Is Logic Normative?

Last year I read a paper by Gillian Russell with the self-explanatory title "Logic Isn't Normative" (Russell 2020). I already had some views about the normativity of logic, and found the paper a bit of a challenge for them, because the picture Russell presents of the relationship between logic and normativity is quite similar to how I see things, even though I think logic is normative and she thinks it isn’t. It seems to me that although our positions are in some ways quite close, there are things we disagree on that would prevent me from framing my position her way, as a view according to which logic isn't normative. I thought it'd be instructive, at least for me, to have a think about why.

Russell’s view, as I understand it, is roughly as follows. The subject matter of logic itself is descriptive.¹ Logic studies which arguments are and aren’t truth-preserving, and this isn’t in itself a normative matter. If we’re taking the constituents of arguments to be sentences, then whether arguments are truth-preserving or not will depend on what sentences mean, the conditions under which they would and wouldn’t be true given what they mean, and which conditions are and aren’t jointly possible. (We'll look at the idea of truth preservation over things other than jointly possible conditions later.) This is all plausibly non-normative (although see note 1). If we take the constituents of arguments to be something else, say propositions, then the story’s a little different but it’s all still descriptive. (There’s some discussion about what the constituents of arguments are in Russell (2008).) These descriptive facts can then be combined with normative bridge principles from outside logic like you should only believe things that are true and you should only reject things that are not true, resulting in norms like if X ⊨ C then you shouldn’t believe all of X and reject C. The fact that you get the norms out at the end doesn’t make logic normative, because you could combine those bridge principles with any claim P and get the norm “you shouldn’t reject P”. So logic combines with the extralogical bridge principles to give you norms, but that doesn’t make it normative, since the normativity all comes from the bridge principles and not from the claims about truth-preservation, and only the latter are the subject matter of logic.

Now, even though I think logic is normative, this view isn’t so different from mine. (Indeed I expect that to some extent her views expressed elsewhere have probably influenced mine on this issue.) I think that there are the non-normative laws that Russell identifies as logical laws, and normative bridge principles that combine with them to give general principles about the relationship between logical consequence relations and how we should think. The difference between my view and Russell’s is that I think the resulting principles, which are normative, are logical laws. To an extent this may seem to be just bookkeeping, but I think it matters. Russell’s view suggests that when people disagree over what the right logic is, either they’re disagreeing over something descriptive, or their disagreement can at least be traced to a disagreement over something descriptive. The bridge principles are seen as either obvious or at least uncontested, so the descriptive stuff is where the action is. I don’t think that’s right; I think there’s plenty of action on the normative side, and that when people are arguing over what the right logic is, their disagreement is often irreducibly normative. First I’ll talk about three objections to Russell’s position that she discusses and one she mentions briefly; then I’ll talk about two potential cases of logical disagreement that I think strengthen the objection she calls the argument from demarcation, and finally I’ll talk a bit about what I think people are arguing over when they argue over what the right logic is.

Three Arguments For The Normativity Of Logic

Russell considers three arguments for the normativity of logic, and argues that they don’t establish anything inconsistent with the picture she’s putting forward, according to which logic itself is descriptive and all the normativity comes from extralogical bridge principles. She calls the three arguments the argument from normative consequences, the argument from error, and the argument from demarcation.

The argument from normative consequences says that logic must be normative because it has normative consequences and you can’t derive ought from is. The Humean premise that you can’t derive ought from is can of course be questioned, but even if we grant it there’s nothing here to undermine Russell’s position. What we think about logic does affect how we ought to think, but maybe that’s only because of the bridge principles. There’s no principle making it impossible to derive normative conclusions from a combination of descriptive and normative premises, and that’s what Russell thinks is going on.

The argument from error says that the laws of logic must be normative laws and not descriptive laws because people violate them. The laws of physics are descriptive, and we can’t break them. But people do believe inconsistent things and make fallacious inferences, and isn’t that breaking the laws of logic? Not according to Russell. On her view, breaking a logical law would be doing something like wearing a red hat without wearing a hat. (She doesn’t use this example.) You can’t break that kind of law, which indicates that they are descriptive, not normative. It seems to me that the question of whether someone with inconsistent beliefs is violating a logical law or not is very similar to the question of whether norms like you shouldn’t have inconsistent beliefs are logical laws or not, and so I agree with Russell that the argument from error isn’t going to get her opponents anywhere. At best it begs the question.

The argument from demarcation is the one that I think is most interesting and poses the biggest threat to Russell’s position. She quotes John MacFarlane:

Logic is often said to provide norms for thought or reasoning. Indeed, this idea is central to the way in which logic has been demarcated as a discipline, and without it, it is hard to see how we would distinguish logic from the disciplines that crowd it on all sides: psychology, metaphysics, mathematics, and semantics. (MacFarlane 2004:1)

Now, as I understand it Russell’s view about where logical laws come from takes them more or less to be reducible without remainder to semantics and metaphysics plus the mathematics involved. (There may be some psychology in there too, perhaps in the semantics.) This means you can see logic as an interdisciplinary thing rather than an autonomous discipline, and it doesn’t need normativity to demarcate a sharp boundary because it doesn’t have one.

For all that’s there in the MacFarlane quote (which is of course not all that’s there in his paper) we could leave it at that, but she rightly doesn’t leave it at that. There’s another worry, which is basically this: when we argue over what the right logic is, what are we arguing about? I suppose my view is that this is the central question in the philosophy of logic. Defenders of classical and intuitionistic logic don’t disagree over which conclusions follow from which sets of premises in classical logic or in intuitionistic logic; those are basically pure mathematical questions and when it comes to investigating them we’re all on the same side. The defenders of different logics disagree over which conclusions follow from which sets of premises full stop.

Russell’s response to this is that while it’s true that the defenders of different logics are not disagreeing about pure maths, that doesn’t mean what they are disagreeing about isn’t purely descriptive. The semantics/metaphysics mashup that is the study of truth-preservation for arguments is descriptive, and that’s what Russell thinks the defenders of different logics are arguing over. Do arguments from (P→Q)→P to P always preserve truth? Classical logicians say yes; intuitionistic logicians say no. According to Russell that’s their disagreement, and descriptive semantics and metaphysics can settle the question. And since this schema is like a version of Peirce’s law and adding it to intuitionistic logic gives you classical logic, settling it would settle all their other disagreements too. (We'll talk more about intuitionistic logic and whether it's really about truth-preservation later.)

A Fourth Way To Argue For The Normativity Of Logic

All three of these arguments have a flavour of indirectness to them; they don’t tell us why or how logic is normative, but just argue that somehow it must be, because otherwise it couldn’t have normative consequences, or we couldn’t break logical laws, or we’d have nothing to distinguish logical questions from mathematical questions. An alternative route would be to argue for a distinctively logical normative reality. Russell (2020: note 15) touches on this possibility when discussing a reviewer’s comment that logic might be normative because truth itself is normative. She accepts the conditional but demurs on the antecedent: she doesn’t think truth is normative (although there are norms involving it, just as there are norms involving many descriptive concepts).

What might this kind of view of truth or logic as normative look like? I’ll make two suggestions. One is a pragmatic theory of truth, where what’s true just is in some sense whatever’s good to believe. Here’s a passage from William James which contains an idea along those lines, although I don’t know enough about James’s views on truth to properly contextualize it:

'What would be better for us to believe'! This sounds very like a definition of truth. It comes very near to saying 'what we ought to believe': and in that definition none of you would find any oddity. Ought we ever not to believe what it is better for us to believe? And can we then keep the notion of what is better for us, and what is true for us, permanently apart?" (James 1907, Lecture VI §16)

A second possibility is suggested by the Aristotelian line that you have to believe the law of non-contradiction if you want to say anything at all.² If his argument works (and I won't attempt to explain how it is supposed to work) then that arguably gives a reason for believing the law of non-contradiction directly, rather than with reference to its truth. Presumably this could also be contested as not entailing that logic itself is normative — so far all Aristotle’s argument does is give a reason for believing something — but both Aristotle’s argument and the pragmatist suggestion offer ways of embedding normativity deeper into the subject matter of logic than it is on Russell’s picture. But we won’t pursue these ways of way of arguing for the normativity of logic further here, and instead we’ll go back to the argument from demarcation.

Irreducibly Normative Disagreement

Consider the argument form called explosion (aka ex falso quodlibet or ex contradictione quodilibet), where you derive an arbitrary conclusion from contradictory premises. For example: Australia is big, and Australia is not big, therefore the sky is green. It’s classically (and intuitionistically) valid, and logics where it fails are called paraconsistent. There are two main objections to it that you hear from people who think the right logic is paraconsistent.

