Simple Quantified Deontic Logic

1. Introduction

Deontic logic replaces the alethic modalities of necessity and possibility with modalities of moral necessity and moral possibility, i.e., obligation and permissibility (O, P). A principal part of the domain includes moral agents, and the predicates would principally be actions. Given a possible world semantics, O and P quantify over a subset of possible worlds, deontic alternatives, in which the agents act or otherwise satisfy conditions designated permissible, obligatory, or optimal in the world they are alternatives to. (Typically the actual world.) In this paper, I argue that the local character of deontic alternatives, together with modifications suggested by Lou Goble to the deontic operator and two deontic axioms[1], permits a simple quantified deontic logic (SQDL), complete with a fixed domain and deontic analogues to the Barcan Formula and its converse,[2] while still being compatible with actualism.

2. Why Quantification?

Much of informal deontic discourse employs quantifiers. Some examples: “Someone should call 911.” “If something is bad, and someone could prevent it without sacrificing anything morally significant, they ought, morally, prevent it.” “None may pass.” “Sarah is obligated to help everyone on the list.” Adding quantifiers to formal deontic logic therefore seems a natural step; and was explored by Hinttika as early as 1957.[3] Yet today, though a quantified deontic logic would seem to have great potential for reflecting and clarifying ordinary deontic speech, standard deontic logic remains stolidly propositional. This is largely owing to the empiricist idea that modality attaches primarily to propositions, together with the assumption that propositional ‘oughts’ would be intensional and thus opaque to quantification and substitution, like their alethic and doxastic cousins. This presumption of deontic opacity is commonly accepted almost without argument, even though most deontic contexts are clearly extensional, dealing with duties of actual agents or the states of affairs they bring about.[4] Let us examine, then, such arguments for deontic opacity as we can find.

3. Arguments against quantification in deontic contexts

James Forrester has offered three. First, Forrester argues that when a deontic operator has scope over a proposition, existential generalization fails. “If there ought to be a law, there need not be such a law now. If it ought to be the case that we preserve resources for our remote posterity, then no doubt both we and the resources exist, but the posterity need not.”[5]

Second, Forrester argues that while the subject terms in de re ‘oughts’ are extensional, the predicates and object-terms need not be; again failing the test of existential generalization. Forrester: “If Christopher ought to build St. Swithin’s Cathedral, it might or might not be the case that there is such a thing as St. Swithin’s Cathedral that Christopher ought to build.”[6] 

Third, Forrester claims that universal instantiation also fails. He asks, “If Jane ought to help all the children whose name is on the list, and Tomas’s name is on the list, does it follow that Jane ought to help Tomas?”[7] Forrester argues that this plausible-seeming inference commits an operator-shift fallacy, since while Tomas’ name is on the actual list, his name might not be on the list in the deontic alternatives where Jane helps everyone on it, making such inferences analogous to confusing the necessity of consequence with necessity of the consequent like:

In all possible worlds, if Tom is a bachelor, then Tom is unmarried.

But Tom is a bachelor_____________

So, in all possible words, Tom is unmarried.

4. Deontic Alternatives and Simple Deontic Actualism

To answer these arguments, I propose an actualist interpretation of deontic alternatives making possible a simple quantified deontic logic with a fixed domain but without any appeal to exotic ontologies such as haecceitism or contingent Nonconcretism.[8] My basic claim is that deontic alternatives, properly understood, exclude non-actual individuals altogether, making talk of unexemplified essences, contingent nonconcretea, or other proxies for possibilia unnecessary in deontic contexts.

