The Content and Logic of Imperatives

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This paper articulates an account of imperatives that sensibly supports the idea of a logic of imperative inferences. We rebuke common objections to the very possibility of such a logic, from a perspective based on recent linguistic work on the morphosyntax of imperatives. Specifically, we develop the notion that the content of an imperative sentence includes both a force operator alongside an imperational content to which the force applies. We further argue that this account of the content of imperatives constitutes a plausible and flexible framework to develop a logic of imperative by examining a number of reconstructions of this idea using semantical analogs of widespread modal semantics. After studying the performance of those approaches, we conclude that progress in imperative logic has been hindered by the failure to adopt conflict-tolerant and resource-sensitive semantics, but suggest that such considerations can be incorporated in this flexible framework. Finally, we also propose a simple account of the difference between operators applying to the content of imperatives and imperative operators, in a way that sheds light on some of the issues underlying the usual antinomies.

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  1. 1.

    One might think that this definition of imperative inference is a bit awkward, in that it includes inferences with only declarative sentences. Another reasonable definition would be to require that an imperative inference contain at least one imperative sentence. Despite the potential awkwardness, we adopt the more inclusive definition, because we want to have a term to refer to arguments formulated in a language containing imperatives in general, without having to specify each time whether it is purely declarative or not. Nothing hinges on that stipulation.

  2. 2.

    In a recent paper, Recanati (2013) argues that the “analysis of imperative sentences remains, to this day, very much an open issue, and further investigations into their syntax and semantics are needed before we can adjudicate between the various proposals currently on the table.” Though we believe this judgement to be somewhat exaggerated, it remains the case that no systematic interface between linguistics, philosophy of language, and logic has formed as a paradigm.

  3. 3.

    “On ne peut pas concevoir un syllogisme où les deux prémises seraients à l’indicatif et la conclusion à l’impératif; mais on peut en concevoir qui soient bâtis sur le type suivant: Fais ceci, or, quand on ne fais pas cela, on ne fait pas ceci, donc fait cela. Et de pareils raisonnements ne sont pas hors de la portée de la science.” (Poincaré 1920, p. 236) Note that, where Poincaré speaks of indicative sentences, we prefer ‘declarative’, since we need a term that includes all truth-apt sentences, which include sentences expressed in the subjunctive mood.

  4. 4.

    Some of the controversial elements arise from the difficulty of maintaining that all moral normative judgements reduce to a (system of) prescriptive sentences in the imperative mood, since other forms of normative judgements (e.g., evaluative and comparative) are not of a prescriptive nature.

  5. 5.

    This view is not new (see, e.g., Williams 1963; Wedeking 1970; Harrison 1991). As an example, Wedeking points out that the “constructions ‘Since open the window’ and ‘For sit down and wait’ are not grammatical expressions” and asks: “If there are imperatives which are used as premises in arguments, why may we not make the argument explicit by preceding a premise with ‘since’, as we may in normal arguments?” (p. 162)

  6. 6.

    He acknowledges the objections raised by Vranas (2010), but briskly dismisses them. We side with Vranas and supplement his case.

  7. 7.

    The term ‘mood’ is often used in relation to speech acts, rather than syntax. For instance, it is not uncommon to distinguish the imperative mood from the commissive mood, so as to distinguish cases such as ‘Come in!’ from cases in which a command is given. Though such a notion is obviously important to linguistics and philosophy of language, we will focus on the strictly syntactic notion. Accordingly, we would say that ‘Come in!’ is an imperative, even though it typically doesn’t perform the act of giving a command. We thus distinguish the question of determining the mood from that of determining the illocutionary force.

  8. 8.

    In the case of the imperative sentence “Come in!”, we could come to the conclusion that it performs a permissive speech act as follows. Suppose a knocks on the door. a’s knocking on the door manifests a wish to come in. In this context, commanding a to come in would be redundant, and perhaps infelitous, and so the imperative sentence is performatively interpreted as granting a permission rather than giving a command.

  9. 9.

    We can make a similar distinction with declarative sentences and their logic treatment; in this case, we could have a logic of statements (or assertions), and a logic of declarative sentences. Moreover, since declarative sentences express propositions, we could have a logic of propositions. This decomposition appears at the very beginning of modern logic with Frege (1879), who distinguishes between the proposition that A, denoted \(\text {---}A\) (with the content stroke), and the assertion that A, denoted \(\vdash A\) (with the judgement stroke). We introduce terminology below that extends the parallel to what we refer to as propositional logic.

