1. The non-triviality strategy

Broadly speaking, this paper is about the nature of scientific theories and the prospects for reasoned debate about their underlying logical principles. More narrowly, it is about ex contradictione quodlibet (ECQ), a form of argument that proceeds from contradictory premises to literally any conclusion you like.Footnote ¹

$$P,\neg P \therefore Q$$

ECQ is valid according to classical logic, but invalid according to paraconsistent logics (this is the defining feature of paraconsistency), hence advocates of paraconsistent and classical logics must adhere to radically different accounts of logical entailment. How do we know which side gets things right? Lewis (Reference Lewis, Priest, Beall and Armour-Garb2004), for one, is skeptical about the prospects for settling such disagreements rationally. He thinks there is no common ground between interlocutors.Footnote ²

Brute intuitions certainly won't help settle disagreement over ECQ, but perhaps there is another way forward. Perhaps the ‘knock-on effects’ of different logics, e.g. the way they interact with certain theories might indicate which is correct. Call a logic explosive if, like classical logic, it judges ECQ to be valid. According to explosive logics, an inconsistent theory is trivial in the following, technical sense: its logical closure contains absolutely every sentence whatsoever.Footnote ³ “[This] can be regarded as a theory without discrimination: it fails to discriminate between any sentence and its negation by including them all” (Michael Reference Michael2013: 351). Under a paraconsistent logic, inconsistent theories are considered non-trivial because ECQ is deemed invalid.

Some advocates of paraconsistency draw attention to inconsistent theories that working scientists found to be both interesting and productive. They contend that the underlying logic of such theories is paraconsistent. Michael (Reference Michael2016) refers to this method of trying to motivate paraconsistent logic as the non-triviality strategy (NTS). What makes NTS particularly interesting is its promise to motivate a logical system on contingent, empirical grounds rather than on the basis of philosophical intuition.Footnote ⁴ Michael argues, however, that this strategy fails in its purpose.

In this paper, I will argue that Michael's criticism significantly over-reaches. The fundamental problem is that he seems to be pulled in two directions, one of which is dialectically unhelpful. Michael nominally treats NTS with a measured hand, but more often than not he places more of a burden on the advocate of paraconsistency than on the advocate of explosive logic. This is most evident when he suggests that for the purposes of his critique, it suffices to show how ‘the classical logician, can countenance non-trivial inconsistent theories’ (Michael Reference Michael2016: 3349). This is wrong. The compatibility of classical logic with a given theory does not show that the underlying logic of that theory is, in fact, classical. The weaknesses in Michael's argument are symptomatic of this preferential treatment of one viewpoint in the debate over another. While he does make important observations in the course of his discussion, he draws the wrong conclusions. My goal is not so much to settle the debate over ECQ as it is to argue that this debate is still very much open.

The plan is as follows. ${\rm \S}$2 describes some allegedly inconsistent theories and argues that we should grant that these theories are inconsistent as per standard interpretation. ${\rm \S}$3 summarizes the dilemma that Michael presses against NTS and raises the possibility of a ‘third horn’ that may characterize some motives for paraconsistency. Michael claims that when NTS is understood as either a deductive or abductive argument, it fails either way. ${\rm \S}$4 evaluates the deductive horn of the dilemma and ${\rm \S}$5 evaluates the abductive horn of the dilemma. So as not to bury the lead, the core of the paper is the fifth section. Prior sections are largely devoted to setup as well as a fairly congenial evaluation of the first horn of Michael's dilemma. But the second horn of his dilemma fares poorly. In particular, I offer stringent resistance to the claim that abductive renderings of NTS fundamentally rest on confusion about the subject matter of logic. This is simply not the case and, as such, Michael's critique does not undermine efforts to support paraconsistency on the basis of inconsistent science. ${\rm \S}$6 concludes the paper and highlights the need for further debate.

2. Inconsistent science

NTS appeals to inconsistent theories as ‘drivers’ of the paraconsistent agenda. The thought that such logical lessons can be learned by paying attention to inconsistent theories crops up time and again. For example, Routley et al. (Reference Routley, Plumwood, Meyer and Brady1982: 58) reject classical logic ‘primarily on the grounds that there are non-trivial inconsistent theories which just cannot be discounted’. The best candidate theories of this kind are drawn from the formative history of science and mathematics. It is not crucial for present purposes to go into great detail about these theories, but I will briefly describe some of the important examples. The calculus in its earliest form is often said to be an inconsistent theory as is Bohr's theory of atomic structure that built upon the work of Rutherford and Planck. What makes these examples compelling, in part, is their undeniable contributions to the advancement of science.

