Kevin Scharp (2005; 2013) argues that truth is an inconsistent concept and so should be replaced.⁵ The case of truth is not our concern here. We just want to know what he means by ‘inconsistent concept’. To illustrate, Scharp invents the helpful artificial example of a ‘rable’ (2013: 36):

In what sense is rable inconsistent? In some cases, no inconsistency arises. Brown table? Rable. Red couch? Not a rable. But what about a red table? Here ‘rable’ applies, since it’s a table. But it also disapplies, since it is a red thing. So—and here’s the inconsistency—the red table must be a rable and must not be a rable. So a rable-user must either accept a contradiction or else deny the manifest fact that there are red tables. Having rable in one’s conceptual repertoire, Scharp concludes, “corrupts it in a certain way” (2013: 36). We’ll call a concept S-inconsistent if its correct application with respect to some actual facts (like the fact that there is a red table), leads to contradiction.⁶

S-inconsistency may be familiar from another artificial example: ‘tonk’ from Prior (1960). Given the ‘or’-like introduction rule of ‘tonk’, from “Arthur Prior was a philosopher” we can infer “Arthur Prior was a philosopher tonk it’s raining and it is not raining”. And from that, given ‘tonk”s ‘and’-like elimination rule, we can infer “It’s raining and it’s not raining”. So ‘tonk’ let’s us go from a true sentence to a contradiction.

But S-inconsistency should not be assumed to be limited to artificial examples. There’s a plausible case to be made that it applies to real concepts not invented by philosophers, such as concepts associated with certain normatively laden terms, like boche (Dummett 1973: 454) or chaste (Eklund 2017: 175), as well as some theoretical terms from incorrect scientific theories, such as (Newtonian) mass (Scharp 2013: 37). Supposing, for example, that boche disapplies to x if x is not barbarous, but applies to x if x is of German nationality, then the fact that there are Germans who are not barbarous will mean application of the concept in accordance with its constitutive rules leads to contradiction.

Before we move on, there are two things to emphasize about S-inconsistency and Scharp’s understanding of it. The first is that it’s understood by Scharp in an inferentialist way, in the sense that the concept and its possession is constituted by rules of use and inference that possessors of the concept are related to in some special way. Scharp says of (1-a) and (1-b) that they

Later in the chapter he proceeds to refine this proposal, ending up with a more sophisticated account of what it takes to possess the concept in terms of defeasible entitlement, but it is still an inferentialist account.

The other thing to note about S-inconsistency is that it involves a predicate having a bilateral meaning. The predicate ‘rable’ is given both application conditions and disapplication conditions.

S-inconsistency is a good candidate for a concept’s being semantically defective. It does seems like a defect, and it applies to concepts construed in the thin, content-grasping-ability sense we are using. But what if we reject inferentialism?

The main competitor to inferentialism is what I’ll call standard representationalism, which consists of two components: representational semantics and messy metasemantics. The former involves characterizing contents in representational terms—such as in terms of propositions with truth conditions. The latter involves denying that rules of inference or use play any straightforward role in determining concepts’ contents or their possession conditions. On standard representationalism, the meaning of a word or concept is just what that word or concept contributes to a representational content,⁷ not anything directly involving rules for its application, disapplication, or inferences involving it. And given messy metasemantics there’s no particular rules that an individual must accept (or be disposed to infer in accordance with, or what have you) in order to use that word or possess that concept.

It is plausible that S-inconsistency is unavailable to the standard representationalist, as Cappelen (2018: 85, 142) concludes. After all, if there are no constitutive rules, then there are no inconsistent constitutive rules. It seems that if representationalists want to call concepts semantically defective, they will need an alternative account.⁸ As I’ve said, I think this is incorrect. I’ll argue one can be a certain kind of standard representationalist, but still go in for S-inconsistent concepts. However, I do think S-inconsistency is unavailable to some common forms of standard representationalism, and it will be helpful to see why.

According to what I’ll call unilateral representationalism, contents are exhausted by a single, unilateral entity. On one way of doing this in terms of possible worlds, we take contents to be intensions:⁹ for a declarative sentence, the content is a set of worlds (those worlds where the sentence is true), for a predicate, a function from (possible) individuals to sets of worlds (the set of worlds where the given individual satisfies the predicate), and so on. Applying this to concepts, the content of a predicative concept like cat is a function which takes an individual and returns the set of worlds where that individual is a cat. If the actual world is in this set, the individual in question is actually a cat, otherwise it is not.

