Monday, January 18, 2021

On the Characterization Postulate

I often mention the characterization postulate on this blog, and since it is such an important tool in object theories of all sorts, I figure it is time to provide a brief overview of just what it is supposed to be.

So why do object theorists need a characterization postulate (CP)? Well, the answer is because such a postulate is required for the epistemological adequacy of object theory. Existent objects do not pose a problem, because we can discover the properties of these by extensional means, i.e. through the use of empirical evidence. But such a procedure is for the most part unavailable for us when it comes to nonexistent objects. I say "for the most part" because we can indeed discover some properties of nonexistent objects through such means as dreams or hallucinations, but these procedures are not at all exhaustive.

So what we require is a logical, e.g. a priori, means to discern the properties of nonexistent objects, and this is where the CP comes in. In essence, the CP is a logical tool which allows us to do that. To express how important it is to object theory, the CP appears quite early in the process of logical construction. Indeed, once we have added descriptors to zero-order logic we can already bring the CP into play (but we needn't go into the technical details of that here).

But I should note that I have been writing as if the CP is one unique thing. This, however, is untrue; for we have many different CPs. The most natural one is the Unrestricted Characterization Postulate (UCP). This runs as follows:

UCP: An object has exactly those properties it is characterized as having.

This is quite natural and does a lot of work. Indeed, it is surely the first CP that comes to mind for the object theorist, and it is no doubt used in much argument and informal reasoning. But unfortunately, the UCP cannot be true tout court. This is for one very simple but devastating reason: namely, it allows us to prove the existence of any object whatsoever.

Consider the following object: "The existent non self-identical spider-eyed lamb". Let's call this object L. By the UCP, it follows that L is existent. But it is obvious that L is non-existent (it violates the law of identity). Therefore, the UCP is false.

Now one might think we can get around this problem by somehow weakening our logic, by analogy to how we can avoid the paradoxes of naive set theory by weakening the underlying logic. But this option is not available to us, for the problematic consequences of the UCP do not depend upon any axioms or inference rules; rather, they only depend upon the presence of descriptors (e.g. term forming operators like "a", "an", "the", etc.) And since eliminating descriptors from our logic is completely out of the question, we must look elsewhere for solutions.

So it is clear the UCP doesn't work. What is the object theorist to do? Well, he could very well just persevere with the UCP. In effect, he would have to use heuristic rules in order to avoid the untoward consequences. No doubt this can certainly be done (and indeed there is an analogy with how many of the textbooks for classical logic make use of naive set theory, even though naive set theory paired with classical logic leads to triviality), but such a route puts the object theorist on unsure logical footing.

Thus, it would seem that a better option would be to suitably restrict the CP. A radical restriction is what we might call the Existential Characterization Postulate (ECP).
This is as follows:

ECP: If an object exists, then it has exactly those properties it is characterized as having.

The ECP is surely true and quite unobjectionable. Indeed, it is true under the mainstream philosophical theories such as empiricism, idealism, and materialism. But for a full object theory the ECP will not do. For it is both far too restrictive (in that it tells us nothing about nonexistent objects) and it is technically redundant (since we already have empirical means at our disposal for discerning the properties of existent objects). So we will need to look elsewhere to find a CP that does some real work.

One way to do so is by expanding the ECP to what we might call the Possibilist Characterization Postulate (PCP). This runs as follows:

PCP: If an object is possible, then it has exactly those properties it is characterized as having.

This will no doubt appear quite attractive to philosophers of a rationalist persuasion. But while it might seem to be a real advance upon the ECP (since now we are able to do real work in discerning the properties of nonexistent objects),this is merely illusory. For, in one sense, the PCP is too restrictive; but it another sense, it is far too permissive.

Let us first consider how it is too restrictive, by referring back to our old friend L. What does the PCP tell us about this? Well nothing at all; because L is an impossible object, and the PCP tells us only about possible objects. Now of course, the rationalist object theorist won't actually consider this to be a weakness, since for him no object is impossible. But the advantages that consideration of impossible objects bring (which are too numerous to go into fully here, but they include such benefits as a resolution of the semantic paradoxes) makes this in my opinion an unacceptable stance to take.

Secondly, the PCP is too permissive because it still allows for the unacceptable ontological arguments mentioned earlier, although of course only restricted to possible objects. For the existent golden mountain (call it M) is certainly a possible object. So by the PCP, M exists. Indeed, it seems that something like the PCP is at work in both Descartes' ontological argument and in the principle of plenitude (viz. the notion that every possible object exists).

Of course, we can duly restrict the PCP, leading to what we might call the Qualified Possiblist Characterization Postulate (QPCP), which runs as:

If an object is possible and does not exist, then it has exactly those properties it is characterized as having.

The QPCP certainly gets rid of the untoward ontological consequences of the PCP, but it is still too restrictive. The classical rationalist who wants to avoid the ontological argument and the principle of plenitude will no doubt rest easy with it. But I think we can do better,

Now, instead of restricting the CP by only applying it only to certain types of items (as the previous postulates do), we can restrict it in other ways too. One quite natural way is by applying it only to certain types of properties. A familiar distinction among object theories is that between nuclear and extranuclear properties. In brief, nuclear properties are ordinary properties of individuals. In other words, they are just those features which delineate what we might call the 'nature' or the 'essence' of an object, while extranuclear properties do not. Alternatively, we might say that nuclear properties apply directly to the object, whilst extranuclear properties in some sense depend upon the object's nuclear properties.
Such a distinction may appear ad-hoc to some, but it actually has a clear pedigree within the philosophical tradition; refer to Kant's distinction between determining and non-determining predicates, or to the Frege-Russell distinction between first-level and second-level functions.

