Abstract
Besides pure declarative arguments, whose premises and conclusions are declaratives (“you sinned shamelessly; so you sinned”), and pure imperative arguments, whose premises and conclusions are imperatives (“repent quickly; so repent”), there are mixed-premise arguments, whose premises include both imperatives and declaratives (“if you sinned, repent; you sinned; so repent”), and cross-species arguments, whose premises are declaratives and whose conclusions are imperatives (“you must repent; so repent”) or vice versa (“repent; so you can repent”). I propose a general definition of argument validity: an argument is valid exactly if, necessarily, every fact that sustains its premises also sustains its conclusion, where a fact sustains an imperative exactly if it favors the satisfaction over the violation proposition of the imperative, and a fact sustains a declarative exactly if, necessarily, the declarative is true if the fact exists. I argue that this definition yields as special cases the standard definition of validity for pure declarative arguments and my previously defended definition of validity for pure imperative arguments, and that it yields intuitively acceptable results for mixed-premise and cross-species arguments.
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Notes
If one understands propositions as sets (e.g., of possible worlds), then one can define the conjunction of infinitely many propositions or prescriptions (see Vranas 2008, p. 545). Those who do not want to talk about the conjunction of infinitely many propositions or prescriptions can restrict their attention to arguments with finitely many premises.
One might argue that the sentence “I understand why you hit him, but now apologize” expresses the conjunction of the proposition that I understand why you hit him with the prescription “now apologize”. But what kind of entity is the conjunction? If the conjunction is true exactly if I understand why you hit him, then the conjunction is not a prescription (since prescriptions cannot be true or false); and if the conjunction is satisfied exactly if you now apologize, then the conjunction is not a proposition (since propositions cannot be satisfied or violated). Maybe, then, the conjunction is the unordered pair whose members are the proposition and the prescription. But then no simplification is achieved by conjoining the single declarative with the single imperative premise.
One might claim that an imperative argument is fully successful only if its conclusion is obeyed (e.g., only if one donates to charities, if the conclusion is “donate to charities”). I reply that an imperative argument can be fully successful even if its conclusion is not obeyed, for example due to weakness of will (or because you die). For example, if I adduce an argument which (correctly) convinces you that the balance of reasons supports the prescription “confess” but you fail to confess due to weakness of will, it does not follow that there is anything wrong with my argument. Compare: if I adduce a sound pure declarative argument for the conclusion that there are irrational numbers but you irrationally refuse to believe this conclusion, it does not follow that there is anything wrong with my argument.
One might argue that the property of being true is very different from the property of being supported by reasons, so calling both properties “meriting endorsement” achieves only a terminological—but no real—unification of declarative and imperative arguments. I reply that the two properties are not so different after all because a proposition (that can be believed) is true exactly if there is an objective epistemic reason to believe it. To see what I mean, recall first the distinction between subjective and objective normative reasons for action: if I am thirsty and the liquid in the bottle is petrol but my evidence unambiguously indicates that it is water, then there is a subjective reason for me to drink it but an objective reason for me not to drink it. (If there is no objective reason for me to drink it, then no argument with the conclusion “drink the liquid in the bottle” is fully successful; this suggests that an imperative argument is fully successful only if its conclusion merits endorsement in the sense of being supported by objective reasons.) One can make a similar distinction between subjective and objective epistemic reasons: in the above example, there is a subjective epistemic reason for me to believe that the liquid is water, but there is an objective epistemic reason—namely the fact that the liquid is petrol—for me to believe that the liquid is petrol (although I have no evidence that it is petrol; cf. Gibbard 2005, pp. 340–341). Given that a proposition (that can be believed) is true exactly if there is an objective epistemic reason to believe it, there is a sense in which, both for propositions and for prescriptions, meriting endorsement amounts to being supported by objective reasons (although it is objective epistemic reasons for propositions, but objective reasons of any kind for prescriptions; see note 16). Nevertheless, my claims about objective epistemic reasons might be considered controversial, so for propositions I will stick to defining meriting endorsement as being true.
I understand D2, like D1 and all definitions in this paper, as prefixed with “necessarily”. (Necessity and validity are understood logically or conceptually throughout this paper—cf. note 30 and corresponding text—but could also be understood metaphysically; cf. Vranas 2011, pp. 376–377 n. 7.) If one assumes that (1) the accessibility relation between possible worlds is reflexive and transitive (i.e., whatever is necessary is both true and necessarily necessary) and (2) necessarily, every valid argument is necessarily a valid argument, then one can show that D2 is equivalent to: necessarily, if an argument is valid and its premises merit endorsement, then its conclusion merits endorsement (cf. Vranas 2011, p. 375). (In this paper, I assume that the accessibility relation between possible worlds is an equivalence relation—see Burgess 1999, 2003 for a defense of the claim that the correct system of (propositional) modal logic for logical necessity is S5—but I specify weaker assumptions about accessibility that are sufficient for particular results.)
Except perhaps if P and \(P^{\prime }\) are both impossible, so that the prescription whose satisfaction proposition is P and whose violation proposition is \(P^{\prime }\) is empty (i.e., its context is impossible; e.g., “if \(2+2\) is 5, dance”): as it turns out, adopting the convention that empty prescriptions are necessarily supported by every reason considerably simplifies certain results. I will not elaborate because in the sequel (except in notes 47 and 63) I ignore empty prescriptions: whenever I (implicitly) use a quantifier ranging over prescriptions, I assume that it ranges only over non-empty prescriptions.
I define guaranteeing (and supporting) so that, necessarily, only facts guarantee propositions (and only facts support prescriptions). I assume that, necessarily, if something is a fact then, necessarily, it is a fact exactly if it exists. (So, necessarily, every fact exists, and a fact could not have existed without being a fact.) If one accepts what Rodriguez-Pereyra (2006, p. 958) calls a “traditional definition of truthmaking” (namely: e is a truthmaker for a proposition P exactly if (1) e exists and (2) the proposition that e exists entails P), then a fact guarantees a proposition exactly if the fact is a truthmaker for the proposition. But that definition of truthmaking is controversial, so I avoid any further talk of truthmakers in this paper.
I grant, however, that a disanalogy remains: for prescriptions, but not for propositions, meriting endorsement is relative to times, agents, and normative perspectives (moral, legal, etc.), as I explain in note 13. I discuss some implications of this disanalogy in notes 19, 23, 41, and 66.
For the sake of simplicity, I assume that no distinct facts exist at exactly the same possible worlds (i.e., no distinct facts f and \(f^{\prime }\) are such that, for every possible world w, f exists at w exactly if \(f^{\prime }\) exists at w). This “modal criterion for the identity of facts” is accepted by several but not all theories of facts (Mulligan & Correia 2007/2009), but for my purposes the assumption is innocuous: if distinct facts existed at exactly the same possible worlds (e.g., the fact that Socrates is human and the fact that Socrates is human and every number is a number), they would sustain the same—or necessarily equivalent—propositions and prescriptions.
Rather than saying that (1) a prescription merits all-things-considered endorsement exactly if it is undefeatedly supported by some fact, one might say that (2) a prescription merits all-things-considered endorsement exactly if it is supported by the balance of reasons, understood as the conjunction of all facts that are reasons (Vranas 2011, pp. 374–375). If all defeaters are reasons, then (1) and (2) are equivalent: (a) if a prescription is undefeatedly supported by some fact, then it is supported by the balance of reasons (because the conjunction of the balance of reasons with a fact that undefeatedly supports a prescription and thus is a reason (i) just is the balance of reasons and (ii) supports the prescription), and (b) conversely, if a prescription is supported by the balance of reasons and all defeaters are reasons, then the prescription is undefeatedly supported by some fact, namely by the balance of reasons (because if by hypothesis the balance of reasons supports a prescription but one assumes for reductio that the conjunction of the balance of reasons with some fact does not support the prescription, then that fact is a defeater and thus a reason if all defeaters are reasons, and then the above conjunction is just the balance of reasons and by hypothesis supports the prescription, and the reductio is complete). But if some defeaters are not reasons, then the support provided by the balance of reasons to a prescription may be defeated, and in such a case it seems inappropriate to say that the prescription merits all-things-considered endorsement; so I prefer (1) to (2). One can also show that, regardless of whether all defeaters are reasons, (1) is equivalent to the claim that (3) a prescription merits all-things-considered endorsement exactly if it is undefeatedly supported by the balance of reasons, and (1) is also equivalent to the claim that (4) a prescription merits all-things-considered endorsement exactly if it is supported by the conjunction of all facts.
Interestingly, the “all-things-considered” understandings of D2S and of D2J are equivalent. Indeed, a proposition and a prescription merit all-things-considered endorsement separately exactly if they merit all-things-considered endorsement jointly: if some fact f undefeatedly sustains a proposition and some fact g undefeatedly sustains a prescription, then some fact (e.g., the conjunction of f with g) undefeatedly sustains both the proposition and the prescription (mainly because the conjunction of any fact with the conjunction of f with g is the conjunction of g with some fact and thus sustains the prescription), and—trivially—vice versa.
One might argue that D2JP (in contrast to D2JA) is not in any interesting sense a desirable consequence of a definition of argument validity because D2JP is almost trivial: at almost every possible world, almost every prescription is supported by some reason—for example, by the fact that someone somewhere prefers the satisfaction over the violation proposition of the prescription. In reply, I grant that almost every prescription is possibly supported by some reason (see Vranas 2011, pp. 433–434), but the above reasoning does not show that almost every prescription is actually supported: it is probable that no reason actually supports the prescription “mutilate yourself” (because, among other things, no one happens to prefer the satisfaction over the violation proposition of the prescription), and indefinitely many further examples could be adduced. Moreover, here is a way to see that D2JP is indeed a desirable consequence: if, on a given definition of argument validity, the property of meriting pro tanto endorsement were not transmitted from the premises to the conclusion of a valid argument, then adducing (for example) a recognizably valid pure imperative argument and strong reasons supporting its premise would not be enough to convince rational people that any reason supported its conclusion—an unpalatable result. Note finally that it will not do to drop D2JP and say that transmission of meriting all-things-considered endorsement is both necessary and sufficient for an argument to be valid: as I argue in §4 (see Argument 14), it is not sufficient.
