April 4, 2026
The following is the second part of an edited transcript of a debate between the authors held at the Oxford Socratic Society on March 3, 2026 (see here for Part I). A PDF of the whole debate is available here.
Audience Questions
Student 1: Just a question for Kit Fine. Is there a normal form in general for ‘if it were that P, then if it were that not P, then it would be that Q’?
Kit Fine (KF): Well, you just conjoin. You’ve got ‘if P were to be the case then if Q were to be the case then R would be the case’. Now we take that to be equivalent to ‘if P and Q were to be the case, then R would be the case’. So, in the case when ‘Q’ is ‘not P’ – this is the case that Tim was discussing – then of course you would end up with ‘if P and not P were to be the case, then R would be the case’, which is actually, I think, a way of reading the English. But, as Tim pointed out, there’s another way of reading it. So, my way of doing things is not getting at the other reading. But as I said, my interest wasn’t in natural language. My interest was in the kind of things we’re trying to say when we use counterfactuals but in a regimented way, which might depart from some of the ways in which we use natural language – just as classical first-order logic and its treatment of the quantifiers and the connectives departs from natural language, I would say.
Student 2: I was wondering if you could return to the point of the debate earlier about the difficulty of extending the truthmaker framework to negation and universal claims, and I think Tim brought up a quite interesting point that we didn’t really get into very much.
KF: Yeah, well let me make two points. One is, I think of truthmaker semantics as a semantics, not as a metaphysics. It may be that Russell was interested in metaphysics, in trying to understand ultimately how things work. So I’m perfectly happy with the thought that there’s a so-called state space which tells you what the facts are, and I’m perfectly happy with the idea that that is to some extent a contextual matter – what it is we regard as the truthmakers is to some extent a contextual matter. It depends on the discourse: if you’re an Eskimo, what will make it true that something is white? Would it be this white or that white? The rest of us, we’re just happy with white. You might also just be happy with – you ask, what makes it true it’s not raining? – you may just be happy with: it’s not raining, that negative fact. You think, oh my God, negative fact in the world? When I’m doing semantics, I’m not necessarily interested in the world. So, that would be what I could say.
Now, when it comes to the negative existential ‘there are no dragons’ – there’s a question that’s actually how you regiment this – but one thing you could say is that you go through everything in the universe, and, in each case, there’s the fact that it’s not a dragon, and then there’s also the fact that these are all the things that there are. So it’s that big thing: ‘this is not a dragon, that’s not a dragon’, blah blah blah blah. Or maybe, if it’s not being a dragon, it’s a matter of some positive fact, but anyway, just stick with this, the thing not being a dragon. So the truthmaker would be: this is not a dragon, that’s not a dragon, and these various things are all the things that there are. That would be the truthmaker, at least on one standard approach. I actually have a highly unorthodox approach, where I have these generic truthmakers. They’re not in the world, but I think they do the job of explaining what it is that’s making these things true.
I should mention here that there’s a lack of fit between what I was saying earlier, about having an intensional conception of reality. As I said, for the purposes of doing semantics, I’m not even sure that the semantics should directly relate the languages to the world. But anyway, that’s a separate issue.
TW: Could I just make a couple of comments about that? One of them is just that I think Russell simply was not very clear about when he was doing semantics and when he was doing metaphysics. As we would now put it, he was pretty wobbly on the use-mention distinction, and I think people used the word ‘logic’ for something that was sort of semantics and sort of metaphysics.
But on the specific thing about going through some inventory of the contents of the universe and saying ‘that’s not a dragon’, ‘that’s not a dragon’, and then finally saying ‘and there’s nothing else’: I think Russell did actually consider a view like that, and what worried him was that the ‘and nothing else’ was just another negative existential of the very same kind that you’re dealing with. It’s saying that it’s not the case that there is something which isn’t on this list. So there’s a kind of circularity in giving a truthmaker of that kind, because it doesn’t really seem that the negative fact – that there is not something which is not on the list – should be anything special; it’s just one more negative existential. I think that was one of the alternatives that he considered and rejected for circularity – and again, he wasn’t really able to find anything that would satisfy him.