One objection is from people who are dialetheists, which means they think that contradictions can be true. Regular readers may recall that I think dialetheism is probably true. Dialetheists can object to explosion on the grounds that it doesn’t preserve truth. If contradictions can be true (without everything else also being true), then some arguments with contradictory premises can have all true premises without having a true conclusion. This fits into Russell’s model just fine: the disagreement over whether the right logic is paraconsistent stems from a disagreement over whether explosion preserves truth, which here turns on the entirely non-normative matter (bracketing the suggestion that truth is itself normative) of whether contradictions can be true.

Another objection to explosion is about relevance. Even if contradictions can’t be true, and so explosion (vacuously) preserves truth across all possible situations, maybe it still isn’t logically valid because the conclusion might have nothing to do with the premises. This is one of the main motivating thoughts behind relevant logic (also known as relevance logic). Not all fans of relevant logic are dialetheists, although some are. My own view is that the complaint about dialetheism and the complaint about relevance have basically nothing to do with each other, but even if you don’t think that, you don’t have have to be a dialetheist to want your logic to be relevant.

As I see it, the relevantist objection to explosion is a counterexample to Russell’s positive view about what the subject matter of logic is. Here we have a disagreement over what the right logic is which isn’t a disagreement about truth-preservation. This doesn’t yet establish that the disagreement is about something normative, although I think that in fact it probably is and will say more about why later. But how might Russell respond to this potential counterexample?

One possible response is just to maintain that logic is about truth-preservation, and so if the relevantists and non-relevantists agree about which arguments preserve truth then they agree about logic and their disagreement must be about something else. Of course, nothing prevents us from saying this, and indeed nothing prevents us from defining the word ‘logic’ so that what we say is true. But as I understand Russell’s position it’s more ambitious and interesting than that. It’s not meant to be just a recipe for taking a particular side in a verbal dispute. As I understand her position it’s meant to be more like a diagnosis: what all those people arguing over the right logic are arguing about boils down to a disagreement over something descriptive, specifically something about truth-preservation.

Hartry Field (2015) says something related to this that I feel I ought to mention but don’t properly understand. Here he is:

While it is correct that there are logicians for whom truth-preservation is far from the sole goal, this isn’t of great importance for my purposes. That’s because my interest is with what people who disagree in logic are disagreeing about; and if proponents of one logic want that logic to meet additional goals that proponents of another logic aren’t trying to meet, and reject inferences that the other logic accepts only because of the difference of goals, then the apparent disagreement in logic seems merely verbal. (Field 2015: 35, his emphasis)

I think that to see relevantists and non-relevantists as only having different goals isn’t quite right. It’s not just that relevantists want both truth-preservation and relevance while their opponents only want truth-preservation. At least sometimes, I think they’re both targeting the same notion of logical consequence but disagree over whether validating irrelevant arguments is a sign that something has gone wrong. If all you wanted was truth-preservation and relevance you might be happy with a kind of filter logic³, the most flatfooted of which would be classical logic minus any sequents with no variables appearing in both the premises and the conclusion. That logic would be non-transitive, since it would validate (Q&¬Q) ⊨ (Q&¬Q)∨(R&¬R) and (Q&¬Q)∨(R&¬R) ⊨ (R&¬R) but not (Q&¬Q) ⊨ (R&¬R). But relevantists usually aren’t happy with this, because they want an integrated account of logical consequence that is both truth-preserving and relevant. You can still see this as a difference of goals if you like, but even then it seems wrong to me to describe it as a mere verbal dispute. If verbal at all, it’s something along the lines of what Carrie Jenkins (2014) calls a serious verbal dispute; but even without committing to the possibility of there being such things we can note that differences of goals are things we can have serious arguments about. What confuses me even further about Field’s position is that he ends up arguing that we should understand even the arguments about logic that he’s discussing as arguments about how we should regulate our degrees of belief, and not as arguments about truth-preservation. His paper is complex and I think I’m probably missing something, but I just wanted to note that if he’s saying what he sounded to me like he was saying, then that’s a point of disagreement between us.

Supposing we don’t want to say that the disagreement between relevantists and non-relevantists is a (non-serious) verbal dispute turning on two different uses of the term “logic”, another possible response is to diagnose the relevance objection to explosion as not being about truth-preservation but still being descriptive. Maybe logic is about relevance and truth-preservation, or some kind of composite of the two, and the disagreement is over whether explosion meets that (descriptive) condition. Now this option might be available if we all saw relevance the way David Lewis did. Lewis (1988) argues that explosion is relevant after all: asserting a contradiction is equivalent to asserting everything, so a contradiction in the premises asserts whatever the conclusion is, and so it is relevant to the conclusion. The disagreement between Lewis and the relevantists could be thought of as a descriptive disagreement over whether explosion is relevant or not. The problem with this diagnosis is that it would put a lot of non-relevantists on the wrong side. Plenty of people, at least as I understand it, think that explosion is not a relevant inference but it is still part of the correct logic because the correct logic need not be relevant. They agree with the relevantists on the descriptive point but disagree about whether explosion is valid, so their disagreement must be about something else. In the next section we’ll reconsider whether the disagreement over relevance can be made out as being about truth-preservation after all, but for now it doesn’t seem possible to diagnose the dispute over explosion between relevantists and most of their opponents who aren’t David Lewis as being over the descriptive matter of whether explosion is relevant. There is general agreement that it isn’t relevant, even where there is disagreement over whether it’s valid.

Model Theory

So, what exactly is truth-preservation? The basic idea is that arguments preserve truth iff whenever the premises are all true the conclusion is true. This could be understood in terms of possibility: it’s not possible for the premises to be true and the conclusion not. That could be metaphysical possibility, or it could be logical possibility, or perhaps it could be some other kind of possibility. It seems to me that for Russell’s purposes it’d be better not to say logical possibility, because of the risk of circularity. If we explain logical possibility in terms of truth-preservation (since that’s what logic is about) and then explain what we mean by truth-preservation by appealing to logical possibility, it seems there’s a circle. It’s a bit like the problem Quine talked about in ‘Two Dogmas Of Empiricism’ (Quine 1951) of all the proposed accounts of analyticity appealing to some notion or other that’s similarly unexplained; but leaving aside what Quine says there, it seems to me there would be a circle in our case, and not in a good way. So we might be better off going with metaphysical possibility than with logical possibility. That way people who disagree over e.g. the law of non-contradiction can be disagreeing over whether all of its instances are metaphysically necessary. People who argue over that law often do disagree about whether all of its instances are metaphysically necessary, and might be happy to agree for the record that this is what they’re arguing about, at least sometimes. But since the relevance objection to explosion is sometimes being posed by people who think that contradictions are not metaphysically possible, truth-preservation over metaphysically possible worlds can’t be what they’re arguing about. The same goes if we understand truth-preservation as just relating to the actual world, in which case they’d be arguing over whether any contradictions are true. The non-dialetheist relevantists agree with the classicists that no contradictions are true, so that’s not what they’re arguing about either. Now, if you think actual-world truth-preservation is a ridiculous thing for logic to be about, you can have some fun ridiculing Quine:

...the business of formal logic is describable as that of finding statement forms which are logical, in the sense of containing no constants beyond the logical vocabulary, and (extensionally) valid, in the sense that all statements exemplifying the form in question are true. Statements exemplifying such forms may be called logically true. (Quine 1953: 436, his emphasis)

We should note that Quine isn’t saying that everything true is logically true; he’s saying that every instance of a schema that contains no non-logical constants and all of whose instances are true is logically true. The paper is a review of a book by PF Strawson (1952), and he attributes Strawson the same view but with the instances having to be analytic rather than merely true. This makes sense in the context of Strawson and Quine’s differing views over the tenability of the notion of analyticity, and Quine’s avoidance of necessity here is also in keeping with his views about that. Quine’s position does mean putting some weight on a distinction between logical and non-logical vocabulary, but it’s worth noting that fans and detractors of the law of non-contradiction can have their argument over whether all its instances are true without needing to use such a distinction. We only really need such a distinction if we’re trying to distinguish logical truths from non-logical truths.

(As an aside, I really do recommend reading Quine’s review if you’re interested in Quine and have a sort of middling knowledge of his ideas; he says what he thinks about a lot of different things in a fairly casual way, and it’s also the place where the slogan “philosophy of science is philosophy enough” appears in print.)

