Rules and Possible Worlds

Following my discussion of the rule concept that I started in previous posts, I will here briefly explore an intriguing possibility consisting in conceptualizing rules on the basis of “possible worlds” semantics. More specifically, I will define rules as (soft) constraints on possible worlds. For the intereste reader, this approach is pursued in more details in several papers by the philosopher Jaap Hage (see here and here).

In the previous posts, I have concluded that Searle’s distinction between constitutive rules and regulative rules is a linguistic rather than an ontological one.[1] I have also suggested that Epstein’s frame principles are formally analyzable as rules, already on the basis of a (informal) possible worlds reasoning. As a reminder, Searle and Epstein respectively suggest the following syntax for their constitutive rules and frame principles:

SCR         This X counts as Y in circumstances C

EFP         For any z, the fact “z is X” grounds the fact “z is Y”

By contrast, regulative rules have the generic “if… then” form:

R              If X, then Y

One may be easily lead astray by the fact that though formally equivalent, these definitions leave several things implicit. Consider EFP first. Its statement includes what I will call an object (z), facts or states of affairs (“z is X” and “z is Y”) and properties (X and Y). Quite the contrary, SCR only contains objects (X and Y) as well as a set of conditions C (which is only implicit in EFP). Finally, R is only about facts, though there must be an implicit statement about the set of conditions where R obtains.

The disentanglement of objects, facts and properties is required to see the formal identity of these three definitions. If I use upper letters X, Y, Z to denote properties and small letters a, b, c to denote objects then we have something like

∀a[X(a) → Y(a)]

This statement reads as ‘for any object a, if a has property X then it has property Y’. Consider for instance these two examples:

(a) Pieces of paper that have been engraved by the Federal Reserve are money.

(b) John has made six fouls, therefore he is fouled out.

In example (a), objects are (some) pieces of paper and properties are ‘to have been engraved by the Federal Reserve’ and ‘being money’. In example (b), the object is John and properties are ‘to be ascribed six fouls” and ‘to be fouled out’. Moreover, each X(a) and Y(a) denotes a fact of the type ‘this object has this property’. Finally, both statement are expressed against a background of conditions (e.g. in (a), we assume that we are in the USA, in (b) that John participates to a basketball game). This indicates that rules only work as parts of larger institutions, i.e. are connected with other rules that may specify conditions and implications. For instance, the property “being fouled out” implies something like “one is forbidden to return on the court for the game”. Ultimately, all rules are reducible to statements about what is possible and necessary (and conversely, impossible). This is at this point that it is useful to introduce a possible world framework.

Informally, a possible world can be seen as an exhaustive description of some counterfactual reality[2]. More formally, a possible world corresponds to a set of sentences (propositions and formulae) describing states of affairs, each sentence being ascribed a truth value (e.g. in some world, it may be true that the Golden State Warriors are the 2015 NBA champions but false that the finalist were the Cleveland Cavaliers; both statements are of course true in the actual world). Several kinds of constraints may restrict the range of possible worlds[3]. The most obvious one are logical constraints: for instance, it is not possible to p and not-p to be both true in the same world. More generally, the logical constraints are defined by the various axioms that are imposed on the underlying syntax. Similarly, physical constraints (e.g. a piece of metal cannot being heated without expanding) and conceptual constraints (e.g. an object cannot be a triangle and a circle at the same time) may be imposed on possible worlds. The point is that a set of constraints determine whether or not two or several states of affairs are compatible in a given world. Now consider the general statement of a rule

∀a[X(a) → Y(a)]

Suppose we define an axiom such that if the statement is valid (i.e. the rule actually exists), then the whole sentence combining the antecedent and the consequent is necessarily true (i.e. true in all possible worlds). For instance, take the following rule which is supposed to hold in any official basketball game:

“Any player who makes six fouls is fouled-out”.

Because this sentence is necessarily true, then there cannot be a possible world where a player has made six fouls but is still playing the game. The combination of these two states of affairs is ruled out because we have imposed as an axiom that if the above rule exists, then they cannot hold together. We can be more subtle however by adding an accessibility relation between possible worlds. The interpretation of the accessibility relation depends on the kind of logic we are using but in the present, it would state that any world w’ that is accessible from a world w should have the same set of valid rules (but the reverse needs not to be true – it depends on the property of the accessibility relation). But worlds non accessible from w could have a completely different set of rules.

This formal approach is thus helpful to see that virtually all rules work as constraints on possible worlds irrespective of their syntax. It can also serve as a basis to study the conditions for a set of rules for being consistent (see the papers linked to above). This is an important issue if we acknowledge that institutions are sets of rules. It is likely that for any given institution, some rules cannot be changed easily at the risk of leading to inconsistency, while others may be modified without engaging the coherence of the whole institutional edifice.

Notes

[1] In his book Making the Social World, Searle suggests an alternative criterion for distinguishing between these two kinds of rules. Regulative rules are identified to “standing directives” while constitutive rules consist in “standing declarations”. The former have only for function to bring about some form of behavior, while the latter make something the case by representing it as being the case. The distinction is interesting as an account of the different ways rules may be generated. It is not so clear that is helpful to distinguish different forms of rules though, if we consider that being self-referential is more or less a necessary condition for any rule to holds.

[2] Philosophers disagree regarding the nature of possible worlds. David Lewis was the most prominent proponent of a realist account according to which possible worlds are true worlds that exist in some alternative reality and that can discovered. Others like Saul Kripke hold that possible worlds are theoretical constructions where it is stipulated what is true according to them.

[3] Because they are build on the basis of a truth value function, Possible worlds models are said to be ‘semantic’. Most of the time (but not so much in economics), they are combined with a language (a syntax) responding to a logic articulated around several axioms: modal logic, deontic logic, epistemic or doxastic logic are the most discussed.