# 94th Joint Session of the Mind Association and Aristotelian Society

### James Skinner (St Andrews)

Abstract

This paper articulates a novel argument showing that many species of logical pluralism are inconsistent with the normativity of logic for reasoning. I begin by arguing for a ‘thick’ conception of the normativity of logic on which logic is not only normative for the combinations of beliefs we may have, but also for the methods by which we may form them. I then develop an argument – the normative contradiction objection – which shows that a wide variety of logical pluralisms are inconsistent with this thick conception of logic’s normativity. This is because together they entail contradictory claims about how one ought to reason whenever one ought to believe a set of propositions, X, and C follows from X on one of the pluralist’s logics but not another. Accordingly, if logic is normative for reasoning, these pluralisms are untenable.

Biography

James Skinner is a second year MPhil student at the University of St Andrews, where he is writing a dissertation defending logical monism – the thesis that there is exactly one correct logic – under the supervision of Kevin Scharp and Francesco Berto. Besides the philosophy of logic, James takes a keen interest in formal epistemology, political philosophy, and ethics. Prior to his MPhil, James did his undergraduate degree at the University of Oxford, where he started life as an Economics and Management student before seeing the light and switching to Philosophy, Politics, and Economics.

• Gavin Thomson

11th July 2020 at 4:10 pm

Hi James, great talk. I learned a lot!

I have a couple of questions along proof-theoretic/inferentialist lines.

You note in your talk that different logics/consequence relations may be correct for different domains. Classical logic for physical objects, intuitionistic logic for computable proofs, maybe linear logic for some kind of resource-based situation. Could someone who is a pluralist about truth – maybe someone who thinks that truth-talk is just expressive of certain features of reasoning and not “metaphysically weighty” – reject NCO on those grounds? Or is that really just re-stating either Reply 1 or Reply 2 in terms of truth rather than logical consequence?

My second question was about the formulation of NCO. If I understand this rightly, this argument is formulated using a “global” consequence/deducibility relation (above and beyond the “object logics” |=_1 and |=_2). Which models a “logic of admissible belief formation”?
What if this global relation is a multiple-conclusion entailment relation, along the lines of sequent calculus, such that \Gamma |– (12), (14) should be understood as saying that at least one of (12) and (14) is assertible if all of \Gamma is assertible? That is, conclusions are understood disjunctively (and premises conjunctively). This seems to me to block the move to (15).

1. James Skinner

11th July 2020 at 7:01 pm

Hi Gavin,

Thanks for watching and your questions – I’m glad you enjoyed it!

I’m not sure I’ve entirely understood the grounds on which this species of truth-pluralist is rejecting NCO — would you mind elaborating, please? For someone who is a domain-relative truth pluralist, and builds domain-relative logical pluralism on top of their truth pluralism (e.g. Lynch, Pedersen), their position will be immune to NCO because there will be no argument to which two logics can be correctly applied.

Turning to your question about the appropriate metalogic which the NCO uses, there are of course logics in which NCO fails. I’m afraid I’m not familiar with multi-conclusion logics and so can’t say anything too specific in response – but here are some more general thoughts.

(1) For any pluralist who accepts that logic is formal (e.g. Beall & Restall, Varzi etc) and who accepts an explosive logic as one of the correct logics, NCO is a problem because it follows that the explosive logic is correct with respect to NCO, and NCO still goes through.

(2) Supposing the pluralist rejects formality, they must first give a principled reason why the explosive logics they accept elsewhere are incorrect in a metalogical setting, and this must be consistent with the rest of their pluralism. For instance, this move isn’t available to Blake-Turner and Russell — according to their pluralism logics are correct relative to one’s epistemic aims (e.g. classical logic when one’s aim is to draw true conclusions, intuitionistic logic when one’s aim is to draw true and demonstrable conclusions etc) — because if one’s aim is to draw true conclusions in the metalogical setting then classical logic is the correct logic and so NCO’s conclusion is true.