The key to my proposal is understanding the narrow locality of deontic alternatives. Deontic alternatives are commonly mis-described as utopias in which every agent fulfills their every duty. But deontic alternatives must be ways the actual world could be, now or in the near future. They can be “ideal” only in the sense of being the best scenarios we could actually hope for. If they were perfect worlds, they would be inaccessible and practically useless to us in deliberating about our actual duties. Recognizing this, Forrester attempts to combine the “ideal” understanding with a branching world model: “If deontic alternatives are branching worlds in which people will always do as they ought, it is not necessarily true that they have done as they ought. The past of a deontic alternative is as checkered as that of the real world—for it is the real world’s past.”[9] But this compromise won’t work either. The idea of a world with our world’s past suddenly becoming one where everyone always does as they ought itself verges on impossibility. Inevitably, any future with our past will carry moral conflicts and dilemmas into the present and future, cases where it will be impossible to meet one obligation without violating another, or without making it impossible for another party to fulfill theirs. And alternatives continually become inaccessible to us as we make more choices. Regarding the world’s natural resources, it is already too late to leave “as much and as good” to our remote posterity, and we are forced consider what residual obligations we still have towards future generations.[10] Our real deontic alternatives are gritty and full such residual, second-best obligations.

Chellas formulated Kant’s Law (“ought implies can”) simply as ~O^[11] while Forrester opts for ~O(p & ~p). Both virtuously avoid mixing modalities from different logics but give the misimpression (to which Forrester succumbs) that only the logically impossible is excluded from the obligatory.[12] This is simply too narrow. One can’t be obligated to do the physically impossible, nor even the circumstantially impossible, any more than the logically impossible. Just because there is a possible world where I am a billionaire able to end world hunger does not imply my obligations can extend so far beyond my actual means.[13]

Deontic alternatives are also materially co-extensional with the actual world, containing all and only the same stuff. While there may be maximally consistent states of affairs such that pigs fly, magic works, and Ponce de Leon finds the Fountain of Youth, possible but non-actual people exist, or even simply ones where I solve world hunger, such worlds are not ways the actual world could be, and will not be found among its deontic alternatives. Deontic logic, thus interpreted, is uniquely suitable for a fixed domain logic, complete with deontic analogues of not only the Barcan formula and its Converse, but the Buridan formula as well.[14]

It has been objected that this will obliterate the de re / de dicto distinction, generally thought essential to distinguishing between personal oughts of obligation and propositional oughts of optimality. This overstates the consequences.[15] First, the Deontic Converse Buridan Formula remains invalid even in a fixed domain, so we can’t infer Someone is actually obligated to satisfy A from It ought to be the case that there is someone who satisfies A (although we can make the inference in opposite direction). Second, it still makes sense to read  “It ought to be the case that Jane helps everyone on the list” differently from “Jane’s duty is to help everyone whose name is on the list.”  The truth conditions for singular two-place relations come out differently depend on whether it is read in an agentive rather than propositional mode. On a de dicto reading, the truth-conditions for active- and passive-voice readings of It ought to be that Jane helps Tomas are the same, whereas on a de re reading, the truth-conditions of It is Jane’s duty to help Tomas versus It is Tomas’s duty to be helped by Jane are distinct. [16].

5. Forrester Rebutted

Returning to Forrester, Forrester offered “There ought to be a law” as an example where existential generalization fails, saying, “…there need not be such a law now.” When someone utters something like “There ought to be a law,” they usually have in mind reasonably specific normative proposition n that they think should have the force of actual law. Taking the ‘ought’ here to be de dicto, this translates to:

In every deontic alternative, normative proposition n has the status of positive law.

Which is straightforwardly generalizable to:

There is at least one actual thing that has the status of positive law in every deontic alternative.

That actual thing is the normative proposition that the speaker has in mind.  So even though Op doesn’t entail p, it turns out that if there ought to be a law, there is something that ought to be that law!!

The most direct response to the argument from duties to remote posterity in Forrester’s second example is to simply deny that such obligations exist. Being the object of obligation is a property, albeit an accidental one, and having a property implies existence. Conversely, non-existence entails ineligibility for being an object of obligation. Talk of duties to distant progeny is therefore figurative. It has to be cashed out either in terms of an obligation to maximize utility, or else in terms of duties to immediate progeny. Each generation has a collective and immediate obligation to come as close as possible to leaving “as much and as good” for their children, those children will likewise have a similar obligation to their children, and so on. Sadly, with non-renewable resources and climate stability, such obligations may already be impossible to fulfill (sorry kids!)