The fact that at 7am Adam promises to cook dinner for Eve can support at 8am but not at 6am the prescription “Adam, cook dinner for Eve” (since at 8am the promise has been made but at 6am the promise has not yet been made), so a fact can support (i.e., sustain) a prescription at one time but not at another. Similarly, the above fact can be a reason for Adam but not a reason for Eve (since it is Adam, not Eve, who promises), so a fact can support a prescription relative to one agent but not to another (cf. Horty 2012, p. 41). Moreover, the above fact can be a moral but not a legal reason (usually, promises to cook dinner are morally but not legally binding), so different kinds of support (moral, legal, prudential, epistemic, etc.) can be distinguished. To deal with these complications (which I ignore in the text), distinguish (1) relativized from (2) quantified validity: (1) an argument is valid relative to a time t, an agent j, and a kind of support k exactly if, necessarily, every fact that k-sustains at t relative to j every premise of the argument also k-sustains at t relative to j the conclusion of the argument, and (2) an argument is valid exactly if it is valid relative to all possible times, agents, and kinds of support. (I do not think that atemporal support exists. If impersonal support—which takes the perspectives of all agents into account—exists, include it in the range of the relevant quantifier by saying that it corresponds to a dummy agent \(j^{*}.\) Similarly, if all-inclusive support—which takes all specific perspectives (moral, legal, etc.) into account—exists, include it in the range of the relevant quantifier by saying that it corresponds to a dummy kind of support \(k^{*}.\)) Similar remarks apply if support is relative to a norm of rationality (Pigden 2012, pp. 29–31). Relativized validity may appear strange, but in note 19 I argue that in certain cases it is more useful than quantified validity.
It can also be shown that, if the accessibility relation between possible worlds is transitive, then the General Definition has as a further consequence what one gets by replacing “undefeatedly” with “indefeasibly” (and dropping the parenthetical remark) in my second formulation of D2JA in §2.2.3, so the property of being indefeasibly sustained—just like the properties of being sustained and of being undefeatedly sustained—by some fact is transmitted from the premises to the conclusion of a valid argument.
The above informal proof of the equivalence between the standard definition and Definition 1 can be formalized, and is then seen to rely on the assumptions that (1) the accessibility relation between possible words is reflexive and transitive and (2) necessarily, if a proposition is true, then it is guaranteed by some fact (e.g., the fact that the proposition is true). By using the same assumptions, one can show that Definition 1 is also equivalent to: a pure declarative argument is valid exactly if, necessarily, if some fact guarantees the premise of the argument, then some fact guarantees the conclusion of the argument.
Similarly, the argument form “the bottle contains petrol” to “believe that the bottle contains petrol” is not valid. Indeed, the proposition that the bottle contains petrol does not entail that there is an undefeated reason for you to believe that the bottle contains petrol: even if that proposition entails that there is an objective epistemic reason for you to believe that the bottle contains petrol (see note 4) and that reason cannot be defeated by any epistemic reason, that reason can still be defeated by a non-epistemic reason (for example, by the fact that your daughter’s life will be saved exactly if you do not believe that the bottle contains petrol; cf. Vranas 2011, p. 410).
See: Bergström (1962, p. 41), Castañeda (1975, p. 122), Chaturvedi (1980, p. 477), Clarke (1975, p. 418; 1985, pp. 102–103), Hare (1977, p. 468), Lemmon (1965, pp. 55–56), MacKay (1971, p. 94), and Sosa (1964, p. 69; 1967, p. 60); cf. Ross (1941, p. 61; 1941/1944, pp. 37–38) and Stalley (1972, p. 26). Contrast: Boisvert and Ludwig (2006, p. 882) and Katz (1977, pp. 235 n. 58, 241).
One might claim that the assumption leads to an infinite regress: it has the consequence that, if a fact \(f_{0}\) undefeatedly supports a prescription I, then the fact \(f_{1}\) that \(f_{0}\) undefeatedly supports I also undefeatedly supports I, and the fact \(f_{2}\) that \(f_{1}\) undefeatedly supports I also undefeatedly supports I, and so on. I reply that I do not consider the regress vicious: if \(f_{0}\) undefeatedly supports I, then (by the definition of undefeated support in §2.2.3) the conjunction of \(f_{0}\) with any fact undefeatedly supports I, so in general infinitely many facts undefeatedly support I.
Recall that support is relative to times (see note 13; to simplify, ignore here the fact that support is also relative to agents, and also ignore the existence of different kinds of support). By adapting my proof of Equivalence Theorem 1, one can derive the following results for (1) relativized and (2) quantified validity (see note 13): (1) P entails \(I^{\prime }\) relative to time t exactly if P entails that some fact whose existence follows from P undefeatedly supports \(I^{\prime }\) at t, and (2) P entails \(I^{\prime }\) exactly if P entails that some fact whose existence follows from P undefeatedly supports \(I^{\prime }\) at all possible times. But then quantified validity is not very useful (at least for cross-species imperative arguments): if P entails (for example) “teach at 2pm”, then P is in general false, because P entails that some fact supports “teach at 2pm” even at times before you are born. To see that relativized validity is more useful, note that “teach at 2pm” is entailed relative to 8am by the proposition (\(P^{*}\)) that the fact that at 8am you promise to teach at 2pm undefeatedly supports at 8am “teach at 2pm”. In contrast to P, \(P^{*}\) is not in general false; if \(P^{*}\) is true, then “teach at 2pm” merits all-things-considered endorsement at 8am, and this is useful to know even if “teach at 2pm” does not merit endorsement later on (e.g., because at 9am you are released from your promise).
Morscher (1972) points out that Poincaré (1913, p. 225) only talks about the two premises of a syllogism, but Weinberger (1976) replies that Poincaré’s reasoning shows that Hare is correct to attribute the rule formulated in the text to Poincaré. On possible objections to Poincaré’s Principle based on arguments similar to Argument 7 or Argument 9, see: Castañeda (1960a, p. 46; 1960b, pp. 173–174; 1968, pp. 39–42; 1975, pp. 198–199), Hamblin (1987, pp. 90–92), Morscher (1974, pp. 22–23), Popper (1945, p. 205), and Prior (1949, p. 71) (contrast: Clarke 1973, p. 217; Hoche 1995, pp. 341–342; Lemmon 1965, p. 68; MacIntyre 1965, p. 520). On rejections of the principle on other grounds, see: Geach (1958, pp. 55–56) (cf. note 22; Clarke 1970, p. 100; Morscher and Zecha 1971, p. 209), Borchardt (1979, p. 202), and Gibbons (1960, pp. 209–210) (cf. also Bohnert 1945, p. 311; Zellner 1971, p. 21). On (at least tentative) endorsements of (versions of) the principle, see: Bergström (1962, pp. 46–47), Castañeda (1963, p. 234), Dubislav (1937, p. 342), Frey (1957, pp. 438, 465), Grue-Sörensen (1939, p. 195), Jörgensen (1938, p. 288; 1938/1969, p. 9), Lalande (1963, pp. 136, 169), Lemmon (1965, p. 69), Moritz (1954, p. 79), Popper (1945, p. 53; 1948, p. 154), and Weinberger (1972, p. 151).
This suggestion coheres with Hare’s claim that the rule under consideration is the “basis” of Hume’s is/ought thesis—which can be formulated as the thesis that no normative conclusion follows from a consistent set of non-normative premises (Hare 1952, p. 29; cf. Castañeda 1960a, p. 49; Duncan-Jones 1952, p. 199; Moritz 1954, p. 79). An alternative attempt to solve the puzzle begins by noting (following Mavrodes 1968, pp. 356–358; cf. Bergström 1962, p. 43) that saying that a set of premises “does not contain at least one imperative” can be understood either as claiming that (1) no premise is an imperative or as claiming that (2) no premise is an imperative and the set of premises does not implicitly contain an imperative. On the former understanding, which is common in the literature, one gets a version of Poincaré’s Principle which is indeed false if Argument 4 or the argument from “I ought to do X” to “let me do X” is valid. On the latter understanding, however, one can get a version of Poincaré’s Principle which need not be false if the above two arguments are valid: arguably, the premise of Argument 4 (similarly for “I ought to do X”) implicitly contains an imperative. But if it implicitly contains an imperative because, as Mavrodes suggests, saying that a set of premises “does not implicitly contain at least one imperative” means “does not validly entail at least one imperative”, then the corresponding version of Poincaré’s Principle is “completely trivial” (1968, p. 357).
To defend the validity of (a variant of) Argument 11, Geach uses certain “rule[s] whereby, given one valid inference, we may derive another” (1958, p. 52); for example, “p, q, ergo r // p; ergo, either not q or r” (1958, p. 55). Castañeda replies that “Geach nowhere establishes that [these rules] also apply to imperatives” (1958, p. 45). For further discussion of Geach, see: Bergström (1962, pp. 45–46; 1970, pp. 423–424), Borowski (1977, pp. 458–460), Espersen (1967, pp. 99–100), Lemmon (1965, p. 56) (cf. Sosa 1966), Parsons (2013, p. 68), and Stalley (1972, pp. 24–25).
(1) For the sake of simplicity, in this informal proof I implicitly assume that the accessibility relation between possible worlds is universal; a formal proof can be given which assumes only that the accessibility relation is symmetric and transitive. (2) For (a) validity relative to a time t and (b) quantified validity (see note 13), replace in the theorem “supports I” with (a) “supports I at t” and with (b) “supports I at some possible time”, respectively. (Remarks similar to (1) and (2) apply to Equivalence Theorem 3 in §5 and to its proof. By contrast, a formal proof of Equivalence Theorem 1 can be given which assumes only that the accessibility relation is reflexive.) (3) Because I assume that, necessarily, every fact exists (see note 7), it turns out that for the purposes of this paper it does not matter whether quantifiers are or not restricted to range only over existing objects; this is why in the formulation of the theorem I can switch between “some fact” and “there is a fact”.
In response one might propose a different analogy, to the effect that there is a necessary connection between a prescription I and the proposition that I is undefeatedly supported by some fact. On this analogy, the argument from “marry me” to “there is an undefeated reason for you to marry me” is valid (and I cannot claim that its converse argument is not valid if I claim that Argument 9 is valid), and thus Argument 14 is also valid. I reply that this analogy also fails: although, necessarily, every fact that guarantees a proposition P also guarantees that P is true, it is false that, necessarily, every fact that supports (or even undefeatedly supports) a prescription I also guarantees that some fact undefeatedly supports I. Indeed: possibly (i.e., at some possible world w), the fact f that you have promised to run (1) undefeatedly supports (at w) “run” but (2) does not guarantee that some fact undefeatedly supports “run” because, possibly (i.e., at some possible world \(w^{\prime }\)), f exists but no fact undefeatedly supports (at \(w^{\prime }\)) “run”—e.g., because the fact that your leg is now broken also exists (at \(w^{\prime }\) but not at w).