On such a view, there’s no way to get S-inconsistency going. For any given thing, the actual world is either in the set that the concept returns or it isn’t. So applying rable to a given red table either returns a set with the actual world or it returns a set without the actual world. If the former, then it’s true that the table is a rable. If the latter, then it’s false that the table is a rable. Either way, it won’t end up both true and false that it’s a rable. So either way, rable is not S-inconsistent. And the same goes for any other concept, as far as this framework is concerned. So if you’re a unilateral representationalist of this sort, you cannot appeal to S-inconsistency as the source of semantic defects.

A familiar variant of standard representationalism does not allow for S-inconsistency. This may be taken to confirm that standard representationalism itself excludes S-inconsistency. But in fact, as we’ll see in the next section, if a standard representationalist gives up unilateralism, they can allow for S-inconsistency.

Bilateralism is the view that meanings come with two components: a positive part and a negative part. An inferentialist can be a bilateralist by claiming meanings are composed of application and disapplication conditions, or, for sentences, assertibility and deniability conditions. This flavor of inferentialism has a variety of uses.¹⁰ Scharp relies on just this kind of bilateralism to give an inferentialist treatment of S-inconsistent concepts like ‘rable’. It is having application and disapplication conditions applying in the same case that makes for inconsistency.

Bilateral representationalism is less common than bilateral inferentialism, but it is not without its proponents and has considerable attractions. For the representationalist, the meaning of an expression is just some representational content, or a contribution to a representational content. To be a bilateralist, then, what the representationalist needs is a kind representational content with a positive and negative part. There are various ways to do this, but the one I find most attractive is based on the truthmaker semantics of van Fraassen (1969) and Fine (2017b). The meaning of a sentence, on this view, is not the set of worlds where the sentence is true, but rather the set of potential facts which would make that sentence true, were any of them to obtain. The fact that I have the specific red couch I do is a truthmaker for (2).

But there are also plenty of other potential facts—actual or otherwise—that make this sentence true, or would make it true if they obtained. As a first pass, we take the meaning of (2) to be the set of all those potential truthmakers. Since some of them are actual, the sentence is actually true.

So far, there’s nothing bilateral here. To get that, we want to talk about falsehoodmaking in addition to truthmaking. (3), for example, is false.

But what makes it false? Well, the fact that I have a red couch does, as well as lots of other similar facts about others and their red couches.¹¹ A bilateral truthmaker semantics takes falsehoodmakers—actual or potential—to be included in the meaning of a sentence. So the meaning of (3) will include its potential truthmakers, which are presumably a bunch of facts that do not actually obtain, but it will also contain its potential falsehoodmakers, among which are the fact that I have a red couch.

If we are to have a reasonable theory, though, we cannot just lump the truthmakers and falsehoodmakers together. There’s a big difference between whether some fact is a truthmaker for a sentence or a falsehoodmaker for it. The meanings need to specify, then, which facts are the would-be truthmakers, and which are the would-be falsehoodmakers. So instead of taking the meaning of a sentence to be a set of potential facts, we take it to be a pair of such sets: the set of truthmakers and the set of falsehoodmakers. Meanings, on this view, are bilateral, and provide both a sentence’s truthmaking conditions and its falsehoodmaking conditions.

This view is different from the more familiar unilateral representationalism, of course, but it is still a bona fide version of standard representationalism. The meanings it posits are representational: the meaning of a sentence represents how things are if the sentence is true, it just also represents how things are if the sentence is false, and uses finer-grained entities than possible worlds to do so. And there is no specification of constitutive rules and no more need here to deny messy metasemantics than there is for the unilateral representationalist. But more on that momentarily. First let’s see how this view can accommodate S-inconsistent concepts.

Recall that an S-inconsistent concept is one whose correct application with respect to some actual facts leads to a contradiction. From a truthmaker semantics perspective, what this amounts to is some actual facts implying that some sentence S both has an actual truthmaker (and so is true) and an actual falsehoodmaker (and so is false).¹² Given the standard truthmaker semantics for conjunction and negation, this would entail that S ∧ ¬S has a truthmaker, and so is true.¹³ Now we ask: could a concept have a content that results in this?