  10. 10.

    This is not only for natural language. In programming languages, for instance, we could find expressions such as ‘Do proc()’; ‘Do’ would play the role of the imperative operator, and the procedure would here be an act-type.

  11. 11.

    Brink (1994) convincingly demonstrates the importance of developing a framework that includes forces pulling in different directions—i.e., conflicting forces—in order to account for moral reasoning.

  12. 12.

    In this sense, aspects of the logic of imperatives could be regarded as being fundamentally about deliberation over our actions and ways of realizing actions. This seems to be in line with the approach developed by Charlow (2014).

  13. 13.

    There may be one exception, namely, declarative sentences containing deontic modals.

  14. 14.

    We note that that it is not entirely clear whether the idea should be understood as a conditional or a biconditional. If the trick is interpreted merely as a criterion for validity of imperative inferences, then a conditional interpretation is more natural. However, as is the case in other papers (e.g. Hansen 2014), we interpret the trick as a biconditional. This interpretation takes Dubislav’s trick as a substantive definition of validity which postulates the absence of logically relevant non-indicative structure in imperative inferences. Which interpretation is adopted depends on whether one regards Dubislav’s trick as a reductive or non-reductive account of validity for imperative inferences.

  15. 15.

    Though there may be sets \(X\in N(w)\) such that, for no \(\varphi \in {\mathscr {L}}\), \(X=\llbracket \varphi \rrbracket\), we will continue to call N(w) a set of propositions for convenience and intuitive appeal, as is common in the literature.

  16. 16.

    There is another equivalent requirement on N: for all \(X,Y\in \wp (W)\), if \(X\cap Y\in N(w)\), then \(X,Y\in N(w)\), not to be confused with its converse, closure under intersection. Naturally, this corresponds to the rule \(!(\varphi \wedge \psi ) / !\varphi\).

  17. 17.

    Jennings (1974) provided another early application of neighborhood semantics to deontic logic, but the resulting system differs from the one discussed here.

  18. 18.

    One caveat is that, if we take a deontic modal to be also expressing that a certain content has directive force, then one could obtain a valid argument with an imperative conclusion that has no imperative premise, provided it contains a deontic modal.

  19. 19.

    Han (1997) suggests that the systematic infelicity of certain grammatical patterns warrants making this hypothesis.

  20. 20.

    A possible exception would be cases in which A’s necessity is itself salient.

  21. 21.

    Possible exceptions would include cases in which one is commanded and trying to do the impossible (e.g. Payette 2018).

  22. 22.

    We point out, in passing, that some of Hansen’s examples are fairly easy to handle as part of the logic relating the content of imperatives. He refers to the following argument as ‘Weinberger’s paradox’ (p. 169): “Close the window and play the piano! Therefore: Play the piano!” The context of the argument is such that the neighbours have already made a noise complaint, so that the addressee is not in fact directed to play the piano with the window opened. Such a case is easily dispatched by introducing the so-called temporal conjunction as an operator on the content of imperatives, in which case !B does not follow from \(!(A\wedge _T B)\). The suggestion that temporally ordered sequences of actions can have directive force should be uncontroversial.

  23. 23.

    Of course, in addition to adding C or D to the list, one needs to take the logical closure under a logic described in Sect. 3.

  24. 24.

    Another case, that we believe was first articulated by Williams, is the case of “imperative disjunctive syllogism.” It is usually presented as ‘Do A or B. But don’t do B. So do A.’ We have already pointed out that the argument form \(!(A\vee B),!\lnot B / !A\) is perfectly fine. However, we propose to symbolize Williams’ imperative disjunctive syllogism as \(!A\vee _I!B, !\lnot B\>/\> !A\). Taken as such, the problem with this inference is that the premises are said to have inconsistent implied permissions. However, there is no difficulty accounting for this case with the proposed framework.


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Correspondence to Matthew Lynn.

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We would like to thank Gillman Payette and two anonymous reviewers for their excellent comments.

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Fillion, N., Lynn, M. The Content and Logic of Imperatives. Axiomathes (2020) doi:10.1007/s10516-019-09471-w

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  • Imperative inference
  • Poincaré’s principle
  • Jørgensen’s dilemma
  • Ross’ paradox
  • Dubislav’s trick
  • Conflicting obligations
  • Resource-sensitive reasoning