Reasons: Logic and Hyperintensionality

Abstract

In this chapter we argue that normative reasons are hyperintensional, and put forward a formal account of this thesis. That reasons are hyperintensional means that logically equivalent propositions may be different reasons for the same thing (be it an action, a duty, an ought, etc.), and therefore cannot be substituted for each other.

Appendix: Horty’s Default Logic

John F. Horty (especially in Horty 2012) has provided not only a fully developed theory to bridge “oughts” with reasons, but also an interesting nonmonotonic logic based on reasons he claims is able to explain much of our normative reasoning and its puzzles (for instance normative conflicts) .

The aim of this section is to give a (as far as possible) neutral account of (i) Horty’s logical system based on defaults; (ii) his proposal to bridge reasons and oughts; (iii) his solution to deal with normative conflicts; and to discuss and evaluate his theory in the broader framework of the philosophy of normative domain especially in comparison to the system sketched in the present chapter.

Classical logic is monotonic, i.e. adding “new information” does not change what is already established: If $\Sigma \vdash _{M } C$ then $\Sigma \cup A \vdash _{M } C$ (where $\vdash _{M }$ stands for monotonic consequence).

Horty is convinced that monotonic classical logic, based on deduction, cannot properly model everyday normative reasoning. Why? Because normative reasoning eschews exceptionless generalizations, in favor of defeasible ones. A logic supporting normative reasoning should therefore be, contrary to classical logic, nonmonotonic: new information should never be able to withdraw previous conclusions.

Horty goes on to formulate a default logic (based on Raymond Reiter’s proposal — cf. Reiter 1980) augmented with priorities. I will recap the major traits of a fixed priority default theory here.

A fixed priority default theory $\Delta$ is a structure $\langle W, \mathcal {D}, < \rangle$, where W is a set of ordinary logical formulae, standing for what is accepted as true (in other, slightly inappropriate, terms: what is known to be a fact); $\mathcal {D}$ is a set of default rules, and < is a strict partial ordering on $\mathcal {D}$.

Defaults Horty’s theory stands on the very notion of ‘default’ or ‘default rule’. What are defaults? Defaults are rules (rules like modus ponens) holding by default, that is, they can be defeated provided there are exceptions, or new information is learned.

Using a quite battered example, a standard default $\delta$ can be considered $Bird(x) \Rightarrow Fly(x)$.

Priority The notion of priority helps Horty (later on) to convey the idea of the different strength of different reasons, as modeled by different default rules.

As we have seen, there is a strict partial ordering ‘<’ between defaults $\delta _{1}, \dots , \delta _{n}$.

A strict partial ordering satisfies the transitivity property: $\delta < \delta '$ and $\delta ' < \delta ''$ then $\delta < \delta ''$ and it satisfies the irreflexivity property: it is not the case that $\delta < \delta$.

A strict partial ordering does not satisfy the property of connectivity, that tells us that given any two rules, one is always stronger than the other: either $\delta < \delta '$ or $\delta ' < \delta$.

Horty gives two informal arguments to refute connectivity (these arguments are based on an understanding of default rules as reasons, although at this point a full theory to link the two has yet to be developed): first, some reasons are incommensurable (p. 20), because are based on very different sources, for instance.¹⁵

Second, even if two reasons are comparable in strength, they may have equal strength, so that we cannot rank them.

The issue of connectivity, and in general of whether it is always possible to rank two given rules has an enormous bearing on the question of (normative) conflicts. If two rules can always be prioritized, then conflicts are only apparent, or prima-facie, rather than actual, irresolvable conflicts. Without connectivity the possibility of genuine (normative) conflicts remains open.

Now we have a set of known facts, a set of rules to apply, and some notion of priority for these rules. But what are the criteria, if any, to rank these rules in case of conflict? Informally, Horty lists four: (i) specificity; (ii) reliability; (iii) authority; (iv) our own reasoning.

Other technical concepts: scenarios, extensions A scenario of a theory $\Delta$ is some subset $\mathcal {S}$ of the set $\mathcal {D}$ of defaults, and it is supposed “to represent the [...] defaults that have actually been selected by the [...] agent as providing sufficient support for their conclusion (p. 23).”

An extension is the belief set generated by the union of W with the conclusion generated by a given (proper?) scenario, closed under logical consequence.

Of course not all defaults in $\mathcal {D}$ are always in use; quite on the contrary, they must be triggered by some already given propositions. To use the default $\delta \in \mathcal {D}$ that “birds fly”, you first need to have, in W, that t is a bird. Once you have this information, then $\delta$ is triggered, and you can conclude that t flies.