In addition to the ordinary interpretation function ⟦·⟧ that will take an expression to its bilateral meaning, let’s introduce a function ⟦·⟧+ that takes an expression to the positive part of its meaning (in the case of a sentence, the set of truthmakers) and a function ⟦·⟧− that takes an expression to the negative part of the content (in the case of a sentence, the set of falsehoodmakers).

The meaning of a predicate should be the kind of thing that when combined with an individual denoting expression (or generalized quantifier) returns a sentence meaning, a bilateral truthmaker/falsehoodmaker proposition in the system we’re considering. Applying ⟦·⟧+ to a predicate F should give us a function from (possible) individuals to sets of potential facts—the potential facts of the given individual’s being F. Applying ⟦·⟧⁻ should again give us a function from individuals to sets of potential facts, but this time those of the given individual’s not being F.

So, starting with an ordinary predicate, ⟦table⟧+ is the function from individuals x to the set of truthmakers for “x is a table”. ⟦table⟧− is the function from individuals x to the set of falsehoodmakers for “x is a table”. Assuming that table is not an inconsistent concept, there is no individual x such that “x is a table” has both an actual truthmaker and an actual falsehoodmaker. When applied to my kitchen table, ⟦table⟧+ returns a set of potential facts of my kitchen table’s being a table, at least one of which is actual. But applying ⟦table⟧− to my kitchen table returns the set potential of facts of its not being a table. None of these are actual facts, since there are no actual facts making it false that my kitchen table is a table. So, as expected, we don’t get a contradiction by applying the concept table to my kitchen table—it does not have both actual truthmakers and actual falsehoodmakers, only actual truthmakers. Similarly, I assume, for other tables and for non-tables. Nothing both actually is and is not a table.

Applying an inconsistent concept to some object, on the other hand, will return both an actual truthmaker and an actual falsehoodmaker, resulting in a sentence that is both true and false. In this system, it’s possible to give a meaning to rable so that (4-a) has both an actual truthmaker and actual falsehoodmaker, which would mean (4-b) has an actual truthmaker.

In other words, we can give a meaning to rable that makes it S-inconsistent.

Suppose the fact that r is a table is included in ⟦rable⟧+(r). In other words, r’s being a table would make (4-a) true. Now suppose the fact that r is red is included in ⟦rable⟧−(r). In other words, r’s being red would make (4-a) false.

There’s nothing in our theory that prevents there from being such a meaning for rable.

Now suppose that r is a red table. Then both of the facts in question are actual, and (4-a) has an actual truthmaker and an actual falsehoodmaker, so is both true and false. And given standard truthmaker semantics for negation and conjunction, this means that (4-b) has an actual truthmaker, and so is true. This meaning of rable, together with the facts about some red table r, lead to a contradiction. On this theory, rable is S-inconsistent. But this is also a representationalist meaning. So a representationalist can say that rable has this content. Thus, by appeal to bilateral meanings, the representationalist can accommodate S-inconsistent concepts.¹⁴

But is the required bilateralism really a representationalist view? Perhaps, one might object, it is a kind of inferentialism in disguise. How are we to make sense of these ‘positive’ and ‘negative’ parts of the meanings? Surely this must be something to do with assertibility and deniability. But once we’ve agreed to that, it’s not clear how the supposedly representationalist bilateralism is different from the inferentialist bilateralism in the style of Timothy Smiley (1996), Ian Rumfitt (2000), and others.

The most prominent defenses of bilateralism, I admit, do come from an inferentialist perspective. They take the positive part of sentence meaning to be assertibility conditions and the negative part deniability conditions (or acceptance and rejection conditions). But this does not mean that bilateralism must take on inferentialist commitments.