According to a prevalent interpretation, when Leibniz and Newton first invented the calculus, they posited ‘infinitely small’ or ‘infinitesimal’ quantities with a curious dual nature.Footnote ⁵ Their methods required that distinct infinitesimals have different angles and slopes, but it also required that when two quantities are ‘infinitesimally far’ apart they are identical. In other words, it is important for some purposes that infinitesimals can be distinguished from zero, while it is important for other purposes that infinitesimals are indistinguishable from zero, so if δ is an infinitesimal, then δ = 0 and δ ≠ 0.Footnote ⁶ Yet as we know, this theory eventually spawned the entire field of analysis and stimulated the development of contemporary foundations of mathematics. According to an even more well-known interpretation, Bohr's atomic theory was a result of fusing together classical electrodynamics and the burgeoning quantum theory of radiation into an inelegant kludge.Footnote ⁷ Atoms were conceived of as miniature ‘solar systems’ with electrons orbiting a nucleus, coupled with an additional postulate taken over from quantum theory, viz. that electrons can only occupy discrete orbital states. The result is that the classical electrodynamic part of the theory says that an orbiting electron constantly accelerates and loses energy, while the quantum part of the theory says that it does not. Despite its awkward structure and ad hoc construction, Bohr's atomic theory successfully predicted the spectral lines of ionized helium and paved the way for the Standard Model of physics.

If you find it hard to believe that any scientist would knowingly endorse such blatant contradiction, you are not alone. Nonetheless, the readings sketched above are not of my own making, but are quite commonplace. It is, of course, still possible that such interpretations are mistaken. Since the early calculus and Bohr's atomic theory are amongst the most impressive examples of allegedly inconsistent scientific theories, it is important to know whether or not they are really inconsistent theories. There are philosophers on both sides of the interpretive fence. Hettema (Reference Hettema1995) and Ghins (Reference Ghins and Magnani2014), for example, argue that Bohr's theory is actually consistent on a careful reading of the evidence. There are two main reasons I will not address the question of ‘real’ inconsistency head on: first, it may be a badly formed question, and second, it is not relevant to the aspects of the debate I want to discuss.

Regarding the first point, Vickers (Reference Vickers2013, Reference Vickers2014) argues that any representation of the sentences of a theory will be a formal reconstruction that brings its own assumptions into play. Theories are living, growing things, so there will inevitably be a number of different, but equally good reconstructions of ‘the early calculus’ or ‘Bohr's atomic theory’ available. We can take a specific article, book, or set of correspondences as our touchstone for the contents of a theory, but such choices will always be somewhat arbitrary. Furthermore, Vickers holds that the interesting philosophical questions only come into play after we have identified a candidate set of sentences as ‘the contents of the theory’. Then we can ask whether those sentences played a role at some particular stage of scientific development, what work they did, and what attitudes scientists had toward them. Most importantly, we can answer these questions without a settled criterion for individuating theories or identifying the ‘real’ theory. It is, thus, ill-formed to ask whether a theory was ‘really’ inconsistent.

Regardless of whether we accept every facet of Vickers’ methodology, we can happily embrace the following lesson. There is no fact of the matter about the referent of a label like ‘the early calculus’ or ‘Bohr's atomic theory’. We can locate various reconstructions of the sentences of the theory and consider how those sentences may have been used, but other reconstruction will be possible. We should just grant that there is a perfectly good sense in which the target theories can be said to be inconsistent and move on to consider what this teaches us.

Seen in this light, the debate over NTS can be understood as a debate about whether the following conditional is true: ‘If such-and-such theory is reconstructed as an inconsistent body of information, then its underlying logic is paraconsistent.’ This is what proponents of NTS accept and their critics reject. We are concerned with the relation between a theory's content and its logic, not primarily with the content itself. This is my second reason for evading the question of ‘real’ inconsistency: it is largely irrelevant to the topic I want to discuss. As such, I will just assume that the common interpretation of the target theories is in good standing and focus on whether this has any bearing on logic.

Vickers claims that there is no impetus for such questions once we accept his outlook on theories, but this is puzzling. He claims that when scientists are aware that they are working with a theory that harbors inconsistency, they handle it with caution. Since they cease to trust what they infer, the logic of the theory does not inform their inferences: “ECQ does not threaten, because explosion can only come in via contradictories, and reasoning will be halted as soon as contradictories are derived. … At this point a scientist assumes, quite reasonably, that any further inferences will carry zero scientific value” (Vickers Reference Vickers2013: 104).Footnote ⁸ It is worth noting that this part of his view does not flow directly from his methodological stance, which simply prioritizes questions of how theories are used over their content. Putting that aside, however, I fail to see how the alleged skeptical attitude of scientists toward a theory obviates questions about its logical properties.

