Heisenberg in one of his excursions into philosophy (Physics and Philosophy) discusses epistemology, logic, and ontology. In developing his ideas on how the innovations of quantum mechanics impact on philosophy he introduces (or rather reintroduces with reference to Aristotle) the concepts of potentiality.
He comes to potentiality through considering the logic required to refer to things that quantum mechanics describes. That is, it ...
... concerns the ontology that underlies the modified logical patterns. If the pair of complex numbers represents a "statement" in the sense just described, there should exist a "state" or a "situation" in nature in which the statement is correct. We will use the word "state" in this connection. The "states" corresponding to complementary statements are then called "coexistent states" by Weizsäcker. This term "coexistent" describes the situation correctly; it would in fact be difficult to call them "different states," since every state contains to some extent also the other "coexistent states." This concept of "state" would then form a first definition concerning the ontology of quantum theory. One sees at once that this use of the word "state," especially the term "coexistent state," is so different from the usual materialistic ontology that one may doubt whether one is using a convenient terminology. On the other hand, if one considers the word "state" as describing some potentiality rather than a reality —one may even simply replace the term "state" by the term "potentiality" —then the concept of "coexistent potentialities" is quite plausible, since one potentiality `may involve or overlap other potentialities.
In the version of quantum mechanics that I have been developing these "coexistent potentialities" should be understood in relation to the \(\sigma\)-complex. Despite his hints, Heisenberg does not succeed in including potentiality in the required logic. What is needed is a modal logic that includes a POTENTIALITY operator. Barabara Vetter constructs such a modal logic and defends it with modal metaphysical arguments in her book Potentiality: From Dispositions to Modality. There is a need for a theory going beyond the naive position that there are objects that simply have the potential to have certain properties. In quantum mechanics in particular it is not sufficient to say than an electron has the potential to being in some place. If this is elaborated to saying that it has some position with a certain probability, we get a deceptive oversimplification. The theory of potentiality developed by Vetter give potentialities the status of properties of objects. Contrary to Heisenberg 's view that "one considers the word "state" as describing some potentiality rather than a reality". Vetter's theory allows us to enlarge the domain of what is real to potentialities.
In standard quantum mechanics observables can take possible values. The possibility is to be described by the theory and the values are represented by the spectrum of the self-adjoint operator that in turn represents the observable. Vetter proposes, in general, to define possibility as follows:
POSSIBILITY: It is possible that p \(=_{\mbox{df}} \) Something has an iterated potentiality for it to be the case that p.
We have now moved on to potentiality and more generally its iteration. I have discussed dispositions in previous posts and now explore Vetter's proposal that we call those properties which form the metaphysical background for disposition ascriptions potentialities. In the definition the something also need explanation. Something can be an object or any collection of objects.
Potentialities are individuated, not by a pair of stimulus and manifestation and a corresponding conditional, but by their manifestation alone. The manifestation of a potentiality is the property that the potentiality’s possessor would have if the potentiality were to be manifested. A manifestation can be a property of and object (or collection of objects) or a relation that the object (collection) has with something else.
Joint potentiality
The relation between a potentiality and its manifestation, the relation ‘...is a potentiality to ...’. Manifestations individuate potentialities and therefore the classification of manifestations as relations or properties provides a classification of the joint potentialities:
Type 1 joint potentialities are joint potentialities whose manifestation is a relation between, or a plural property of, all its possessors.
Type 2 joint potentialities are joint potentialities whose manifestation consists in a property or relation of only some, but not all, of its possessors.
In type 1 joint potentialities, the manifestation is a relation (or plural property) holding non-trivially between all its possessors, such as for an example of the door key's joint potentiality with the door for the former to unlock the later. In type 2 joint potentialities, however, the manifestation of a joint potentiality concerns only some of its possessors; such are the cases of
- a uranium pile (with a potential to go unstable) plus the rods and fail-safe mechanism
- a glass (with a potential to brake) but protected by styrofoam,
- a city (with a potential to be destroyed) and its defence system.