First, note that even if we follow the inferentialists in taking meanings to be assertibility and deniability conditions, this does not yet imply we must reject messy metasemantics. Suppose we say that declarative sentences in some language L have as their meanings pairs of assertibility and deniability conditions. In virtue of what is someone a speaker of L? On one view, it is through knowing/accepting/being disposed to act in accordance with/etc. the assertibility and deniability conditions for the sentences in L. Such a view would be an inferentialist one. But this is not the only possible view. Instead, we can say that it is due to some complex web of interactions and dependencies that hold the linguistic practice together, even in the absence of a common rules that all participants are required to know, accept, or be disposed to follow.¹⁵ If we take inferentialism to require non-messy metasemantics, then even taking meanings to be be combinations of assertibility and deniability conditions does not make one a full-blown inferentialist.¹⁶

But we can go further than this. Bilateral meanings simply need not be taken to consist of assertibility and deniability conditions. On the truthmaker semantics framework I used above to introduce bilateralism, the meanings are not assertibility and deniability conditions, but truthmakers and falsehoodmakers. These may, of course, have some connections to assertibility and deniability, but this need only be loose connections that representationalists should be happy with, the same kind of connections that standard representationalists would take there to be between truth conditions and assertibility. And such connections need not be taken to be part of the meanings themselves, regardless of how strong the connections are.

But how, one might worry, are sentences supposed to get the positive and negative parts of their meanings, if not through direct relations to patterns of assertion and denial (and sanctions on assertions and denials)? And how can someone be a speaker of the language if they don’t know or act in accordance with these patterns?

This is just a worry about whether there can be a good metasemantics for representationalists. To fully address it would be to recapitulate the longstanding debate between representationalists and inferentialists, which I will not do here. Instead, I will just note that it seems to me that metasemantics no more forces the bilateralist to accept inferentialism than it does the unilateralist. The standard representationalist needs to hold that the true metasemantics is messy and is able to determine representational meanings. Moving to bilateralism may somewhat increase the explanatory burden here—the metasemantics needs to fix both positive and negative contents, not just single unilateral contents—but I see no reason to think that there should be insurmountable trouble for the bilateral representationalist if there isn’t such trouble for the unilateral representationalist. In any case, turning this into a serious problem for the bilateralist representationalist specifically would require a more detailed and well-supported metasemantic theory than is currently available.

So it seems there can be a bilateral representationalism which allows for S-inconsistency. Perhaps some representationalists will be hesitant to take it up, though, due to metaphysical scruples. Before, as unilateral representationalists, they could get by with the familiar sets of worlds and the truth-at-a-world relation.¹⁷ Now we’re in a less familiar land of actual and potential facts, truth-making, and falsehood-making.¹⁸ And not only that, these facts include dreaded negative facts like the fact that the table is not a couch. Such a view may be a representationalist one, but shouldn’t it still be excluded by a robust sense of reality?

I myself am not troubled by an ontology of facts, even negative and potential ones, nor by the relations of truthmaking and falsehood making. They don’t seem especially alien and they can be put to a lot of good theoretical use.¹⁹ That said, for the comfort of the those more scrupulous than myself I will show that a more pared down bilateral representationalism is sufficient for S-inconsistency.

We can start by staying within the general approach of truthmaking and falsehoodmaking, but dropping talk of facts and reverting to sets of worlds. On this view, we use what Fine calls ‘loose’ truthmaking and falsehoodmaking. A sentence is loosely made true by any world where the sentence is true, and loosely made false by any world where the sentence is false. Meanings, then, will still be sets of truthmakers and falsehoodmakers, but these will just be sets of worlds. So we can keep our bilateral meanings by using truthmakers and falsehoodmakers, but drop the ontology of facts and keep to the comfort of possible worlds.

Some may be worried not just about the facts, but about the relations of truthmaking and falsehoodmaking themselves. However, even an extra cautious representationalist who wants to avoid any such talk of ‘making’ true or false can still be a bilateralist. They can take the loose-truthmaking view just sketched and then interpret the relations between the sentences and sets of worlds not as truthmaking and falsehoodmaking, but as truth conditions and falsehood conditions. Truth conditions, I trust, are not too exotic. And if you’re okay with truth conditions, and you’re okay with falsehood, you should be okay with falsehood conditions.

So on this most conservative bilateralist representationalism, the meaning of a sentence is a pair of sets of worlds. One set is the sentence’s truth conditions, the other is the sentence’s falsehood conditions. An S-inconsistent sentence is one where the actual world is in the intersection of these two sets and an S-inconsistent predicate is one which maps some individual to a pair of sets that intersect in this way.²⁰ I find the truthmaker semantics approach more natural, but if you’re a representationalist who wants to be able to appeal to S-inconsistency without changing much about your view, you can do so.