(3) Suppose this hurdle is cleared and the correct logic in the metalogical setting is non-explosive so that reductio fails. Although this renders NCO’s final step invalid, it will still be necessarily truth-preserving providing that we are consistent context, which metalogic is ordinarily thought to be.

1. Gavin Thomson

13th July 2020 at 11:40 pm

Thanks for the response James, that clears things up for me! In my first question I was muddling mono and poly-pluralism.
In my second Q, what I was aiming for was the question of whether the argument relies on a particular understanding of logical consequence — so I should perhaps have aimed it at the set-up of the bridging principles rather the metalogic.
As you mention Restall, he has a paper “Multiple Conclusions” online which is very readable and might be of interest.
Thanks again for the talk!

• Liam Kofi Bright

11th July 2020 at 6:22 pm

Thanks for the talk, James! I was wondering if you think there’s any possible logic for equivocators style mental fragmentation that might save things for the poly-pluralist here. I can evaluate one and the same argument, but I am running mental tracks for each logic that I consider pertinent to the topic at hand, and hold separate the belief states they induce. (I think this would be to deny transmission in your framework.) I haven’t really thought this through, but it seems to cohere with some ideas in the previous philosophy of logic literature so might be feasible. What do you think? In any case, thank you again, very interesting! (Sorry if I misunderstood!)

1. James Skinner

11th July 2020 at 7:53 pm

Hi Liam,

Thanks so much for taking the time to watch, and for your question!

I’m not sure that appealing to mental fragmentation will help the poly-pluralist defend against NCO. Suppose that the pluralist runs different mental tracks for their different logics and that their contradictory beliefs about what they ought to believe are (for lack of a better word!) quarantined. Although this compartmentalisation may stop the poly-pluralist from seeing the contradiction, it does not banish the contradiction itself (i.e. they are still forbidden from and permitted to believe the conclusion). And this is all that NCO needs — that is, NCO centres on poly-pluralisms entailing contradictory claims about how agents ought to reason, irrespective of whether the agents realise that this is so.

1. Liam Kofi Bright

12th July 2020 at 2:52 pm

Yes, thank you, and I think that is quite right!

• Arvid Båve

12th July 2020 at 12:27 pm

Hi James,

thanks for this talk, very interesting! I was wondering about your claim that there is something wrong with someone believing and , but not . Isn’t it a standard response to the preface paradox that this is not true? This paradox is of course quite central to the question about the normativity of logic, as is the more general idea that it is sometimes rational to hold inconsistent beliefs when we don’t know which one should be dropped (as argued by Lewis).

A second question concerns what I thought would be the obvious response to your argument on behalf of pluralists, which is to go relativist about normative claims, so that, according to logic L, one ought to so and so, but not according to L’, and there is no such thing as its being absolutely true that one ought to so and so. Do you think such relativism is the only way out for pluralists?

1. Arvid Båve

12th July 2020 at 12:29 pm

Sorry, my pointy brackets weren’t formatted correctly, so I’ll try with parentheses: …something wrong with someone believing (p) and (q) but not (p and q)…

2. James Skinner

12th July 2020 at 3:37 pm

Hi Arvid, I’m glad you found it interesting!

The Preface Paradox is of course very central to the normativity of logic, and some have argued that it can be rational to hold inconsistent beliefs in some situations. However, VBP* is an evaluative norm — that is, one which states an *objective and ideal* standard for correct reasoning. As Steinberger points out in his paper Three Ways in Which Logic Might Be Normative’, the Preface Paradox only arises because we are non-ideal reasoners who lack the cognitive resources to figure out where we went wrong. But if VBP* is in the business of stating an objective and ideal standard for correct reasoning — rather than directing non-ideal agents how to reason — we need not worry that it demands that our belief sets be consistent. After all, from an objective and ideal standpoint, it surely is incorrect to have inconsistent beliefs! Another way of circumventing worries about VBP*’s strength is simply to note that, for NCO to get off the ground, VBP* need not hold unrestrictedly for all arguments — if it only held for simple arguments where it is within our cognitive capacities to infer the premisses from the conclusion, that would be fine as the arguments which the pluralist’s logics conflict over are all fairly basic (e.g. disjunctive syllogism).