Extrinsic potentiality
An object has an extrinsic potentiality if the object has a joint potentiality whose co-possessors are the dependees of the extrinsic potentiality. To return to the key and door example, in the case of the key’s extrinsic potentiality to open the door, the dependee and co-possessor (the door) was part of the potentiality’s manifestation (unlocking the door). In general, where a potentiality’s manifestation consists in a relation to a particular other object—such as opening this particular door, as opposed to opening some door with a lock of a specific shape—the potentiality will be extrinsic, and the object involved in the manifestation will function as its dependee and as the co-possessor of the relevant joint potentiality.
Iterated potentiality
As well as joint or extrinsic the potentialities can be iterated. That is, objects can have the potential to have potentials.
Objects and collections of objects have potentialities to possess properties. Potentialities themselves are properties. So, things should have potentialities to have potentialities and the latter potentialities might themselves be potentialities to have potentialities. So, there is nothing to prevent things from having potentialities to have potentialities to have potentialities, or potentialities to have potentialities to have potentialities to have potentialities ... and so forth. These are iterated potentialities.
Formalising potentiality
In Vetter's theory potentialities include dispositions and abilities as properties of individual objects or collections of objects. However, in this theory POTENTIALITY cannot be defined because it is the metaphysical underpinning of dispositions and consequently possibilities. Vetter uses examples such as those I have discussed in explaining dispositions in earlier posts. Ordinary language semantics is also used as when the potential to do or be something is discussed. It is on this bases that potentiality is then posited to be metaphysically fundamental.
Although POTENTIALITY cannot be formally defined it can be formalised to show how it operates. Just as there are modal operators for necessity and possibility that act on propositions, the modal potentiality operator is called POT.
POT must [therefore] be a predicate operator which takes a predicate—specifying the potentiality’s manifestation—to form another predicate, which can then be used to ascribe the specified potentiality to an object.
Formally, where upper case Greek letters are predicates,
$$ \mbox{POT} [\Phi](t)$$
ascribes to t the potentiality to \(\Phi\), where \(\Phi\) is an \(n\)-place singular predicate and \(t\) is a singular term, or \(\Phi\) is an \(n\)-place plural predicate and \(t\) is a plural term. Potentialities can be described by more general sesntences rather than just predicated. For the POT operator to work with sentences rather than predicates, let the sentence \(\phi\) be for something to be such that \(\phi\) is true is equivalent to it have the potentiality to \(\Phi\). To express logically complex predicates and to turn closed sentences into ‘such that’ predicates, we need to introduce a standard predicate-forming operator, \(\lambda\).
$$ \mbox{POT} [\Phi](t) \rightarrow \mbox{POT} [\lambda x.\phi](t)$$
Where \(\phi\) is a sentence, open or closed, and \(\phi[t/x]\) is the result of substituting a term \(t\) for any free occurrence of \(x\) in \(\phi\), the sentence \(\lambda x.\phi (t)\) is true just in case \(\phi[t/x]\) is true. Intuitively, \(\lambda x.\phi\) turns the sentence \(\phi\) into a predicate meaning ‘is such that \(\phi\)’.
Although there are further subtleties it is now possible to formulate a simple form of iterated potentiality.
$$ \mbox{POT}[\lambda x. \mbox{POT}[F](x)](a) $$
which ascribes to \(a\) an iterated potentiality to be \(F\). This formalisation can be iterated further.
The electron again
Returning to an electron described by quantum mechanics. An electron has the potential to be somewhere, to have a momentum, to have charge and a mass. But the electron description in quantum mechanics has even more structure. It appears somewhere with a certain probability in certain circumstances. Quantum mechanics does not just provide bare probability distributions for each potential property but has those probability models combined in a \(\sigma\)-complex; corresponding, I propose, to the "coexistent states" of Weizsäcker. The electron has a joint potentiality with other objects to manifest one of the probability distributions from the complex, in certain circumstances. Then that probability distribution weights the appearance of a specific set of values of the property with a numerical probability. Therefore, the probability that the electron will appear in a specific region of space in certain circumstance is an iterated potentiality. The electron has the potential to manifest one of a complex of probability distributions and a probability distribution captures in mathematical form the potential for the electron to appear in a region of space.
This example indicates how the concept of potentiality, as formalised, can be used to describe quantum processes. It remains to be seen how useful this will be when we introduce considerations of spin, entanglement, interference etc. This will be explored in future posts.