There is, I admit, some cost in adding complexity to our meanings. Better a set than a pair of sets, all else equal. My aim is not to show that accommodating S-inconsistent concepts comes at no price. Rather, it’s that the price is not inferentialism, but rather bilateralism. And this price, it strikes me, may well be worth paying.²¹

I’ve argued that representationalists can accommodate S-inconsistency by going bilateral. If S-inconsistency is a defect—and it seems plausible that it is—then it’s a semantic defect, so concepts with such contents are semantically defective. And so one can be a representationalist and hold that concepts can be semantically defective. This would be a way to make sense of a concept’s being defective, even for someone who (i) is a representationalist, (ii) construes concepts in a thin, relatively non-committal way, and (iii) is skeptical that practical defects in themselves can make concepts defective. There remains a deeper question, though. I say it’s plausible that S-inconsistency is a defect. But what is defective about it?

A first thought: it is defective because its use, if one recognizes the facts, will lead one to contradiction. Yes, but what’s wrong with that? If some content really has an actual truthmaker and an actual falsehoodmaker, one way to react is to take there to be a defective concept like rable responsible and revising our conceptual schemes so we can no longer think this content. But couldn’t we also react by taking this to be a vindication of dialetheism?²² After all, it would mean that there’s some content that is both true and false, which would seem to be reason not to get rid of the concept. Doing so would just prevent us from appreciating the truth (and falsity) of the content.

I’ve just put the problem in representationalist terms, but it’s worth observing the same issue arises for the inferentialist. It is natural to say—and it is Scharp’s view—that being put in a position where one is required both to affirm and deny something means something has gone wrong. But couldn’t we say instead that sometimes one should affirm and deny something? Why not just keep the content around, believe both it and its negation, and adopt a paraconsistent logic to avoid explosion?²³

For a satisfying account of concepts’ semantic defectiveness, we need to say not just how S-inconsistency is possible, but what is wrong with it. And given that the presence of S-inconsistent concepts would mean there are some contents that are made both true and false by actual facts, it’s not clear what problem there would be in being able to grasp such contents, and accepting the attendant contradictions.

I do not know what the best response to this problem is. It does seem like there’s something wrong with rable and the like, so I am not ready to embrace S-inconsistency as non-defective. But what to do instead? We could go back to square one, rejecting bilateralism and searching for some other account of concepts’ defectiveness. But I think this would be to give up on the S-inconsistency proposal too soon.

The path that I currently favor is not to reject either bilateralism or accept dialetheism in a straightforward way. Instead, I propose we make a distinction: according to descriptive dialetheism, there are true contradictions graspable with certain sets of concepts (including, perhaps, our own actual concepts). According to normative dialetheism, one rationally may have a conceptual repertoire that allows one to grasp these true contradictions. The view I like best, though I cannot yet offer an extensive defense of it, is to accept descriptive dialetheism while denying normative dialetheism.

“What! And reject the Law of Non-Contradiction? Absurd!”

The view I’m suggesting does indeed involve giving up one form of the Law of Non-Contradiction, according to which no contradiction is true. But distinguish, as Beall (2004: 3) does, the following:

I am proposing that we reject Simple Non-Contradiction, due to the existence of S-inconsistent concepts, but still accept something like Rationality Non-Contradiction. The proposal can go either way, as we’ll see, on Ontological Non-Contradiction.

If we take logic to be primarily a normative subject, concerned with ideals of reasoning, we need not stray from classical logic in order to accept this kind of rejection of Simple Non-Contradiction. We don’t even need to stray from it if we take logic to be concerned with the ‘laws of truth’, so long as we understand this to be limited to truths concerning non-defective concepts.

This kind of view has a distinguished pedigree: Frege himself held something like it, if only because he thought that sentences with defective concepts like empty names lack truth values altogether (Frege 1892/1997: 157, 163–64). Russell came even closer, agreeing with Strawson that “ordinary language has no exact logic” (Russell 1957: 389) and focusing on “an ideal logical language” partly to investigate “what structure we may reasonably suppose the world to have” by “by inquiring what logic requires of a language which is to avoid contradiction” (Russell 1924/2010: 144). And Tarski (1935/1983: 165, 267) despairs of giving a definition of ‘true sentence’ which is both “in harmony with the laws of logic and the spirit of everyday language”.