This passage is rather important because Horty identifies reasons as propositions, rather than rules, and in particular with “the premises of triggered defaults (p. 27).”

Proper scenarios, informally defined, are those scenarios made of binding defaults, that is, (also informally) those defaults that are triggered, not conflicting, and not defeated (cf. pp. 29–33).

A theory may have multiple or no extension. I shall examine these two cases in turn.

Multiple extensions First, there can be multiple extensions based on different scenarios in a given theory. There are three ways, Horty tells us, to make sense of multiple extensions: the choice option, the credulous option, the skeptical option.

Under the choice option, different extensions are interpreted as “different equilibrium states that an ideal reasoner may arrive at on the basis of the initial information from that theory (p. 35).” If the choice is arbitrary, then it is difficult to see how there can be a formalized notion of consequence: which extension is the actual conclusion set of the original theory?

Under the credulous option, whenever a proposition is in at least one extension it is taken to be a conclusion of the default theory. Under this option, a consistent default theory can yield inconsistent conclusions. Horty discusses some strategy to avoid inconsistency on pp. 36–7.

Under the skeptical option, if any proposition is treated differently in more than one extension, it cannot be taken to be a legitimate conclusion of the original theory. The intersection approach considers conclusions to be those propositions belonging to every extensions of the theory.

No extensions A theory with no extension is just a theory without a proper scenario (since extensions are generated only by proper scenarios). This may be taken to mean that there is no coherent interpretation of the theory (p. 39). Under the intersection approach (an approach that considers conclusions to be those propositions belonging to every extensions of the theory), since there is no proposition in common (there is in fact no extension at all), then every proposition can be considered as a conclusion. This resembles ex contradictione quodlibet in classical logical: from a contradiction everything follows.

1.1 Reasons as Triggered Defaults Premises

Horty turns to explain how (and why) a theory of defaults can serve well in interpreting reasons.

“X is a reason for Y in the context of $\mathcal {S}$ just in case there is some default $\delta : X \Rightarrow Y$ from the underlying set $\mathcal {D}$ that is triggered in the context of $\mathcal {S}$ (p. 41).”

Horty characterizes his account as austere and relative: austere because the number and quality of reasons are limited to the defaults of the underlying theory; relative because the number and quality of reasons depend on a particular scenario: “what counts as a reason for what might vary from one scenario to another (p. 45).”

It is possible to define an absolute reason relation just when there is a unique proper scenario in the underlying theory; or it is a reason in all proper scenario of that theory.

Conflicts Conflicted reasons, reasons strength, and defeated reasons are defined via the corresponding definition of conflicted defaults (supporting contradictory conclusions), priority of defaults through the strict partial ordering <, and defeated defaults (defeated reasons can be defined also using the notion of conflicted reasons and their strength: reason W defeats reason X iff they support contradictory conclusions and W is stronger than X).

X is good reason for Y iff X is a reason for Y and it is not conflicted or defeated.

1.2 Oughts

We have seen Horty’s conception of defaults as basic, and how defaults are related to reasons (reasons are the premises of triggered defaults). I have not critically discussed Horty’s idea in philosophical terms.

However, it seems that one of the fundamental concepts used in normative questions is that of “ought” . How are reasons and defaults related to oughts?

Horty provides two “procedures” to derive oughts from reasons and defaults. These two procedures depend on whether or not we want to allow normative conflicts. Horty then builds two deontic logics, each corresponding to one or the other procedure. The first allows conflicts among all-things-considered, or strong, oughts, whereas the latter does not, invoking instead a disjunctive solution.

A normative conflict is “a situation in which an agent ought to perform an action X, and also ought to perform an action Y, but in which it is impossible for the agent to perform both X and Y (p. 65).”

Horty introduces a conditional deontic operator O to represent the all-things-considered ought: O(Y / X) “is the statement that, all things considered, Y ought to be the case under the circumstances X (p. 68).”

An unconditional ought is just a conditional O, conditional on verum, a trivially true proposition: $O(Y) =_{def } O(Y/\top )$.

The orthodox view about reasons and oughts is the following: “an agent ought to perform an action if there is an undefeated reason for the agent to perform that action (p. 69)” (according to Horty’s reading of Chisholm) or “an agent ought to perform an action when the reasons that favor performing that action outweigh the reasons that oppose doing so (p. 70)”, as Horty reads other major theories like Baier’s, Harman’s, Schroeder’s.

However, Horty notes that this a very direct relationship between reasons and oughts; one that he finds hardly acceptable. One possible solution (discarded in Horty 2012 , but possibly taken up in Horty and Nair forthcoming [a]) is to develop a logic of reasons; another is to consider obligatory actions not only those the agent has a reason to do, but also those entailed or required by such reasons.