The conclusion of Argument 15 entails that some fact is possibly a reason, so another consequence of the theorem is that “marry me” entails that some fact is possibly a reason. Arguably, this consequence is intuitively acceptable because the proposition that some fact is possibly a reason is necessary (and thus follows from any prescription). Opponents of facts may well disagree, but recall that I can replace talk of facts will talk of true propositions (§2.2.2). Nihilists about reasons, who hold that it is impossible for reasons to exist, may also disagree, but clearly my project in this paper is not addressed to such nihilists (since my appeal to reasons is crucial to my account).
If this assumption is true, then a consequence of Definition 4 is that any proposition follows from a necessarily violated prescription. For example, “you smiled” follows from “don’t tell the truth and tell the truth”, so Argument 2 (§2.2.2) is valid. One might claim that Argument 2 is not valid according to Definition 4 because it is possible that some fact (e.g., the fact that if you tell the truth your mother will die but if you do not tell the truth an innocent person will be imprisoned) supports both “don’t tell the truth” and “tell the truth” but does not guarantee “you smiled”. I reply that (1) this is not possible (given my assumption that favoring is asymmetric) and (2) even if it is possible, Argument 2 is still valid: on my account (see §2.1 and the paragraph right after the formulation of the General Definition in §2.3), Argument 2 is valid exactly if, necessarily, every fact that supports the conjunction of its premises (as opposed to supporting every premise) guarantees its conclusion—and this is so if, necessarily, no fact supports “don’t tell the truth and tell the truth”.
A similar reasoning shows that the converse entailment (from the conclusion of Argument 16 to the conclusion of Argument 12) holds if the following plausible assumption (which is the converse of the above assumption) is true: necessarily, if it is necessary that no fact supports a given prescription, then the prescription is necessarily violated (see Vranas 2011, pp. 433–434).
(1) One can similarly show that the argument from “kiss him” to “I permit you to kiss him” is not valid. This result is intuitively acceptable because the premise of the argument, namely the prescription I express by using the imperative sentence “kiss him”, can also be expressed by someone else (using the same imperative sentence), and thus intuitively it does not entail that I grant you permission to kiss him. Admittedly, it would be strange for me to say “kiss him, but I do not permit you to kiss him”, but this does not show that “kiss him” entails “I permit you to kiss him” (compare: it would be strange for me to say “it is raining, but I do not believe it is raining”, but this does not show that the proposition that it is raining entails the proposition that I believe it is raining; cf. Moore 1942, p. 543; 1944, p. 204; 1993). (2) Similarly, the argument from “kiss him” to “you are permitted (i.e., not forbidden) to kiss him” is not valid: possibly, some fact (e.g., the fact that he will be happy if you kiss him) supports “kiss him” but does not guarantee that you are permitted to kiss him. This result is intuitively acceptable because, if that argument is valid, then (given that adding declarative premises preserves validity; see §6.2) the argument from “kiss him” and “you are not permitted to kiss him” to “you are permitted to kiss him” is also valid—but intuitively it is not. (The premises of the latter argument are consistent; see note 43.) (3) One might argue that “kiss him” presupposes that you are permitted to kiss him, and similarly “marry me” presupposes that you can marry me, so it is a purely verbal issue whether the argument in (2) or Argument 18 is valid. I reply that the existence of large literatures on the validity of deontic analogs of these arguments (i.e., on whether obligatoriness entails permissibility, and on whether “ought” implies “can”) suggests that whether these arguments are valid is not a purely verbal issue.
Following MacFarlane (2005/2009), distinguish “four general attitudes” towards logical validity. (1) Demarcaters hold that “logicians who investigate the (non-formal) kind of ‘validity’ possessed by [arguments like ‘this is yellow; so this is colored’] are straying from the proper province of logic into some neighboring domain” (e.g., lexicography or metaphysics). (2) Debunkers hold that “logic is concerned with validity simpliciter, not just validity that holds in virtue of a limited set of ‘logical forms.’ ” (3) Relativists agree with Demarcaters that “logical consequence must be understood as formal consequence”, but relativize “logical consequence to a choice of logical constants”. Finally, (4) Deflaters hold that logical validity is not a relative notion but “logical validity” is a “family resemblance” term, like “game”. I sympathize with Relativists: in the context of imperative logic, I take “fact” and “favor” to be logical constants, like “and” and “or” (cf. Warmbrōd 1999, pp. 534–536). But I do not need to insist on this; instead I want to note that Debunkers and Relativists need not deny that my project in this paper is important for logic, and that Demarcaters and Deflaters who deny this (by using the reasoning of the above objection) seem committed to the unpalatable claim that typical projects of deontic and epistemic logicians are unimportant for logic (contrast Harman 1972, p. 81).
More precisely, the claim that no declarative (i.e., indicative) conclusion can be validly drawn from only imperative premises follows from Hare’s Thesis (which is then refuted by the validity of Argument 12) if (1) no imperative (premise) is also declarative (Gibbons 1960, p. 209) and (2) no declarative conclusion can be validly drawn from no premises whatsoever (cf. MacKay 1969, p. 147). Rescher (1966, p. 73 n. 1) notes that Hare (1952, p. 34)—implicitly—restricts the rule to premises that are not what Hare calls “hypothetical” imperatives. Bergström (1962, p. 47) and Espersen (1967, p. 101) argue that, to avoid trivial counterexamples, the rule should be restricted to consistent sets of premises and to non-necessary conclusions.
Indeed, this prescription is satisfied exactly if Dan has only one daughter and you marry her, is violated exactly if Dan has only one daughter and you do not marry her, and is avoided exactly if Dan does not have only one daughter—and it is natural to say that these are precisely the satisfaction, violation, and avoidance conditions of the prescription expressed by the composite sentence “Dan has only one daughter; marry her” (although it is also natural to say, alternatively, that the composite sentence expresses a prescription—namely the premise of Argument 20—which is violated if Dan does not have only one daughter). This specification of the prescription expressed by the composite sentence avoids Castañeda’s (1963, pp. 228–229) objection that “[‘marry her’] is not by itself a complete imperative, since it does not contain the referent of the pronoun [‘her’]”, and “cannot, therefore, be a premise”. Note the analogy between (1) the suggestion that “Dan has only one daughter; marry her” amounts to “Dan has only one daughter; if Dan has only one daughter, marry her” and (2) the equivalence between \( S\, \& \,T\) and \( S\, \& \,(S\rightarrow T),\) where S and T are any declarative sentences. Note also that the sentence “marry Dan’s only daughter” can also be understood as expressing the prescription expressed by “make it the case that Dan has only one daughter and you marry her”; but this understanding is atypical (cf. Grice 1981/1989, p. 270), and in any case does not correspond to a valid or intuitively valid argument. Finally, one might propose understanding the sentence “marry Dan’s only daughter” as expressing either the prescription expressed by (a) “marry the x who is Dan’s only daughter” or the prescription expressed by (b) “let it be the case that you marry the x who is Dan’s only daughter”, where “the x who is Dan’s only daughter” corresponds to a restricted quantifier (cf. Pietroski 1999/2014). Concerning (a), I reply that, absent a theory of imperative restricted quantification, it is unclear what exactly (a) expresses. Concerning (b), I reply that, even if the prescription that (b) expresses entails “let it be the case that Dan has only one daughter”, it does not entail “Dan has only one daughter”.
Consider also the argument from “marry me” to “you are not already married to me”. In contrast to the prescriptions typically expressed by “marry Dan’s only daughter” (understood as “marry sooner or later the person who now is Dan’s only daughter”), which cannot be satisfied if Dan does not (now) have only one daughter, the prescription typically expressed by “marry me” (understood as “marry me sooner or later”) can be satisfied even if you are already married to me: we can get a divorce and then get remarried. Once this is noticed, the illusion that the argument from “marry me” to “you are not already married to me” is valid (and thus refutes Hare’s Thesis) should vanish. Note that I do not need to take a stand on whether “marry me” presupposes that you are not already married to me (cf. Adler 1980, pp. 105–107; Lemmon 1965, p. 57; Sosa 1964, p. 5; Warnock 1976, p. 294; Wedeking 1969, p. 38; see also Clarke 1975, p. 418). (Primarily to avoid unnecessary controversy, in this paper I rely on standard declarative logic; so I do not examine views on which, for example, “you will marry Dan’s only daughter” is neither true nor false if Dan does not have only one daughter. So I take it that “you will marry Dan’s only daughter” entails that Dan has only one daughter; otherwise, at some possible world, Dan does not have only one daughter but you marry Dan’s only daughter—which is absurd.)
For similar examples, see: Geach (1958, p. 52), Pigden (2011, pp. 3, 5–6), and Rescher (1966, p. 96). For further endorsements of the view that such arguments are valid, see: Adler (1980, pp. 102–103), Bergström (1962, pp. 40, 47), Castañeda (1958, p. 45), Clarke (1970, p. 100), Gombay (1967, p. 150), Morscher and Zecha (1971, p. 209), Parsons (2013, pp. 86–87), and Sosa (1964, pp. 88–89; 1970, p. 221).
One might wonder why the conjunction of the conditional prescription \((I_{1})\) “if he comes, leave the files open” with \((I_{2})\) “do not leave the files open” is an unconditional prescription, namely the premise of Argument 22. To see intuitively why, note that \(I_{2}\) is (necessarily) equivalent to the conjunction of \((I_{3})\) “if he comes, do not leave the files open” with \((I_{4})\) “if he does not come, do not leave the files open”. But \(I_{3}\) & \(I_{4}\) is equivalent to \(I_{3}\) & (\(I_{3}\) & \(I_{4}\)), so \(I_{2}\) is equivalent to \(I_{3}\) & \(I_{2}.\) Then \(I_{1}\) & \(I_{2}\) is equivalent to \(I_{1}\) & (\(I_{3}\) & \(I_{2}\)), and thus to (\(I_{1}\) & \(I_{3}\)) & \(I_{2}.\) But \(I_{1}\) & \(I_{3}\) is equivalent to “if he comes, leave the files open and do not leave the files open”, so (\(I_{1}\) & \(I_{3})\) & \(I_{2}\) (and thus \(I_{1}\) & \(I_{2}\)) is equivalent to the conjunction of “let it be the case that he does not come” with \(I_{2},\) and thus to the premise of Argument 22. (The above equivalences can be rigorously justified by using Definition 5, but here I am appealing to their intuitive plausibility. For the sake of simplicity, throughout this paper I assume that necessarily equivalent propositions are identical, and thus that so are necessarily equivalent prescriptions—i.e., prescriptions whose satisfaction propositions are necessarily equivalent and whose violation propositions are also necessarily equivalent. Dropping this simplifying assumption would not affect my main claims; for example, I would say that the conjunction of the premises of Argument 21 is necessarily equivalent to the premise of Argument 22, but this would not affect my main claim that Argument 21 is valid exactly if Argument 22 is.)
