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.
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• 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.
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• 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
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