The problem of the unbuilt cathedral in Forrester’s third example can be approached from two directions. First, if it ought to be the case that Christopher build St. Swithin’s, surely it ought to be the case that there is something that Christopher builds. In other words, in every deontic alternative, there is at least one thing that Christopher builds. But this is unsatisfactory because it introduces the existential quantifier inside the scope of the deontic operator.

The second approach is to include, in addition to spatio-temporal objects, both abstract and intensional objects in the overall extension of the world. We may then think of construction as the act of taking an intensional object and making it extensional; the idea of St. Swithin’s. meeting the raw materials the architect employs and becoming a hylomorphic whole. Given such an understanding of construction as the bringing together of matter and form, it isn’t a stretch to say something already exists corresponding to the object of Christopher’s obligation.

Finally, given Goble’s definition of the deontic operator, Forrester’s argument for the failure of universal instantiation collapses. It ought to be the case that if Tomas’ name is on the list, then Jane will help Tomas will be true just in case the list designated by l in the actual world is in the extension of N in every deontic alternative. The list, therefore, does not vary from world to world. This eliminates the possibility that Tomas’s name might not be on the list in the deontic alternatives where Jane helps everyone on it.

6.0.     Closing Remarks

I have argued for a simple quantified deontic logic with a fixed domain that does rely upon exotic ontologies like haecceitism or contingent concretism. I would like to close by remarking on two advantages and one suggestion for further research.

6.1 If one could not existentially generalize from “Christopher is obligated to build St. Swithin’s,” to “There is something Christopher is obligated to build,” a quantified deontic logic would have to be a free logic. The same holds if one is committed to a variable domain.[17] The alternative I have proposed therefore has the virtue of preserving classical logical intuitions in deontic contexts.

6.2 If deontic contexts were generally intensional and opaque, inferring particular moral judgments from general principles would frequently involve a fallacious form of universal instantiation. This might well invite skepticism about moral reasoning on the part of logicians, or skepticism about logic on the part of ethicists, or both. But if deontic discourse is largely extensional, deductive moral reasoning is relatively unproblematic.

6.3 Since propositions are abstract objects they are presumably necessary existents, and so exist in all possible worlds, so also constituting a fixed (albeit uncountably large) domain. Some propositions are true in all worlds, some are false in all worlds, and contingent propositions, being true in some worlds and false in others, individuate the possible worlds. It follows that a second-order version of the same simple quantified deontic logic should be able to handle internal relations between propositions in general, and normative propositions in particular, again without appeal to any ontology more exotic that what has been attributed to propositions for over a century

[1] Goble, Lou. “`Ought' and Extensionality.” Noûs, 30:3 (Sep. 1996), 330-355. See also Goble, Lou. “Quantified Deontic Logic with Definite Descriptions,” Logique et Analyse 37:147/148 (Sep-Nov., 1994), 239-253. To deal with cases where the denotation of definite descriptions could seemingly vary across possible worlds causing failures of substitutivity, Goble offers a reinterpretation of the deontic operator O such that OCa is true if and only if the individual denoted by a in the actual world satisfies C in every deontic alternative, essentially building fixed designation directly into the operator; this definition will also apply where a designates individuals with proper names, The two restricted axioms are (1) If it is a logical truth that A implies B, then it is likewise a logic truth that  OA implies OB and (2) If A is logically equivalent to B then C is equivalent to C’, where C’ is the result of replacing one or more occurrences of A in C by B. The restriction: “These principles may not hold when the statements involved contain. . . descriptions . . . essential to the validity of their hypothese” Goble 1996, 349. I adopt all of these changes in this paper.

[2] Basically the simplest modal logic K combined with the axioms and rules of classical quantification theory and the basic axioms of standard deontic logic, as modified by Goble.