One might argue that the validity of Argument 12 does not refute a version of Hare’s Thesis restricted to consistent sets of premises and to non-necessary conclusions (see the end of note 31): the conclusion of Argument 12 is true (it is definitely possible that there be a reason for you to marry me) and thus is necessary (assuming that whatever is possible is necessarily possible). More subtly, one might argue that the validity of Argument 12 does not refute the restricted version of Hare’s Thesis because Argument 12 either (1) has a necessary conclusion or (2) has an inconsistent premise: either (1) the conclusion of Argument 12 is true and thus necessary, or (2) the conclusion of Argument 12 is false, in other words it is necessary that no fact supports the prescription “marry me”, and then this prescription—which is the premise of Argument 12—is necessarily violated (given the assumption in note 28) and thus is inconsistent (given the definition of imperative inconsistency that I have defended in Vranas 2008, pp. 545–548: a set of prescriptions is inconsistent exactly if it is necessary that at least one of the prescriptions be violated). I reply that Argument 15 (as opposed to Argument 12) does not have a necessary conclusion, so the validity of Argument 15 refutes even the restricted version of Hare’s Thesis.
Proof of (1). (a) Suppose that, necessarily (i.e., at every—see note 23—possible world), every fact that both supports I and guarantees P guarantees \(P^{\prime }.\) Take any possible world w at which some fact f which guarantees P (so that, necessarily, if f exists, then P is true) possibly supports I. Then, at some possible world \(w^{\prime },\,f\) both (i) supports I (since f possibly supports I at w) and (ii) guarantees P (since, necessarily, if f exists, then P is true), and thus—by the supposition—f guarantees \(P^{\prime };\) i.e., necessarily, if f exists, then \(P^{\prime }\) is true. Since f exists at w, \(P^{\prime }\) is true at w. (b) Conversely, suppose that, necessarily, if some fact which guarantees P possibly supports I, then \(P^{\prime }\) is true. Take any possible world w. If a fact f both supports I and guarantees P at w, then f guarantees \(P^{\prime }\) at w because, necessarily, if f exists, then f (and thus some fact which guarantees P) possibly supports I (since f supports I at w) and thus—by the supposition—\(P^{\prime }\) is true. (2) can be similarly proved.
It might be thought that the validity of Argument 25 refutes the restricted version of Hare’s Thesis I examined in note 36. One might reply, however, that either (1) it is impossible for you to repent, and then the premises of Argument 25 are inconsistent (by Definition 7 below in the text and the assumption in note 28), or (2) it is possible for you to repent, and then the conclusion of Argument 25 follows from the declarative premise of that argument alone (because then it is necessarily possible for you to repent, so the antecedent of the declarative premise of Argument 25 is necessary, and then that premise is necessarily equivalent to its consequent, which is also the conclusion of Argument 25).
One might propose the following alternative definition: a proposition and a prescription are inconsistent exactly if, necessarily, no fact both guarantees the proposition and undefeatedly supports the prescription (equivalently, the proposition entails that no fact undefeatedly supports the prescription; to see the equivalence, use a reasoning similar to that in note 11). If a proposition and a prescription are inconsistent according to Definition 7, they are also inconsistent according to the alternative definition. Not conversely, however: the proposition that no fact undefeatedly supports “run” and the prescription “run” are consistent according to Definition 7 (possibly, the fact that no fact undefeatedly supports “run” but you have promised to run both guarantees the proposition and supports the prescription) but are inconsistent according to the alternative definition. I have no clear intuition on whether the above proposition and prescription should count as inconsistent, but I think that Definition 7 is preferable to the alternative definition because I think that the General Definition, to which Definition 7 corresponds, is preferable to the most plausible definition of argument validity to which the alternative definition of inconsistency corresponds (see “Undefeated sustaining” in the Appendix).
Given this assumption and the converse assumption that, necessarily, if it is necessary that no fact supports a given prescription, then the prescription is necessarily violated (cf. note 28), defining a prescription to be inconsistent exactly if it is necessarily violated (see Vranas 2008, pp. 545–548 for a defense of this definition) is equivalent to defining a prescription to be inconsistent exactly if it is necessary that no fact supports it (and one can show then that a prescription I is inconsistent exactly if, for some necessary proposition P, P and I are inconsistent). There is then no important disanalogy between my previously defended (Vranas 2008) definition of inconsistency for prescriptions (a definition that did not appeal to meriting endorsement) and Definition 7 (which does appeal to meriting endorsement).
(1) If support is understood weakly (see §6.1), then for the above reasoning to go through I must be unconditional (see Vranas 2011, p. 389 n. 27). (2) The above reasoning, in conjunction with what I said in §3 (after I introduced Argument 4), fulfills the promise I made in §2.2.2 to defend the claim that the premises of Argument 3 are inconsistent. (3) Corresponding to relativized and quantified validity (see note 13), relativized and quantified consistency and inconsistency can be defined; quantified inconsistency is inconsistency relative to all possible times, agents, and kinds of support, whereas quantified consistency is consistency relative to some possible time, agent, and kind of support.
It can be shown that the first three cases (but not the fourth one) can be subsumed under the fifth case. For example, concerning the first case: if P is impossible, then P (trivially) entails that no fact supports I. So the five cases are not mutually exclusive. To see that the five cases are not collectively exhaustive either, consider the prescription I expressed by “don’t tell the truth” and the proposition P that (1) you have sworn to tell the truth and (2) every fact that guarantees that you have sworn to tell the truth defeats the support that any fact provides to I. According to Definition 7, P and I are inconsistent: necessarily, every fact that guarantees P guarantees that you have sworn to tell the truth and defeats the support that any fact provides to I, and thus does not support I (because a fact that supports I cannot defeat that support). But P entails neither the negation of I (because P does not entail that any fact supports that negation) nor that no fact supports I, so this is not an example of the fourth or of the fifth case (and thus not of any of the first three cases either).
Here are also five examples on which Definition 7 returns a verdict of consistency (cf. Vranas 2008, p. 562 n. 49). (1) The proposition that I do not permit you to kiss him and the prescription “kiss him” (cf. note 29) are consistent: possibly, they are both sustained by the fact that I do not permit you to kiss him but he will die if you do not kiss him. (2) The proposition that you are not permitted to kiss him and the prescription “kiss him” (cf. note 29) are consistent (contrast Hare 1967, p. 311): possibly, they are both sustained by the fact that you are required not to kiss him but he will be happy if you kiss him. (3) The proposition that you are already married to me and the prescription “marry me” (cf. note 33) are consistent: possibly, they are both sustained by the fact that you are already married to me but we have agreed to get a divorce and then get remarried. (4) The proposition that Dan has no daughter and the prescription expressed by “marry Dan’s only daughter” (understood as “if Dan has only one daughter, marry her”; if it is understood instead as the premise of Argument 20, see the last paragraph of §6.2.2) are consistent: possibly, they are both sustained by the fact that Dan has no daughter but you have promised that, if Dan has only one daughter, you will marry her. (5) The proposition that you will not marry me and the prescription “marry me” are consistent (cf. Williams 1966, p. 5): possibly, they are both sustained by the fact that you have promised to marry me but you will not marry me.
Given Definition 7, the third consequence of Definition 6 that I examined above is equivalent to: if \( P \& \sim P{^{\prime }}\) and I are consistent, then I and P do not jointly entail \(P^{\prime }.\) The converse does not hold (contrast Sosa 1970, p. 221); for example, the argument from (I) “marry me” and (P) “the sky is blue” to \((P^{\prime })\) “there is a reason for you to marry me” is not valid (cf. Argument 14), but \( P \& \sim P^{\prime }\) and I are inconsistent because \(\sim P^{\prime }\) is (and thus \( P \& \sim P^{\prime }\) entails) the proposition that no fact supports I (i.e., there is no reason for you to marry me).
To show this, suppose that (1) at some possible world w, the fact—call it \(f_{p}\)—that P is true supports I, and that (2) at every possible world (and thus at w), every fact that both supports I and guarantees P guarantees \(P^{\prime }.\) Since, necessarily, \(f_{p}\) guarantees P, by (1) \(f_{p}\) both supports I and guarantees P at w, so by (2) \(f_{p}\) guarantees \(P^{\prime }\) at w. Then, necessarily, if \(f_{p}\) exists, then \(P^{\prime }\) is true. But, necessarily, if P is true, then \(f_{p}\) exists. So, necessarily, if P is true, then \(P^{\prime }\) is true; i.e., P entails \(P^{\prime }.\)
Similarly, if the proposition that Grimbly Hughes is the largest grocer in Oxford entails that every reason for you to go to the largest grocer in Oxford is a reason for you to go to Grimbly Hughes, then the argument from “go to the largest grocer in Oxford” and “Grimbly Hughes is the largest grocer in Oxford” to “go to Grimbly Hughes” is valid (cf. Geach 1958, p. 53; Hare 1952, p. 35; Pigden 2011, p. 2; Stalley 1972, pp. 24–25). On arguments similar to Argument 33, see: Åqvist (1967, p. 23), Bergström (1962, p. 42), Grue-Sörensen (1939, pp. 196–197), Hansen (2008, p. 4), Ledent (1942, pp. 268–269), Poincaré (1913, p. 236), Rand (1939, p. 318; 1939/1962, p. 249), and Rescher (1966, pp. 100–101). One might claim that Argument 33 is not valid because the invalidity of its pure subargument (from “disarm the bomb” to “cut the wire”), which arises from the possibility that you disarm the bomb without cutting the wire, is not “rectified” by adding the premise that you can disarm the bomb only if you cut the wire, because that premise does not preclude the above possibility: maybe, although it is possible that you disarm the bomb without cutting (but rather, for example, by melting) the wire, you cannot do so (because, for example, you have no way to heat the wire). I reply that precluding the above possibility is not necessary for “rectifying” the invalidity: it is enough to add instead, for example, a declarative premise that entails “cut the wire” or that entails “don’t disarm the bomb”.