The salient fact is that scientists exhibit discriminate inferential behavior even when they work with inconsistent theories. They do not just infer everything from a contradiction. Vickers suggests that although this may look like a violation of ECQ, we needn't view it that way. We can accept an explosive account of logic and still explain scientists’ inferential behavior by appeal to their skeptical attitude toward inconsistent theories. A skeptical attitude could prevent the theorist from deploying the full force of logic in the context of a given theory. So, discriminate inferential behavior in the face of inconsistency could be compatible with the validity of ECQ. This is fair, but even if scientists were skeptical toward the early calculus and quantum theory, the suggestion above at best makes room for the possibility that such theories lived harmoniously alongside an explosive logic. The proposed explanation is worth further consideration, but in its current form it is hardly conclusive.

As far as I can see, then, even if we are primarily interested in how inconsistent theories are used, it may be worthwhile to think further about their logical properties. Several competing views of the matter are possible. Proponents of NTS contend that interesting and productive inconsistent theories are non-trivial, their underlying logic paraconsistent. Priest et al. (Reference Priest, Tanaka, Weber and Zalta2018) refer to such theories as ‘a most telling reason’ to accept paraconsistent logic. Some authors go further, claiming that paraconsistency is unavoidable once we recognize the usefulness of inconsistent theories: “[W]hat do we take to be the ontological commitments of an inconsistent theory? It is clear, and well-known, that in inconsistent settings like this, a paraconsistent logic is required” (Colyvan Reference Colyvan, Bueno and Linnebo2009: 162). This is not just a niche debate, either. Logical entailment is a theory-neutral relation, hence if the entailment relation lacks some feature locally with respect to certain theories, it lacks that property in general. If paraconsistent theories exist ‘in the wild’, then logical entailment as such is paraconsistent.Footnote ⁹

Michael (Reference Michael2013, Reference Michael2016) argues that this strategy for motivating paraconsistency fails because it rests on questionable assumptions about the nature of theories and logic. His critique takes the form of a dilemma directed at two possible interpretations of the argument involved in NTS. In the next section, I describe Michael's criticism in more detail and briefly consider a way out of the dilemma before moving on to a more careful analysis of the challenge posed to NTS.

3. Michael's dilemma

How are theories like the early calculus and Bohr's atomic theory meant to bear on our understanding of logic? When advocates of paraconsistent logic draw attention to these theories, what purpose does it serve? Given the surrounding rhetoric, it is natural to think that these theories feature as evidence in an argument that is meant to support the paraconsistent conclusion that ECQ is invalid. Michael considers two options: NTS might be a deductive argument or an abductive argument. He argues that if it is a deductive argument, then in the worst case scenario it is blatantly circular and in the best case scenario it relies on false assumptions about the epistemic value of theories. If it is an abductive argument, then it relies on false assumptions about the nature of logic. I elaborate on this dilemma in a moment, but I first want to explore the idea that some versions of NTS might not express an argument at all.

I have in mind dialetheic assertions of non-triviality that often seem to be intended not in the spirit of an argument, but in the spirit of an intuition pump. For example, Weber et al. (Reference Weber, Badia and Girard2015: 544) offer the following ‘proof’ that their preferred inconsistent set theory is non-trivial: “Either naive set theory is trivial or not. If not, we are done. If trivial, then, since this very proof is in naive set theory, it follows that the system is not trivial – since, after all, anything follows. QED.” They remark that while this might seem like a joke, it is not. They are simply ‘calling attention to the faith that we all put in the integrity of logical systems.’ But why emphasize these limitations, especially if it will not persuade anyone to accept their theory?

Perhaps certain kinds of attention to the properties of inconsistent theories are simply meant to pump our ‘paraconsistency intuitions’. Suppose that agents perceive any blatantly trivial theory to be nonsensical, i.e. that it is futile to cognitively engage with such a theory because it is ineligible to be true on structural grounds analogous to a word-salad sentence like ‘colorless green ideas sleep furiously’. This is an empirical conjecture on which I officially take no stand, but if it is the case, then drawing attention to a theory and elaborating on its commitments or metatheoretic properties – in short, stimulating the phenomenology of cognitive engagement – will make the theory seem sensible and, thus, non-trivial. Note, I am certainly not saying that cognitive engagement with a theory guarantees that it really is non-trivial. What I am saying is that cognitive engagement with a theory might cause us to feel as if it is non-trivial, which may impact our logical judgments about what is compatible with or follows from the theory. Just as the invitation to imagine a horse invokes our visual intuition, so the invitation to work with a theory may invoke our logical intuition. In that case, asserting that a theory is inconsistent but non-trivial may not be part of an argument, but more like a window through which to glimpse the dialetheic worldview.Footnote ¹⁰ This may even catalyze a gestalt shift in the receptive reader.

