by Corry Shores
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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own.]
Part 4: Extensions of Classical Logic
Chapter 11: Leibnizian Modal Logic
11.3 A Natural Model?
A natural model is one that is sufficient to what it is modeling, meaning that its “domain consists of the very objects we mean to talk about,” and its “predicates and names denote exactly the objects of which they are true on their intended meanings” (Nolt 325). In modal logic, a natural model will have an actual world where the domain of objects are actual objects and the predicates are assigned to objects that actually have those predicates. The other possible worlds are variations on the actual world. But name designations remain the same in all worlds in the model. Possibility in possible world modeling can be understood as informal possibility, meaning that neither syntactic nor semantic contradictions are allowed, or formal possibility, where only syntactic contradictions are prohibited. This means that the extensions of predicates can be inconsistent with one another, and for now we use this formal notion of possibility in our possible world modeling. Related to this distinction is the one between the realist and the nominalist view of essences. Realists believe that essences are real things in the world. So in terms of possible worlds, something’s real essence in our actual world will determine what sorts of variations it can undergo while still being identical to itself in other worlds, at least insofar as it is essentially the same even if certain properties vary. The nominalist view thinks that essences are artifically created through linguistic practices, so from this perspective there is no certain way to know what sorts of variations something can undergo while still being essentially the same. To avoid this issue, we do not deal with metaphysical possibility, which would be concerned with the ways and extents something’s essence allows it to vary, but rather we are concerned with logical possibility, which is concerned just with logical contradictions that might hold in a world. And as we noted, we are concerned with formal possibility rather than with informal possibility. That is to say, we only care about syntactical contradictions of the form A and not-A. And we are not concerned with semantic contradictions of the form, for example, of Ra & Ca, where R means ‘is red’ and C means ‘is colorless.’ In other worlds, in some world, ‘a’ can be in both the extensions of R and of C, even though their semantic meanings are incompatible.
In the prior sections we have been learning about how to construct models for possible world semantics. Our examples so far were very simplistic so to introduce us to the basic structures and principles. But these models hardly resemble anything we would consider a real world, as they consist just of very small sets of numbers (Nolt 325).
[Nolt refers to how in section 7.2 we saw that even though models in predicate logic are similarly unrealistic, we can still produce a natural model by giving appropriate meanings to predicates and names. I have not summarized this section, but he will define natural model again here. A natural model it seems can be quite limited, as it may be far more restricted than the real world we live in, but so long as it is sufficient to the domain we are discuss, like geometry or subatomic physics, it can be a natural model. For modal logic the natural model will consist of possible worlds.]
A natural model is a model whose domain consists of the very objects we mean to talk about and whose predicates and names denote exactly the objects of which they are true on their intended meanings. A natural model for geometry, for example, might have a | domain of points, lines, and planes. A natural model for subatomic physics might have a domain of particles and fields.
A natural model for modal discourse will consist of a set of possible worlds – genuine worlds, not numbers – each with its own domain of possible objects. And that set of worlds will be infinite, since there is no end to possibilities.
We next wonder, what is a possible world [in a natural model for modal logic]? For Leibniz, possible worlds are universes that are more or less like our world. In one such world, Nolt can be a farmer. Can he be a tree, he wonders? It all depends on our philosophical view of essences. Nolt gives two philosophical views on essence, the realist and the nominalist view. Realists think that essences really exist out in the world and that they are things we might discover. Nominalists however think that essences are not discovered but are rather created by our linguistic practices. [Nolt’s point here seems to be that for the realist, there would be a way to determine Nolt’s essence and to see if he remains essentially the same were he a farmer and were he a tree. The nominalist however would see these possibilities as arbitrarily generated in the mind, and so there is really no way to give a definite answer.]
Philosophers who think that the nature of things determines the answers are realists about essence. Realists believe that essences independent of human thought and language exist “out there” awaiting discovery. (Whether or not we can discover them is another matter. ) Opposed to the realists are nominalists, who think that essences – if talk about such things is even intelligible – are not discovered, but created by linguistic practices. Where linguistic practices draw no sharp lines, there are no sharp lines; so if we say increasingly outrageous things about me (I am a farmer, I am a woman, I am a horse, I am a tree, I am a prime number ...), there may be no definite point at which our talk no longer expresses possibilities. For nominalists, then, i t is not to be expected that all questions about possibility have definite answers. (Extreme nominalists deny that talk about possibility is even intelligible.) The realist-nominalist debate has been going on since the Middle Ages; and, though lately the nominalists have seemed to have the edge, the issue is not likely to be settled soon.
For our purposes here, we will not need to settle whether or not it is metaphysically possible for Nolt to be a tree, as we can at least say it is logically possible, meaning that it implies no contradiction.