In the above example, the fact that you have promised to resign today strongly and weakly supports the prescription “resign today”, and weakly but not strongly supports the prescription “resign”. For the sake of simplicity, the above definition of strong support differs in two respects from my previously proposed definition of strong support (Vranas 2011, p. 386): the above definition (1) omits the “satisfaction indifference condition”, which is not needed for my main results, and (2) has the consequence (cf. note 6) that any fact that strongly supports a prescription favors an impossible proposition over itself (because an impossible proposition entails both the satisfaction and the violation proposition of any prescription). Note that strong support entails both weak support and support, but weak support does not entail support (see Vranas 2011, p. 389 n. 27); nevertheless, I assume throughout the paper that every fact that weakly supports a prescription is a reason.
Elsewhere (Vranas 2011, pp. 433–437) I have proved the following Equivalence Theorem: if S, V, and C are, respectively, the satisfaction proposition, the violation proposition, and the context of the conjunction of the premises of a pure imperative argument and \(S^{\prime },\,V^{\prime },\) and \(C^{\prime }\) are similarly defined for the conclusion of the argument, then (1) the argument is s / s valid exactly if either V is necessary or both \(S^{\prime }\) entails S and \(V^{\prime }\) entails V, and (2) the argument is w / w valid exactly if both \(C^{\prime }\) entails C and \(V^{\prime }\) entails V. A corollary of this theorem is that every s / s valid pure imperative argument is also w / w valid, so for pure imperative arguments I prefer “strongly valid” to “s / s valid” and “weakly valid” to “w / w valid”. I do not use this alternative terminology for mixed-premise imperative arguments, however, because some s / s valid mixed-premise imperative arguments are not w / w valid; an example is the argument from (a) the prescription “if you sinned, don’t repent” and (b) the proposition that the fact that you have sworn to repent if you sinned undefeatedly weakly supports “if you sinned, repent” to (c) the prescription “run” (one can show that the premises of this argument are weakly but not strongly consistent). One can show that every s / s valid cross-species imperative argument (see Table 1) is also w / w valid, and that every w / w valid cross-species declarative or mixed-premise declarative argument is also s / s valid.
To be explicit, Theorem 5 is about weak support. Before I prove Theorem 5, I introduce a definition: a pure imperative argument is non-conjunctively w / w valid exactly if, necessarily, every fact that weakly supports every premise of the argument also weakly supports the conclusion of the argument. By Definition 5 and the Equivalence Theorem in note 48, Theorem 5 is equivalent to the following theorem: given any two-premise pure imperative argument such that the context of one of its premises entails the context of the other, the argument is non-conjunctively w / w valid exactly if it is w / w valid. This theorem is a consequence of the conjunction of two claims: (1) necessarily, every fact that weakly supports the conjunction of two prescriptions also weakly supports both conjuncts (see Vranas 2011, p. 398 n. 38 for a proof), and (2) if the context of one of two prescriptions entails the context of the other, then, necessarily, every fact that weakly supports both prescriptions also weakly supports their conjunction. To prove (2), suppose that \(C^{*}\) entails C, and take any fact f that (at some possible world) weakly supports both I and \(I^{*}.\) Then, by Definition 9, f strongly supports some prescriptions \(I^{\prime }\) and \(I^{*\prime }\) such that \(C^{\prime }=C,\,C^{*\prime }=C^{*},\,S^{\prime }\) entails S, and \(S^{*\prime }\) entails \(S^{*}\)—and thus V entails \(V^{\prime }\) and \(V^{*}\) entails \(V^{*\prime }.\) By Definition 5, the context of \( I\, \& \,I^{*}\) is \(C\vee C^{*};\) this is just C, since \(C^{*}\) entails C, and thus is \(C{^{\prime }},\) since \(C^{\prime }=C.\) Given that f strongly supports \(I{^{\prime }}\) and that \(C{^{\prime }}\) is the context of \( I\, \& \,I^{*},\) to show that f weakly supports \( I\, \& \,I^{*}\) it is enough (by Definition 9) to show that \(S^{\prime }\) entails the satisfaction proposition of \( I\, \& \,I^{*}\)—or equivalently that the violation proposition of \( I\, \& \,I^{*},\) namely \(V\vee V^{*},\) entails \(V^{\prime }.\) Given that V entails \(V^{\prime },\) it is enough to show that \(V^{*}\) entails \(V^{\prime }.\) Given that \(V^{*}\) (entails \(C^{*}\) and thus) entails \(C^{\prime },\) it is enough to show that \(V^{*}\) entails \(\sim S^{\prime }.\) To show this, suppose for reductio that \( V^{*}\, \& \,S^{\prime }\) is possible. By Definition 9, f favors \( V^{*}\, \& \,S^{\prime }\) over \( V^{\prime }\, \& \,S^{*\prime }\) (because \( V^{*}\, \& \,S^{\prime }\) entails \( S^{\prime },\,V^{\prime }\, \& \,S^{*\prime }\) entails \(V^{\prime },\) and f strongly supports \(I^{\prime }\)), and f also favors \( V^{\prime }\, \& \,S^{*\prime }\) over \( V^{*}\, \& \,S^{\prime }\) (because \( V^{\prime }\, \& \,S^{*\prime }\) entails \( S^{*\prime },\,V^{*}\, \& \,S^{\prime }\) (entails \(V^{*}\) and thus) entails \(V^{*\prime },\) and f strongly supports \(I^{*\prime }\)). But this contradicts the asymmetry of favoring, and the reductio is complete.
To be explicit, Theorem 6 is about weak support and w / w validity. I will prove the following generalization of Theorem 6: if \( I \& \sim I^{\prime }\) is unconditional, \(\sim V\) entails \(C^{\prime },\) and P is consistent with the proposition that some fact undefeatedly weakly supports \( I \& \sim I^{\prime },\) then the argument from I and P to \(I^{\prime }\) is not w / w valid. To prove this, note first that, necessarily, if P is consistent with the proposition that some fact f undefeatedly weakly supports \( I \& \sim I^{\prime },\) then, possibly, some fact (e.g., the conjunction of f with the fact that P is true) both guarantees P and weakly supports \( I \& \sim I^{\prime }.\) So to show (under the conditions of the theorem) that, possibly, some fact guarantees P and weakly supports I but does not weakly support \(I^{\prime }\) (i.e., that the argument from I and P to \(I^{\prime }\) is not w / w valid), it is enough to prove the following lemma: if \( I \& \sim I^{\prime }\) is unconditional and \(\sim \) V entails \(C^{\prime },\) then, necessarily, every fact that weakly supports \( I \& \sim I^{\prime }\) also weakly supports I but does not weakly support \(I^{\prime }.\) To prove this lemma, suppose that \( I \& \sim I^{\prime }\) is unconditional and \(\sim \) V entails \(C^{\prime },\) and take any fact f that (at some possible world) weakly supports \( I \& \sim I^{\prime }\) and thus (by Definition 9) strongly supports some unconditional prescription \(I^{*}\) whose satisfaction proposition \(S^{*}\) is possible (given my standing assumption that, necessarily, a necessarily violated prescription is necessarily not supported by any fact) and entails the satisfaction proposition of \( I \& \sim I^{\prime },\) namely \(\sim (V\vee S^{\prime })\)—and thus also entails \(C^{\prime }\) (since \(\sim (V\vee S^{\prime })\) entails \(\sim \) V, which entails \(C^{\prime }\)). By claim (1) in note 49, f weakly supports both I and \(\sim \) \(I^{\prime }.\) To show that f does not weakly support \(I^{\prime },\) suppose for reductio that it does. Then f weakly supports both \(I^{\prime }\) and \(\sim \) \(I^{\prime },\) and thus (by Definition 9) strongly supports some prescriptions (with context \(C^{\prime }\)) \(I_{1}\) and \(I_{2}\) such that \(S_{1}\) entails \(S^{\prime }\) and \(S_{2}\) entails \(V^{\prime }\)—and thus \(V^{\prime }\) entails \(V_{1}\) and \(S^{\prime }\) entails \(V_{2}.\) Then, by Definition 9, f favors \(S_{1}\) over \(S_{2}\) (because \(S_{2}\) entails \(V_{1}\) and f strongly supports \(I_{1}\)) and also favors \(S_{2}\) over \(S_{1}\) (because \(S_{1}\) entails \(V_{2}\) and f strongly supports \(I_{2})\), so \(S_{1}\) and \(S_{2}\) are both impossible and the violation proposition of \(I_{1}\) (and of \(I_{2}\)) is \(C^{\prime }.\) Since \(S^{*}\) entails \(C^{\prime }\) and f strongly supports \(I_{1},\,f\) favors any impossible proposition over \(S^{*}.\) But since f strongly supports \(I^{*},\,f\) favors \(S^{*}\) over any impossible proposition. This contradicts the asymmetry of favoring (given that \(S^{*}\) is possible), and the reductio is complete.
Theorem 6 shows that Argument 35 is not w / w valid. To see that Argument 35 is not s / s valid either, note first that the proposition that some fact f undefeatedly strongly supports I is consistent with P. Therefore, (1) possibly, some fact (e.g., the conjunction of f with the fact that P is true) both guarantees P and strongly supports I. But, (2) necessarily, every fact that strongly supports I does not strongly support \(I^{\prime }\) (because, by Definition 9, necessarily, every fact that strongly supports I favors the proposition that you marry him and you do not dump him, which entails the satisfaction proposition of I but the violation proposition of \(I^{\prime },\) over any impossible proposition, which entails the violation proposition of I but the satisfaction proposition of \(I^{\prime }\)). (1) and (2) jointly entail that, possibly, some fact guarantees P and strongly supports I but does not strongly support \(I^{\prime };\) i.e., Argument 35 is not s / s valid.
One can similarly show that the “Catch-22” argument in §1 and the mixed-premise imperative argument in Table 1 (cf. Argument 38) are not valid. Concerning the remaining mixed arguments in Table 1, in my view the cross-species imperative argument is valid (if Argument 9 is valid), the cross-species declarative argument is ambiguous (cf. Argument 19 and Argument 20), and the mixed-premise declarative argument is not valid (cf. Argument 27).
