On Philosophy

September 13, 2007

The Logic Of Change

Filed under: Logic — Peter @ 12:00 am

Propositional logic is well-suited to talking about the way things are, but it has problems dealing with change. This is not to say that time can’t be tacked on, but such improvements tend to be unwieldy, and lead to a strange metaphysics. For example, one way to handle time is to add temporal “quantifiers”, which we might write as a:[φ], meaning that at the time designated a the proposition φ is true. Now this isn’t an invalid addition, but it doesn’t make capturing change easily. To assert that a<b, a:[φ], and b:[ψ] says nothing intrinsically about the relationship between φ and ψ. Consider trying to represent the idea that the water currently in a stream became part of the ocean and that the stream still exists as a stream. We might start with something along the lines of 1:[Sa] ∧ 2:[Oa], indicating that a, I guess some water, is a stream at time 1 and part of the ocean at time 2. But how then do we represent the idea that the stream still exists? We might be tempted to say that 1:[Sb] ∧ 2:[Sb], that b, the stream, is a stream at time 1 and still a stream at time 2 (versus, Rb, which might indicate that the stream became a river). But what is this b, and what is its connection to a? Some might say that it is an abstract object, and that 1:[Iab] ∧ 2:[~Iab]; in other words, the water instantiates this abstract object at time 1, but not at time 2. Such an abstraction is fine by itself, but it is hard to translate such ideas back into terms we can understand as describing the world. Normally we can interpret the predicate-object scheme as indicating that there is some physical object which has some properties, represented by the predicate, in virtue of its causal dispositions. But what does predication indicate with respect to abstract objects? It is simply not clear what exactly such formalism mean, although we have some vague ideas about it being analogous to the situation with physical objects and their predicates.

Another approach is to treat the distinction between physical objects and abstractions (including predicates) as fundamental, and build a description of change on the basis of that. To do this we have to begin by tossing out the idea that change is to be captured by allowing the same object to have different properties over time. Our objects instead are collections of physical stuff that exists at a particular moment of time (sets of particles), and thus in the logic we develop predicates will apply to sets. For example, the teapot at 10:00am is one object, as is that teapot minus its handle, also at 10:00am. However, under this approach the teapot at 10:01am is a logically different object (even if it is, in another sense, the same teapot). In physical terms the set of all objects for a given moment of time is the power set of the set of all the fundamental particles at that moment of time. Of course any ordinary object is really best defined not as a particular set of fundamental particles, but as a predicate. For example, the property of being the teapot could be the predicate T. Now even though the teapot is supposed to be a singular object there are many sets of particles which the predicate T could apply to, because in general taking a few atoms away from the a collection doesn’t stop it from being a teapot, even a particular teapot. Thus, for convenience, we introduce the maximal operator, Μ. ΜxTx asserts that x is a maximal set that satisfies the predicate T. Or, in other words, ~∃y(x⊂y ∧ Ty). Obviously in most cases there is more than one such x, because each distinct time will contain its own such maximal set. And if we are working with a general description (such as “is a teapot” in general, and not the property that picks out my teapot), there may very well be more than one such maximum set even within a single period of time.

Of course while this puts the interpretation on a bit of a better footing we haven’t captured change yet. First let’s consider the easy cases, those in which the “same” physical stuff undergoes changes and becomes something different. To handle these cases all we need to do is add a successor relation, §ab, which indicates that the laws of physics have changed the set of particles a into b (and thus that b belongs to a later time than a), either directly or indirectly (there is nothing in principle preventing b to be part of a time millions of years later than a). Thus to say that my teapot will be re-forged into a pot all that we need to say is that ∃x∃y(Tx ∧ Py ∧ §xy). Obviously I have not used the maximal operator here because some of the particles that composed the teapot may be lost somewhere in the process. And if the pot is a large one, and we wish to claim that the teapot instead became part of it we can assert that ∃x∃y∃z(Tx ∧ ΜzPz ∧ y⊆z ∧ §xy). And the assertion that part of the teapot was changed into something else can be written similarly.

The harder cases are when the “thing” in question isn’t really a stable thing at all, but rather a persistent description. I think it is best to admit that in such cases the persistent description doesn’t really exist in the same way my teapot exists, and thus that we can’t talk about its changes in quite the same way. Consider the stream that becomes a river. In such a situation there really is nothing that changes over time into the river, rather I see water flowing in the same channel day after day. Over time I stop applying the descriptor “stream” to it, and start describing it as a river instead. Key in this process is the fact that there is a third description, that of a flowing body of water in a certain area, which unites them. Without this third description there might be nothing in common between them at all. And, with that in mind, we can formalize this kind of change. All we need to assert is that ∃x∃y(Μx(Sx ∧ Fx) ∧ Μy(Ry ∧ Fy) ∧ ∃u∃w(x⊆u ∧ y⊆w ∧ §uw)). Note that all ∃u∃w(x⊆u ∧ y⊆w ∧ §uw) asserts is that x is prior in time to y, x and y themselves don’t have to be physically related in any way. With that assertion we capture the idea that the flowing water at one time is a stream, but at a later time is a river, and has thus changed from one to the other. This I think captures quite elegantly our common-sense attitudes regarding change, without introducing any baffling metaphysical objects (such as abstract objects).

As a final note I will mention that we can assert that something will fail to exist sometime in the future by asserting that: ∃x(Sx ∧ ∃u∃w(Mu(x⊆u) ∧ §uw ∧ ∀y(y⊆w → ~Sy))), which states that in some future universe of physical stuff, w, there is no set of physical stuff such that S applies to it.

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