[3] Hintikka, Jaakko. “Quantifiers in deontic logic.” Societas Scientiarum Fennica, 23:4 (1957).

[4] As Goble argued. Ibid.

[5] Forrester, James Wm. Being Good and Being Logical: Philosophical Groundwork for a New Deontic Logic.  80-81.

[6] Ibid., 301.

[7] Ibid., 302.

[8] Actualism is an interpretation of quantified modal logic holding that only the actual exists, to be and to exist are synonymous, and thus that the claim, “there are possible objects that don’t exist” is oxymoronic. See Menzel, Christopher. “Actualism.” Stanford Encyclopedia of Philosophy (Summer 2022 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/sum2022/entries/actualism/ . The most well-known version of actualism is probably Plantinga’s haecceitism, in which quantifiers range over individual essences. See Plantinga, Alvin. The Nature of Necessity. OUP, 1974. “Contingent Nonconcretism” is Michael Nelson and Edward Zalta’s name for the actualist system proposed by Linsky and Zalta in the mid-90s. See Bernard Linsky and Edward N. Zalta, “In Defense of the Simplest Quantified Modal Logic” Philosophical Perspectives, Vol. 8, Logic and Language (1994), 431-458. See also, Linsky and Zalta, “In Defense of Contingently Non-concrete objects,” Philosophical Studies (Special Issue: Possibilism and Actualism), 84/2-3 (1996): 283–294; and see Nelson, Michael and Edward N. Zalta, “Bennett and “Proxy Actualism”.” Philosophical Studies. 142:2 (2009) 277-292. Menzel, Ibid. calls the view “New Actualism.”

[9] Forrester, 96. 

[10] See Allen Thompson. “Adaptation, Transformation, and Development: Environmental Change and Rethinking the Human Good.” Environment Ethics 42:1, Spring 2020, 7

[11] Chellas;Brian F. Modal Logic: An Introduction. C.U.P. (1980), 191

[12] “For all that [Kant’s Law] tells us, one might have duties to do what one is physically unable to do.” Forrester, 32.

[13] Carneades.org. “The Paradox of Kant’s Law (Ought Implies Can).” https://www.youtube.com/watch?v=qjNs4t4GQ2A  Accessed on 11/28/2022.

[14] See Hipline, Risto and Paul McNamara, “Deontic Logic: A Historical Survey and Introduction.”  In Gabby et al. eds, Handbook of Deontic Logic and Normative Systems. College Publications (2013), especially 52-53.  According to Hilpinen and McNamara, the validity Deontic Barcan is conditional on the assumption of a constant domain; on expanding domain models, it will be invalid. But on the system we are considering the domain is fixed.

The same reasoning applies to Deontic Buridan: There is a deontic alternative to w0 where all agents satisfy A only if A is permissible for all agents at wa . While in the case of Deontic Converse Barcan: If, in all deontic alternatives to w0, all agents satisfy C, then all agents in w0 are obligated to satisfy C. Hilpinen and McNamara concur on the validity of the formula for both variable and invariant domains. Only the Deontic Converse Buridan [(x)PAx > P(x)Ax], is invalid in both fixed and variant domains. Hilpinen and McNamara offer as counterexample, “Everyone is permitted to have a dinner in Casa Paco, a public restaurant, but no situation in which everyone is having dinner at Casa Paco is permitted . . . because the legal seating capacity of the restaurant is 40 customers.” Ibid., 53.

[15] I am indebted to Lou Goble for pointing to me that the consequences for the distinction were not so severe. Goble, Private Correspondence, 2022

[16] See Chrisman, Matthew. The Meaning of Ought: Beyond Descriptivism and Expressivism in Metaethics. Oxford: OUP, 2016), 111-112. For, more pugilistic framing of the same point, see Geach, Peter T., “Whatever Happened to Deontic Logic?” Philosophia XI (Fall, 1982), 35.

[17] De Coninck, Thijs. An Investigation into the Role of Quantifiers in Deontic Logics. Master’s Dissertation, University of Gent, 2017; 20.