So while we may ultimately wish to throw out the proposal to give up descriptive dialetheism while retaining normative dialetheism, I don’t think it deserves to be left with merely an incredulous stare or an Aristotelian comparison of its proponents to plants. Better to try to develop it and see where that leaves us.

The output of the proposal strikes me as attractive: we get to say concepts like rable are in fact inconsistent, so there are be contents involving them that are both true and false and, because of this, there is some reason to revise or eliminate them.²⁴ It seems plausible to me both that there could be such concepts and that there is some reason to revise or eliminate them. But if descriptive dialetheism is true, why shouldn’t normative dialetheism also be true? If the chief aim of our representational capacities is to be able to represent how things are, and there are inconsistencies among the way things are, shouldn’t concepts which allow us to believe them count as non-defective? It’s hard to see why we should reject Rationality Non-Contradiction if we don’t reject Simple Non-Contradiction.

To defend this proposal, it seems that we will need to provide a criterion for the evaluation of concepts that demands more from them than access to truths. We might, at this point, return to the kind of pragmatic justifications that we put aside at the outset.²⁵ Instead, I will sketch a couple of non-pragmatic options that I think are worth exploring.

Both options take as their starting point that the aim of concept possession is to help one make sense of the world. The options split in what they take this to involve. The first option I’ll call Structure Matching, and consists of two claims:

The first part, (SM1), is closely related to ideas explored by Hirsch (1993) and defended by Sider (2012). It assumes that there is some privileged structure to the world—perhaps some properties (and relations, etc.) are more natural than others,²⁶ or properties are sparse, so that not just every predicate corresponds to a real property.²⁷ Then it says that concepts that express real properties, or more natural properties, are better than those that express some complicated construction of real properties and relations, or less natural properties, or nothing real at all.²⁸

(SM1) depends on a controversial metaphysical claim—that there is such structure—and a controversial evaluative claim—that there is something especially epistemically valuable about limning this structure. Both views have been disputed.²⁹ So the first challenge for pursuing this proposal is to defend (SM1) from various attacks, to put it it on a sound footing.

A more worrisome challenge, in my view, is that of giving a defense of (SM2). This proposal needs to say that S-inconsistent concepts are so bad at carving nature at its joints, expressing something so convoluted or unnatural or empty, that one ought not have them.³⁰ But what reason do we have for thinking that the Book of the World isn’t written with S-inconsistent concepts? We can conjecture that it isn’t, of course, and cite the felt pressure to revise S-inconsistent concepts as evidence that we tend to operate on this assumption, but what grounds do we have to think it’s true? Currently (SM2) seems like mere speculation.³¹

On the Structure Matching picture, what it takes for a concept to help one make sense of the world is for it to sufficiently match some aspect of the world’s structure. The other option treats ‘making sense of the world’ in a more inwardly directed way. The idea is that making sense of the world just involves a certain sort of internal coherence, rather than a successful alignment with external structure. One’s ways of representing the world may fit together into a highly intelligible structure, even if that structure turns out not to correspond to the joints of nature, if there are such things.

This second option I’ll call Internal Intelligibility, which also consists of two claims:

This option lacks the metaphysical commitments of the Structure Matching proposal, but is hardly on more solid footing. What is it for an agent’s representations to be ‘internally intelligible’? Why care about it? And why should we think that S-inconsistent concepts would detract from it?

Classical logical consistency of beliefs is a natural proposal for spelling out at least part of what internal intelligibility amounts to, and it would make S-inconsistent concepts problematic. But what’s so great about classical logical consistency? Maybe this is where the buck stops: perhaps logical consistency is just part of the constitutive aim of concept possession, and there’s nothing more to be said about why that is. But this is unsatisfying and it’s not clear why we should accept it. A better defense of Internal Intelligibility would require a better explanation of why internal intelligibility involves something that would make S-inconsistency into a defect.

What all of this shows is that we still need a better understanding of what could make a concept defective and why. What I have done here is clarify an issue about semantic defects and cut out some work that remains to be done. There is no problem stemming from representationalism in attributing a putative semantic defect like S-inconsistency to a concept. The problem—for representationalists and inferentialists alike—is to say why this should make concepts defective in themselves, as opposed to showing we should be descriptive and normative dialetheists. Mere appeal to S-inconsistency is not enough. I’d put my money on the right theory making S-inconsistent concepts defective, but for now this is just a bet.