The point is that someone may draw attention to an inconsistent theory not as evidence of some further conclusion, but as an end in itself. Doing so may provide the reader with insight or understanding that is otherwise unavailable.Footnote ¹¹ Having said this, however, I set it aside. Since we are concerned with rational debate about paraconsistency, it is somewhat beside the point that non-evidential uses of inconsistent theories are possible. From the perspective of textual interpretation, it also seems wrong to read statements of NTS as intuition pumps meant to coax us into a ‘paraconsistent way of seeing things’. Proponents of NTS do not speak as if they are constructing a new way of seeing the world, but as if they are showing us what is already there. The appeal to the early calculus or Bohr's atomic theory is meant to convince the otherwise unconvinced reader that explosive logics get the ‘logical facts’ wrong. Routley et al. (Reference Routley, Plumwood, Meyer and Brady1982) and Colyvan (Reference Colyvan2008a, Reference Colyvan2008b, Reference Colyvan, Bueno and Linnebo2009) most clearly express such inentions when they suggest that paraconsistency is unavoidable in light of inconsistent science, but related arguments are also advanced by da Costa and French (Reference da Costa and French2003) and Brown and Priest (Reference Brown and Priest2004, Reference Brown and Priest2015) to name a few.

This clarifies the issue that most matters for present purposes, viz. whether an argument that starts from the evidence we have about scientific theories can support conclusions about their logical properties. Framed in such general terms, it may be difficult to reach any fully satisfactory view of the matter. Michael does, however, make a noble attempt to divide and conquer the relevant options, so we will follow his lead and see how far it gets us. The first horn of Michael's dilemma takes aim at interpretations on which the argument involved in NTS is understood to be a deductive argument. In the next section, I elaborate on this horn of the dilemma and consider what it shows us about deductive versions of NTS.

4. The deductive horn

We have seen how some proponents of NTS forcefully assert that paraconsistency is unavoidable in light of inconsistent science. This is highly suggestive of a deductive inference from one to the other. What do the steps of such an argument look like? Michael (Reference Michael2016: 3350) rightly points out that the following argument is problematic.

On this rendering, the non-triviality of theories like the early calculus and Bohr's atomic theory is taken for granted as a premise. Michael does not take issue with this argument on the grounds that its premises fail to support its conclusion. Quite the opposite: if there are inconsistent theories whose logical closure does not contain every sentence, it does follow that ECQ is invalid. The problem is that the premise and conclusion of this argument are just two ways of saying the same thing.Footnote ¹² In the context of a debate about the logical properties of certain theories, this argument begs the question by assuming what it is trying to establish.

This is reminiscent of Lewis’ worry that interlocutors in such a debate always beg the question against one another. Yet, for all that we have said so far, it is not clear that we have reached an impassable stalemate. There may be another way to understand NTS as a non-circular deductive inference to the conclusion of non-triviality based on some other properties of a theory. What property of the theories under consideration could guarantee that they are non-trivial in the logician's technical sense? Michael frames this part of his discussion in terms of an ambiguity between the technical notion of ‘trivial theory’ that we have been using and a more familiar notion of ‘trivial theory’ that means something like ‘uninteresting’ or ‘useless’. He then considers whether the notions are appropriately related. In ${\rm \S}$2, we happily took for granted the standard examples of interesting and productive inconsistent theories, so we are now talking about whether that feature of a theory guarantees it to be non-trivial in the technical sense. Rather than focusing on ambiguity in the word ‘trivial’, however, let's just be clear that we are concerned with the question of whether the epistemic value of a theory is facilitated by properties of its logical closure.

Michael argues that there is no way to bridge the inference from one to the other, thus deductive versions of NTS are doomed to fail. But if we understand NTS as a deductive argument, why assume that its premises must take this particular form? Granted, the epistemic value of the scientific theories under consideration is widely acknowledged and conspicuous. Perhaps the most notable thing about the early calculus and Bohr's atomic theory (apart from their inconsistency) is that they were interesting and productive theories. Still, this does not mean that a deductive version of NTS must use this information as its main premise.

