philosophy of logic, with contemporary defenders such as JC Beall, Greg Restall, Roy Cook and, not least, Stewart Shapiro.
Stewart Shapiro is a professor of philosophy at the Ohio State University. He has written several influential books and articles in areas such as the philosophy of mathematics, philosophy of logic and philosophy of language. In the philosophy of mathematics he is one of the foremost defenders of the view known as structuralism; the view that mathematics primarily is about the study of structures of a certain kind, such as number-theoretic structures, group-theoretic structures, set-theoretic structures, and so on (as opposed to objects such as numbers, groups and sets). Professor Shapiro has also defended pluralism about logic, set out in most detail in his book Varieties of Logic from 2014. In this interview Shapiro tells us more about logical pluralism, the arguments in favor of it, and how it relates to mathematical practice and even a pluralism about mathematics itself.
Navigating the Boundless Ocean of Pluralism: An Interview with Stewart Shapiro
By Adrian Kristing Ommundsen & Hans Robin Solberg
What do you take pluralism about logic to be?
The official definition of pluralism about something is that there is more than one of that thing. So, pluralism about logic is the view that there is more than one equally good, or true, or correct, or best, logic. The slogan of the opponent, the monist, is: “There is one true logic.” Everyone agrees that there are lots and lots of formal systems but the monist says that only one of them gets it right, whatever the “it” is.
There are different ways of characterizing logical consequence. You take this as evidence that ‘logical consequence’ is polysemous or expresses a cluster concept. Could you give some examples of the different characterizations and explain how polysemy or being a cluster concept motivates pluralism?
The two prominent trends on the contemporary scene are model-theoretic and proof-theoretic. The model-theoretic camp defines consequence in terms of interpretations or meanings or something like that. So, an argument is valid if, under all interpretations, the premises are true, then the conclusion is true. The proof-theoretic tradition has it that logic has to do with deduction. They would lay out some rules for the meaning of the logical connectives: “and”, “or”, “not”, “for all”, and so on, and define an argument as valid if its conclusion can be obtained from its premises in accordance with the rules.
Therefore, we have at least two different notions of validity. One might argue that one of them is right – it captures the notion of logical consequence – but I suggest that both are legitimate notions. That is one instance of what I mean by logical consequence being a cluster concept or polysemous. The notion could be sharpened in various ways incompatible with each other.
An alternative way of motivating pluralism is by the logic-as-model view. What is that view?
I guess the idea is that logic is modeling something in the sense that scientific models model things. Usually, whenever you have a model it is almost always going to distort and idealize. Logic does that too; and it can be modeled in different ways. Now, what is it that is being modeled? That is where things get kind of murky. If it is correct inference, then that would lead you to the next question: maybe aspects of correct inference are being modelled, or aspects of reasoning, and that would lead to pluralism. However, if there is no unique thing there, modeling is like a way to sharpen these notions.
You push the fact that there rarely is a uniquely best model. But this is compatible with there being a unique underlying thing we are trying to model.
I think that is right. If the thing being modeled happens to be sufficiently sharp or determinate, then modeling would not lead to pluralism.
Do you think that logicians are trying to model a unique real-world phenomenon and that pluralism enters only at the level of models, or that there are many distinct real-world phenomena, hence the pluralism?
I do not think logicians are trying to model a unique real-world phenomenon, but I have not thought about it that much. There are standards of correct inference in mathematics, and in ordinary language too. As a matter of fact, felt inferences are among the data semanticists use. It is completely up in the air how that relates to logic and what logicians do. I would suspect that there would be many distinct real-world phenomena; either that, or the real-world phenomenon is messy and inconsistent, and could be sharpened in various ways.
What would these phenomena be? Well, one thing it might be is meaning – but as Michael Glanzberg and others have pointed out, the data that linguists use is mostly inference, but we would not recognize it as logic. The inferences would go from things like “x is north of y” to “y is south of x”. Instead, you could sort of just hone in on the logical terms, but then I am not sure why there would be a unique phenomenon there. Another thing that we might be modelling is standards of rigor in mathematics, in line with John Burgess’s recent book. It would be an empirical question whether there is a unique standard there.
Is this a model in the same loose sense that we have scientific models?
Yes, although it might be better for the cases where we are just not confident that we do have a sharp phenomenon that we are modelling or capturing.
Like with biological models of species…?
Maybe something like that or economic models. If there is a single phenomenon there, and it is determinate and sharp and we are distorting it because it is too complicated, then that would not lead to a pluralism.
