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Logic and Semantics for Imperatives

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Abstract

In this paper I will develop a view about the semantics of imperatives, which I term Modal Noncognitivism, on which imperatives might be said to have truth conditions (dispositionally, anyway), but on which it does not make sense to see them as expressing propositions (hence does not make sense to ascribe to them truth or falsity). This view stands against “Cognitivist” accounts of the semantics of imperatives, on which imperatives are claimed to express propositions, which are then enlisted in explanations of the relevant logico-semantic phenomena. It also stands against the major competitors to Cognitivist accounts—all of which are non-truth-conditional and, as a result, fail to provide satisfying explanations of the fundamental semantic characteristics of imperatives (or so I argue). The view of imperatives I defend here improves on various treatments of imperatives on the market in giving an empirically and theoretically adequate account of their semantics and logic. It yields explanations of a wide range of semantic and logical phenomena about imperatives—explanations that are, I argue, at least as satisfying as the sorts of explanations of semantic and logical phenomena familiar from truth-conditional semantics. But it accomplishes this while defending the notion—which is, I argue, substantially correct—that imperatives could not have propositions, or truth conditions, as their meanings.

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Notes

  1. Exemplary references include [12, 17, 24, 32–34, 41, 48, 50, 62, 87].

  2. See, e.g., [57, 75, 84–86]. In addition to this, there is a great deal of work on imperatives and similar topics that originates from meta-ethical concerns; a small sampling: [3, 18, 37, 49, 65].

  3. See, e.g., [2, 5, 40, 47, 56, 58, 59, 68, 83].

  4. See, e.g., [6] and subsequent work in the STIT tradition.

  5. For views in this tradition, see [33]; [[88] Section 23]; [22, 50, 76, 92].

  6. On the notion of clause-type and the universality of the imperative, see especially [58].

  7. Since the standard interpretation of \(\wedge \) is commutative, this would seem to commit me to some degree of non-substitutivity-of-logical-equivalents principle for imperative contexts. In reality, I (along with a large contingent of working semanticists) think natural language conjunction is not in general commutative. Thus, if \(\wedge \) is to represent natural language conjunction, \(\wedge \) should not be given a commutative interpretation; if it is not to represent natural language conjunction (but, instead, the familiar Boolean connective), it is not generally correct to represent natural language conjunction using \(\wedge \). In particular, I think that and, when it coordinates distinct descriptions of actions by way of describing a complex action, should be understood as non-commutative: lighting the stunt double on fire and putting him in a protective suit is a much different action from putting the stunt double in a protective suit and lighting him on fire. It is, therefore, misleading to represent an imperative like put the stunt double in a protective suit and light him on fire using an expression in which the imperative operator takes scope over a sentence whose conjuncts can be reversed without affecting that sentence’s interpretation. For a logic that has a smooth time capturing such distinctions, see [70, 71]. (I would like to say more about this, but doing so would mean taking a stand on how to resolve the paradoxes I am describing here—something I would like to avoid.)

  8. There has been, for various reasons, a healthy amount of doubt, some of which I am sympathetic to, about the possibility of (valid) imperative inferences in the literature; see [35, 39, 87]. For general defenses of imperative inference, see [23, 85] (although, I should note that I do not find [85]’s defense persuasive). I should note that my own positive view will suppose that imperatives have a logic in only a relatively weak sense: their semantics should guarantee that certain constellations of attitudes that are constitutively connected to an agent’s acceptance of imperatives display inferential relations to other attitudes that are constitutively connected her acceptance of imperatives. I know of few people who deny that imperatives have a logic in this sense.

  9. Strictly speaking, the data I appeal to here do not support the claim of irreducible embeddability, since (i) the inability to represent a superficially embedded natural language imperative with a sentence of \(\mathcal {L}_{\mathtt {imp}}^+\) does not imply that (ii) there is no more expressive language that successfully represents that imperative, but represents it as embedded. Indeed, the discussion of Section 5.5 implicitly relies in the falsity of (ii). I hope the reader will forgive some imprecision here in the service of exposition. For clearer examples of embedding, see Section 6.2.