But I think the difficulty is that what gives a nice initial motivation to the truthmaker idea is that we have these kinds of non-trivial example, as with disjunctions and conjunctions and positive existentials. Because if they’d just gone around saying: look, we need to talk about what makes statements true, and the proposition that grass is green is true because of the fact that grass is green, the proposition that there are more than twenty people in the room is true because of the fact that there are more than twenty people in the room, and it just went on like that; I think the audience would be wondering, what’s the point of all of this? We’re not really getting any bang for our buck. So it’s the initial non-trivial examples that give us a sense that we’re getting some real action with this theory.
But if it turns out that the non-trivial examples are just a sort of special case, then that undermines the initial promise of the theory, because what we were hoping for was that it could deliver by producing similar kinds of non-trivial truthmakers for these other statements that were going down towards the simpler and simpler facts or truthmakers, whatever you call them. If we’re not getting that, then it makes it look as though the original examples were just a false promise.
KF: An ad hominem argument just occurred to me, which is this. Consider ‘everything is not a man or mortal’. You might ask: what makes that true? Each thing, either not being a man, or its being mortal – or perhaps a man and a mortal – and these things being all the things there are. That last thing was the thing that was worrying us.
But if we’re necessitists, we don’t need to say these things are all the things there are, because this is something that holds as a matter of necessity. So we don’t need this further fact if we’re necessitists. Under truthmaker semantics for a constant domain, what makes a universal statement true is the same as what makes the conjunction of all of its instances true. So the semantics for a universal quantification is no more problematic than the semantics for conjunction.
TW: I think it’s easier for an intensionalist like me to say that than for a hyperintensionalist. I’m speaking on the basis of very faint memories of his lectures on logical atomism, but I think Russell wasn’t thinking ‘oh, it could have been a different set of things’. I think he was looking for some kind of perspicuous explanation of why this statement was true. Just as, if you don’t say anything at all about all these things being an exhaustive inventory of the contents of the universe, then there’s something lacking in your explanation, because there’s a gap in it. Although a hard-line, coarse-grained intensionalist like me isn’t going to be moved by that, I would expect a hyperintensionalist who wants the relations to be a bit more perspicuous than that to feel that there was something lacking in this kind of truthmaking – because something that was super-relevant to the truthmaking just hadn’t been put on the table.
KF: Well, you’ve answered my ad hominem argument with another ad hominem argument.
TW: Yes.
KF: But what I can say is there is a version of truthmaker semantics on which this isn’t going to be a problem, namely that we don’t insist upon this explanatory demand: as long as the truthmaker necessitates what is made true, that should be enough, and this further explanatory demand doesn’t have to be met. So there would be a version of truthmaker semantics where this isn’t a problem.
TW: Yeah, but it doesn’t seem very much in the spirit of the way that you’ve developed it, because in developing truthmaker semantics you’ve insisted that this is a non-modal theory, and that these sorts of thing that are going on in the truthmaker framework are just not to do with modality. So to appeal to modality in terms of the necessitation at that point – of course you’re better placed than I am to say what’s in the spirit of your view – doesn’t seem very much in the spirit of your view as you’ve expounded it.
KF: I’m trying to get you to meet me halfway!
TW: A politician – it might have been Aneurin Bevan – said: we all know what happens to the man who takes a position in the middle of the road – he gets run over.
Student 3: You talked about this point towards the end of the debate about the sort of Kripke of hyperintensional logics, and I’m wondering what sorts of characteristics would the work of such a figure, hypothetically speaking, be expected to have, or what sort of goals would it have to meet in order for it to sort of achieve something like what Kripke achieved?
TW: Maybe this future Kripke is somewhere in the audience.