Similar remarks apply to the argument—call it Argument 33*—from “disarm the bomb” (I) and “you will disarm the bomb only if you cut the wire” (P) to “cut the wire” \((I^{\prime }).\) (Contrast Argument 33.) By Theorem 6, Argument 33* is not valid: \(I^{\prime }\) is unconditional, and the proposition that some fact undefeatedly supports “disarm the bomb without cutting the wire” (which is \( I \& \sim I^{\prime }\)) does not entail that you will disarm the bomb without cutting the wire (i.e., does not entail that P is false) and thus is consistent with P. The result that Argument 33* is not valid is intuitively acceptable because (1) the arguments (a) from “disarm the bomb” and “you will not disarm the bomb” to “cut the wire” and (b) from “disarm the bomb” and “you will cut the wire” to “cut the wire” are intuitively not valid (cf. MacKay 1969, p. 153; 1971, p. 94; Parsons 2013, p. 68; Bennett 1970, p. 318) but (2) one gets arguments equivalent to the above two by adding to the premises of Argument 33* the declarative premises “you will not disarm the bomb” and “you will cut the wire”, respectively, so—it is intuitively clear that—the above two arguments are valid if Argument 33* is valid (because adding declarative premises preserves validity). On arguments similar to Argument 33*, see: Bennett (1970, p. 318), MacKay (1969, pp. 150–151), Rescher (1966, p. 99), Sosa (1964, p. 89), Stalley (1972, pp. 23–24), and Vetter (1971, pp. 75–76).
Some people might accept the standard definition of validity for pure declarative arguments and still reject Declarative Monotonicity because (1) they reject the claim that adding imperative premises preserves validity—call this claim Imperative Monotonicity—and (2) they reason as follows that Declarative Monotonicity entails Imperative Monotonicity: if Declarative Monotonicity holds, then, for any prescription I, adding to the premises of a valid argument a declarative premise that entails I preserves validity, but then adding instead the imperative premise I also preserves validity, and Imperative Monotonicity also holds. Concerning first (2) above, I reply that the above reasoning is mistaken: even if an argument whose premises include a proposition P that entails a prescription I is valid, the argument that one gets by replacing P with I need not be valid. For example, the argument from the proposition “the fact that you have promised to marry me is an undefeated reason for you to marry me” to the proposition “there is a reason for you to marry me” is valid, and the former proposition entails “marry me” (cf. Argument 4), but the argument from “marry me” to “there is a reason for you to marry me” (Argument 14) is not valid. So those who reject Imperative Monotonicity should not reject Declarative Monotonicity on the basis of the above reasoning. Concerning now (1) above, by using claim (1) in note 49 one can show that Imperative Monotonicity is a consequence of the General Definition (for weak support; Imperative Monotonicity fails for strong support), so let me address an objection to Imperative Monotonicity. One might claim (cf. Horty 1997, pp. 35–36) that (a) the argument from “don’t eat with your fingers” to “if you eat asparagus, don’t eat with your fingers” is valid but (b) the argument from “don’t eat with your fingers” and “if you eat asparagus, eat with your fingers” to “if you eat asparagus, don’t eat with your fingers” is not valid. I reply that the conjunction of the premises of the second argument is “don’t eat asparagus, and don’t eat with your fingers” (as one can show by using Definition 5), and the argument is w / w valid (as one can show by using the Equivalence Theorem in note 48). Perhaps those who deny the validity of the second argument understand its first premise as “except if you eat asparagus, don’t eat with your fingers”, and thus as different from the premise of the first argument; but then we have no counterexample to Imperative Monotonicity (contrast Horty 1994, pp. 58–60). Note finally that, it Imperative Monotonicity holds and Argument 35 is valid, then the argument that one gets—call it Argument 35*—by adding to the premises of Argument 35 the imperative premise “marry him” is also valid. But the conjunction of “either marry him or dump him” with “marry him” is just “marry him” (as one can show by using Definition 5), so Argument 35* is equivalent to the argument from “marry him” and “you are not going to marry him” to “dump him” and thus is (intuitively) not valid. It follows that, if Imperative Monotonicity holds, then Argument 35 is (intuitively) not valid either.
Theorem 6 shows that Argument 38 is not w / w valid. To see that Argument 38 is not s / s valid either, note first that the proposition that some fact f undefeatedly strongly supports the prescription \((I^{+})\) “if you drink, don’t drive, and if you don’t drink, drive” is consistent with P. Therefore, (1) possibly, some fact (e.g., the conjunction of f with the fact that P is true) both guarantees P and strongly supports \(I^{+}.\) Moreover, (2) necessarily, every fact that strongly supports \(I^{+}\) also strongly supports I (as one can show by using the Equivalence Theorem in note 48) but does not strongly support \(I^{\prime }\) (because, by Definition 9, necessarily, every fact that strongly supports \(I^{+}\) favors the proposition that you drive and you do not drink, which entails the satisfaction proposition of \(I^{+}\) but the violation proposition of \(I^{\prime },\) over the proposition that you do not drive and you do not drink, which entails the violation proposition of \(I^{+}\) but the satisfaction proposition of \(I^{\prime }\)). (1) and (2) jointly entail that, possibly, some fact guarantees P and strongly supports I but does not strongly support \(I^{\prime };\) i.e., Argument 38 is not s / s valid.
One might claim that Argument 38 is so intuitively compelling that there must be an interesting and satisfactory definition of validity on which the argument is valid. I reply that I cannot exclude this, but I have yet to see such a definition. Alternatively, one might claim that Argument 38 is valid even on my definition of validity because P entails that every fact that supports I also supports \(I^{\prime }\) (see §6.1): necessarily, if you are going to drink, then every reason for you not to drive if you drink is a reason for you not to drive. I reply that this is false: P is consistent with the proposition (O) that some fact supports I but does not support \(I^{\prime }.\) This is because (1) O follows from the proposition that some fact supports \( I \& \sim I^{\prime }\) (see the lemma in note 50) and (2) the latter proposition is consistent with P: the proposition that some fact supports “drive and don’t drink” does not entail that you are not going to drink.
(1) One might claim that Argument 40 is trivially valid because its conclusion is irrelevant given its declarative premise. I do not see how the irrelevance results in validity, but in any case the objection can be bypassed by replacing in the text the conclusion \(I^{\prime \prime }\) with “if there is a conclusive reason for you to drive, don’t drive”—a prescription which is not irrelevant given the premise that you are going to drink. (2) One might note that the pure imperative argument from “if you drink, don’t drive” and “drink” to “if you don’t drink, don’t drive” is—weakly but not strongly—valid (as one can show by using the Equivalence Theorem in note 48); how does this cohere with my claim that Argument 40 is intuitively not valid? I reply that the above pure imperative argument fails to be intuitively invalid because its conclusion is redundant (see Vranas 2011, p. 396), in the sense that its conjunction with the conjunction of the premises is the same as the conjunction of the premises (namely “drink and don’t drive”). I see no similar reason for saying that Argument 40 fails to be intuitively invalid. (3) One might note that the pure declarative argument from “if you drink, you won’t drive” and “you are going to drink” to “if you don’t drink, you won’t drive” is intuitively not valid but nevertheless valid; why not say the same thing about Argument 40? I reply that this suggestion (a) grants that Argument 40 is intuitively not valid, which is what I am claiming (in order to argue that the result that Argument 38 is not valid is intuitively acceptable), and (b) fails to refute the claim that Argument 40 is in fact not valid: in the Appendix I point out that some arguments are not valid although their “corresponding” pure declarative arguments are valid. (On the other hand, I have no proof that Argument 40 is not valid: Theorem 6 cannot be used because \(I^{\prime \prime }\) is conditional, and one can show that the generalization of Theorem 6 in note 50 cannot be used either.)
Interestingly, the “inconsistency response” does work against an attempt to argue that, contrary to what I said in §6.1, Argument 32 is not valid. More specifically, consider the argument—call it Argument 32*—from “disarm the bomb” and “every reason for you to disarm the bomb is a reason for you to cut the wire \((P^{+}),\) and the fact that someone will die if you don’t disarm the bomb is an undefeated reason for you to disarm the bomb \((Q^{+}),\) but the fact than ten other bombs will explode and ten people will die if you cut the wire is an undefeated reason for you not to cut the wire \((R^{+})\)” to “cut the wire”. By Declarative Monotonicity, Argument 32* is valid if Argument 32 is valid. But Argument 32* is not valid, so Argument 32 is not valid either—or so one might argue. I reply that Argument 32* is valid because its declarative premise is inconsistent: \(Q^{+}\) and \(R^{+}\) jointly entail that some reason for you to disarm the bomb is not a reason for you to cut the wire, which is the negation of \(P^{+}.\)
In this example, f guarantees P and weakly supports I but not \(I^{\prime }.\) For an example in which a fact guarantees P and strongly supports I but not \(I^{\prime },\) suppose that the prescription “if you drink, don’t drive, and if you don’t drink, drive” is strongly supported by the fact that (1) you are going to drink and (2) you have promised that, if and only if you drink, you will not drive. Then this fact guarantees P and (as explained in note 55) strongly supports I but not \(I^{\prime }.\)
The point is that, although the above example (and, more generally, the possibility that some fact—like f—guarantees P and supports I but does not support \(I^{\prime }\)) falsifies the claim that, (1) necessarily, every fact that both supports I and guarantees P also supports \(I^{\prime },\) it does not falsify the claim that, (2) necessarily, if some fact both supports I and guarantees P, then some (maybe different) fact supports \(I^{\prime }\): in the above example, although f does not support \(I^{\prime },\) maybe some different fact does. I reply that a different example falsifies (2). Suppose that “don’t drink” (and thus also I; see the Equivalence Theorem in note 48) is supported by the fact g that (a) you have promised not to drink, (b) you are going to drink, and (c) there is no reason for you not to drive. Then some fact (namely g) both supports I and guarantees P, but no fact supports \(I^{\prime },\) so transmission of meriting pro tanto endorsement jointly (and thus also separately) fails for Argument 38. Similarly, to see that transmission of meriting pro tanto endorsement fails for Argument 35, suppose that “marry him” (and thus also “either marry him or dump him”; see the Equivalence Theorem in note 48) is supported by the fact that (i) you have promised to marry him, (ii) you are not going to marry him, and (iii) there is no reason for you to dump him. Then some fact both supports “either marry him or dump him” and guarantees that you are not going to marry him, but no fact supports “dump him”.