  10. For a substantially similar case involving deontic conditionals, see the Miner Paradox of [42]. Given that it will either rain or not, cases like these may feel puzzling in that they can seem, in a certain light, to support challenges to the validity of modus ponens (understood as a general rule sanctioning the inference of an indicative conditional’s consequent if the antecedent is true). But, as [28, 91] emphasize, and as I failed to properly appreciate in my [16], it is possible to formulate the notion of consequence that underwrites valid inference in such a way that insisting on the consistency of (13a)–(13c) is compatible with endorsing a rule of modus ponens.

    A common strategy for debunking this sort of example appeals to the idea that the antecedents of the CIs here are elliptical for sentences containing reference to the knowledge of the addressee. That is to say, (13a), on the salient reading, is actually given the reading if it rains and you know it, take the umbrella; if it is not given this reading, (13a)–(13c) are inconsistent. This is something I just deny. The salient reading of the CIs in question is not one that makes explicit reference to anyone’s epistemic states. Insofar as the salient reading yields a consistent interpretation of (13a)–(13c), assigning (13a)–(13c) a consistent interpretation does not require positing covert reference to anyone’s epistemic states in the antecedents of indicative conditionals. (For more direct arguments against the strategy I am considering here, see [11].)

  11. To forestall a possible response: it might be thought that this claim cannot be right, on the grounds that (i) \(!(\phi \rightarrow \psi )\) and \(!(\neg \phi \vee \psi )\) are “equivalent”, (ii) imperatives of the form \(!(\neg \phi \vee \psi )\) have permission content: disjunctive imperatives permit either disjunct, and this is why the Ross inference is bad. Whether or not this is true for disjunctive imperatives, it is irrelevant here. Clearly (15) does not permit robbing Jones gently.

  12. Two-valued approaches collapse into wide-scope-ism because a ci like (16) is either satisfied or not. Obviously it is not satisfied only if: you turn the A/C on and do not shut the window. So, since the semantics is bivalent, it is satisfied only if: you either do not turn the A/C on or do shut the window. In other words, it has the same satisfaction-conditions as an imperative of the form ! (you turn the A/C on \(\rightarrow \) you shut the window).

  13. This criticism also arguably applies to the Satisfaction-Conditional theory of [23] (although certain complexities Fox’s theory, in particular his treatment of Free Choice Permission, make the argument more complex). I am grateful to an anonymous referee for bringing this reference to my attention.

  14. See [[84], 547] for an argument that the sentences should be regarded as consistent (best as I can tell, the argument is a restatement of the claim that the sentences should be regarded as consistent because it is possible not to violate both). In reply, I am inclined to simply insist on the intuition—there is no argument that could persuade me of the consistency of (16) and (17). They are inconsistent in all of the ways that count (cf. Section 2.1).

  15. Possibly the study of Vranas’ imperative conditional could be motivated by appeal to the claim that the logic of this conditional is a topic of sui generis interest. I am doubtful. Imperatives do have a logic (and, like Vranas [84], I think there are valid imperative inferences). So far as I can tell, however, there are no logical or inferential phenomena about imperatives that are both interesting and proprietary to them. The phenomena Vranas cites (like, for instance, the fact that conditional imperatives can be avoided) tend only to loom large in the context of an approach, like his, that takes satisfaction conditions to be an appealing way to theorize about the semantics and logic of imperatives. Since I do not think this, I do not find the phenomena in question particularly interesting.

  16. Schroeder focuses his objections on Expressivist accounts of normative language, but they extend straightforwardly to both Expressivist and speech act accounts of imperatives.

  17. There are serious costs to relinquishing the canonical way of explaining object-language inconsistency—costs that far outstrip its canonicity. You can, for instance, forget about reductio ad absurdum reasoning in the metalanguage about object-language inconsistency. For the same reason, it seems that the familiar Tarskian technique of exploiting disquotation—e.g., inferring from the fact that ‘\(a\wedge b\)’ holds at X the fact that both ‘a’ and ‘b’ hold at X—as a tool of proof in the metalanguage will be unavailable. If X is a state of mind, disquotation in this sense is simply invalid.