KF: Kripke gave something like a canonical formulation; as Tim said, people had already moved in that direction without quite getting there. It just seemed right, the framework he developed. But it’s hard to say in the abstract what would be required.
Also, I think it’s not just a question of it being a nice theory, but seeing that there’s this range of applications as well. So there’s a kind of systematicity and unity; but I don’t know what exactly that would mean.
TW: One of the nice features of Kripke’s intensional framework is that it has had lots and lots of applications outside philosophy: in mathematics, in computer science, in linguistics, in theoretical economics through epistemic logic and so on. So maybe you could say something about what you think are the most promising applications of the hyperintensional framework outside philosophy.
KF: Well, actually, there have been a significant number of applications. People are using that formalism to develop various AI products. I’m afraid this is not something I really know much about, but people in AI have found it useful. Just today, I came across an article, I can’t remember its title, but it had to do with the safety of AI procedures. Indeed, for me, one of the principal motivations behind truthmaker semantics was something called the frame problem in AI, but I don’t know whether anyone’s pursued that particular direction. Also, a fair number of people are now applying truthmaker semantics within linguistics. In fact, one of my colleagues in the linguistics department was an advocate of the possible world semantics and has now come over to the other side, the dark side. So it is catching on within linguistics.
I myself would like to think that there should be significant applications to confirmation theory and that sort of thing, and I know a number of people who have worked in that direction, but I can’t speak for how good the work is. There certainly has been a gathering of momentum in the applications that have been made.
TW: One thing to say about confirmation theory is that most of the work in it is done within a probabilistic framework. I’m not sure if it’s obvious, but it is in fact the case that the state spaces that probability theory is based on, mathematically, are pretty much exactly the same as Kripke’s possible worlds framework. The terminology is all different: for a start, in possible world semantics, the set of worlds is an uppercase ‘W’, and in the probability case you have to look for an uppercase ‘Ω’; and then the probability theorists talk about outcomes rather than worlds, but basically the formal properties are exactly the same. In possible world semantics, sets of worlds are called propositions, whereas in probability theory they’re called events; but again, it’s basically formally exactly the same.
In a way it’s not a surprise because Kripke was partly inspired to his possible world semantics – he told me – by looking at the kinds of state space you get in natural science and physics, for example – sometimes called phase spaces, sometimes called dynamical systems, depending on the extra structure that they have. That’s a kind of very standard approach in the natural sciences; it’s also the approach that probability theory, as a mathematically well-developed theory, is based on. So Kripke was actually just applying the same framework to modal logic. It’s not some kind of weirdo science-fictionish thing about possible worlds as some kind of strange alternative realities or anything like that; it’s in fact an application of a very well-developed approach that you get in lots of the natural sciences – and, as I said, it’s what is going on in probability theory.
The kind of work that I’ve seen on confirmation theory has mainly been within a probabilistic framework, and so it’s just, as it were, more intensional activity. There may be room for something else, but at least the default in that area, I think, is still an intensional approach.
Student 4: This question is sort of in the spirit of Professor Williamson’s last question: what do you both think are the very most important issues, theoretical and practical, that are likely to turn on this dispute?
KF: There are lots of small issues, so it’s hard to say. Take something like the atomic theory of gases. It took some time for that to be accepted, and it was an incremental process. It just turned out other views didn’t work; the view did work and it was a beautiful theory.
It’s somewhat similar. How is the issue to be resolved? I think the applications are very significant in this regard: to the extent that one can show that this theory actually is good for something, any reasons one might have to doubt it would seem like being philosophic reasons in a bad sense of the term. If it can really do some important work – I mean through the applications, it could be linguistic, or what have you – that would be the best evidence it seems to me that we can have that there was something worthwhile there.