I don’t think that distinguishing between different kinds of deontic operators is the only (or the best!) response the pluralist can give — I think there are many others (e.g. whether normative contradictions are the kinds of things that can be used in a reductio, what the appropriate metalogic is, change in logic change in meaning considerations etc). But I also have various thoughts as to why none of these alternatives are particularly attractive either! I hope these comments speak to your questions, and if they don’t please let me know!

1. Arvid Båve

13th July 2020 at 4:51 pm

Hi again and thanks for the full response! I think your point about the norms being objective and ideal probably settles that worry. As for relativism, I think I agree that it’s not an attractive option, but I guess it might still be the best one (best does not entail good!). On the other hand, I wasn’t sure about the argument involving deductive vs. inductive oughts, since that is a little different form the kind of relativism I had in mind, which just says that there is the classical ought, the intuitionistic ought, etc., but no absolute ought (as far as logical norms is concerned). But maybe that, too, would somehow fall prey to the kind of objection you raise.

• Joel Yalland

12th July 2020 at 1:51 pm

Hi James, thank you so much for your contribution, it’s been really informative. This isn’t an area I’m familiar with by any stretch, but I’m interested in how the acceptance of logic or logics, as well as logic/s itself bears on dialogue and particularly dis/agreement. That being said, hearing about disagreement within or between logics is a new avenue that hadn’t really occurred to me, so I found this a really fascinating and useful talk.

Considering that there are some areas where disagreements are irresolvable and others where we might desire not to resolve them or unify belief/attitude (ethics and aesthetics come to mind), it strikes me that we might want (even have) to maintain more than one logic to rule out closure or uniformity, which is either undesired or (perceived as) unnecessary.

With that in mind, I find myself most drawn to the CLCM objection and particularly its suggestion that pluralists could simply stick with being a mono-pluralist. Do you have any thoughts on the matter, or could you say a little more on this?

1. James Skinner

12th July 2020 at 4:08 pm

Hi Joel,

Good to hear that you found it useful and interesting! Since you said that your being drawn to the CLCM reply is based on the idea that we might not want to resolve logical disagreement, I’m going to focus on this idea — I hope that in doing so I’m not misunderstanding you!

I’m not super familiar with the epistemology of disagreement literature, but here are some thoughts. I agree that we might not want to resolve disagreements in some areas. The areas you mentioned were ethics and aesthetics and (I’m guessing!) the reason why we might not want to resolve disagreements in these areas is because it is commonly thought that, when it comes to matters of aesthetics and ethics, there is no fact of the matter. (I’m going to focus on aesthetics because I’m a moral realist and think that there is a fact of the matter in ethics!). So, for example, supposing that you think the Mona Lisa is beautiful and I find it revolting, there’s no point trying to resolve our disagreement because there is no fact of the matter whether the Mona Lisa is *actually* beautiful or revolting. By contrast, as soon as we do think there is a fact of the matter, it looks like we will want disagreements to be resolved. If you and I disagree over how many moons Jupiter has, at least one of us has a false belief, and given that beliefs aim at truth we ought to resolve the disagreement by figuring out what the truth is and believing it.

*If* what I have said above is correct, then we will not want to get rid of logical disagreements just in case there’s no fact of the matter whether a logic is correct. However, both the logical monist and pluralist are united in thinking that there is a fact of the matter — that is, both think that some logics are correct and others false (though they disagree over which and how many). The position that there is no fact of the matter, logical instrumentalism (logics are not the kind of things which can be correct/true, just more or less useful), is a separate position and one which pluralists stand in opposition to, and so this avenue of reply isn’t open to pluralists. All this is *very* tentative because I’m not that familiar with the disagreement literature, but I’d be really interested to hear what you have to say in response!

• Matthew McClure

12th July 2020 at 4:10 pm

Thanks for the very interesting talk! I have a couple of things to raise—hope they’re coherent and clear.