Having said that, I do think Michael is right to interpret the argument in this way, but he doesn't say why this is the case, so let me try to fill in that gap. Remember, we are thinking about how NTS can be understood as a deductive argument leading to the paraconsistent conclusion that ECQ is invalid. This argument can either proceed directly or indirectly. By a direct argument, I mean a proof that a given inconsistent theory is non-trivial and, thus, must be closed under paraconsistent logic. Since logic is under debate, however, any such alleged proof will be epistemically circular, i.e. it will rely upon the very methods it is trying to vindicate.Footnote ¹³ After all, such a ‘proof’ would have to comply with the paraconsistent judgment that ECQ is invalid, otherwise it would easily be able to show that the target theory is, in fact, trivial. What about an indirect tact? Are there any premises about inconsistent theories that could feature in such an argument apart from the ones we already mentioned, i.e. those that appeal to the epistemic value of such theories? Perhaps a route from semantics is possible. If the meanings of expressions and their logical roles are tightly inter-woven, then semantic data can support conclusions about logical principles. The present evidence, however, does not seem to indicate anything special about the linguistic resources of inconsistent theories. Put another way: did the innovative contributions of the calculus or quantum theory essentially revolve around how they treat the meaning of negation? I am not saying that there is no semantic data anywhere that would support non-classical logical conclusions. What I am saying is that the data from inconsistent science does not seem to be semantic data of this kind.

That leaves one option. If NTS does express a non-circular deductive argument, its premises must appeal to the only noteworthy, impartial evidence we have about inconsistent scientific theories: some of them are interesting and productive. The key steps of a deductive argument along these lines are as follows.Footnote ¹⁴

On this interpretation, the claim of non-triviality is a sub-conclusion in a longer line of argument. Again, Michael does not take issue with this argument on the grounds that its premises fail to support its ultimate conclusion. From the sub-conclusion above, it follows that our exemplary interesting and productive inconsistent theories are, in fact, non-trivial, which shows that ECQ is invalid.

Michael claims, however, that the first premise of the argument above is false. He argues that features that imbue a theory with epistemic value can come apart entirely from its logical closure. For one thing, the conceptual repertoire of a theory in and of itself can be of great interest to scientists.Footnote ¹⁵ For another thing, science often makes advances by comparing and contrasting how different theories are defined. None of this requires theories to be non-trivial. We can find value in dissecting a set of concepts even if they turn out to conflict with one another, and we can find value in comparing different axiom sets even if they ultimately result in the same predictions. This shows two important things about trivial theories. First, a trivial theory can be interesting in some respects well after it has become clear that the theory cannot be true. Second, equally trivial theories can differ in ways that makes one of them more interesting to us than the other. Michael concludes that properties of a theory's logical closure are far from the only thing that contributes to its value.

He and Vickers are both especially interested in idealizations, which make a good case for scientific interest in theories that are literally false. For example, we might construct a model of the ocean in a way that entails it is infinitely deep because the actual limits on its depth do not matter. Perhaps we want to model the path of currents and the exact depth of the ocean will make no difference to our calculations. Then it is unimportant how our model represents the ocean's depth, even if it absurdly represents it as infinitely deep. As we know, the only perfect map of a country is the entire country itself, but this does not make for a useful map.Footnote ¹⁶ Idealizations always distort something and, as such, have literally false commitments, but we can still make predications with them. The trade off in simplicity is often worthwhile because it makes applications of the model more efficient.

What does this have to do with the present debate? These observations about idealizations are meant to show that scientific theories can be valued for many reasons, not all of which have to do with the truth of their commitments. In particular, idealizations are theories that we value purely for their predictive strength and, as such, we are willing to trade in some degree of accuracy. This is meant to show us that the first premise of the argument above is false.

These observations, however, center entirely around the engineering virtues of theories, which prompts a further question. A theory can, of course, be valued because it gives definitions that encode certain intuitions or because it gives an inaccurate model that is accurate enough for some application, but these are highly localized features. What this fails to address is the veritistic virtue or ‘truth-likeness’ of theories that accrues from the accuracy of their overall picture of the world. What differentiates theories in this respect? Notably, the veritistic virtue of a theory is a global property that intimately depends on its logical closure. In this respect, the comparison with idealizations is a misleading analogy because aspects of an idealization that can safely be ignored are explicit, but that is not the case with live, scientific theories. This aspect of epistemic value is important to the present debate because interest in the early calculus and Bohr's atomic theory did not arise from the way those theories were engineered. They were the best candidates at the time for capturing the literal truth about their target phenomena. Even if we think that these theories are not actually true, this is irrelevant. The point is that working out what was true according to such theories played a significant role in their epistemic value.Footnote ¹⁷

The pressing question, then, is: Why do we find the majority of trivial theories so useless in this regard, while others rank highly in veritistic virtue? No answer to this question is offered in his (Reference Michael2016) paper, but in an earlier paper, Michael claims that the difference lies in the immediate consequences of theories: “[I]ntroduce the notion of immediate consequence on the basis of explicit rules of proof. … That such a distinction is necessary is all the classical logician needs to explain how false and even inconsistent theories can be distinguished and so how some among them can be useful” (Michael Reference Michael2013: 360).Footnote ¹⁸ The thought here is that a few steps of deliberate, incremental inference may lead to interesting predictions in the context of one theory, but lead directly to outright absurdity in the context of another theory. This is meant to show that the immediate consequences of trivial theories can differ and this may contribute to a difference in their veritistic virtue. If so, then Michael's criticism of deductive versions of NTS is spot on: there are no principles linking the epistemic value of a theory to properties of its logical closure.