Horty discards, however, any theory that directly links reasons for actions and oughts. He proposes to isolate a coherent subset of all the reasons an agent has, and only from there to derive the relevant oughts.

(The problem here is with “coherent”. If Horty with coherent means “consistent”, then this seems question-begging.)

There are two ways of doing this: the conflict account, and the disjunctive account.

According to the conflict account, a proposition is an all-things-considered ought when it is supported by one of the theory’s proper scenarios. This account allows normative conflicts.

According to the disjunctive account, a proposition is an all-things-considered ought when it is contained in each one of the theory’s extensions. This account doesn’t allow normative conflicts, but only yields disjunctive obligations.

Ditto for conditional oughts, with the proviso that this time it’s not only just the theory’s extensions, but the theory’s extensions supplemented with the premises of the conditional oughts.

Both the conflict and the disjunctive accounts don’t allow for strengthening the antecedent (in classical logic, if $A \vdash B$ then $A \wedge C \vdash B$): from $\Delta \vdash O(Y/X)$ it doesn’t necessarily follow that $\Delta \vdash O(Y/X \wedge Z)$.

The conflict and the disjunctive account do not have a monotonic notion of logical consequence, and “can be classified as nonmonotonic logics. They cannot be accommodated in any simple way within the modal, or intensional, framework that is so often appealed to as a foundation for deontic logic [...] (p. 81).”

With a simple stipulation, Horty makes these logic “noncontradictory, in the sense that neither will allow a consequence statement of the form $\Delta \vdash \phi \wedge \lnot \phi$ (p. 82).”

In this way the conflict account can handle normative conflicts (of the form: O(Y) and $O(\lnot Y)$), whereas logical contradictions (of the form: O(Y) and $\lnot O(Y)$) are ruled out.

Moreover, all-things-considered oughts are closed under logical consequence (if $\Delta \vdash O(Y/X)$ and $Y\vdash Z$ then $\Delta \vdash O(Z/X)$ — cf. pp. 82–83).

Let’s suppose that in the example above (if $\Delta \vdash O(Y/X)$ and $Y\vdash Z$ then $\Delta \vdash O(Z/X)$)

Both the conflict and the disjunctive accounts support the principle ought implies can (p. 83).

Neither the conflict and the disjunctive accounts allow inconsistent all-things-considered oughts, but in different ways.

The conflict account blocks agglomeration (if $\Delta \vdash OY$ and $\Delta \vdash OZ$ then $\Delta \vdash (OY \wedge OZ)$), so that if we have two inconsistent oughts OY and $O(\lnot Y)$ we cannot conclude that $O(Y \wedge \lnot Y)$.

The disjunctive account allows agglomeration but “there is no risk that such agglomeration would lead to an individually inconsistent ought since the entire collection of supported oughts is itself jointly consistent (p. 85).”

1.3 A Comparison Between Horty’s and Classical Semantics

Horty (forthcoming) puts forward a very useful comparison between his approach and the classical semantics for deontic modals based on the Kratzerian paradigm.

It turns out that many of the details and results of the default framework can be translated back in Kratzerian terms, either (more or less) straightforwardly, or with some additional technical work.

First, default deontic logic allows conflicting oughts to be derivable, but so does the classical framework, without any substantial modification if not for the evaluation rule for deontic necessity.

Second, while default deontic logic allows for an explicit ordering of “norms” , “rules”, or “reasons” (i.e., of the defaults), tracking perhaps their importance, such a feature can be recovered by lifting the ordering source.

Third, default deontic logic seems to be at an advantage in working with a very intuitive notion of conditional ought, in a sense that does not seem straightforwardly recoverable in the classical framework.

However, it is in reasoning about reasons, or higher-order norms (i.e. in considering the relative importance of various reasons, or thinking which reasons are to be excluded from considerations), that Horty thinks default deontic logic is superior to the classical framework for the very simple fact that this kind of feature is not even expressible in the latter.

While this critique is hardly deniable, I think it is unfair for the following reason: it just switch the subject matter, introducing norms and reasons explicitly in the object system, and having oughts grounded in them. While this is totally acceptable (and, I believe, a huge step in the right direction, philosophically speaking), it is too much to ask from the classical semantics, which is just concerned with truth-conditions, deals with oughts as if they were part of the “descriptive” language, and does not aim to a metaphysical explanation of oughts. Horty’s critique is as unfair as someone criticizing his system for not being able to settle the truth of factual discourse with the same means used for grounding normative discourse (remember that in the classical framework worlds also serve to determine completely the truth values of the propositions under consideration) .