Getting back to our main thread, the argument between non-relevantists and non-dialetheist relevantists seems like it can’t be about either truth-preservation at the actual world or truth-preservation across all metaphysically possible worlds, because the parties to the disagreement don’t disagree about those things. What other truth-preservation-based options are there? Well, a good place to look is at the model theories for the proposed logics. With relevant logic it’s sometimes a bit tricky how exactly truth-preservation figures in the definition of logical consequence, but to keep things simple let’s look at first degree entailment, or FDE, which is the extensional base for a lot of relevant logics. Suppose the relevantist and the classicist are arguing over what the One True Logic is for sentences with no conditionals or modal operators: just sentence letters, negation, conjunction and disjunction. (The example should work equally well with or without allowing quantifiers.) They might both agree that metaphysical possibility is represented by classical models, and that classical logic describes truth-preservation across metaphysically possible worlds. Nonetheless, the relevantist thinks that the (extensional fragment of the) One True Logic corresponds to truth-preservation across FDE models, in which sentences can be true, false, both, or neither, and the classicist thinks it corresponds to the classical models. Classical logic validates explosion and FDE doesn’t.

To describe their disagreement as being about truth-preservation we want some neutral description of the class of things they think the One True Logic corresponds to truth-preservation over, and then they can disagree over whether that class of things corresponds to the class of FDE models or the class of classical models. What might that class of things be? It can’t be metaphysically possible worlds, as we’ve seen. I think the most promising option is probably something along the lines of information states. (Sometimes people actually do talk about FDE as modelling information states; I didn’t just make it up.) It’s characteristic of information states that they can be incomplete, and on the FDE picture they can also be inconsistent, even if the world itself can’t.⁴

For these purposes, an information state isn’t supposed to just be any old set of sentences. The classicist obviously thinks that’s too broad, but so does the relevantist. If an FDE model says P is true then it also has to say P∨Q is true, for example. Now, it’s plausible enough that if you already have the information that P is true, then in a sense it wouldn’t be new information to learn that P∨Q is also true. In a sense it might be new information, since you might have the information that P but not have inferred that P∨Q. But it still seems there’s a sense in which it wouldn’t be new information, and that’s roughly the sense that delimits the notion of an information state that the classicist and relevantist are arguing over, if they're arguing over information states at all.

If we want to use this as part of an argument that logic isn’t normative, we need to think a bit more about what this sense of an information state is, because the sense in question might itself be normative. Basically, we might be dealing with a notion of an information state that’s good in some way. Maybe it’s good in that it contains all the information that might be rationally permissible for someone in a given situation to use, or just in that it’s closed under rationally permissible inference, where this is understood independently of the model theory, perhaps in terms of proof rules. (The information states may also need to be prime; although see note 3 above.) But these notions involving rational permissibility are normative. If we want a descriptive notion, the most promising place to look seems to me to be psychology. Maybe people have some operations that they perform to round out their information states, and logic is about truth-preservation over information states closed under those operations. That seems to make sense, and it seems plausibly non-normative, but it’s psychologism, and psychologism isn’t in fashion. Some people will tell you Frege refuted it. I wouldn’t go that far and expect it’ll get rehabilitated sooner or later; I read a New York Times review by James Ryerson (2018) that made it sound like Irad Kimhi (2018) has been trying to rehabilitate psychologism in some form, although possibly quite a different form from the one I gestured at a moment ago. (I haven’t read Kimhi’s book myself and get the impression I might not fully understand what he was up to in it if I did, if only because I’m pretty clueless about Hegel.) Nonetheless, my understanding is that psychologism isn’t back in fashion yet and I think that psychologism is where we’ve ended up if we want to diagnose the relevance objection to explosion as being both descriptive and about truth-preservation.

Old Fashioned Intuitionism

In her discussion of the demarcation argument, Russell says something about intuitionistic logic that suggests she might think there actually are at least some disputes about what the right logic is that aren’t descriptive disputes about truth-preservation. I’ll quote the whole paragraph so that if I missed some important context you don’t end up missing it too:

But there is also a stronger, more worrying, version of the demarcation argument. It asks what the results of logical inquiry are actually telling us. What do E-sentences like ¬¬P ⊨ P mean? What makes them true or false? The answer, as it often does, seems clear at first: it means P is a logical consequence of ¬¬P. There are no interpretations on which P is true and ¬¬P is not [sic; I think this should be "no interpretations on which ¬¬P is true and P is not" - MBC]. But now we find that intuitionists and classical logicians disagree about whether this is so. The classical logician says that ¬¬P ⊨ P is true. Asked to support this claim she may offer a short model-theoretic argument: on any interpretation on which ¬¬P is true, P is true as well and this is all that is required for logical consequence. The intuitionist disagrees. They say that ¬¬P ⊨ P is false and (let’s assume that they are a modern sort of intuitionist, happy to characterise their view in terms of truth-preservation across interpretations in Kripke-models) denies the classical logician’s claim about interpretations. They say that there are Kripke models and interpretations on such models on which ¬¬P is true but P is false. (The classical logician denies that these are interpretations – i.e. denies that these are genuine counterexamples to DNE.) (Russell 2020: 377–8, emphasis mine)

Let’s grant that Russell’s account of the disagreement between intuitionists and classicists as being a descriptive disagreement about truth-preservation is correct for modern sorts of intuitionist. What about the other sort of inutitionist? Would the disagreement between them and the classicists be a counterexample to Russell’s view?

Maybe the old-fashioned intuitionist can say the same kind of thing about ¬¬P ⊨ P that our non-dialetheist relevantist says about P, ¬P ⊨ Q: it isn’t metaphysically possible for the premises to be true and the conclusion false, but it takes more than that for an inference to be valid, and this inference doesn’t meet the other criteria. The logic that captures all the criteria for validity is intuitionistic logic, and this can be mathematically defined by its proof theory, or with Heyting algebras, or with Kripke frames. The fact that the Kripke frame definition is extensionally correct doesn’t point to any particular interpretation of the frames as possible states of the world. Perhaps we can think of them as information states while maintaining that what counts as an information state is itself normative, just as we could with relevant logic. As evidence that the intuitionists are serious that they really don’t think it’s possible for ¬¬P to be true and P not be true, they can even point to the fact that ¬(¬¬P & ¬P) is a theorem of intuitionistic logic.

I think that if you can find an old-fashioned intuitionist then they may well have every right to say this sort of thing. Even so, there are at least two ways for Russell to argue that even if old-fashioned intuitionism is technically a counterexample to her account of logical disagreement, it isn’t a very good one. One option is to point out that old-fashioned intuitionism is old-fashioned for a reason, and while they may have been arguing with the classicists over something normative, it’s not something we consider worth arguing over anymore. Of course there’s nothing to stop someone raising their hand in the Q&A after someone defends a logical schema on grounds of truth-preservation, and saying “I don’t care if all instances of this schema preserve truth; you still shouldn’t be making these inferences, and in my book that means it’s not valid!” Perhaps a debate will ensue; perhaps it will get shut down. It is in any case possible to have these debates; the question is whether they are worth having, and whether the debates logicians deem worth having are of this kind. Perhaps old-fashioned intuitionism is old-fashioned because logicians in their wisdom decided that this was not something worth arguing over.

Another option arguably doesn’t even concede that old-fashioned intuitionism is a counterexample. Sometimes philosophers have debates and don’t fully understand what they’re debating about. That’s why they write papers with titles like “What we disagree about when we disagree about ontology” (Dorr 2009), and why I can wind up thinking that “when we argue over what the right logic is, what are we arguing about?” is the central question in the philosophy of logic. It’s why we so often take seriously the possibility that the participants in a debate may be talking past each other, and it’s why Peter van Inwagen can write papers explaining that he doesn’t understand something and he doesn’t think anyone else does either (e.g. van Inwagen (1981)). In this light, maybe what’s old-fashioned about old-fashioned intuitionism isn’t that they weren’t arguing about truth-preservation; it’s that they were working before Kripke had helped them figure out that they were. Of course an intuitionist could refuse to get with the programme (if indeed this is the programme), but then they’d be dealt with as in the previous paragraph, getting told by the wider community of logicians that they don’t have to go home but they can’t stay here.

Now I don’t think this kind of response is available in the case of non-dialetheist relevantism, simply because the relevantist so plainly wants more out of a notion of validity than necessary truth-preservation. Anderson et al (1975 & 1992) was called Entailment: The Logic of Relevance and Necessity because it was about the programme in logic that tries to capture a notion of entailment that is both relevant and necessarily truth-preserving. Maybe a time will come when information states get thoroughly psychologized and we’ll be able to understand relevance non-normatively and in terms of truth-preservation, and then those old-fashioned relevantists that remain may be told to go home too. But we’re not there yet, and if psychologism’s poor reputation is earned, we should hope that we never will be.