To avoid an impasse at this point, we shall invoke a distinction that enables us to sidestep the problem of essence. Whether or not it is metaphysically possible (i.e., possible with respect to considerations of essence) for me to be a tree, it does seem logically possible (i .e., possible in the sense that the idea itself-in this case the idea of my being a tree-embodies no contradiction). Contradiction is perhaps a clearer notion than essence; so let us at least begin by thinking of our natural model as modeling logical, not metaphysical, possibility.
Now that we will not be concerned with metaphysical possibility but rather with logical possibility, we will think of objects as being essenceless. So now a possible world is any set of objects, which can possess any combination of properties and relations. [The next idea seems to be the following. Nolt does not mention it, but he seems to be implying that a possible world understood in terms of logical possibility is one with no contradictions of the form A and not-A. But now let us suppose we have the propositions Ra and Ca. So far, there is no formal contradiction. But a contradiction can arise on the semantic level when for example Ra means “object ‘a’ is red,” and Ca means “object ‘a’ is colorless.”]
In confining ourselves to logical possibility, we attempt to think of objects as essenceless. What sorts of worlds are possible now? It would seem that a possible | world could consist of any set of objects possessing any combination of properties and relations whatsoever.
But new issues arise. Some properties or relations are mutually contradictory. It is a kind of contradiction, for example, to think of a thing as both red and colorless. Similarly, it seems to be a contradiction to think of one thing as being larger than a second while the second is also larger than the first. But these contradictions are dependent upon the meanings of certain predicates: ‘is red’ and ‘is colorless’ in the first example; ‘is larger than’ in the second. They do not count as contradictions in predicate logic, which ignores these meanings (see Section 9.4).
On the basis of these two ways that contradictions can exist in possible worlds, we will distinguish two sorts of possibility. The first is informal logical possibility, which excludes these cases where the semantic meanings lead to contradictions. Formal logical possibility is only concerned with structural (syntactic) contradictions of the form A and not-A. The extensional meanings of predicates can still be odd or contradictory.
If we count them as genuine contradictions, then we must deny, for example, that there are logically possible worlds containing objects that are both red and colorless. If we refuse to count them as genuine contradictions, then we must condone such worlds. In the former case, our notion of logical possibility will be the informal concept introduced in Chapter 1. In the latter, we shall say that we are concerned with purely formal logical possibility.
Only if we accept the purely formal notion of logical possibility will we count as a logically possible world any set of objects with any assignment whatsoever of extensions to predicates. If we accept the informal notion, we shall be more judicious – rejecting valuations which assign informally contradictory properties or relations to things. We shall still face tough questions, however, about what counts as contradictory. Can a thing be both a tree and identical to me? That is, are the predicates ‘is a tree’ and ‘is identical to John Nolt’ contradictory? The problem of essence, in a new guise, looms once again. Only by insisting upon the purely formal notion of logical possibility can we evade it altogether.
In the next chapter we will no longer be dealing with the relatively simple system of Leibnizian semantics. For now, we will just use the formal notion of logical possibility (327).
[Nolt then describes how to build a natural model in modal logic. We first take some set of sentences. Next we formalize them using modal predicate logic. Then the natural model for these sentences is an “infinite array of worlds.” Any one world has as its domain some set of actual and/or possible objects. Then for each world we assigned extensions to the predicates. (He says we do that “in all possible combinations (so that each domain is the domain of many worlds)”. I am not exactly sure how that works without an illustration in mind, but it might be something like, for example, for the predicate “is red” we would figure out every possible extension for this, such that one world gets one such set and another world gets another. But I am not sure.) One of these domains consists of just actually existing objects, and the predicates are assigned to extensions that they actually do have. This then of course is the actual world. In the actual world, names are assigned to their proper objects, and in all other worlds, these names track their proper objects, meaning that they continue to name them.]
Now, take any set of sentences you like and formalize them in modal predicate logic. The natural model for these sentences is an infinite array of worlds. Any set whatsoever of actual and/or merely possible objects is a domain for some world in this array. The predicates of the formalization are assigned extensions in each such set in all possible combinations (so that each domain is the domain of many worlds). Among these domains is one consisting of all the objects that actually exist and nothing more. And among the various assignments of extensions to predicates in this domain is one which assigns to them the extensions they actually do have. This assignment on this domain corresponds to the actual world. (Other assignments over the same domain correspond to worlds consisting of the same objects as the actual world does, but differing in the properties those objects have or the ways they are interrelated.) If our discourse contains any names, on the intended interpretation these names name whatever objects they name in the actual world; but they track their objects (i.e. , continue to name them) through all the possibilities in which they occur.
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.