  18. I have argued elsewhere [15] that the sort of semantics developed in this paper, designed in part to skirt the Frege-Geach problem, has a rightful claim to the Expressivist mantle, despite declining to assign speech acts or states of mind as the meanings of sentences. In light of this, here is a more precise statement of my claim here: Expressivism, as ordinarily understood, has a hard time with the Frege-Geach Problem. The issues here are complex, but see my [15] for discussion.

  19. See my [13] for an argument that this sort of wide-scope conjunction is obligatory.

  20. For this reason, the Modal Analysis of imperatives has frequently been seen as a contender in the historical discussion of imperatives. [29] in fact dates the Modal Analysis to Kant’s Groundwork! The Modal Analysis was certainly seen as a contender in the Hare-era debate on imperatives (see [24, 33]), but was generally dismissed on the grounds that imperatives, unlike deontic modals, were “two-valued” with regard to their prejacents. As Geach explained, “In answer to a request for orders, ‘am I to do P?’, only two answers are possible—‘do P’ and ‘do not do P’, which are contradictories. No other order is a direct answer to the question; and to say ‘you may either do P or not’ would not be an answer but a refusal to answer—I was asked for an order and I refuse to give any. But there are three relevant answers to the moral enquiry ‘ought I to do P?’” [[24], 49]. Geach’s observation is questionable—I think ‘it is up to you’ counts as a perfectly good answer to the question ‘what am I to do?’ (unless the interrogator is supposing that only an order will do). Even if it were right, it would show only that some imperative questions—some questions of the form am I to do P?—were distinct in meaning from deontic questions—questions of the form ought I to do P? This is irrelevant to the question of whether the answers to imperative questions are distinct in meaning from the answers to deontic questions.

  21. See also [2, 4, 31, 67].

  22. Exactly this fact is enthusiastically exploited by [67, 68] to support a modal account of imperative semantics.

  23. Something close enough, anyway. See [42, 91] for discussion.

  24. I assume, as is plausible, that (30b) can be represented with an imperative of the form \(!\neg \phi \).

  25. On the importance of accounting for this dimension of imperative meaning, see [59].

  26. Strictly speaking, the tests suggested here for whether a sentence has a representational function are not decisive. There are many philosophers who think that, although things like indicative conditionals, epistemic modals, and moral claims certainly can be called true/false and can be asserted and believed, they nevertheless lack both a representational use and a propositional semantics. Ultimately, I think the proper way to draw the representational/non-representational distinction for a sentence appeals to cognitive phenomena—whether, say, Bayesian probability functions are defined over its semantic value—rather than crude linguistic tests of the sort suggested here.

  27. Not to be confused with the syntactic mood of its main verb. The verbal mood of an interrogative is typically the same as its (declarative) answers. But interrogatives and declaratives are different kinds of clauses.

  28. Portner [58, 59] is a forceful defender of this sort of principle.

  29. This sort of idea is inspired by and closely related to the approach to semantic theorizing advocated by [77, 78], and to a somewhat lesser extent the approach of [89, 90].

  30. Disjunction is defined in terms of negation and conjunction as follows. \(S \models (\phi \vee \psi )\) iff \(S \models \neg (\neg \phi \wedge \neg \psi )\). This holds iff \(\forall w \in S: \{w\} \not \models \neg \phi \wedge \neg \psi \) iff \(\forall w \in S: \{w\} \not \models \neg \phi \) or \(\{w\} \not \models \neg \psi \) iff \(\forall w \in S: \{w\} \models \phi \) or \(\{w\} \models \psi \). This is the correct notion of disjunction for the sake of our cognitively oriented semantics: for a disjunction \((\phi \vee \psi )\) to hold (“be known”) at S, it is required only that, for every world in S, at least one of \(\phi \) or \(\psi \) holds at that world. This is crucially (and correctly) distinct from a notion of disjunction on which \((\phi \vee \psi )\) holds at S iff \(S \models \phi \) or \(S \models \psi \). Knowing \((\phi \vee \psi )\) does not require knowing at least one of \(\phi \) or \(\psi \); one can know a disjunction without knowing which disjunct is true.

