There is a large, nested, complexly intersecting, multidimensional area of logic known as “modal logic.” Standard (“assertoric” – dealing with comparatively simple assertions) logic essentially forgoes any considerations of the modes (hence, “modal”) in which an assertion is considered to be true or false; it simply is, or it is not (true or false). Modal logics are intended to examine the ways (modes) in which a proposition or assertion might express such truth or falsity. A great deal of very good work has been done in this area of study, but it remains a long way from solving its most basic problems; indeed, most proposed “solutions” do not so much “solve” their problems as strangle them.i I am at once deeply impressed by the technical sophistication of contemporary work on modality, and profoundly dissatisfied with the narrowness of its vision. Because one of the “modes” in which an assertion or proposition might be true or false is whether it is possibly true or false.
I can certainly inundate any interested party with citations, but anyone capable of following those citations would most likely already be familiar with them. It takes years of dedicated study to bootstrap one’s self up through propositional, into quantificational, and finally on to modal logics. On the other hand, it takes nothing more than the most elementary capacity for cognition to instantly see that there is a difference between saying that “X is the case,” and “X might be the case.” Just as we can talk about Jazz without mastering the saxophone, or relativity without deriving proofs related to the Ricci tensor, we can talk about possibility without becoming research mathematicians in formal logic. One might even argue that mastering such mathematics would not ideally equip us to talk about possibility which is, after all, a metaphysical, rather than a mathematical topic.
It is important to make a distinction here between probabilities and possibilities. Probability is a kind of possibility, but it is one that is limited to calculable chances (I’ll use the word “chances” specifically for probabilities). While there are various techniques for recursively generating meaningful probabilities from relatively inchoate data (Bayesian methods are particularly relevant here), it remains the case that developing calculable estimates of such chances is a remarkably small part of what happens when our inquiries engage the slippery concept(s) of the possible. It is not just that things which might be vastly out run those chances we can calculate. Things that might have been, but whose chances have long since passed, remain present in our world as the context and background for that which is, and the projective source for that which still might be.
Probability (and, conjointly, statistics) are difficult enough topics in their own right, but trying to wrap our heads around the characteristics of possibilities leads us into a much more complicated constellation of issues. I am personally dissatisfied with most of the attempts to create a logic of such possibilities, largely because such attempts begin from a generally inadequate logic of the actual (assertoric logic, recall) and then attempt to shoehorn possibility into a construct from such models. The thing to realize here is that any logic will entail metaphysical commitments, and these commitments become vastly more pressing when one moves from the assertoric to the modal.
There are two very broad categories of logic: extensional and intensional. Extension (extensional logic) works from the premise that the basic units of reasoning are self-identical “things” which can be unambiguously individuated – basically, they can be “pointed out.” This quality of unique individuation is the extension of the “thing.” Intension has to do not with unique and individual things, but with the generic comprehension and meanings of concepts. Extensions are sharply defined and easy to work with; intensions are often vague and nearly impossible to formalize. A great many people spent a great many years trying to make intensional logics work, without a lot of success. Meanwhile, in the last quarter of the 19th C., and well into the 20th, extensional logics began enjoying an almost (almost) unbroken string of successes. As a result, intensional approaches were essentially abandoned in favor of exclusively extensional one. But there are two problems with this choice: (1) the move to purely extensional logics came at the cost of an enormous amount of metaphysical baggage that no one wanted to acknowledge or discuss, and (2) just because something is difficult does not mean that it is wrong.
The problem is that thinking about possibilities (qua possibilities) is not at all like thinking about rocks or houses, or pumpkin lattes or the location of the Lesser Antilles. We come closer to the mark by considering the differences between the breakfast we actually ate, and the lunch we are yet to put together. One is an established fact – a “thing,” if you will, which will always be just and only the thing it is/was. The breakfast that we ate has achieved a kind of uneraseable absoluteness as a truly self-identical “thing,” and it will always be just and only the thing it was. (Our impressions and interpretations of that thing might vary with time: the delicious breakfast I happily consumed at 8:00 AM may turn into a bloated belly and upset stomach by the time noon rolls around.) But thinking about things that are not yet, or things that once were but now only present themselves in the mode of what they might have been – the lunch that might yet be made, the breakfast that could have been but was not – requires dealing with things not as they “are” but in the fullness of their relatedness. All the little “bits” – NOT binary digits, but parts and pieces that are what they are, here and now – that might go into lunch have a fairly rigid “isness” about them. But how those tasty bits might be put together! Ah! What possibilities!
Another imperfect analogy might be this: extensional reasoning understands what each piece on the chess board is and how that piece can move; intensional reasoning understands the game of chess and how to win. Digital computers have become so powerful that, using extensional logic, they can brute force their way through the process of mechanically projecting a huge number of allowable combination of positions that can be reached from any given position of pieces on the board. But that is not how people play chess. A good player (to say nothing of a great one) will look at the board and, without calculating the almost innumerable extensional combinations, SEE the game as a single, relational and intensional whole. That is how possibilities function, and that is how a logic of possibility must ultimately operate if it is to be of any real value. To stay mired in extensional approaches while attempting to get a real and effective grasp on the nature of possibilities would be like trying to cross interplanetary distances with a steam engine. Perhaps it could be done, if one had unlimited access to brute forces of unimaginable scales; but the result would not be elegant, and it would (in point of fact) be achieved only at the expense of misrepresenting the real, underlying relations involved.
Programming computers to play chess – even to win games against Grand Masters – has not taught us a thing about how people actually play chess. Well, that’s not strictly true; it has taught us that people playing chess do nothing even remotely similar to what computers do. The same is true with our contemporary (modal) logics of possibility. Some of these modal approaches claim to be intensional, but they are really nothing more than extensional approaches on steroids, like computer solutions to chess. These faux “intensional” approaches are akin to being charged by a Cape Buffalo and declaring, “My, isn’t that subtle!”
One of the projects I’ve set for myself, for this next and the following years, is to work on a genuinely intensional logic of possibility. This involves collaborating with other fellow scholars, recovering ideas from the past that have long been ignored, and being satisfied – even excited! – by the thought that I am doomed to “failure.” This is not a puzzle that will be solved in my lifetime (smarter people than myself have already failed.)
But we “fail forward;” that is how inquiry works. We ask questions, so that we might learn to ask better questions.
Baby steps: so full of possibilities.
i Here I gloss one of my favorite sayings of Ernst Cassirer, in a reference he makes to Descartes’ appeal to God in order to fix the dualism between mind and extension. If you actually care about a specific citation, then go read the entire three volumes of the Philosophy of Symbolic Forms, and then come back to me.