Logical Pluralism

I’ve said a lot about why I don’t think logic is just a descriptive subject concerned with truth-preservation, and it’s coming from the perspective of my positive view about what we disagree about when we disagree about logic, but I’ve kind of danced around that positive view so far and now I’ll be a bit more explicit about how it goes.

Suppose we have a claim about logical consequence, just considered as such and not further interpreted as being about normativity, necessity, truth-preservation or anything else. Just a claim about, as they say, what follows from what. Claims like that are thought to have implications of various kinds, principally for how we should reason and for metaphysical possibility but perhaps for other things too. With a bit of imagination you could probably find all sorts of places to plug a logical consequence relation into a philosophical theory.

I think that this picture leaves logical consequence itself a bit mysterious, and so I’d prefer to argue over the supposed implications directly. If you want to argue over whether contradictions are metaphysically possible, just go ahead and argue over it. If you want to argue over which inferences people should be making, then argue over that. We don’t need to displace these debates to the question of what the right logic is tout court. And in particular, we don’t always have to give the same logic as our answer. The non-dialetheist relevantists are already doing this, giving a different answer to the questions of whether it’s metaphysically possible for explosion’s premises to be true and its conclusion not and whether it’s rationally permissible to infer explosion’s conclusion from its premises.

We also don’t just have two questions, one metaphysical and one normative. At least on the normative side, there are many questions to ask. MacFarlane (2004: 7) has a long list of possible bridge principles getting us from logically valid sequents to norms on thought. Some examples:

• If A, B ⊨ C, then . . .
  • Co+: If you believe A and you believe B, you ought to believe C
  • Bp+: if you may believe A and believe B, you may believe C
  • Wr+: you have reason to see to it that if you believe A and you believe B, you believe C

MacFarlane adds that these can also all be replaced by versions beginning “If you know that A, B ⊨ C, then…”. For each of those bridge principles we can take a logic and ask whether the corresponding rule always holds for it. Maybe in practice a logic that works with one bridge principle will usually work with the others, or at least the others we’re interested in, but the point is that they’re all meaningful questions, and we might expect that more than one of the questions is interesting. And perhaps MacFarlane’s list, long as it is, doesn’t exhaust all the possibilities. Indeed, I don’t think it does. Here are two other avenues. First, the rules in MacFarlane’s list are all based on material conditionals, (the conditionals here are to be understood materially) but there are other conditionals to be had and these might give different rules. Second, Kwasi Wiredu (1973) makes a distinction between deducibility and inferability, and a related distinction between hypothetical arguments that bring out the consequences of something we may not believe, as in a conditional proof or a reductio ad absurdum, and categorical arguments where we accept some premises and draw a conclusion from them. He argues that some logical schemas are good for one of these notions and not for the other. The distinctions don’t appear in any obvious way in MacFarlane’s list, so that gives us some more questions to explore.

Something to note is that if you want to ask these questions without a robust notion of logical consequence itself, you’ll want to have some independent understanding of metaphysical necessity, or good inference, or whatever it is the question is about. And in fact I do think that it’s good to have notions of those things that aren’t confined to what could be explained by something resembling logical consequence; for example I think metaphysical necessity is also useful for explaining the difference between a physical law and a mere regularity, and perhaps we should have a notion of good inference that applies to both deductive and non-deductive inferences. I don’t expect everyone to agree with me about this, and I see how if a robust notion of logical consequence really could explain both the metaphysical side and the normative side then that would be an attractive thing to have. But we don’t have to decide this now. We debate the first order issues, we put forward accounts of what the debates are really about and debate those too, and we don’t need to prejudge the question of whether the debates are all really about the same thing.

I should spell something out because people have sometimes been unhappy about it when I’ve talked about it before. The different questions we can ask about logical schemas like “are all their instances true?”, “are all their instances metaphysically necessary?” and “is it always permissible to accept the conclusion when it’s permissible to accept all the premises?” are not supposed to give candidate necessary and sufficient conditions for notions of logical truth or validity. When we ask if any instances of “Fn&¬Fn” are true we’re asking exactly the same question about that schema as we can ask about the schema “n is a cat”. In practice the schemas we ask about will often contain no paradigmatically non-logical vocabulary simply because the questions will otherwise tend not to be both interesting and amenable to the methods logicians use. But there’s no need for a demarcation between the logical and the non-logical, or between the questions of this form that are and aren’t legitimate. This relieves the theoretical burden of attempting to give candidate necessary and sufficient conditions for logical truth or validity (since we can have a lot of the debates we want to have without doing so), but it also has the practical benefit of legitimating logics of things like belief, knowledge, obligation, obviousness or whatever that seem to use concepts that aren’t clearly logical but do seem to have a logic in a way that the concept of a cat perhaps does not. Rather than taking a logic as a whole and asking if it’s the One True Logic, we can look at an individual schema, a collection of schemas or even the schemas defining a whole logic and ask whether those schemas satisfy a particular condition. Other (perhaps less interesting) schemas will probably satisfy the condition too, and that’s fine: the inquiry is open-ended.

These days there’s a fair bit of logical pluralism about, considered as a broad category of views united by the idea that there may be more than one correct logic. The picture I’ve described above is the version of logical pluralism I like. Now, considered from a distance it seems like on this picture there’s a whole array of different questions which might have different logics as the answers to them, and different combinations of positions one might adopt. I have thought quite a lot about some of the questions, but I do sometimes worry that a lot of them will end up collapsing into each other, leaving a lot less fun to be had and leaving a lot of logics sitting in the textbooks unused. But even if we end up landing on one true logic with all the metaphysical and normative implications you might expect, it still seems to me that we should start out with the approach I’ve described, and I still hold out hope of ending up somewhere interesting.

The Organon

So I’ve proposed that we break down the question of what the right logic is into a bunch of different questions, some normative and some descriptive. (The normative side seems to have a lot more, but the descriptive side has material and necessary truth-preservation, and there’s no telling what questions psychologists may come up with.) There are well formed questions of both kinds, and even when you hold the answers to the descriptive questions fixed, the answers to the normative questions don’t become obvious or trivial. But I think that even if we accept all this, there’s a worry that the normative questions might be somehow second-rate.

Think about what the people investigating the descriptive side get to do. For the semantics aspect they get to engage with linguistics, psychology, maybe computer science; for the metaphysics they might get to engage with physics, and both camps get to to do the pure maths side of things. The descriptive camp's findings thus stand to have an air of impressiveness to them. The normative side seems to risk being rather less impressive, offering recommendations and expressing preferences. Instead of proofs and experiments and citations of science journals, we have handwaving, table-thumping, and appeals to the great importance of believing the true and nothing but the true. It might seem like a serious scholar would prefer to stick to the hard-nosed descriptive stuff and let people use that information how they will.

I don’t think this is something we should panic about, but we don’t have to write it off as a concern altogether either. The first thing to note is that even if philosophers haven’t got much business arguing for one or other logic as the answer to a normative question, they still need to develop the logics for people to choose from. If the non-dialetheist relevantists are any indication, the logics motivated by normativity rather than (mere) truth-preservation may also be rather more complicated. This means that the philosophers will need to think about the normative questions so that they know what kinds of options people are going to want. Even if the normative questions are not the target of the research, they can still motivate the research. However, in practice we can expect that this thinking about what kind of normative options people might want will sometimes be elaborate enough that the normative questions do become the target of at least some of the research.

This doesn’t strike me as a problem, but that may be because I’m on the other side anyway. I think the normative questions are serious philosophical questions in their own right and it is the business of philosophers to investigate them. In short, it’s epistemology. Back in the day people used to use the term ‘logic’ to cover a lot of what we’d now think of as epistemology, and Aristotle’s so-called logical works include the Posterior Analytics as well as the Prior Analytics (plus the Categories, Topics, Sophistical Refutations and On Interpretation, and sometimes the Rhetoric and the Poetics). This isn’t for nothing, since in many ways the normative side of logic is about the justification of beliefs (typically by inferences from other beliefs), and that makes it epistemology. In the MacFarlane quote earlier he said that logic was crowded on all sides by psychology, metaphysics, mathematics, and semantics, and I wouldn’t be sorry to see it crowded on a fifth side by epistemology if it meant the normative logical questions were getting taken seriously.