I. Are you too quick in dismissing the L₁ obligation vs. L₂ obligation reply? (Maybe you say more on this in the paper?) The polypluralist might grant that this doesn’t provide a satisfactory solution in this case & analogous cases, but that the trouble lies with the bridging principles not being rich enough: we might have it that S has certain L₁-obligations and L₂-obligations but the real normative question here is what obligations-sans-subscripts S has, and to get that, we need certain normative principles—e.g. a straightforward one that would do the work in this case, Oₙφ⊨Oφ. The worry at bottom is that dismissing this reply without motivating such a subscript-removing principle unfairly queers the polypluralist’s pitch.

II. Perhaps the argument is only going to work against a certain sort of polypluralist. One objection some polypluralists could make to the NCO argument is to reject the move from 12 and 14 to 15 as a valid instance of reductio/¬-in/&c. Since 12 and 14 are inconsistent (granting they aren’t subscripted), this is only going to work if all the correct logics are paraconsistent, but polypluralism qua polypluralism doesn’t commit one to the validity of an inference along the lines of φ ··· ψ, ¬ψ / ¬φ. Such a polypluralist might go so far as to say: ‘Yes, where we’re obligated to believe Γ, and Γ⊨₁φ but Γ/⊨₂φ, we’re obligated and not obligated to infer φ. Sure, we might fail to do this because it’s pragmatically irrational, very difficult, or indeed impossible, but _evaluative_ normativity requires it (just like it might be that we’re rationally obligated both to one-box and two-box in the Newcomb case, to use Graham Priest’s example). Rationality is difficult, and may give rise to dilemmata.’ (Hope this is relatively clear!)

Thanks again for the fab talk!

1. James Skinner

12th July 2020 at 8:51 pm

Hi Matthew,

Thanks for these great questions! Please correct me if I’m misunderstanding your first question, but I think it’s the same as Arvid Båve’s second question — if it’s not, would you mind clarifying for me, please? I’m not sure that I need to defend a subscript-removing principle because my rejoinder to this response was that positing the distinction was unfounded in the *first instance*, as it would be if one gave it in response to the Preface or Lottery paradoxes. For my money, positing distinctions between different kinds of deontic operators in response to NCO or the other paradoxes seems like a case of positing ambiguities when in trouble!

1. Matthew McClure

13th July 2020 at 12:08 am

Hi James,

My questions are indeed related to Gavin Thomson’s and Arvid Båve’s (hadn’t seen the comments when I wrote my question—never occurred to me that the comments might be _below_ the question form lol), though they are a little different, and perhaps importantly so.

With respect to (II), the analogy I made between the quagmire 12 and 14 land the polypluralist in and a classic dilemma is misleading—apologies for any confusion and thanks for clearing it up. The thrust of a (particular sort of) polypluralist’s objection to NCO as (II) has it is that all of the correct logics (including, of course, the metalogic NCO is formulated in) invalidate φ, ⸳⸳⸳ ψ, ¬ψ / ¬φ (or this sort of reductio). The polypluralist might then not have much to worry about logically here, since 12 and 14 don’t logically undermine polypluralism. She may coherently go so far as to just swallow the contradiction as a peculiar fact about evaluative normativity. (The closer analogy is with a case as the following: the law has it that: 1. all road users are obligated yield at yield signs, but 2. cyclists are permitted not yield at yield signs—imagine, for sake of plausibility, that these provisions came into being decades apart and noöne noticed the inconsitency. Then a cyclist is both obligated and not obligated to stop at a yield sign.) Worth stressing that this doesn’t quell the force of the argument against polypluralism when there is a correct explosive logic (as Beall and Restall have it, e.g.). I’m not denying that ‘[i]f one accepts that in (one of) the appropriate metalogic(s) reductio [of the form above] is valid, and that logic is formal, then one cannot make this reply’, but rather asking: what if one doesn’t accept reductio in the above form as valid? Perhaps this is little solace for polypluralists in general, but might be solace for a particular sort of polypluralist (one I feel some sort of sympathy for).