  31. There is a familiar worry about treating these properties as full duals, i.e., about turning these ‘only if’ claims into ‘iff’ claims. For one way for a plan to fail to allow \(\neg \phi \) (without thereby requiring \(\phi \)), as well as to fail to require \(\neg \phi \) (without thereby allowing \(\phi \)), is for it to be undecided about \(\phi \) (see [21, 65, 66, 79, 80]).

    In my view, this is basically parallel to the problem of distinguishing strong permission that \(\neg\phi\) (allowing \(\neg\phi\)) from weak permission that \(\neg\phi\) (being undecided about \(\neg\phi\), i.e., failing to require that \(\phi\), while also failing to “actively” permit that \(\neg\phi\)). It is an old problem, and one on whose resolution I would like to avoid taking a stand on here. However, a natural idea here would be to imbue plans with more structure, so that the distinction between strong and weak permission can be represented (cf. Yalcin’s [90] addition of the sort of structure to information states allowing the representation of the attitude of treating something as an issue).

  32. Notice that the \(\lambda \)-notation here is to be read non-standardly (since describing the function using ‘only if’ instead of ‘iff’ means that we fail to pick out a unique such function). A \(\lambda \)-expression of the form \(\lambda z Fz\) will, therefore, refer to a (rather than the) function mapping z to \(Fz\).

    A referee for this journal has helpfully worried that this leaves the resulting semantics quite incomplete—that I have only described a class of functions that an imperative or permission sentences might denote, rather than the particular functions they do denote, and that the cognitive instruction an utterance of an imperative will, on my view, proffer is, thus, quite indeterminate. Quite so, and I agree that this is a shortcoming of the view as stated. But the view as stated is to be understood provisionally: the semantics I would, in the final instance, endorse does not assign families of functions as semantic values; a unique function will be assigned, and the cognitive instruction thus proffered in expressing such a function will be fully determinate. Since stating such a semantics would, of necessity, require taking a stand on the above mentioned paradoxes, I have declined to do so here. Instead, I have opted for stating precise necessary conditions on the function expressed by an imperative. I have done this for the sake of showing that the framework, no matter how it is filled out, will be able to give satisfactory explanations of the phenomena I am interested in here.

  33. For a similar view of the preferential content of imperatives, see [75].

  34. I am here ignoring the complication that truth may ultimately be relative to other things than plans.

  35. For a variety of reasons, I think the Kratzer semantics is inadequate (cf. [28, 91]), but to simplify I will work with it here.

  36. If we’re working in a Categorial or Variable-Free Grammar (see [38]), we could alternatively treat the open argument place as expressing the identity function \(\lambda X_{\langle st,t\rangle }.X\) and generate the correct meaning for the sentence by combining this semantic value with that of its sister node by the composition operation \(\lambda f\lambda g \lambda X.f(g(X))\).

  37. For strong criticisms of Han and Mastop, see, respectively, [60] and [75].

  38. Barker [5] concurs that imperatives express actions, although his broader theory is quite different from Portner’s.

  39. This isn’t to say there is no way to model the update that CIs might “execute” on a context; for such a model see [[14], 178ff]. But adopting such a model would require a rather major reworking of Portner’s theory—one that is not obviously available to him. Why? The idea that the semantic value of an unembedded imperative is an action fails to generate a plausible semantic value for a ci: saying that a ci’s semantic value is typed as an action collapses the theory into wide-scope-ism (since Portner’s theory would have this action be added to the To-Do List), while saying it is typed as something else is, insofar as (i) we expect clauses of the same type to have semantic values of the same type (see [58]) and (ii) conditionals are, per [7], typed as clauses according to the type of their matrix, rather than subordinate, clause (so that CIs are, in fact, typed as imperatives), prima facie undesirable.

    This said, Modal Noncognitivism in no way requires an account of how CIs update the context, contra Portner’s [60] remark that an “issue for [Modal Noncognitivism] is that it does not define a particular update to the context for a given imperative.” Modal Noncognitivism’s theory of imperative meaning is given, not by a theory of how imperatives update the context, instead by the semantic identities proposed in this section.

  40. This is something of which Portner is aware—indeed, enthusiastic. See his [61], which uses this fact in an analysis of permission interpretations of imperatives (e.g., go ahead, have an apple!). I have argued elsewhere that this is not a satisfactory treatment of permission imperatives [[14], 15ff]. Even if it were, this feature of the analysis would be undesirable in light of the argument I am giving here.