One last thing I thought I’d mention is about a debate between Aristotelians and Stoics a long time ago. Stoics divided philosophy into ethics, physics (which included a lot of what we’d now call science but also some of what we’d still call philosophy), and logic (which included a lot of epistemology). The Aristotelians didn’t think of logic (and epistemology) as a separate branch like that; they thought of it as a tool used by the other branches. They accordingly called Aristotle’s logical works the Organon, which means ‘tool’. That’s my understanding of how it went, anyway. I don’t really know the history and the similarity between what happened then and what I’ve been talking about here may be superficial, but I’ll continue. (Regular readers may recall me talking about this dispute over the status of logic before.) Suppose that the normative questions really are either straightforward given the answers to the descriptive questions or somehow not amenable to philosophical argument. Then we’re left with logic as Russell conceives it, a descriptive discipline involving maths, linguistics, probably psychology, but perhaps not a great deal of philosophy as such, at least as we currently think of it. (There’s admittedly probably some metaphysics in there.) This corresponds to the Aristotelian position: on this model logic is news philosophers can use, but it’s not really philosophy proper. If we like Quinean naturalized epistemology, we might even want to view the rest of epistemology the same way, recapturing the Aristotelian vision of the relationship between logic/epistemology and the rest of philosophy. On the picture I favour, the normative questions aren’t straightforward, are amenable to philosophical argument, and aren’t reducible to descriptive questions about something else. That puts me on the side with the Stoics.

As I say, I may have the history wrong and the parallel may be superficial in any case. But even if it wasn’t the debate the Stoics and Aristotelians were having, I think there’s still a debate to be had. Is logic normative? I think it is, at least sometimes, and when it is, it blends into epistemology. But if the interesting part of epistemology isn’t normative, then the interesting part of logic probably isn’t normative either.

Notes

[1] I’ll mostly be talking as if there was a straightforward normative/descriptive distinction, and Russell’s argument is framed more or less in those terms, but I think a lot of what she says and what I’m saying would probably still make sense even if there wasn’t a straightforward distinction to be had.

[2] The argument comes in Metaphysics IV 4, and I understand there's a large literature on it, which I regret to report I have not read, aside from Dutilh Novaes (2019). I'm not a fan of the law of non-contradiction and my hope is that what's correct in Aristotle's argument can be explained in terms of a conflict between accepting and rejecting the same proposition, rather than between accepting both a proposition and its negation. (My view, following e.g. Smiley (1996) is that rejecting a proposition and accepting its negation aren't the same thing.)

[3] Filter logics, so called because you get the consequence relation by taking the consequence relation of an irrelevant logic and applying a relevance filter of some kind to it, are mentioned in Priest (2008:173–4). Aside from as a curiosity or just to illustrate a bad approach to securing relevance, I don't really know what they're for. It possible they're not for anything else, but I expect they do have some independent interest, although as I say I don't know what it is.

[4] The apparent fact that information states can be incomplete makes it a bit unsatisfactory to portray the classicist as defending a view about truth-preservation over information states. We might portray the disagreement between fans of FDE and the three-valued logic K3 as being about information states, and K3 still validates explosion, but we’re still left asking where that leaves the classicist. We can get some of the way by saying that information states include classical tautologies for free. You might also wonder why the only admissible information states are prime, in that they must include at least one of P and Q whenever they include P∨Q: can’t someone have the information that P∨Q without having either the information that P or that Q? It’s a fair point, although it can be levelled at the FDE models as much as the classical ones. If both sides expand the class of information states to include any set of sentences closed under their favoured logic, that should give alternative model theories for the same consequence relations, but including non-prime information states. The classicist will also get the information state containing all sentences — FDE had it already — but that doesn’t change the consequence relation either and if they want they can exclude it by fiat. Allowing non-prime models may end up giving the classicist a model theory for classical logic that they can credibly defend as being the appropriate class of information states, although it’s possible there’s a straightforward objection to it that I’ve overlooked. Of course the option including the non-prime models does at least seem to use the logics themselves to define the classes of admissible information states, and this relates to an issue we’ll talk a bit about below.

References

• Anderson, Alan R. & Belnap, Nuel D. (1975). Entailment: The Logic of Relevance and Neccessity, Vol. I. Princeton University Press.
• Anderson, Alan Ross ; Belnap, Nuel D. & Dunn, J. Michael (1992). Entailment: The Logic of Relevance and Necessity, Vol. II. Princeton University Press.
• Dorr, Cian (2005). What we disagree about when we disagree about ontology. In Mark Eli Kalderon (ed.), Fictionalism in Metaphysics. Oxford University Press. pp. 234--86.
• Dultilh Novaes, Catarina (2019). Aristotle’s Defense of the Principle of Non-contradiction: A Performative Analysis. In D. Gabbay, L. Magnani, W. Park, and A.V. Pietarinen (eds.), Natural Arguments: A Tribute to John Woods. London, College Publications.
• Field, Hartry (2015). What Is Logical Validity? In Colin R. Caret & Ole T. Hjortland (eds.), Foundations of Logical Consequence. Oxford University Press.
• James, William (1907). Pragmatism: A New Name for Some Old Ways of Thinking. Duke University Press.
• Jenkins, C. S. I. (2014). Serious Verbal Disputes: Ontology, Metaontology, and Analyticity. Journal of Philosophy 111 (9/10):454-469
• Kimhi, Irad (2018). Thinking and Being. Harvard University Press.
• Lewis, David (1988). Relevant implication. Theoria 54 (3):161-174.
• MacFarlane, John. (2004). In What Sense (If Any) Is Logic Normative for Thought?. Draft https://johnmacfarlane.net/normativity_of_logic.pdf
• Priest, Graham (2008). An Introduction to Non-Classical Logic: From If to Is. Cambridge University Press.
• Quine, Willard V. O. (1951). Two Dogmas of Empiricism. Philosophical Review 60 (1):20–43.
• Quine, W. V. (1953). Mr. Strawson on logical theory. Mind 62 (248):433-451.
• Russell, Gillian (2008). One true logic? Journal of Philosophical Logic 37 (6):593 - 611.
• Russell, Gillian (forthcoming). Logic isn’t normative. Inquiry: An Interdisciplinary Journal of Philosophy 1:1-18.
• Ryerson, James (2018). Unpublished and Untenured, a Philosopher Inspired a Cult Following. New York Times, 26 September 2018. https://www.nytimes.com/2018/09/26/books/review/irad-kimhi-thinking-and-being.html
• Smiley, Timothy (1996). Rejection. Analysis 56 (1):1–9.
• Strawson, P. F. (1952). Introduction to Logical Theory. Routledge.
• van Inwagen, Peter (1981). Why I Don't Understand Substitutional Quantification. Philosophical Studies 39 (3):281-285.
• Wiredu, J. E. (1973). Deducibility and inferability. Mind 82 (325):31-55.

Rationalism And Science

I've been thinking a bit about rationalism, empiricism and science, and I tried writing some notes to organize my thoughts about it. It's all pretty half-baked and unsatisfactory, but I've turned it in into something I can put here anyway, in case it's the kind of thing anyone likes to read.

Rationalism vs Empiricism

• Roughly: empiricists think knowledge only comes from experience, while rationalists think pure reason has an important role.
• Rationalists tend to think experience has some role too, although they might think that the kind of knowledge that involves experience isn't so good, or perhaps isn't knowledge strictly speaking at all.
• Empiricists sometimes think it's OK to get mathematical and/or logical knowledge from pure reason. Maybe you can get some knowledge of analytic truths that way too, although under the influence of Quine (and Morton White, usually via Quine) they might not think there are any properly analytic truths.
• If you want to firm up this second source of knowledge as part of your position, you can appeal to Hume's Fork. Hume's Fork is the idea summed up in this quote from the end of his Enquiry Concerning Human Understanding (emphasis in original):

  "If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion."

  It takes some interpreting to get this passage to be arguing for exactly the empiricism-plus-maths-and-logic position, but it's in the ballpark. If you do adopt this package, you might even be able to get away with calling yourself a logical empiricist, like Rudolf Carnap. Carnap is very in at the moment, and is also generally considered to have been a great guy.

Descartes And Leibniz

• They were scientists and mathematicians, but they were also metaphysicians.
• As philosophers, we tend to look more at their metaphysics than their science, unless we're specialists. The science is more obsolete than the metaphysics (although perhaps not more wrong), science isn't really taught via 300-year-old texts even when it isn't obsolete, and we're not studying science anyway. And we're probably even less interested in reading their maths than in reading their science.