As regards (I), my question is distinct from Arvid’s. I’m not suggesting we relativize normativity to particular logics, nor denying that (as in the analogous preface case) qualifying the oughts doesn’t itself provide a satisfactory solution—best to think of it, as you suggest, as labelling the problem (though my intuition is that this itself might be helpful: a problem disambiguated is a problem halved). The suggestion _is_ that—if we’re to avoid begging the question against the polypluralist—we really need to make these oughts (in NCO) L₁- and L₂- oughts, and (thus) richer normative claims are required to get a contradiction of the sort the conjunction of 12 and 14 is supposed to be. One way to see it is by consider a bridging principle, e.g. if φ⊨ψ then O(sBφ⊃sBψ). For the polypluralist, this is ambiguous: ‘which turnstile do you mean?’, she asks (quite smugly, one imagines). Most natural reading of this sort of principle when we’re considering multiple consequence relations seems to me something like: if φ⊨ₙψ then Oₙ(sBφ⊃sBψ). _Then_ (once we’ve subscripted) we can consider what s is obligated-sans-subscripts to believe. For example, we might say that if there’s a valid argument from a set Γ of s’s beliefs to φ in _any of the correct logics_, s oughta (sans-subscripts) believe φ (I believe Stephen Read has articulated the gist of this); on the above principle, she’s already Lₙ-obligated to believe φ on her belief of everything in Γ, so the further normative principle goes something like: if ∃n(Oₙσ) then Oσ. Something along this lines seems to me necessary to properly articulate logical normativity for polypluralists.

Hope that clears the questions up. (Please don’t feel obligated to respond. I am an undergraduate and broadly ignorant of the literature, so there’s a fair chance my concerns here are misguided.)

1. James Skinner

13th July 2020 at 9:57 am

Hi Matthew,

Thanks for the clarifications. I think you’re right that there’s a very specific kind of polypluralist for whom NCO might not be a problem, so thanks for highlighting that! I do, however, think that this does little to attenuate the force of NCO. First, as you’ve noted, every pluralism on the market accepts some explosive logics and so NCO remains a problem for them. I’m also somewhat sceptical about how attractive a paraconsistent pluralism would be, on which only non-explosive logics are amongst the correct ones — many of the considerations usually used to motivate pluralism (e.g. resolving logical disputes so both sides can be right) seem to militate against such a position. Second, NCO *may* still be a problem for the paraconsistent pluralist: although reductio is invalid in paraconsistent logics, it may still be necessarily truth-preserving if one is in a consistent context. I’ll leave whether this is so as an open question. Third, not all paraconsistent logicians are dialetheists, and so simply swallowing the normative contradiction as you suggest may not be a price that all are willing to pay.

Turning to your other question, I’m afraid I’m still not quite seeing the difference between yours and Arvid’s questions. You say, “I’m not suggesting we relativise normativity to particular logics” but later say, “to avoid begging the question against the polypluralist—we really need to make these oughts (in NCO) L₁- and L₂- oughts”. The latter seems to be relativising the deontic operators to logics, which is what I am arguing is unfounded (the turnstile should of course be indexed to the appropriate logic, but we are talking about the RHS of the bridging principle). *If* the distinction between different deductive oughts were founded, and not simply an instance of labelling the problem, then some subscript-removing principle would be required and is perhaps defensible.

1. Matthew McClure

13th July 2020 at 1:17 pm

Hi James,

I think that satisfactorily answers my question about the move from 12 and 14 to 15.