The problem is that it seems unlikely that the notion of immediate consequence can do the work it is called upon to do. What makes one sentence a ‘more immediate’ consequence than another? This is a notoriously thorny question to which we have not been offered a wholly satisfying answer.Footnote ¹⁹ Michael says that a more immediate consequence is one that results from a shorter proof, where the length of a proof depends on how it is constructed from a set of primitive rules. The same extensional account of provability can, however, be axiomatized with different rules in different frameworks such as Hilbert systems, Fitch systems, and Gentzen systems. Which rules in which framework are primitive for these purposes and why do those rules play such an important epistemic role? The plausibility of Michael's account of veritistic virtue depends on an answer to this question since different rules will identify different consequences as more or less immediate by the above definition. The claim that trivial theories vary in truth-likeness is meant to undermine deductive versions of NTS by severing the link between epistemic value and logical closure, but without further details it is hard to judge the success of this critique.

To sum up, Michael argues that if NTS expresses a deductive argument, it is either viciously circular or relies on false premises about the epistemic value of theories. His argument is not airtight, but he has raised problems that must be addressed by any attempt to deductively infer that the underlying logic of interesting and productive inconsistent theories is paraconsistent. The second horn of Michael's dilemma takes aim at interpretations on which the argument involved in NTS is instead understood to be an abductive argument. In the next section, I elaborate on this horn of the dilemma and consider what it shows us about abductive versions of NTS.

5. The abductive horn

Some proponents of NTS envision a much a looser argument from inconsistent science to paraconsistent logic. For example, Brown (Reference Brown1992: 398) admits to having modest aspirations for his paraconsistent model of ‘old quantum theory’ (OQT), i.e. Bohr's atomic theory: “What I aim to do is to present some evidence that there really was (on the part of some scientists, at least) a substantial cognitive commitment to OQT, and that some of its characteristics have a simple and straightforward explanation in terms of the model I propose.” The claim is not that inconsistent theories must be underwritten by paraconsistent logic, as if we can envision no alternative view about their logical properties. Rather, the claim is simply that a paraconsistent model is better than an explosive one. This is a clear expression of an abductive argument or inference to the best explanation, but what do the steps of such an argument look like? What can possibly be explained by logic? The salient fact that Brown fixes upon and that came up once earlier in our discussion is the fact that scientists exhibit discriminate inferential behavior when they use inconsistent theories. Roughly speaking, an abductive interpretation of NTS proceeds as follows.

The conclusion is put forth as the best, most likely explanation of the data we have about inferential practices in science. There is obviously a lot packed in here, so we need to think further about the strength of this inference.

The conspicuous ‘gap’ in this argument is the move from inferential behavior to conclusions about logic.Footnote ²⁰ This move presupposes that logic regulates reasoning in some sense (otherwise there is no explanatory relation between them). Stated thus, however, we have only a slogan: ‘logic regulates reasoning’. What does it really mean? What is required to warrant the conclusion of such an argument? A quick and dirty way to think about it is as follows: deductive logic is supposed to tell us about correct deductive inference; explosive logics specifically tell us that we should infer any and all conclusions from inconsistent premises; but this appears to be wrong because scientific practice bears witness to discriminate inference in the face of inconsistency. Michael thinks that abductive versions of NTS fail precisely because they depend upon the kind of assumption just sketched and “… it is not true that if the underlying logic is classical then a rational agent will simply infer any and all consequences of a theory they believe is inconsistent” (Michael Reference Michael2016: 3360).

Useful inference serves our epistemic goals. What one should infer, thus, depends in part on one's goals. Agents can have different goals in different contexts, so any principle that implies otherwise is inherently questionable. When we say that deductive logic tells us about correct deductive inference, is that correctness always and everywhere? It is questionable that such absolutely general norms exist, but there are more concrete problems as well. Consider addition, the form of argument that proceeds from a premise P to the disjunctive conclusion P∨Q, where Q can be anything you like. A wide range of paraconsistent and explosive logics agree that addition is valid, so this is a nice, neutral example. Michael points out, however, that it is widely acknowledged that addition never serves our epistemic goals:

There seem to be conclusions that are entailed by certain premises, yet it is not the case that any agent who believes the premises are true should infer that conclusion is true. In fact, the assumption that entailment is truth-preserving in itself is enough to see that this is the case. Suppose that P entails Q and, until recently, I believed that P is true, but I just learned that Q has been refuted beyond any reasonable doubt. In such circumstances, I obviously should not insist that Q is actually true just because it follows from my current set of beliefs; I should revise my beliefs and, in particular, reject P. If it is not always rational to infer the logical consequences of a theory, then nothing about logic can be learned from how a theory is used.