TW: I agree with that. One thing I would say is that what you’re looking for in a framework – which is what intensionalism is for a lot of work – is something settled and robust, so that you don’t have to keep making it up as you go along; you just have this well-defined framework which is understood, and then you can apply it. You’re just using it, you’re not tweaking it; so it has its own autonomy. That’s what you want from a framework: to take it out of the dispute, so it’s just something there that we can use without having to reinvent it every time we use it, or to make extensions of it – we’ve got something that is general enough that we’re genuinely applying it rather than inventing some new theory which is kind of similar to something else. One advantage of the intensional framework has been that you can quite easily use it in that spirit. Of course, it’s been generalized in various ways, but it is basically pretty stable and robust.
Student 5: Back to the question of what it is for a sentence to be true, such as the sentence ‘grass is green’. I’m just curious what to make of the view that for a sentence to be true is for the predicate to apply to the subject, such as the subject grass, and the subject is among the extension of the predicate.
TW: You seem to be assuming that every sentence is of subject–predicate form, but that isn’t the case. For example, if you take a disjunction, like ‘Either Mary is guilty or John is guilty’, that doesn’t have an overall subject-predicate form. One of the ways in which semantics has had to get much more sophisticated in the 20th – and 21st – century is simply that we’re much more aware of the fact that we have to deal with a huge range of sentences, which differ from each other a lot in their structure, so that you can’t assume in advance that what you’re dealing with is a sentence of some particular structure.
It took quite a long time for this to be fully accepted in logic. For example, John Stuart Mill didn’t like the idea that you can have conjunctive propositions of the form ‘A and B’ – he said ‘such a sentence is no more a conjunctive proposition than a street is a conjunctive house’. He just hadn’t got the kind of generality of the category of proposition-expressing sentences. A lot of the reason why the way people talk about truth now is somewhat different from the way they used to is just appreciation of the fact that we really need a general theory that will work for declarative sentences of all kinds of structure – that we just shouldn’t take it for granted in advance that we know in advance what the structure of a given sentence will be.
Student 6: May I take this debate to the arena of normativity? How would either framework apply? From what I understand, you’ve written recently on intensionalism as applied to problems in normativity, moral realism, etc., and one of the implications of the intensional framework is that you end up with some kind of naturalism. For any necessitated normative thesis, there will be a co-extensional, non-normative…
TW: You mean, co-intensional.
Student 6: Co-intensional, sorry – non-normative fact. You wouldn’t want to fill in the details as to how we should carve normative space by giving it some kind of Canberra Plan analysis of concepts.
TW: God, no! I don’t want to have anything whatsoever to do with the Canberra Plan.
Student 6: Right. But neither would you want to, if you’re a naturalist, in your case…
TW: I don’t think you’ve got the position quite right. It’s certainly the case that I think there are examples where a predicate formulated using normative expressions expresses the same property as a predicate that doesn’t contain normative expressions.
For example, let’s assume that torturing babies for fun is necessarily wrong, it’s just impossible for it not to be wrong. Then that means that ‘torturing babies for fun’ and ‘wrongly torturing babies for fun’ pick out the very same property, but one of them picks it out using a normative expression and the other one doesn’t.
But the conclusion I’m drawing from that is not that the underlying property is natural, is non-normative, or something like that; it’s simply that a distinction that can be made at the level of linguistic expressions – whether they contain normative material – just doesn’t project onto a distinction between the things that they express. It’s not part of my agenda that somehow the natural or the non-normative vocabulary is more basic than the normative vocabulary or anything like that; it’s just that the idea is that this normative–non-normative distinction, which we can make at the level of linguistic expressions, shouldn’t be applied to what the expressions are picking out in the world because these differences just don’t project down. It’s not that I have some kind of agenda where I think that let’s say, the language of fundamental physics is somehow more basic than anything else. It’s more simply a matter of: be careful about whether distinctions are being made at the linguistic level or at the metaphysical level.
Student 6: Right, but in that case then my question is: if the distinction between the normative and the non-normative depends only on the level of our representations – so there are no irreducibly normative facts or some moral…
TW: I’m not working with facts anyway, but you may use true propositions.