On the other question, perhaps I can make it a little clearer how this is importantly different from Arvid’s question. On the presupposition I suggested, logic still yields unqualified oughts, just the ought _given by the bridging principle_ is indexed to the particular (correct) logic in question; we then supplement this with a principle connecting the qualified oughts to unqualified ones (what I called a subscript-removing principle). This allows us to articulate disagreements about what pluralists are committed to while keeping a simple and intuitive bridging principle. For example, consider case where: s believes everything in \Gamma, and \Gamma\models_{L_1}\varphi but \Gamma\not\models_{L_2}\varphi—is s obligated to believe \varphi? One might say ‘yes’ on the basis of the principle that if \exists L(O_L\varphi) then O\varphi. One might say ‘no’ (at least, not obligated by these validity/invalidity facts) on the basis of the principle that if \forall L(O_L\varphi) then O\varphi. I suppose we _could_—if we so choose—build this normativity into a bridging principle, e.g.: if \exists_{L}(\Gamma\models_L\varphi) then O(\forall\gamma_{\in\Gamma}(sB\gamma)\supset sB\varphi), but (as I said above) I think this distinction allows us to better articulate disagreements, and to have two simple intuitive principles rather than one complicated one. To have no quantification over correct logics at all just seems to me to ignore pluralism (not refute it).

To contrast, the relativist denies unqualified oughts altogether: on relativism, one is only ever L_1-obligated or L_2-obligated—never just obligated—to believe the conclusion of a single-consequence valid argument whose premisses one believes. On the distinctions drawn above, this is to say that the relativist rejects subscript-removing principles.

Hope that clarifies the distinction

2. James Skinner

13th July 2020 at 2:19 pm

Hi Matthew,

Glad that we’ve resolved your first question. Turning to your second, what you’re suggesting regarding indexing the deontic operators is the third objection I consider in the paper. To index the ought to’ operator to classical logic is to say something along the lines of, one classically ought to…’. (Maybe there’s a distinction to be drawn between indexing and relativising as you suggest, and I’ve been muddling them together. Apologies!). Either way, I think my original rejoinder that indexing the deontic operators is unfounded still stands, and so there’s no need to become embroiled in a dispute over the plausibility of a subscript-removing principle.

Your suggestion about quantifying over logics seems to me to be an altogether different worry, but a very interesting one! Loosely, if I’m understanding correctly, the idea is that the pluralist can circumvent NCO by maintaining that normativity supervaluates over correct logics (or, equivalently, that the subscript-removing principle quantifies over all correct logics). This is something I cover in the paper but unfortunately had to omit from the talk due to time constraints. There are two closely related reasons why I don’t think this will work. First, it seems to be incongruent with the very idea of pluralism and its motivations — that is, if the pluralist’s logics *really* are correct, they should be normative for reasoning (if logic is normative for reasoning at all). Second, and more worryingly for the pluralist, making this move collapses their position into monism (or nihilism, depending on how many logics they accept). If normativity supervaluates over correct logics, this is equivalent to saying that the only logic that is normative is the logic which is the intersection of all the logics the pluralist accepts. At this point, however, the pluralist has relapsed back into monism (this is an analogue of the collapse argument against domain-relative pluralism, except using normativity as the conceptual constraint rather than formality).

If you’re interested in the idea that normativity supervaluates (or subvaluates) over correct logics, I’d suggest reading Ferrari & Moruzzi’s Logical Pluralism, Indeterminacy, and the Normativity of Logic’! Hope that addresses your worries (finally!) — sorry it took me a while to figure out what you were getting at (if indeed I have!).

3. Matthew McClure

13th July 2020 at 3:19 pm

Hi James,
Thanks for the discussion—very interesting! Thanks also for the paper recommendation.

From what I get from what you say here, our disagreement about qualifying vs. relativising and over quantifying seems primarily verbal; seems to me little value in dragging it out. I see the force of your worries about the subscript-removing principle that requires qualified obligation according to all logics to get unqualified obligation.

One example that might be illuminating as to whether we are substantially disagreeïng is this: against quantification, you say that ‘if the pluralist’s logics *really* are correct, they should be normative for reasoning’—but this sounds to me like you’re implicitly assuming a quantified v.b.p. along the lines of: if \exists L(\varphi\models_L\psi) then O(sB\varphi\supset sB\psi). (And whether to quantify directly in VBPs/IBPs vs. in a standalone subscript-remover seems a matter of taste.)

Thanks again for the discussion, and your talk!