Michael argues that abductive versions of NTS make this mistake because they falsely presuppose that logic regulates reasoning. He diagnoses this as the result of conflating the theory of logical entailment with the theory of ‘correct inference’. One is about the static relation of truth-preservation between sentences, the other is about the dynamics of belief-maintenance: “[L]ogic does not govern the dynamics of commitment but rather its statics. Logic is not a theory of what I should infer but rather what I am committed to, that is, what follows from my explicit commitments” (Michael Reference Michael2016: 3358). In saying this, he echoes Harman (Reference Harman1984, Reference Harman1986) who famously challenged the normative bearing of logic on reasoning.Footnote ²¹

Much of this is right and important. In particular, it is good to be clear about the conceptual difference between entailment and inference. Contrary to what Michael says, however, these observations do not undermine abductive versions of NTS. The mere fact that entailment and inference are distinct does not show that there is no explanatory relation between them. Indeed, the foregoing discussion has at most shown that one specific explanatory assumption regarding logic and inference is false. The phenomena of useless-to-infer consequences and rational rejection of prior beliefs are counterexamples to the following principle:

But this (Ought) principle represents just one possible way to understand the slogan ‘logic regulates reasoning’ and there may be other, more nuanced, and more plausible ways of understanding that slogan. In order to undermine abductive versions of NTS as such, Michael would need to establish the falsity of every relevant principle relating entailment to inference, which he has not done.

Since he explicitly draws on Harman's celebrated work, perhaps Michael thinks there is no need to argue this point himself because Harman already established long ago that logic has no normative bearing on reasoning. That, however, would be a mistake. With respect to the abductive argument we are considering, the important question is whether there is any relevant explanatory relation at all between entailment and inference. In a widely read unpublished paper, MacFarlane (Reference MacFarlaneMs) identifies a variety of putative normative principles that could hold between logic and reasoning. They vary with respect to the kind of mental states that are subject to logical norms, as well as the kind and scope of deontic modality that applies those states. Steinberger (Reference SteinbergerForthcoming) argues that the plausibility of these principles depends, in part, on logic's normative role: logic might provide first-personal directives for belief revision, it might provide standards by which to evaluate belief states, or it might provide the grounds from which to apportion epistemic praise and blame. While slightly different from one another, each of these types of norms is pertinent to rational belief-maintenance. If logic is unfit to play one of these roles in the cognitive economy, it may yet be fit to play another. The description he gives of his project in Harman (Reference Harman2009: 333) makes clear that he has always been concerned with directive norms, i.e. principles intended to capture the way that recognition of logical entailment guides the process of belief revision. He specifically challenges principles of this kind, so at most his arguments show that logic does not regulate reasoning in the specific sense of providing first-personal directives for belief revision. Despite this, some other kind of normative relation between entailment and inference may hold.

When he first introduces NTS, Michael describes it as ‘a challenge to the [explosive] logician’ to show how they can ‘countenance non-trivial inconsistent theories’ (Michael Reference Michael2016: 3349).Footnote ²³ Unfortunately, this way of framing the issue betrays a biased perspective from the start. Michael grants that NTS may express an abductive argument. So understood, however, the appeal to inconsistent science poses a far greater challenge to the explosive logician than that described in the quote above. The real challenge is to show that explosive methods of handling inconsistent theories are the best explanation of the way those theories are actually used. It is not enough to show that inconsistent theories could in some way be countenanced in the setting of an explosive logic. That does not settle anything. At best, it establishes one explanatory option in the debate. Michael sometimes carries this thought so far that it leads him to make claims like: “[T]here are ways in which a rational agent can use an inconsistent theory even when their underlying logic is classical. For that reason it is not true that the explanation in terms of a paraconsistent underlying logic is the best explanation of the inferential practices” (Michael Reference Michael2016: 3360). This is simply wrong and deeply problematic. If we take the contrapositive of this inference, it literally says: the only way in which a paraconsistent explanation can be the best explanation is if there is no alternative, i.e. no way in which a rational agent can use an inconsistent theory when the underlying logic is classical (or explosive). That is asking too much in the context of an abductive debate: the best explanation needn't be the only possible explanation. Michael seems to draw hasty conclusions about the viability of abductive versions of NTS because he gives preferential treatment to explosive logics.