Maybe at the level of the models?
Maybe at the level of the models it would, but not at the level of the phenomena. The idea would be that all the models are distorting. Some advocates of non-classical logic, paraconsistent folks, think that classical logic is a reasonable model, up to a point. But most of them insist that there is a richer and more complex phenomenon. Again, if the target phenomena you are modelling were definite, sharp and clear, then that would not give rise to pluralism. If it is something that is loose, sloppy, context-sensitive in all kinds of interesting ways, like meaning, then you would have a pluralism there, because what you are doing now is proposing different sharpenings.
And the last option is you’re conflating a lot of different phenomena; they might be sharp, but there are several distinct phenomena there.
That could be the case as well. There could be several that are all getting mixed up and we need to sort them out.
You point out that there are what you take to be interesting and legitimate mathematical theories that employ non-classical logics.
Yes, that is really where my thinking is these days.
And these would be rendered inconsistent if classical logic were imposed. What are some cases of this, and how do you argue from this to pluralism?
The cases that I mention in the book, or develop in the book, are well-known intuitionistic theories. One case is intuitionistic analysis, which is a theory of the real numbers. It starts with the same idea that classical analysis does, that real numbers are equivalence classes of Cauchy sequences of rational numbers. But with intuitionistic analysis we insists that the logic is intuitionistic and because of that it ends up proving theorems that are classically false and even classically inconsistent. For example, they show that all functions on a bounded interval are uniformly continuous. On the other hand, when you use excluded middle for the logic of classical analysis, it is easy to define discontinuous functions. A second case involves a principle that was prominent in some schools of intuitionism, where you add a version of Church’s thesis to the theory. It is put like this: “If for every x there is a unique y such that such and such, then there is a recursive function that computes the such and such.” A third example is smooth infinitesimal analysis, which is, I think, a lovely theory of infinitesimals that, again, is inconsistent with classical logic but still rather robust.
Alright, that is all intuitionism, and those are the non-classical cases that I am most confident of, but there could presumably be others using paraconsistent logic. Various theories of truth and some interesting mathematics could develop out of that. Those are obviously inconsistent — they are designed to be — but non-trivial. And a lot of the work on truth uses a non-classical logic of various kinds and that could give rise to interesting mathematics. But so far the jury is out on that. There is a bit of an industry, mostly in Australia and New Zealand, of trying to develop inconsistent mathematics, where you just accept contradictions, in set theory or something, and then build the logic to keep the thing from becoming trivial.
How do you go from that to pluralism? Well, if these are interesting mathematical theories, then you sort of have a choice at this point. So let us just look at the intuitionist ones. If you think that this is a legitimate theory and not an incoherent one, then excluded middle would not be part of the “one true logic”—if you think there is but one true logic. So, you could just say “OK, well then the one true logic is intuitionistic” and then you can do classical mathematics by adding excluded middle as a non-logical axiom. If the other logics pan out, then you still could hold on to monism, by insisting that the one true logic is just the intersection of all of the logics. But I suspect that would be a logic which is so weak that no one would be very interested in it.
At that point, the discussion becomes a bit terminological: “What you want to call ‘logic’?” If you by ‘logic’ mean ‘standards of inference that are correct everywhere’, which of course the monist would insist on, then it would be a sort of quasi-empirical matter as to what that would be, because who knows what theories people are going to concoct. And it is probably going to be amazingly weak and not very interesting. Or you could call it pluralism and call these things different logics and just leave it at that. But I think that is mostly terminological, nothing to get bent out of shape over.
So someone who is adamant that classical logic is the one true logic would exclude those theories as incoherent or not mathematics at all. It is an interesting argumentative strategy, to start with these theories and observe that these are legitimate and interesting.
I sort of just rely on quasi-empirical data: They are published in math journals, people get PhDs and publications working on them, and so on.
So it is a kind of naturalism?
In a sense, I guess. So, the philosopher says “classical logic is the one true logic, these things are inconsistent.” Okay, fine, you can say that. If you say, “The one true logic is first degree entailment,” then that also does not seem to make that much room for intuitionism. So then they would have to say that the intuitionistic theories are illegitimate from their philosophical armchair. They will not be the first to have done that.
You also mention the view of taking the consistency of some mathematical axioms as sufficient for there to exist a structure satisfying those axioms or the legitimacy of a theory employing those axioms. Given this, you mention that a relativism or pluralism about mathematics itself would follow from logical pluralism. Could you say something more about this?