Speaker 6: But how would you carve normativity at the level of language? How would you know which are the true normative predicates, if not by doing some kind of conceptual analysis?
TW: Definitely not by doing conceptual analysis! I think we have some capacity to recognize which linguistic expressions have some kind of evaluative valence to them, like ‘right’ and ‘wrong’, ‘good’ and ‘bad’, and so on, and deontic modal expressions like ‘permitted’ versus ‘forbidden’ and so on; obviously these are expressions which have certain kinds of connection with action, assessments of action, and so on.
There will be some story along those lines, but I don’t think that these expressions can all be somehow analysed and then there will be some particular component in them which is the evaluative one. I don’t think those kinds of program of analysis are likely to work out. In that sense, we’re not using conceptual analysis, we’re just using our ordinary understanding of these expressions and some ability to recognize expressions which have a valence versus ones that don’t. Then, of course, the specifically moral ones are a subcategory of that, because lots of forms of evaluation are not moral evaluation.
KF: If I could just interject my own view on this question? I think it is necessary that torturing babies for fun is wrong. But for me, this is what I call normative necessity, not metaphysical necessity. Intensionalism is always relative to a certain modality; normally people take this to be the metaphysical modality. I want to deny that it’s a metaphysical necessity that torturing babies for fun is wrong, even though there’s another notion of necessity on which it is necessary. So even if I were an intensionalist with respect to metaphysical necessity, I wouldn’t go the way of Tim. But that’s just my opinion about the kind of necessity that’s involved in these statements.
Student 7: This is perhaps broadening things out a bit, but I’m interested in where you both see the heart of your disagreement lying, because I feel like there’s been a number of different areas which have been discussed in which kind of your theories – you see them as being more useful individually in that area. Do you think your disagreement lies in just the fact that you find your theories more useful in specific areas, or do you think it’s more fundamental and can be pinpointed? And do you agree on where the disagreement is, or not?
TW: I think it is in some ways a methodological disagreement, which has to do with the kind of comparative weight that we give to different sorts of criteria in evaluating theories. I’m not sure Kit would agree with this characterization, but I tend to be suspicious of complicating theories. I’m very wary, for example, of just abandoning some simple theory that seems to be working pretty well just because we come across something which feels like a counterexample to it.
I’m as keen on counterexamples as anybody; I have been called a counterexample machine, so I can’t be too hostile to them, but I think when we’re assessing them, we’re using what psychologists would call heuristics, that are quick, easy ways of assessing things, but are not 100% reliable. So when we have a conflict between a simple, so-far well-working theory and a counterexample, I think we should be pretty suspicious of the counterexample. I’m putting a lot of weight on criteria like simplicity and elegance in what’s sometimes called the abductive comparison of theories, whereas I think Kit is less averse to complicating things and is more inclined than I am just to take what feel initially like compelling counterexamples at face value and think that these really just are counterexamples.
But if there is such a difference between us, that’s just at the methodological level, and then of course there are more specific disagreements between us, e.g., on intensional versus hyperintensional frameworks, which I would be inclined to explain by this. I don’t think it’s a huge methodological difference between us; my feeling is that we’re actually not that far apart on the spectrum of philosophers, where our views and approaches are really not radically different; it’s just that when we zero in on something specific – like using an intensional framework versus a hyperintensional framework – then we have some specific disagreements.
KF: I think you’re right about one thing about my own view, wrong about another. I hate complications; I’m much in favor of simple theories as you are; and I think simplicity is a guide to the truth. But you’re right in thinking that I’m much less suspicious of intuition than you are. The way I see it, the question is how best to make progress in philosophy.
One way is to pursue a theory and to give it your all, but another is to expose yourself to the complexity of the data and so on and so forth. In the case of science, the first method worked very well. Newton came up with a beautiful, simple theory; it was just amazing. If he’d been bothered about all the complexities and so on and so forth, he wouldn’t have got where he did.