To some extent, what I have just done is attempt to diagnose the cause of hastily drawn conclusions. I may be wrong. This is all speculative. And the quality of the argument itself is what really matters, not the psychological underpinnings of its author. Even if my diagnosis is off the mark, however, I hope to have made clear that Michael's critique does not undermine abductive versions of NTS as such.

To drive this home, I will briefly sketch an abductive version of NTS that entirely resists Michael's critique. We need two main ingredients for this argument. The first ingredient is the claim that logic regulates reasoning in science according to the following, conservative epistemic norm:

For my envisioned argument to work, this principle would have to be an objective background fact about scientific practice. I say that (Reason) is a conservative principle because, in virtue of the fact that every sentence entails itself, this principle implies that if an agent believes P, they have some reason to believe P. Hence, according to this principle, current scientific belief always has some degree of epistemic support.

(Reason) does not conflate entailment and inference, it simply states that they are related in an interesting way. It also respects the previous observations about rational inference, viz. that it can be useless to infer some of the logical consequences of our beliefs and that sometimes we ought to revise our current beliefs rather than accept their logical consequences. (Reason) does not conflict with these observations, so it does not imply that agents have unexpected or untenable epistemic burdens. Having defeasible reason to believe a sentence is much weaker than being rationally obligated to believe it and certainly does not override all considerations that might count against believing it. To the extent that (Reason) is independently plausible, it is perhaps best understood as a source of epistemic appraisals. According to this principle, when P entails Q and I believe P, then in virtue of my beliefs, I apparently have some responsibility to respond to evidence weighing against the truth of Q. If I am in such a position and someone calls my attention to countervailing evidence against Q and I am indifferent, i.e. behave as if I have no stake in the issue, then I can be blamed for shirking my epistemic responsibilities.

The second ingredient we need is the claim that scientists’ discriminate inferential behavior when using inconsistent theories can be explained by the following posit: that competent scientists recognize that even if they believe an inconsistent theory, there can be sentences that they have no reason whatsoever to believe. Call this the ‘no reason’ explanation. It is not difficult to show that this is at least a plausible explanation. The claim of epistemic competence can be contested if we cast the net of ‘science’ too widely, but Newton and Bohr seem to count amongst the best thinkers in history. As for the second part, this is simply a claim of independence and can be motivated by considerations that were already raised by Michael. Useful inference serves our epistemic goals. Scientific theories usually have explicitly narrow subjects, e.g. the membership of the State Council of South Korea is of no interest to the particle physicist. If a sentence does not fall within the remit of a theory, then the theory can neither give reason for or against believing that sentence.

What makes the envisioned version of NTS truly abductive is that the inference to paraconsistency goes by way of implicitly selecting the ‘no reason’ explanation as best amongst the alternatives. To fill out the details of this argument, I would say more about the range of alternative explanations on offer. Vickers (Reference Vickers2013) proposal that scientists’ inferential behavior is explained by their skeptical attitude is one alternative worth considering. Since it is not my goal to settle the debate over ECQ in this paper, I am not going to argue that one of these explanations is the best all things considered, but we can imagine the factors that could weigh in one direction or the other. Take an arbitrary sentence. Do scientists who endorse inconsistent theories behave as if they are unhappily committed to this sentence or are they indifferent to it as if they have no reason to entertain it one way or the other? All that matters for present purposes is that we can appreciate the overall strategy of the envisioned argument: there are various possible explanations for why scientists exhibit discriminate inference in the face of inconsistency, but the proponent of NTS sees the ‘no reason’ explanation as the best explanation on grounds such as its simplicity and depth.

This, in turn, warrants the conclusion that the underlying logic of such inconsistent theories is paraconsistent if (Reason) is independently true. Given that we have evidence that one can believe an inconsistent theory without having reason to believe just anything, the most likely truth of the matter is that such theories are non-trivial, their underlying logic paraconsistent.Footnote ²⁵ While this argument does presuppose that logic regulates reasoning in some sense, we have shown that it does not rely upon assumptions that are vulnerable to Michael's specific line of criticism.

To sum up, Michael argues that if NTS expresses an abductive argument, then it relies on false assumptions about the nature of logic. He argues that scientists’ inferential behavior in the face of inconsistency does not rule out explosive logic, but this is irrelevant. The best explanation needn't be the only possible explanation. Michael accuses proponents of NTS of confusing the theory of logical entailment with the theory of ‘correct inference’, but he is confused on this matter: an explanation needn't assume that its explanandum and explanans are identical. He has not shown that abductive versions of NTS fail in their purpose.