Well, that is a long trend within mathematics itself, since around the beginning of the twentieth century. It traces to back David Hilbert. The slogan is “consistency implies existence”. To bring the inconsistent theories into the fold, we would have to change that into something like ‘non-triviality implies existence’. Now, of course, it does not follow from that that every theory is going to be interesting, important, and so on.
The development in geometry is a good example. The way it finally emerged, it is not a theory of space. Euclidean and non-Euclidean geometry both ended up surviving and they are both legitimate branches of mathematics.
But all those geometries were brought into the fold under classical logic.
Yes, so I am extending the Hilbert and non-Euclidean program to include other logics.
So your view is a more extensive pluralism about mathematics?
Yes. Hilbert and Ackermann did write a logic book and it was classical logic. It would be a natural extension of the program. But you are right that it is an extension, especially when you are going beyond consistency to non-triviality.
And the kind of theories you would include would be the examples from earlier?
Yes, certainly the intuitionistic ones.
And maybe some paraconsistent ones?
Maybe, but again those just have not been developed as well. That could just be because there are not a whole lot of people working on them; because it has not caught hold yet. But I do not think that philosophers should rule them out.
In the book you distinguish between tolerance, relativism and pluralism.
Well, tolerance is consistent with monism. Tolerance just means, “Okay try it, I am not going to stop you.”
And the added weight of pluralism is somehow that these theories are equally good or equally correct, and so on, which is a more substantial claim about the subject matter of logic?
Yes, or at least potentially so if it pans out. Tolerance is more like a psychological attitude; for example, you are not going to reject a candidate for a job because they are using an incoherent logic. Although, you normally would reject a candidate if they were saying incoherent things all the time. So there is something to tolerance. If we had a candidate and they were arguing that the earth is flat for these and these reasons, we would probably say that we are not interested in hiring them. You can still be tolerant and maintain that there is just one true logic; it is just that you are open-minded about what the logic is, or you are collegial.
Earlier we talked about this semi-empirical argument for pluralism from what is being published in math journals. It could be difficult to say whether that is mathematicians being pluralist or being tolerant.
Yes, you are right. Burgess is more of a monist here. Whenever you use a non-classical logic nowadays you have to label it as such. Well, that is fair. But you are right, it could just be tolerance. In any case, they still regard intuitionistic mathematics as legitimate mathematics. I take that to give some support.
How radical is your pluralism?
People ask me that: “Does anything go?” I do not know how to put this: There is always going to be the pragmatic side of things, there is going to be tons of stuff which is just plain uninteresting, boring, unfruitful or unhelpful. Now, somebody could develop that stuff and call it mathematics. Of course they are not likely to get it published because it is not very interesting. But that can happen with classical mathematics too. So in effect anything goes in the sense that there is no philosophical line to be drawn between legitimate and illegitimate. It is all going to be on the pragmatic side of things. I kind of suspect that I am moving in a Carnapian type direction.
You have also discussed matters concerning the metatheory. Could you introduce some of the ideas here?
One thing mathematicians or logicians tend to do when they get a bunch of systems thrown at them, is to study the relations between them and develop mappings between them. And this pluralism is of course no exception. There are a lot of frameworks for studying the relationships between different intuitionistic and classical theories. So that becomes a branch of mathematics in and of itself, and that has to have a logic just like any branch of mathematics would. Then we ask the same question there: Someone might claim that there is just one true foundation, and here is its logic. They might just go with me partway, saying there are all these branches of mathematics and then there is just one foundation. I would be tempted to go pluralist there as well, but again this is just a matter of which is the most fruitful. Just try it; try with different logics and see if we get interesting insights. And this does not stop: If you have different foundations people are going to want to know how they relate to one another. But that is life. You do not need to have a bedrock place to stand where everything is certain and there is the one true logic.
In many fields of mathematics they have open questions and they have research programs where they are trying to figure out whether a given mathematical claim is true or not.
That is right. Most of the time the claims are classical but not always.
Exactly. And there is also a sense that you can’t just choose a structure.
Yeah, you cannot just solve it by stipulation: “We might not know whether the Goldbach conjecture is true so we just consider a structure where this is considered an axiom.”
But if your pluralism is radical enough you can just say that such a structure is legitimate, that there is such a structure.
There is nothing to rule it out, but it is not very interesting.
But the pluralism doesn’t rule out that as illegitimate?
Well, it might be trivial. Then the pluralism I am interested rules it out. So, in that case, if the Goldbach conjecture is false, then adding it as an axiom would render the whole thing inconsistent and trivial. That is, anything could be proven.