What I call “practical necessity” amounts to one way of understanding what is known as “power necessity” (for an agent at a time), namely the way that corresponds to “C1” in Carlson’s (2000, p. 280) useful taxonomy. (Cf.: Finch and Warfield 1998, p. 525; Huemer 2000, p. 538; McKay and Johnson 1996, p. 120; Speak 2011; see also: Belzer and Loewer 1994, p. 411 n. 11; Carmo and Jones 2002, p. 287.) The operator of practical necessity is a normal modal operator; i.e., if P and \(P\rightarrow Q\) (namely the disjunction of Q with the negation of P) are practically necessary, then Q is also practically necessary. Indeed: if P and \(P\rightarrow Q\) are practically necessary, then (1) Q is true because P and \(P\rightarrow Q\) are true, and (2) Q would be true (i.e., \(\sim \) \(Q\) would be false) no matter what one were to do (among the things that one can do) because otherwise, since P is true and would be true no matter what one were to do (among the things that one can do), one could do something such that, if one were to do it, then \( P \& \sim Q\) might be true (i.e., \(P\rightarrow Q\) might be false), contradicting the practical necessity of \(P\rightarrow Q\). (See Carlson 2000, pp. 286–287 for a similar derivation.)
Suppose \(P^{*}\) is true: it is practically necessary that you drink. (1) It is necessary—and thus practically necessary—that, if I is satisfied (i.e., if you drink and do not drive), then \(I^{\prime }\) is also satisfied (i.e., you do not drive). (2) Since it is practically necessary that you drink, it is also practically necessary that, if \(I^{\prime }\) is satisfied (i.e., if you do not drive), then I is also satisfied (i.e., you drink and do not drive). So \(P^{*}\) entails that it is practically necessary that the satisfaction propositions of I and \(I^{\prime }\) are materially equivalent. Similarly for the violation propositions.
On the basis of my arguments to the effect that Argument 38 is (intuitively) not valid, one might raise three objections to my claim that those who accept (the intuitive acceptability of) the Equivalence Assumption should also accept that Argument 42 is (intuitively) valid. (1) One might take Theorem 6 to show that Argument 42 is not valid. I reply that Theorem 6 does not show this (if the Equivalence Assumption is true). Indeed, (one can show that) \(P^{*}\) entails that \( I \& \sim I^{\prime }\) (namely “drive and don’t drink”) and “drink and don’t drink” are practically equivalent. Therefore, if the Equivalence Assumption is true, \(P^{*}\) entails that a fact supports \( I \& \sim I^{\prime }\) exactly if it supports “drink and don’t drink”, so \(P^{*}\) entails that no fact supports \( I \& \sim I^{\prime }\) (given that, necessarily, no fact supports “drink and don’t drink”) and thus \(P^{*}\) is inconsistent with the proposition that some fact undefeatedly supports \( I \& \sim I^{\prime }.\) (2) If Argument 42 is valid, then the argument—call it Argument 42*—that one gets by adding to the premises of Argument 42 the declarative premises Q and R (see Argument 41) is also valid (by Declarative Monotonicity). But Argument 42* is (intuitively) not valid, so Argument 42 is (intuitively) not valid either—or so one might argue. I reply that Argument 42* is valid because the declarative premise \(P^{*}\) of Argument 42 is inconsistent with \( Q\, \& \,R\) (if the Equivalence Assumption is true). Indeed, (one can show that) \(P^{*}\) entails that “don’t drink” and “drink and don’t drink” are practically equivalent. Therefore, if the Equivalence Assumption is true, \(P^{*}\) entails that a fact supports “don’t drink” exactly if it supports “drink and don’t drink”, so \(P^{*}\) entails that no fact supports “don’t drink” and thus \(P^{*}\) is inconsistent with \( Q\, \& \,R\) (which entails that some fact supports “don’t drink”). (3) If Argument 42 is valid, then (given that Argument 39 is valid) the argument—call it Argument 42* *—from the premises of Argument 42 to “if you don’t drink, don’t drive” \((I^{\prime \prime })\) is also valid (by the transitivity of entailment). But Argument 42** is (intuitively) not valid, so Argument 42 is (intuitively) not valid either—or so one might argue. I reply that, in my view, Argument 42** is valid (if the Equivalence Assumption is true). Indeed, (one can show that) \(P^{*}\) entails that \(I^{\prime \prime }\) and “if you drink and don’t drink, don’t drive” are practically equivalent. Therefore, if the Equivalence Assumption is true, \(P^{*}\) entails that a fact supports \(I^{\prime \prime }\) exactly if it supports “if you drink and don’t drink, don’t drive”, so \(P^{*}\) entails that every reason—and thus every fact that supports I—supports \(I^{\prime \prime }\) (given that “if you drink and don’t drink, don’t drive” is empty and thus is necessarily supported by every reason; see note 6) and thus Argument 42** is valid. It will not do to respond that Argument 42** is intuitively not valid: I take it that we have no clear intuitions about (prescriptions that are practically equivalent to) empty prescriptions.
Cf. Greenspan 1975, p. 267. Deontic analogs of Argument 42 are instances of what has been called “unalterability detachment” (Nute and Yu 1997, p. 9). For contrasts somewhat analogous to the contrast between Arguments 38 and 42, see: Greenspan 1975; Kolodny and MacFarlane 2010, pp. 138–140. Another reason why Argument 38 appears valid may be that Argument 38 is isomorphic to the pure declarative argument from “if you drink, you will not drive” and “you are going to drink” to “you will not drive”. But this cannot be the whole story: a similar reasoning would lead one to expect Argument 41 to appear valid, but it does not.
One can show that, for Argument 44, the declarative premise entails that the imperative premise and the conclusion are practically equivalent (cf. note 62). For other arguments, however, one cannot show this; an example is the argument—call it Argument 44*—from “keep your promises” and “it is practically necessary that you have promised to marry him” to “marry him”. The declarative premise of Argument 44* entails that you can keep your promises only if you marry him; so, if this in turn entails that every reason for you to keep your promises is a reason for you to marry him, then Argument 44* is valid (cf. Argument 33). (Alternatively, one can show that Argument 44* is valid by (1) using Definition 5 to show that “keep your promises” (understood as “do everything you have promised to do”) is the conjunction of “if you have promised to marry him, marry him” with “for everything other than marrying him, if you have promised to do it then do it” and (2) using Imperative Monotonicity (see note 54). But this alternative method of showing validity does not—whereas the previous method does—work for arguments like the one from “push every blue button” and “it is practically necessary that every round button is blue” to “push every round button”.) On arguments similar to Argument 44*, including the widely discussed argument from “take all the boxes to the station” and “this is one of the boxes” to “take this to the station”, see: Bohnert (1945, p. 302), Grelling (1939, p. 44), Grue-Sörensen (1939, p. 197), Hamblin (1987, pp. 87–89), Hare (1952, pp. 25–28), Jörgensen (1938, p. 290; 1938/1969, p. 11), Juárez-Paz (1959, pp. 200–203), Kelsen (1979, pp. 336–337 n. 161; 1979/1991, pp. 401–402 n. 161), Ledent (1942, p. 269), Lemmon (1965, p. 65), MacKay (1969, p. 148), McArthur and Welker (1974, p. 225), Moutafakis (1975, p. 37), Ramírez (2003, pp. 131–134), Rand (1939, p. 318; 1939/1962, p. 249), Ross (1941, pp. 55, 60 n. 1, 68–69; 1941/1944, pp. 32, 36 n. 8, 44), Sosa (1966; 1970, p. 221), Tammelo (1975, p. 39), and Turnbull (1960, pp. 380–381). (Cf.: Bergström 1962, pp. 48–49; Hansen 2008, p. 3; Holdcroft 1978, pp. 85–86; Parsons 2012, p. 51; Sellars 1956, p. 239; Weinberger 1981, p. 21.)
Practical necessity (like support; see note 13) is relative to times and to agents: in general, what an agent can do at a time the agent cannot do at another time, and what an agent can do at a time another agent cannot do at the time. Similarly for practical equivalence, so strictly speaking the Equivalence Assumption is the claim that, necessarily, if two prescriptions are practically equivalent at time t for agent j, then every fact that supports at t relative to j one of the prescriptions also supports at t relative to j the other prescription. So if the declarative premise \(P^{*}\) of Argument 44 is understood as the claim that it is practically necessary now for you that you lied, then Argument 44 is valid relative to now and to you (if the Equivalence Assumption is true). On the other hand, if \(P^{*}\) is understood as the claim that it is practically necessary now for everyone that you lied (and this understanding is reasonable if it is necessary that no one can change the past), then Argument 44 is valid relative to now and to everyone (if the Equivalence Assumption is true).
Here is why I take it to be a consequence of Theorem 6 that Argument 43 is not valid: \(I^{\prime }\) is unconditional, and the proposition (N) that some fact undefeatedly supports “don’t apologize, and let it be the case that you did not lie” (which is \( I \& \sim I^{\prime },\) as one can show by using Definition 5) does not entail that you did not lie and thus is consistent with P. Note, however, that (perhaps surprisingly), if the Equivalence Assumption is true and it is necessary that one cannot change the past, then the proposition (M) that some fact supports “let it be the case that you did not lie” entails that you did not lie (and thus so does N, which entails M). To see this, consider the contrapositive: the proposition (P) that you lied entails that no fact supports “let it be the case that you did not lie”. Indeed, if it is necessary that one cannot change the past, then P entails \(P^{*}\) (as explained in the text); but \(P^{*}\) can be shown to entail that “let it be the case that you did not lie” and any necessarily violated prescription (e.g., “drink and don’t drink”) are practically equivalent, so \(P^{*}\) entails (if the Equivalence Assumption is true) that no fact supports “let it be the case that you did not lie” (given that, necessarily, no fact supports “drink and don’t drink”).
The above remarks are also relevant to an imperative variant of “Chisholm’s paradox” (Chisholm 1963). Consider the argument (inspired by Prakken and Sergot 1996) from “let it be the case that the room is empty”, “if the room is empty, let it be the case that the alarm is on”, “if the room is not empty, let it be the case that the alarm is off (i.e., not on)”, and “the room is not empty” to “let it be the case that the alarm is both on and off”. It is paradoxical to claim that this argument is valid: its conclusion is inconsistent, but its four premises look jointly consistent. However, if the Equivalence Assumption is true and it is necessary that one cannot change the present, then the conjunction of the three imperative premises—which, by Definition 5, is “let it be the case that the room is empty and the alarm is on”—is inconsistent with the declarative premise (“the room is not empty”), so one does not escape the paradoxical result that the above argument is valid. (Much more can be said about this paradox; cf. Carmo and Jones 2002.)