The Appeal Of Empiricism

• When you're reading their metaphysics, sometimes they will invoke principles of pure reason to derive their metaphysical conclusions. For example, Leibniz invokes such useful gizmos as the Principle of Sufficient Reason and the Identity of Indiscernibles. Descartes says things like there must be at least as much reality in an efficient and total cause as in the effect of that cause (from Desmond M Clarke's 1998/2000 translation of Descartes' third Meditation). And Spinoza, the other member of the Big Three rationalists, takes this kind of thing to a whole new level with an axiomatic presentation of his grand metaphysical system in the Ethics. You have to wonder what justifies their principles, and sometimes it can seem like they're just making up premises on the fly to get themselves out of an argumentative tight spot. I suppose this is particularly the case with Descartes, who makes things even worse by appealing to the light of nature, or the natural light of reason, and we often read Descartes when we're young and impressionable. Perhaps harping on about the natural light of reason makes a good impression on some young philosophy students, but it didn't make a good impression on me.
• The metaphysical conclusions that they derive aren't even especially attractive to a lot of us. The existence of God, the non-physical soul, pre-established harmony and so on. We feel like we could easily get by without them, and then we wouldn't have to accept the premises, or the embarrassing Light Of Nature methodology by which they arrived at them.
• Science, on the other hand, is an empirical discipline. We're quite sure of that. So while we might not know exactly what goes on across campus in the science departments, we don't really worry that anything of importance will be lost to science if we become empiricists. So we can keep science much as it is, while taking a suitably unenthusiastic attitude towards metaphysics, saying that metaphysical theses are either unknowable, or probably wrong, or Not Even Wrong.
• We can still do a little bit of metaphysics, criticizing theses for conflicting with our best science, or for being internally incoherent. And we can also sometimes have a go at offering an empiricist critique of some of the things scientists get up to, when we take them to be straying into metaphysics. The interpretation of quantum mechanics is a good source of material there, although it is often difficult for philosophers to understand the relevant science well enough to get taken seriously. (And even if they do understand the science, getting taken seriously still isn't a given.)

Pure Reason In Science?

• All this seems like a very nice package, but there's a fly in the ointment, which is that scientific methodology may actually include an awful lot of rationalism. Sure, it's an empirical discipline, in that scientists gather data and do experiments and so on. But they also spend a lot of time filling whiteboards with equations, and they're even known to have flashes of inspiration in the shower, as if their problem has been suddenly illuminated by the light of nature. (Archimedes' alleged eureka moment in the bathtub is not a good example of this, since his flash of inspiration was about his displacement principle, and one has experiences relevant to that in a bathtub. But it does happen to scientists working on things besides bathtubs too.)
• Empiricists do have resources to push back on this line of thought. They're still allowed to use pure reason for maths, and for logical inferences. Perhaps that's what's going on with the whiteboards and the flashes of inspiration, and so under closer examination it will turn out that scientists aren't doing anything outside the empiricist's rules.
• Now, the closer examination of actual scientific practice is something that has been done an awful lot by other people, under the auspices of history and/or philosophy of science. I blush to confess that I do not have much familiarity with any of this literature. If you do, then please do correct me if I'm wrong, because I'm going to make a naive pessimistic case that under closer examination what we'll find is that science is in fact a bit of a Cartesian free-for-all.

The Problems Of Induction

• Induction, roughly, is when you take some observations, find a general rule that fits the observations, and then apply that rule to make predictions about new observations. If it didn't work, then it'd be hard to see how science could work. But there are two reasons why it is hard to see how induction could possibly work.
• The classic problem of induction is the problem of justifying the idea that old patterns can be expected to persist at all. We set up the dilemma as follows. (It'll be based on AJ Ayer's presentation of the problem in chapter two of Language, Truth and Logic, although Ayer himself thinks the problem's very hopelessness makes it a pseudo-problem as traditionally conceived.) Either inductive reasoning is justified deductively or inductively. An inductive justification would point to how old patterns have persisted after being identified before, and this is itself a pattern we can expect to persist into the future. But this reasoning seems circular. The deductive justification is in bad shape too, on the grounds that there just isn't any logical inconsistency in patterns being broken. A coin landing heads twenty times in a row is consistent with its landing heads the twenty-first time, as an inductive reasoner would presumably predict, but it's also consistent with its landing tails the twenty-first time, providing a counterexample to any attempted deductive justification of induction.
• The new riddle of induction, which we have Nelson Goodman to thank for, is still a problem even if you can solve the classic problem of induction. This time the problem is no longer showing that patterns will persist, but deciding which patterns we should expect to persist. There are lots of patterns consistent with any dataset, and we will get different predictions depending on which patterns we identify. Of course, there are some patterns that humans identify more readily than others. But justifying this as anything more than a cognitive bias, while still working within the empiricist's rules, is difficult.
• Rationalists, as we've learned, can appeal to the light of nature to get themselves out of just this kind of tight spot. Can't justify induction inductively or deductively? Just appeal to the light of nature! Not sure which pattern is more likely to persist? Let the light of nature show you the way.
• This is of course a bit flippant. Rationalists do have resources to invoke premises when an empiricist would be in a tight spot, but they can't do this arbitrarily, at least not by their own lights. Are the resources you need to deal with induction the kind of resources rationalists would appeal to?
• I think they are, or could well be. To get a handle on why, consider Plato's theory of forms. One of the problems the forms are supposed to solve is how we can have knowledge of general things, even though we only experience particular things. Plato's idea is that knowledge of general things is knowledge of transcendent forms, which are somehow reflected in the particular things, and which we apprehend with the intellect rather than with the senses. Mathematical knowledge is knowledge of forms, but scientific knowledge is knowledge of forms too. Aristotle didn't think that forms were transcendent, but (as I understand it) he still thought that scientific knowledge was knowledge about forms, or essences if that's different, and that in any case these are apprehended by the intellect. The exact mechanism by which the intellect apprehends the forms is hard to pin down - Plato suggested we remember them from directly experiencing them before we were born - but the idea of scientific knowledge being knowledge of something general apprehended by the intellect has had real staying power, and is the kind of thing rationalists in particular are into.
• This gives rationalists a bit of purchase on the problems of induction. For the classic problem, the idea is that we can learn something about events before we observe them, because the same forms show up in those events as the ones we have already observed. For the new riddle, the idea is usually that not all rules are equal because some correspond simply to forms, and some don't. Not every gerrymandered property you might think of corresponds to a form, and the ones that don't aren't as likely to fit in robust generalizations.
• You can try to rerun the arguments against the rationalist if you like. How do we know that forms will carry on behaving the same way we're used to? How do we know which properties correspond to forms? Well, because we examine them with our intellects and discover which forms there are and that they behave uniformly. How do we pull that off? Good question! But it seems we do pull it off, since science works. Denying that it could possibly work flies in the face of the evidence, and denying that it could work this way risks begging the question against the rationalist.
• So, we've got an argument that science has a problem that rationalists have the resources to solve and empiricists don't. For that, we didn't need to look at the actual practice of science at all. But now things get a little trickier. I'm going to make two suggestions, neither of which I have the expertise to properly back up.
  • First, a large part of what Descartes and Leibniz liked about being rationalists was that it gave them the resources as scientists to respond to just this kind of problem.
  • Second, not much has changed. Scientists are still leaning heavily on roughly the same rationalist methodology that Descartes was into, and being wildly successful with it too.

How Descartes And Leibniz Did Science

• In the seventeenth century it was possible to feel really good about the prospects for science. The progress they were making made understandable a level of optimism that may only have been equalled before or since by physicists in the late 1920s. They really felt like they would soon have it all figured out, and that when they did the answers would be simple, beautiful, and powerful.
• Descartes and Leibniz also took it more or less as a methodological axiom that the world was nicely ordered along principles that could be understood and discovered by humans. This is an idea that goes back at least to Plato's Timaeus, and on some accounts goes all the way back to Thales and is what sets philosophers like him apart from the storytellers like Homer and Hesiod who came before him. The Stoics were big on the idea, it persisted in some form throughout the middle ages, and in the seventeenth century it was as popular as ever. It's an appealing idea, and people working with the idea keep making discoveries and building cool stuff, while sceptics just bring everybody down. When asked to justify this methodological axiom, neither Descartes nor Leibniz was above talking about God. Both of them thought that God in his goodness had given us rational faculties which when used properly would be able to get us significant scientific knowledge of the world. Leibniz in particular also thought that God's perfection meant that he would make the world excellent - the best of all possible worlds, as they say - and that we could work out what would count as excellent, and use this to guide our theorizing both in science and metaphysics.
• So, how do you do science with this attitude? Well, prompted by observations and guided by the light of nature, you come up with a beautiful, simple, powerful theory of how the world works. You make the theory nice and mathematical, and maybe come up with some new maths like calculus (Leibniz) or co-ordinate geometry (Descartes) especially for the purpose. You do experiments to test the theory. If the predictions are borne out, great! If not, maybe there's something wrong with the experiments. Or maybe there's something wrong with the theory. Eventually you come up with something nice that fits the results you're getting. And because the world does in fact conform at least approximately to the kind of principles that seem simple and beautiful to seventeenth-century optimists, the whole thing went swimmingly.