That method, there’s a certain sense in which it’s not a priori that this was going to work – maybe in some sense it is a priori, but not in any practically meaningful sense. One wasn’t to know that something like Newton’s theory would be so successful, even though it ignored all the complications. Now, there’s a real question as to whether we should, as it were, have the same attitude towards theory that scientists seem to have, and my own inclination is that right now, in the current state of philosophy, this is not the way progress is to be achieved.
My view is very oriented towards very particular intuitions that we have. I’m going to have to exercise judgment about them: is there something here that might be helpful, that might give us a better understanding of something? So I’m much more inclined, as it were, to work upwards from certain intuitions that I feel have something going for them, rather than work top-down from a theory. I hate it when these people have these theories – everything’s physical and so on – and then they see everything in that light. It’s like utilitarianism: progress in moral philosophy is to be gained not by just assuming a particular very general theory is correct; you need to dig deep into the details of very particular things. I guess that’s my attitude right now about metaphysics, and really philosophy in general. We’re not ready for big theories. I’m suspicious of anyone who has big theories.
TW: We’ve got a lot of very good theories just in logic, which are extremely general.
KF: Oh, no, I’m talking about – yeah.
TW: Yeah, but I’m taking it that the philosophical end of logic is part of philosophy. But I don’t see such a big difference here between philosophy and natural science. There are some quite illuminating things that one can get from Thomas Kuhn’s Structure of Scientific Revolutions – which of course became fantastically successful in the ‘60s because it was, as it were, the hippies’ philosophy of science, and people thought that what it was saying was that science is just as irrational as everything else, or something like that. But I think it’s in fact a much more sophisticated picture, and part of Kuhn’s point is that every scientific paradigm at any point has anomalies. It’s pretty obviously true of philosophy, as well, that every theoretical paradigm in philosophy has its own anomalies and people keep working on it despite that.
I think part of Kuhn’s model is that what in a sense may be irrationality at the individual level can add up to rationality at the group level. What he describes is a situation in which it’s typical of scientists to be more committed to their paradigms than is fully warranted by the evidence. He famously mentions Max Planck supposedly saying that progress in science is measured by funerals, which means basically: people, once they’re really invested in a theory, never give up on it, but they do eventually die; where the progress comes from is that the students – who are not so invested already in any one paradigm – make somewhat better judgments about whose research program is more progressive and follow them. That’s where the dynamics come from. But Kuhn’s point is that having people at an individual level irrationally committed to a given paradigm or research program is a very good way of making sure that its explanatory potential is fully explored.
That holds just as much in philosophy. So it’s very important in philosophy that we have both people like me who are fanatically committed to intensionalism, and people like Kit who are fanatically committed to hyperintensionalism, because that’s how the potential of these different research programs gets worked out. Each of us is doing our damnedest for the theory that we like, and then in the long run that will enable people to make informed decisions about what the capacity and limitations of these different paradigms are. So it’s good that we’re disagreeing, and that we’re both willing to go all-out on our conflicting research programs, because, at the level of the whole philosophical tradition, that is how progress gets made.
KF: I wonder if I could just make a further point about how it’s often unclear what is the best method to use. I was in Edinburgh in the early 1970s; I worked a lot with people in cognitive science, and they were very concerned with recognition of speech, speech production, and so on. They tried to work this out on first principles – just really trying to analyse how we speak, what patterns of sounds make sense as productions of language. It was a huge failure! You might have thought a priori: surely this is the way to go! We speak; my words now sound like sentences of English; they flow the way they should. You might think: surely there are very general principles governing how this is so. This approach turned out not to work; and of course now, with pattern recognition techniques where we don’t even understand what the underlying principles are, we’re able to do these tasks. As I said, this is an astonishing fact! One might have thought a priori that we need to look for first principles. So, I think we need to have a properly humble attitude as to what the correct methodology is.