Although the definition is not immediately reductive (i.e., it does not immediately reduce the validity of an argument to the validity of a pure declarative argument), it is usable mainly because the equivalence theorems that I proved do reduce the validity of (1) cross-species imperative arguments, (2) cross-species declarative arguments, and (3) mixed-premise declarative arguments to the validity of pure declarative arguments. The definition is not fully usable, however, given the lack of an equivalence theorem for mixed-premise imperative arguments. (Recall, for example, that I have no proof that Argument 40 is not valid; see the end of note 57.) It is an open question whether there is an equivalence theorem reducing the validity of a mixed-premise imperative argument to the validity of two pure declarative arguments (just as the equivalence theorem in note 48 reduces the (w / w) validity of a pure imperative argument to the validity of two pure declarative arguments).
It can also be shown that w / w (but not s / s) entailment satisfies monotonicity (if \(\Gamma \, w/w\) entails A and \(\Gamma \subseteq \Delta ,\) then \(\Delta \, w/w\) entails A; cf. note 54) and projection (if \(A\in \Gamma ,\) then \(\Gamma \) w / w entails A). (\(\Gamma \) and \(\Delta \) are any non-empty sets of propositions or prescriptions or both, and A is any proposition or prescription.) Moreover, it can be shown that w / w entailment satisfies strong cut (if \(\Gamma \) w / w entails A and \(\Delta \cup \{A\}\) w / w entails \(A^{\prime },\) then \(\Gamma \cup \Delta \, w/w\) entails \(A^{\prime }\)) if the condition holds that, necessarily, every fact that weakly supports two prescriptions also weakly supports their conjunction. (It is not clear whether this condition holds; it is stronger than result (2) of note 49. Cf. Vranas 2011, p. 398 n. 38. Strong cut fails for s / s entailment.) In contrast to reflexivity and transitivity, I do not consider monotonicity, projection, and strong cut necessary for formal acceptability (see Vranas 2011, p. 438 n. 69).
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Acknowledgments
I am grateful to Daniel Boisvert, Joseph Fulda, Alan Hájek, Casey Hart, Ryan Millsap, David O’Brien, Josh Parsons, Manidipa Sanyal, Michael Titelbaum, and especially Aviv Hoffmann, Joshua Schechter, and several anonymous reviewers for comments. Thanks also to Martin Barrett, John Bengson, John Bishop, Anna Brożek, Heather Dyke, Branden Fitelson, Molly Gardner, Paula Gottlieb, Casey Helgeson, Jacek Jadacki, Matt Kopec, Jonathan Lang, John Mackay, Andrew Moore, Jakub Motrenko, Charles Pigden, Gina Schouten, Benjamin Schwan, Larry Shapiro, Elliott Sober, Ivan Soll, Reuben Stern, and Berislav Žarnić for interesting questions, and to my mother for typing the bulk of the paper. Material from this paper was presented at the 7th Formal Epistemology Workshop (September 2010), the University of Otago (December 2011, via Skype), the 2012 Central APA Meeting (February 2012), the University of Wisconsin-Madison (Department of Mathematics, April 2012, and Department of Philosophy, May 2012), the University of Warsaw (May 2012), and the Beijing Normal University (August 2012)..
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Appendix: Alternative definitions of argument validity
Appendix: Alternative definitions of argument validity
In this appendix I briefly examine several alternative definitions of argument validity that have been proposed in the literature or in discussion.
Material conditionals. For any prescription I, let its corresponding proposition be the material conditional whose antecedent is the context of I and whose consequent is the satisfaction proposition of I. For any argument A, let its corresponding pure declarative argument be the argument that one gets by replacing those premises of A that are prescriptions with their corresponding propositions, and also replacing the conclusion of A, if it is a prescription, with its corresponding proposition. Following Clarke (1970, p. 103), one might propose that an argument A is valid exactly if (a) its corresponding pure declarative argument is valid (cf. Smart 1984, p. 17; Pigden 2011, pp. 4–5) and (b) A satisfies the following Principle of Mood Constancy: “The constituent elements of a mixed inference must be in the same mood for every occurrence within the inference” (Clarke 1970, p. 101). This proposal has several problems. (i) Whether an argument satisfies the Principle of Mood Constancy can depend on how the argument is formulated: the imperative sentences “if you kill, kill” and “if you kill, steal or don’t steal” express the same prescription (namely the prescription whose satisfaction proposition is the proposition that you kill and whose violation proposition is impossible), but the argument from that prescription to itself violates the Principle of Mood Constancy if the prescription is expressed by the former sentence and satisfies the principle if the prescription is expressed by the latter sentence. (ii) Arguments 4, 12, 30, and 32 are valid, but their corresponding pure declarative arguments are not valid. (iii) The argument from “kill” and “you kill” to “you kill” is valid (cf. MacKay 1971, pp. 92, 94–95; Morscher and Zecha 1971, p. 211), and so is (more interestingly) Argument 25, but both arguments violate the Principle of Mood Constancy. (iv) Arguments 10, 21, and 38 are not valid, but they satisfy (at least as I formulated them) the principle of Mood Constancy, and their corresponding pure declarative arguments are valid.
Deontic propositions. One might propose that an argument is valid exactly if its corresponding deontic argument is valid (cf. Charlow 2014; Kaufmann 2012), the corresponding deontic argument being defined as in the previous paragraph but by replacing any prescription (not with a material conditional, but rather) with the deontic proposition that the satisfaction proposition of the prescription is all-things-considered obligatory given the context of the prescription. A first problem with this proposal is that many deontic arguments correspond to a given argument, depending on whether obligatoriness is understood morally, legally, epistemically, and so on. This problem is not decisive: there are also many kinds of relativized validity (see note 13), and one might claim that an argument is valid relative to moral (legal, epistemic, etc.) support exactly if its corresponding moral (legal, epistemic, etc.) deontic argument is valid. A second problem with the proposal is more serious, however: the argument from “marry me” to “it is all-things-considered obligatory for you to marry me” is not valid, but its corresponding deontic arguments are trivially valid (and similarly for Argument 14). Similar remarks apply if one replaces a prescription (not with a deontic proposition, but rather) with the proposition that some fact undefeatedly supports the prescription.
Inconsistency. By analogy with pure declarative arguments, one might propose that an argument is valid exactly if the set that consists of the premises and the negation of the conclusion of the argument is inconsistent, in the sense that the conjunction of the propositions in the set is inconsistent with the conjunction of the prescriptions in the set (or, if there are only propositions or only prescriptions in the set, their conjunction is inconsistent; cf. note 40). A first problem with this proposal is that the proposed implication from inconsistency to validity fails for pure imperative arguments (see Vranas 2011, p. 445), for cross-species arguments (see the paragraph after Argument 30 in §5), and for mixed-premise arguments (see note 44). A second problem with the proposal is that the proposed implication from validity to inconsistency fails for pure imperative arguments; for example, the argument from “sing” to “if you dance, sing” is valid (cf. Argument 39) but the conjunction of the prescriptions “sing” and “if you dance, don’t sing” is consistent (the conjunction is “sing but don’t dance”, so it is not necessarily violated). On the other hand, the implication from validity to inconsistency does hold for cross-species arguments (see §5), for mixed-premise declarative arguments (see note 44), and for mixed-premise imperative arguments with unconditional conclusions (as one can show by a reasoning similar to my proof of Theorem 6 in note 50).
Undefeated sustaining. One might propose that an argument is valid exactly if, necessarily, every fact that undefeatedly sustains every premise of the argument also undefeatedly sustains the conclusion of the argument. A first problem with this alternative definition is that, as far as I can see, it does not satisfy transmission of meriting pro tanto endorsement (see D2JP in §2.2.3 and note 12; the definition does satisfy transmission of meriting all-things-considered endorsement). One can show that every argument which is valid according to the General Definition is also valid according to the alternative definition, and that the two definitions are equivalent for cross-species imperative arguments. However, the two definitions are not equivalent for declarative or mixed-premise imperative arguments. For example, the argument from “marry me” to “possibly, there is an undefeated reason for you to marry me” (cf. Argument 12) is not valid according to the General Definition but is valid according to the alternative definition—and I take this to be a second problem with the alternative definition. One might reply that the above argument is valid according to the General Definition on the plausible assumption that, necessarily, if some fact possibly supports any given prescription, then some fact possibly undefeatedly supports the prescription. Indeed, one can show that on this assumption the two definitions are equivalent for cross-species declarative arguments. (One can also show that on a related assumption, namely an analog of Assumption 1 in Vranas 2011, p. 433, the two definitions are equivalent for pure imperative arguments.) Nevertheless, the assumption might be considered controversial, so I think that the General Definition is preferable to the alternative definition. One might alternatively propose that an argument is valid exactly if, necessarily, every fact that indefeasibly (see §2.2.3) sustains every premise of the argument also indefeasibly sustains the conclusion of the argument. This proposal turns out to have the (to mind mistaken) consequence that Argument 14 is valid.
Other definitions. (a) Parsons (2013) defines the content of, e.g., “if you run, smile” as the set of ordered pairs of possible worlds \(<\!w,\,w^{\prime }\!>\) such that either “you run” is false at w or “you smile” is true at \(w^{\prime }\) (and defines similarly the contents of, e.g., “you run” and “smile”), and proposes that an argument is valid exactly if the intersection of the contents of its premises is a subset of the content of its conclusion. According to this proposal, however, the argument from “if you run, smile” to “if you run, run and smile” is not valid, so reflexivity is violated (since the imperative sentences “if you run, smile” and “if you run, run and smile” express the same prescription). Another problem is that, according to the proposal, cross-species arguments like Argument 4 and Argument 12 are not valid. (b) Finally, Sosa (1970, pp. 219–220) proposes a definition of validity for mixed-premise imperative arguments which turns out to have the (to my mind mistaken) consequences that Argument 38 is valid and that neither Argument 32 nor Argument 34 is valid.
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Vranas, P.B.M. New foundations for imperative logic III: A general definition of argument validity. Synthese 193, 1703–1753 (2016). https://doi.org/10.1007/s11229-015-0805-2
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DOI: https://doi.org/10.1007/s11229-015-0805-2