How We Do Science Now

• Basically what I want to say is that we do things much the same way. We might not think that's what we're doing, but it is. More or less.
• When scientists are explaining how science progresses now, they'll sometimes offer an account based on Karl Popper's ideas. I haven't read Popper, and I expect the scientists often haven't either, because the account isn't really very plausible and it'd be uncharitable to attribute it to Popper himself. Here's how it goes. Scientists look at the data they're already aware of, and they come up with a theory that fits. The important thing about the theory is that it be falsifiable, in that there are experiments they could do or observations they could make that would show that the theory was false. Then they do experiments and/or make observations. If the results are what the theory predicts, the theory is more likely to be true. If the results aren't what the theory predicts, the theory is false and so they go back to the drawing board.
• When pressed, people usually admit that this does not really reflect scientists' actual responses to data not fitting their theories. There might be something wrong with the experiment, or maybe the theory does predict it after all but there was something in the initial conditions you weren't aware of. A classic example is the orbit of Uranus: it didn't fit with the predictions people made for it using Newtonian mechanics and gravity, but that's just because they didn't know Neptune existed. Include the gravitational effects of Neptune and Newtonian mechanics and gravity aren't falsified after all. That's how Neptune was discovered, or so I'm told. (The hero of that story is Urbain Le Verrier.) They tried to pull the same trick with a planet inside Mercury, which they called Vulcan, but there it turned out Newtonian mechanics and gravity really were the problem, and Einstein's theory of gravity explains it much better. But there was a long time in between unsuccessfully looking for Vulcan and rejecting Newtonian gravity.
• OK, so let's put aside naive falsificationism. How does science work, then? Well, I'd suggest they do pretty much what Descartes did. You have a sense of the kind of data you're trying to fit your theory into, and then you make something up that you're happy with, guided by the light of nature. Your theory will have some blanks in it like the mass of the electron or whatever, and when you find some way of taking measurements to fill in the blanks, you do. Then you cling on to the theory like grim death in the face of both friendly and unfriendly data until finally you come up with something you're not quite as unhappy with. This account adopts some themes people tell you come up in Kuhn and Lakatos - I haven't read them either - and what it agrees with Popper about is that the whole process works much more smoothly if your theory is the kind of thing that data can be friendly or unfriendly to. (That's why it doesn't work so well for metaphysics, I suppose.)
• The account is a total caricature, of course. But I'm hoping it's a caricature of the actual practice of science, unlike naive falsificationism, which is a caricature of what some scientists imagine they must be doing because rationalism somehow doesn't seem hard-nosed enough. Now, while it might sound like I'm being a bit mean to scientists, that's not my intention. If the rationalists are right, then this could be exactly what scientists ought to be doing. And whatever they're doing, they seem to be doing it very well.

The Moral Of The Story

• Sometimes people give the rationalists a hard time. We do sort of acknowledge that Descartes, Spinoza and Leibniz were great philosophers - I mean, everyone says so, right? - but when we look at their philosophical systems what we see are bad arguments for false conclusions. It's all too easy to attribute this to a bad methodology, saying that the problem is that they were rationalists. Maybe we should ditch the light of nature, stick with empirical methods plus maths and logic, embrace science and be quietist about metaphysics.
• What I'm suggesting is that this line of thought has a great big hole in it, which we don't notice because we don't think of the rationalists as scientists. Or perhaps we do think of them as scientists, but don't think of that as relevant to their philosophy. But it is relevant. They were scientists, they were good at it, they did science as rationalists, and we're still doing that today. If you want to assess rationalism at its best, then don't consider it as a method in metaphysics; consider it as a method in science. That's the moral of the story. The question is, is it a true story?

What's Wrong With This Picture?

• I mentioned earlier that there's a lot of stuff here that I'm pretty ill-informed about. Maybe you thought I was being disingenuous, but I wasn't. There's a lot to learn here, and I haven't learned it. And if you have, you probably don't need me to tell you that. The evidence is right there on the page.
• Nonetheless, this is where I am at the moment with this stuff. From where I'm sitting, it looks like you can't really get anywhere in science as an empiricist, the actual practice of science bears this out, and the rationalists should be given a bit more credit for it. So, what might I be wrong about?
• First, I might be wrong about empiricism's platform. Maybe they've got some clever solutions to the issues with induction that I was worrying about. Or maybe they're willing to embrace a greater level of scepticism than I appreciate.
• Second, I might have Descartes and Leibniz wrong. I've read a fair bit of both, but there's an enormous amount left that I haven't read, and I do find the scientific practice they envisage all a bit mysterious. That's why I describe it in these scathing terms, before insisting that what I described is the tried and tested scientific method that built the modern world. I think we should have a fairly low prior that a method that sounds like that could build a world that looks like this.
• Third, I might have modern science wrong. Who knows what these people get up to? They don't even seem to know. But those historians and philosophers of science I mentioned before probably have some idea, and I could try checking some of that out. I could at least read Kuhn. Everyone reads Kuhn.
• Fourth, I might be wrong about something else. I don't know what. But the whole situation is very unsatisfactory, and I think I must be missing something.

Reading List

• So, I've got some reading to do! If you've got this far and you think I'm a doofus who should calm down and read X and then everything will become clear, then please point me towards X in the comments. In the meantime, here are some things I could look at.
• The logical positivists/empiricists in the first half of the twentieth century were often pretty clued up about science, but they were also pretty thoughtful and self-aware about their empiricism. So perhaps I should read some Schlick or something.
• I found a book in a second hand shop the other day by Daniel Garber called Descartes' Metaphysical Physics. I probably won't read it all, because I am a non-serious person, but it looks relevant and I'll probably read some of it.
• I could have a look at some of Descartes and Leibniz's more scientific stuff, and see how they talk about what they're doing.
• I've only got a pretty sketchy understanding of Plato and Aristotle's understanding of form and how it helps with epistemology, so I should probably read something about that.
• Like I say, I could read Kuhn, at least The Structure of Scientific Revolutions. I should probably read something by Lakatos too. Popper is less of a priority.
• I saw an NDPR review of a book called Platonism at the Origins of Modernity: Studies on Platonism and Early Modern Philosophy, and it looked like some of the papers in that would be relevant.
• Once I've got through that lot I'll probably be down multiple rabbit-holes, so for me to put anything else on the list now would probably be a bad idea. But if you can point me to anything that would sort all this out for me, I'm all ears.

Mathematical and moral deference

Regular readers will remember that once I posted a fallacious proof of the continuum hypothesis. I’m not a mathematician, but I am a philosopher, and the continuum hypothesis is philosophically interesting as well as mathematically interesting, so occasionally I think about it and read about it and generally try to understand it better. The last time I did this I tried to learn about forcing. It didn’t go well.

Forcing is a mathematical technique for producing models of set theory. Paul Cohen used it to produce models of ZFC (Zermelo-Fraenkel set theory with the axiom of choice) in which the continuum hypothesis is false, which showed that it didn’t follow from the ZFC axioms. Kurt Godel had already produced a model of ZFC in which the continuum hypothesis was true, and so this completed a proof that ZFC doesn’t rule the continuum hypothesis in or out. Godel and Cohen’s methods for constructing models of set theory have proved very useful elsewhere in set theory too, and we’re all very grateful.

Anyway, when I was trying to learn about forcing I read a couple of papers with titles like “A beginner’s guide to forcing” and “A cheerful introduction to forcing and the continuum hypothesis”. They sounded like they were about my level. The first of those contained a passage which made me feel a bit better about not having understood forcing before:

All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves. The proofs should be “natural” in Donald Newman’s sense [Donald J. Newman, Analytic Number Theory, Springer-Verlag, 1998]:

This term . . . is introduced to mean not having any ad hoc constructions or brilliancies. A “natural” proof, then, is one which proves itself, one available to the “common mathematician in the streets.”