Philosophy is strewn with attempts to say: this is how we should be pursuing the subject. Look at Descartes, or Kant, and so on. None of us really know how best to pursue the subject. It’s very much an open question, just as it was for speech in Edinburgh – or should have been.
Final Words
KF: One of the things I’ve been trying to say is that, look, what I’ve been calling truthmaking semantics isn’t just a semantics of natural language; it’s also a semantics of formal language. Let me just give one example where I think it has been illuminating. I don’t know how many of you know about intuitionistic logic. Intuitionists don’t accept the law of excluded middle – P or not P. There have been two main semantics for intuitionistic logic: one in terms of constructions or proofs; and another, by Kripke actually, in terms of conditions under which a statement can be true – think of those conditions as bits of partial information. However, it has been hard to see how these two approaches are related.
There’s something I developed, which is a truthmaker semantics for intuitionistic logic, which bridges the gap between the two. It actually does a reasonably good job of explaining how you can go from one to the other. So, I think there’s something genuinely illuminating about this; it’s creating a kind of synthesis, if you like, between two approaches that otherwise do not appear to relate to one another. And it seems to me that the fact that it can do that is symptomatic of the fact that there’s something worthwhile here; there’s some useful techniques here that we can perhaps apply to a wide variety of different fields.
TW: Studying intuitionistic logic can be genuinely illuminating, but what it’s illuminating is an approach to mathematics that is inspired by fundamentally, utterly mistaken views about knowledge and about the nature of mathematics. I’m not saying that your truthmaking vision is implicated in those, but what we’re doing is exploring a way of thinking that is very interesting, but also philosophically mad.
KF: I accept that the initial motivation for intuitionistic logic was not a good motivation. Still, there’s a lot to be said for intuitionistic logic; it can do a lot for us. What we can think of it as is a formalization of the notion of constructive proof. I might be able to show that a contradiction can be deduced from the fact that everything Fs. So classically I know that there’s something that’s not an F, but that’s not to provide an example of something that’s not an F. It’s not a constructive proof because it’s not telling me what the counterexample is. One thing that intuitionistic logic does is it forces you to provide a constructive proof; and that’s something you can accept about the logic without necessarily accepting the kind of epistemological motivation that lies behind it.
TW: Here’s one way in which the difference between the different attitudes to intuitionistic logic comes out. Of course it’s clear that there is a difference between constructive and non-constructive proof, and that it merits exploration, and intuitionistic logic can play a role in that – but what really sorts out who’s on which side is here: if you think that intuitionistic logic is basically a good tool to use in understanding the difference between constructive and non-constructive proof, then you should be happy to use classical logic and non-constructive proof in the metalanguage, because you’re not taking the view that there’s anything wrong with non-constructive proof, just that constructive proof has special characteristics of its own.
Some people investigate intuitionistic logic using a classical metalogic because they’re not buying into the original intuitionistic ideology. But hardline intuitionists insist on using intuitionistic logic in the metalanguage as well as in the object language. As a result of its restrictions – it’s a funny situation – there are nice properties of intuitionistic logic which can be proved if you use classical logic in the metalanguage, but which can’t be proved if you use the intuitionistic logic in the metalanguage so that intuitionism in some respects has virtues which can only be recognized by its enemies.
See here for Part I and here for the whole transcript.
Kit Fine is a University Professor and Silver Professor of Philosophy and Mathematics at New York University. He is the author of books including Reasoning with Arbitrary Objects, The Limits of Abstraction, Modality and Tense: Philosophical Papers, Semantic Relationism, and Vagueness: A Global Theory. Timothy Williamson is Emeritus Wykeham Professor of Logic at the University of Oxford and A. Whitney Griswold Visiting Professor at Yale University. He is the author of books including Vagueness, Knowledge and its Limits, Modal Logic as Metaphysics, Suppose and Tell, and Overfitting and Heuristics in Philosophy.