You mentioned a theorem earlier, about all functions being continuous in intuitionistic analysis — which is provably false in classical analysis. How do you solve the apparent conflict that you have the same statement being true in the one case and false in the other?
Well, first of all it is not clear that it is the same statement. Questions of meaning cannot be taken for granted here. But it is similar to what we would say in the case of geometry. In Euclidean geometry, given a line and a point not on the line there is exactly one parallel line going through that point, in non-Euclidean you have the negation of that, either no line or more than one. How do you resolve that? Well, they are different structures. Is the meaning the same? I do not know. I discussed that in the book too, about how meaning is not as determinate as people seem to think it is.
So the trick is to relativize to a structure?
Yeah. I love that quote that I have from Alberto Coffa in the book:
“During the second half of the nineteenth century, through a process still awaiting explanation, the community of geometers reached the conclusion that all geometries were here to stay . . . [T]his had all the appearance of being the first time that a community of scientists had agreed to accept in a not-merely-provisory way all the members of a set of mutually inconsistent theories about a certain domain…It was now up to philosophers…to make epistemological sense of the mathematicians’ attitude toward geometry . . . The challenge was a difficult test for philosophers, a test which (sad to say) they all failed . . .”
Now, which is the right theory of space? That is an empirical question, or quasi-empirical because presumably you can impose either kind of metric on it; so which is more useful for science, that is an interesting question too.
Some people have tried to argue from the fact that we can apply mathematics in different scientific fields, and therefore that mathematics should be univocal.
What if we are getting applications from two different mutually inconsistent theories? I do not know what to think of that. If they are both applied in the same way, then you could go with the modelling idea: both of them are kind of distorting things a bit but you are still getting decent predictions. That is a possible route. Or they are just applied in different areas. I kind of suspect that intuitionistic mathematics with its insistence on potential infinity will have applications in psychology, perhaps, and areas where you want to make a distinction between something being finite and something being indefinite. In the social sciences and things like that I would expect that there could be applications. Now, let us suppose that for physics, though, everything is classical. I am not sure that there is an ultimate tension here, but maybe there is, if somehow you think that everything is physics. I am not convinced of that: Maybe there could be a pluralism about empirical investigation as well, but I am not ready to articulate and defend it. Still, that is a good question you are raising, about how these different theories which are in some sense incompatible could both find application, and how could we square that, especially if we find application in related areas.
It is interesting, having all these logics and developing those rigorously and having different mathematical projects to which you could try to marry them. For example, people have attempted paraconsistent set theories. And you might prove some interesting stuff. You could imagine that you had a paraconsistent set theory where you proved the negation of the continuum hypothesis. To some mathematicians, that is not really answering the question.
Because they are not interested in that.
And that is the end of the story for the pluralist, to say “It is just because it is not interesting to them.” It is not because it is false or not the “real” truth about, say, the continuum hypothesis?
Now, to what extent are you changing the subject? Actually, that is a theme of the book; that the change in subject is related to changes in meaning. There are times where the meaning just seems to be so different. You do not need to go with the fancy paraconsistent stuff, you get that with intuitionism too. But especially with the paraconsistent stuff, like with Zack Weber’s work where he is developing inconsistent math. He wants to show he gets all the axioms of ZFC. So in one of the cases — it is one that always sticks in my mind — he wants show that you get infinity, and what he gives is a singleton function. If you look at the Dedekind definition of infinity it means you have a one-to-one function which is not onto. So you take the set of all ordinals, Omega, which is inconsistent, so Omega, being an ordinal, is in it and is not in it, then you have a function from it to itself which is one-to-one, yet it is not onto because there is something that is not in its range. That is, there is an injection of Omega into a proper part of itself. That struck me as “No, no, you just changed the subject.” But it is hard to put your finger on that. There is something like “No, that is just distorted so much that I cannot see that as the same subject,” and that could happen with the continuum hypothesis too. Someone might say, “No, that’s not the continuum hypothesis I care about,” but I do not know if you can give rules for whether it is the same subject or a different one.
At one point, Weber even complained about a version of the axiom of choice that Routley put forward in a paraconsistent setting: “Yes, it is classically equivalent, that is, it is a statement which in classical logic is equivalent with the axiom of choice but it does not look or feel at all ‘choichy’.” So, if he is allowed to complain about that, then why does this get to count? But these are interesting interpretative questions, right, whether it is the same or different subject. I suspect that it will not be an all or nothing question, whether it is the same subject or not.