  41. For related reasons, we have reason to prefer a strategy which treats states that accept, for example, \(\Phi \wedge \Psi \) without accepting \(\Phi \) as contradiction-licensing (rather than barely possible, but irrational).

  42. What follows is not to be understood as a critique of dynamic semantics generally (except on the strongest understanding of the dynamic semantic program, on which the semantic value of any sentence is a deterministic program or update function). The dynamic semantic approach to, to pick an example at random, donkey anaphora (see, e.g., [30]) plausibly does not suffer from the problems I describe here.

  43. Here is some detail about Starr’s account to fill this point in. A preference state R is defined as a binary relation on a set of alternative propositions (typed as sets of worlds) such that \(\langle a,a' \rangle \in R\) iff a is preferred to \(a'\). For Starr, imperatives update preference states by strengthening their constituent preferences: updating R with \(!\phi \) yields \(R[!\phi ]\) such that (i) \(R \subseteq R[!\phi ]\), (ii) \(R[!\phi ]\) contains a preference for \(\phi \) to \(\neg \phi \), i.e., \(\langle ||\phi ||,||\neg \phi || \rangle \in R[!\phi ]\), and (iii) if \(\langle a,a'\rangle \in R\), \(\langle a[\phi ],a-a[\phi ] \rangle \in R[!\phi ]\) (where \(a[\phi ]=a \cap ||\phi ||\), i.e., is the result of intersecting a with the proposition that \(\phi \)). But component (iii) of the update amounts to a view about how prior preferences are revised on the adoption of a new preference (one that, in all likelihood, would need to be significantly complicated to yield a realistic account of its target). For a sense of the complexities involved here, see [81].

  44. The main remaining difference is that Starr does not suggest an explicitly modal understanding of notions like requirement and permission. Starr does aim to give accounts of phenomena we’ve decided to set to the side, e.g., Ross’ Paradox, but we do not, I think, disagree about the explanation of these phenomena. (For my treatment of Ross’ Paradox, see [[14], 137ff].) For this reason, I would disagree with his claim that “The preference semantics for imperatives developed above enjoys important empirical and conceptual advantages over existing accounts.”

  45. Perhaps such examples feel forced or contrived. For more persuasive examples; see [[14], 133ff]. There are also, it should be mentioned, cases like:

    1. (49)

      Stop or I’ll shoot!

    2. (50)

      Drink another beer and we’ll win the competition! ([64])

    Following roughly [19, 83], I claim that such sentences plausibly function to express assertions of conditionals:

    1. (51)

      \(\approx \)If you don’t stop, I’ll shoot [perhaps conjoined to an imperative: stop!]

    2. (52)

      \(\approx \)If you drink another beer, we’ll win the competition [perhaps joined to an imperative: drink another beer!]

    While there is a mystery about how to generate such readings, I think there is very little pressure for a semantics of imperatives to account for them. Instead, the pressure is to come up with an analysis of the relevant connectives (‘or’ and ‘and’) that allows them to transform imperatives into subordinate clauses functioning to restrict the domain of quantificational modals like ‘will’.

  46. I am grateful to Andrew Alwood for pressing this worry.

  47. One way to appreciate at least the second claim here is by considering Dreier’s [20] Hiyo example. In a language where the sentence ‘Bob is hiyo’ expresses, by stipulation, the speect act of accosting Bob, this sentence, despite being syntactically declarative, is not evaluable for truth.

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    Nate Charlow

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This work has benefited greatly from the input of others. I would like to thank Andrew Alwood, Chris Barker, Simon Charlow, Allan Gibbard, Benj Hellie, Paul Portner, Will Starr, Eric Swanson, and Richmond Thomason, as well as audiences at the Association for Symbolic Logic’s 2011 APA Group Session on Dynamic Semantics, Cornell University, ´Ecole Normale Sup´erieure, NYU, University of Toronto, and York University. Thanks also to two anonymous referees for this journal.

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Charlow, N. Logic and Semantics for Imperatives. J Philos Logic 43, 617–664 (2014). https://doi.org/10.1007/s10992-013-9284-4

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