I believe that it is an open exposition problem to explain forcing. [Timothy Y. Chow, 'A Beginners Guide to Forcing', p.1]

After reading these papers, however, I still didn’t get it. One stumbling block was that when they sketched how it works, they would invoke a countable standard transtitive model of set theory. A standard model is one where all the elements are well-founded sets and “∈” is interpreted as set-membership. A transtive model is one where every member of an object in the domain (in reality, not just according to the model) is also in the domain. A countable model is one with countably many objects in its domain. And while they don’t construct one of these models, they assure us that the Lowenheim-Skolem theorem shows that there will be such models.

Now, for me, reading this feels the way that spotting an error feels. Because when I learned about the Lowenheim-Skolem theorem as an undergraduate, they told us it said that every theory in a countable first-order language that has a model has a countable model. It doesn’t say that it has a countable standard transitive model. Standardness and transitivity are meta-language things, and you can’t just stick an axiom into the theory so it’ll only have standard and transitive models. So the Lowenheim-Skolem theorem doesn’t guarantee anything of the kind, forcing doesn’t work and Paul Cohen should posthumously give back his Fields medal.

Of course, this isn’t the correct response to my feeling and the accompanying line of reasoning. I’ve just made a mistake somewhere. I think the most likely thing is that there’s a suppressed step in the argument which is obvious to the authors and to their imagined readers but which isn’t obvious to me. It’s normal for mathematicians to leave out steps they think are obvious, which is why maths papers aren’t all ridiculously long. It’s also normal for these steps not to be obvious to everyone, which is why maths papers are often difficult to understand. But in this case I can’t really even imagine what this obvious step might be. It just looks to me the way a silly mistake would look.

The fact I’ve made this error is sad and embarrassing for me, but I’ve no serious doubt that it is a fact. This just isn’t the sort of thing the mathematical community gets wrong. Well, there was that thing with John von Neumann and hidden variable interepretations of quantum mechanics, but the people who spotted the problems with that were mathematicians and physicists, not clowns like me. There’s no serious doubt that forcing works. I’d be a fool not to defer to the experts on this, and I do. (I feel like you won’t believe me about this, but I don’t know how to make it any more explicit.)

Normally when we trust mathematicians, we haven’t inspected the case ourselves. I’ve no significant doubt that Fermat’s last theorem is true, even though I haven’t looked at the proof. I do have significant doubt that the ABC conjecture is true, because not enough people have gone through the proof. Andrew Wiles’s original proof of Fermat’s last theorem didn’t work but could be fixed, but maybe Shinichi Mochizuki’s proof of the ABC conjecture doesn’t work and can’t be fixed. But when there’s a consensus in the mathematical community that they have found a rigorous proof of something, we can all believe it and we don’t have to look at the proof ourselves.

In the case with me trusting mathematicians on forcing and the continuum hypothesis, however, I have inspected the proof and the experience was subjectively indistiguishable from discovering that it didn’t work. That puts me in an odd position. It’s complicated, but the situation is a bit like if I believed that P and that if P then Q but rejected Q. The premises are compelling, the inference is compelling, but the conclusion is impossible to accept.

This is a bit like the situation you get in the presence of a paradox, like the liar paradox or Zeno’s paradoxes about motion. I think a lot of the time we just accept that some paradoxes are weird and there’s no particularly satisfying response to them. But with forcing and the continuum hypothesis, I also don’t think there’s any significant chance that there is any paradox here. It feels to me the way paradoxes feel, but it isn’t really a paradox. It’s a subjective paradox, if you like. An objective paradox arises where the premises are indubitable, the conclusion is unacceptable and the inference from one to the other is compelling. A subjective paradox arises for someone when it just seems that way to them.

I used to think that there weren’t any objective paradoxes and they were all ultimately subjective paradoxes, and I still think that might be right. But there’s a big difference between a subjective paradox where you’re just confused and something like Curry’s paradox where nobody knows how to sort it out. The thing with me and forcing is definitely in the subjective category.

So I’m in a weird situation, but you’re probably not in it yourself. I was reminded of this weird situation when I was reading a paper by David Enoch on moral deference recently. He discusses a couple of cases, one from Bernard Williams and one he came up with himself. Here’s the Williams one:

There are, notoriously, no ethical experts. […] Anyone who is tempted to take up the idea of there being a theoretical science of ethics should be discouraged by reflecting on what would be involved in taking seriously the idea that there were experts in it. It would imply, for instance, that a student who had not followed the professor’s reasoning but had understood his moral conclusion might have some reason, on the strength of his professorial authority, to accept it. […] These Platonic implications are presumably not accepted by anyone. [Bernard Williams, Making Sense of Humanity (Cambridge: Cambridge University Press, 1995): 205.]

Enoch’s own example is about his friend Alon who tends to think wars in the Middle East are wrong, whereas Enoch usually starts off thinking they’re right and then later comes round to agree with Alon. When a new war arises, Alon opposes it and Enoch doesn’t, but Enoch knows he’ll probably end up agreeing with Alon. Alon is likely to be right. So why not defer to Alon?

Enoch wants to explain what’s fishy about moral deference, while disagreeing with Williams about whether moral deference is OK. Enoch says sometimes you should defer. He should defer to Alon and the student should under some circumstances defer to the professor. The fishiness, according to Enoch, results from the fact that moral deference indicates a deficiency of moral understanding or something like it. But two wrongs don’t make a right, and it’s better to endorse the right conclusion without understanding the situation properly, than to endorse the wrong conclusion because you don’t understand the situation properly. I don’t think that misrepresents Enoch’s position too badly, although you can read the paper if you want to know what he actually says.

One interesting thing Enoch does in the paper is to compare moral deference with deference in other areas: prudential rationality, aesthetic assessment, metaphysics, epistemology, and maths. I think that considering the mathematical case can bring to light a source of fishiness Enoch overlooks. I would say there is nothing fishy about me deferring to the mathematical community about Fermat’s last theorem. I’ve good evidence it’s true, I’ve no idea why it’s true, and there’s nothing particularly odd about this situation. But my deference in the forcing case is fishy. People believe ZFC doesn’t imply the continuum hypothesis on the basis of the proof, I’ve inspected the proof, and from where I’m sitting it transparently sucks. To repeat: I don’t think the proof is wrong. But I am in a strange situation.

Now consider two cases of moral deference, adapted from the ones Enoch talks about. First, consider an 18 year old taking Ethics 101. They read a bit of Mill and decide utilitarianism sounds pretty good. They can’t really see yet how the morality of an action could be affected by anything besides its consequences, although they haven’t thought about it much. Professor Williams assures them there’s more to it and things like integrity and people’s personal projects matter. I think the student should probably trust Williams on this, at least to an extent, even before they’ve heard his arguments. (Perhaps especially before hearing his arguments.) Williams knows more about this than the student does, and the student doesn’t have enough information to come to their own conclusion. They’ve just read a bit of Mill.

Now consider Enoch and Alon. Enoch looks at the available non-moral information about the war and comes to the conclusion that this one is right. This time is different, he thinks, and he points to non-moral differences grounding the moral difference. Alon looks at the same information and comes to the conclusion that the war’s wrong. Should Enoch defer to Alon? Well, yes. I think so. And so does Enoch. But how does he go about deferring to Alon? He basically has to reject his inference from the indisputable non-moral premises to the conclusion, even though that inference seemed to be compelling. He’s in the grip of a subjective paradox, just like me.

Williams’s student lacks moral understanding, the way children frequently do. But there’s nothing especially epistemically dodgy about what Williams’s student is doing. He doesn’t have all the information to understand why integrity and projects matter but he has it on good authority that they do. Enoch lacks moral understanding too, but he’s in a worse position. He has what seems to him to be a compelling argument that Alon is wrong this time, and another compelling argument that Alon is probably right. Worse, we might as well stipulate that he has examined Alon’s argument and it doesn’t seem to be much good.

I stipulated that Enoch’s inference from the non-moral facts to his conclusion that the war was right seemed to him compelling. That was deliberate. Maybe you don’t think it’s a plausible stipulation, and if you don’t, then I’m on your side. I think that’s the source of the problem. It’s tempting to think that armchair reasoning has to seem compelling if it seems to be any good at all. What would an intermediate case look like? Either an argument follows or it doesn’t, right? Well sure, but we’ve got no special access to whether arguments follow or not. No belief-formation method is completely reliable. We can go wrong at any step and our confidence levels should factor this in. Nothing should seem completely compelling. Of course, you’d expect me to say that, because I’m a sceptic. But it makes it a lot easier to deal with situations like the forcing thing. By the way, if you think you can explain to me why the Lowenheim-Skolem theorem guarantees that ZFC has countable standard transitive models, feel free to have a go in the comments.