Perhaps the simplest way to lay out this distinction is to list some examples. Standard nuclear properties include such garden variety properties as 'red', 'tall', 'kicked', walked', etc. Extranuclear properties include such things as: ontological properties (viz. 'existent', 'nonexistent'), logical properties ('is consistent', 'is inconsistent'), status properties ('is contingent', 'is impossible',) and converse intentional properties ('is thought about by Larry,' 'is dreamed of by Ron',).

With this distinction in mind we can now formulate a Nuclear Characterization Postulate (NCP), delineated as:

An object has only the nuclear properties it is characterized as having.

It is clear that the NCP allows us to completely avoid the problem of being able to simply define objects into existence (since existence is an extranuclear property) and it is also expansive enough to account for impossible objects. So as a theoretical device the NCP is quite attractive, but it does have its own problems. The first problem is that it leads to untoward consequences concerning relations between existent and nonexistent objects. Consider the fact that Sherlock Holmes lives at 221 Baker Street. By the NCP, Holmes inhabits 221 Baker Street. But Baker Street is an existent object, and it was never inhabited by Holmes, since it is verifiable through empirical means that it never contained Sherlock Holmes as a resident.

For a natural way around this difficulty, we can formulate a Qualified Nuclear Characterization Postulate (QNCP), as follows:

An object has only the one-place nuclear properties it is characterized as having.

Naturally, the QNCP requires that we have some means at our disposal to reduce multi-place predicates to one-place properties. There are several ways to do that, and we needn't go into the technicalities here. But suffice it to say, it is clear that if Holmes has the one-place property 'inhabits-221-Baker-Street', it does not follow that 221 Baker Street has the on-place property 'is-inhabited-by-Sherlock-Holmes', since one-place predicates generally do not imply other one-place predicates, unless we have suitable axioms or meaning postulates in place.

But, as should be no surprise by now, there is yet a further problem lurking in the background, and indeed, it's a problem facing all the previous postulates. Namely, how are we to distinguish between such objects as 'the round square' and 'the existent round square'? The QNCP does not tell us whether these characterizing descriptions denote separate objects or one and the same object. One route we can take is to simply delete the extranuclear property of 'existent' from the second characterization, and conclude that both descriptions denote one and the same object.

But we can also avoid the problem by a new and expanded characterization postulate. We might call this the Suppositional Characterization Postulate (SCP). This is as follows:

An object has the one-place nuclear properties it is characterized as having and for every extranuclear predicate P it is characterized as having, it presents itself as having P.

The idea in its fleshed-out form is due to Routley, but it has roots going all the way back to Meinong's notion of "watered-down properties". Essentially, what we are doing here is systematically producing nuclear analogues of extra-nuclear properties. We can easily see how the above problem is then solved: the existent round square presents itself as existing, while the round square does not.

It seems that we might have pushed the characterization postulate as far as it will go. SCP doesn't appear to run into the types of untoward consequences which the previous CPs ran into, and at a first glance it appears that we cannot extend it any further without running into the triviality problem of the UCP. But that is actually not the case, for there is indeed a CP that is equal in scope to the UCP, but which does not run into triviality. This is the Qualified Characterization Postulate (QCP). It runs as follows:

An item has all the properties it is characterized as having at some world or other.

The QCP really does all the work which the UCP tries to do, except that work is made logically tractable through worlds semantics. It is important to note that the worlds in use here are not merely restricted to the possible worlds of modal semantics; rather, the QCP makes full use of ultramodal worlds, such as incomplete, inconsistent, and open worlds. (We could very well restrict it to only possible worlds, and thus we would have a modalized version of the PCP. Jaakko Hintikka seemed to have just such an idea. But I would still say that this is far too restrictive). Note also how it solves the triviality problem:  we can indeed run an ontological argument to prove the existence of any item, But that does not mean we have proven that the item exists at an actual world. Indeed, it might very well exist only at impossible worlds. Thus it would still be nonexistent at actual worlds.

So that is where our journey ends. To be sure, we have skipped over some CPs one can find in the literature; but these are generally quite technical and beyond the scope of this post. But now we face an important question: which CP should we use? Object-theorists have given different answers to this question throughout the centuries. Meinong held to something like the NCP. Neo-Lockean object theories like that of Parsons tend more towards the QNCP. Classical item theory employs the SCP. Priest and Berto's 'Modal Meinongianism' uses the QCP.

But there's no a priori reason why we should only use one CP; for we can indeed use a variety of them, as the circumstances dictate. Indeed, this is the idea behind the pluralized item theory in Routley's later work; i.e. different sorts of CPs apply to different sorts of worlds. For instance, the SCP might apply at actual worlds, the PCP can apply at possible worlds, and the UCP can apply at some impossible worlds (with triviality now not being a problem, since we should expect some impossible worlds to be trivial). In fact under this approach the QCP becomes redundant, seeing as our plurality of CPs can do everything the QCP can. Indeed, it can do even more, since now we can determine the properties nonexistent objects have at actual worlds, a question Modal Meinongianism leaves unanswered (this is why Priest and Berto have to appeal to existence-entailing properties, as we've discussed in a previous post).

So as we can see, the Characterization Postulate is a deep and fascinating aspect of object theories that is worth careful study. There is much more to be said about the topic, but now is as good a stopping point as any.

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