Spatial Logic

Spatial Logic

Does logic have a real essence, and what might it be? And what exactly is logic? These are not easy questions; they invite us to dig deep. First, what is real essence? The paradigms are material substances and animal species: gold and tigers being the favorite examples. I won’t rehash all this; I will simply observe two things: (a) space is centrally implicated in both, and (b) the real essence doesn’t look much like what it is the essence of. As to (a), real essence is conceived as an arrangement of parts—spatial parts spatially arranged. Thus, we have atomic structure conceived as a configuration of particles in space; and genetic structure consisting of molecules, combined with an anatomical structure of organs. As to (b), the appearance of gold and tigers to the naked eye is not like the real essence as specified under (a), i.e., a particulate corpuscular structure invisible to ordinary perception. So, the real essence is both spatially constituted and hidden from sight; it is not evident to the senses or anything like an ethereal Aristotelian form. It is spatial-material and a matter of theory not observation—hence surprising. It may take an elaborate and bold theory to disclose real essence, and it may not even be possible in some cases. It may not be particularly intuitive or uncontroversial; it isn’t like “bachelors are unmarried males”. Do we, in fact, know of any cases in which real essence is notboth of these things? One might cite mental natural kinds like pain: but it is not at all clear that we have any idea of the real essence of mental kinds, or whether they even have real essences. To put it simply, real essences invariably turn out to be hidden shapes, geometric invisibles—micro-anatomy.

We may therefore entertain the hypothesis that the real essence of logic consists of spatially arranged units of some sort—points or regions or configurations. That is, logical space is, or is analogous to, physical space. Physical space exhibits logical structure in its own right: for example, it is not possible, logically, to move from point a to point b without traversing the intermediate points; and it is necessary that if a point is inside a given region, it must be inside any region that contains that region. Then too, we have all the theorems of Euclidian geometry, as well as the thesis that space is necessarily three-dimensional. And so on.[1] These provide a model for logic as a system of necessary truths regarding propositions or states of affairs or facts. Might the formal properties of space constitute the real essence of the subject matter of logic? Here we run into the question of what precisely logic is: is it a human construction or is it completely independent of the human mind? I won’t adjudicate this question except to say that as a human institution—a symbolic system with an interpretation—we would expect something human to enter in. The natural view, then, is that logic in that sense is a combination of the objective and the subjective—the extra-human and the human. To cut to the chase: logic as it exists today is a combination of space and language. Predicate calculus, say, is a combination, a confluence, of mind-independent modal facts concerning space and its material occupants, themselves modifications of space (strings, knots, holes, what have you), on the one hand, and human language, on the other. Logic is the logic of space (space-matter) and grammar: hence the formulas that crowd the pages of logic texts—those squiggles and spacings. According to metaphysical spatialism, mind and language are both upshots of space as the fundamental stuff of the universe.[2] So, logic is an amalgam of two aspects of space—its own logical structure and the structure of the human mind as it has evolved under conditions of spatial life. The real essence of logic is space in its manifold guises. There would still be logical facts if there were no language and mind, though there would not be logic as we currently think of it. It is the same with geometry: there would be geometrical facts without human minds, though the subject of geometry would not exist under those circumstances. In logical reality, it would still be true that no object can both be in a place and not be in it, but the discipline of logic—what is taught in schools—would not exist (there would be no logical statement of the law of noncontradiction). However, language by itself cannot produce logical facts; it needs space to confer logic on it (and on the world). This is the killer: logic requires space in order to have any reality, because all of reality is ultimately composed of space–in particular, matter and mind are so composed. Logic makes claims about reality, and reality is basically spatial; it turns out that its truths reflect that reality. The logic of space is logic; there is no other logic. Conceptually, it is like this claim: H2O is water; there is no other water. The logic of the world is logic—there is no other logic. Reality’s logic is logic’s reality.

Look at it this way: what else could logic consist in? If we insist on space-logic dualism, we run into the following conundrum: how can logic and space (including matter and mind) be so well suited to each other? How come the logical world and the spatial-material-mental world fit so snugly together? If logic had an entirely separate source, we would expect a lack of fit: space over here, logic over there (each with its own creating deity). It would be a giant coincidence that logic describes concrete reality. But logic seems tailor-made for the spatial universe, as if created for it—by it. The spatial world suggests logic to the mind, if it is properly receptive; space isn’t logically neutral. Space is logical; once it is combined with language we get logical systems, construed as human creations. Space was also material and mental even before these realities showed themselves; the seeds were there before the big bang hit. According to metaphysical spatialism, space is the ultimate reality, the foundation of everything—everything. How could anything not be in space and made of spatial stuff? The only alternative is a non-spatial divine being (whatever that may be) as a source of being. But once we drop this fantasy and embrace cosmological naturalism, we reach the conclusion that space is the engine that brings it all about. That is why Platonism always ends up locating things in a quasi-space; for this is how we imagine universals and numbers to exist. Space is God, to put it bluntly. In any event, space-logic dualism faces a problem of interaction: how do the spatial world and the logical world mesh so tightly together? Does space have a pineal gland of some sort? Metaphysical monism is always superior from this point of view, and spatial monism looks like the way to go. Why has it not been thought of before? Because to the human eye space is an absence, an emptiness, a passive receptacle. But this is to overestimate the physical and metaphysical powers of the human visual system; better to go with the modern view and accept that space is an active medium, the root of all being. Space-matter dualism is a myth, a visual illusion. We need to unify matter and space—matter as knots in space anyone?—and when we do we get a brand new metaphysics. Even logic comes out as a spatial phenomenon (consciousness already went spatial). But this space is not the phenomenal space we see with our eyes; it may not even be fully intelligible to us. Nor is it identical with, or reducible to, the material universe as it emerged after the big bang, since that universe did not even exist at that explosive moment; but it is closely related to the matter that congealed in those early seconds, and to the form it took post-big bang. No doubt the space that now exists owes its nature to the primal space that preceded the big bang, so we have not left that reality completely behind. Perhaps when the universe cools down to absolute zero (or heats up tremendously) the original form of space will re-appear and look more capable of its remarkable feats. In any case, the present point is that space (plus language) constitutes the real essence of logic, appearances notwithstanding.

I have tried to keep this discussion at the ontological level without venturing an opinion about our knowledge of logic, though that inevitably creeps in. Now let me make some remarks about the epistemological question: how do we know logic? We know it by perceiving space and knowing our language. We see that certain things are necessarily true of the spatial world, particularly regarding motion, such as that solid objects cannot pass through other solid objects, and then we use our language (itself a product of space) to state certain logical truths. This introduces the subject-predicate grammatical form into logical cognition, which then influences the form of the formulas familiar from logic texts. That is, perception, particularly vision, makes certain possibilities and impossibilities evident to us, and we express the resulting knowledge in the grammatical forms of our language; thus, we concoct the formulas of standard logic. We combine spatial perceptual knowledge with grammatical knowledge to produce symbolic logical systems. For example, we see that every part of space is necessarily adjoined to another part of space, and this gives us the idea of entailment, when conjoined with a grasp of grammatical structure. Likewise, we understand moving from one place to another, and we then form the idea of moving from one proposition or sentence to another, by relying on our grasp of language. Thus, we construct a conception of logical space modeled on physical space. Space plays an indispensable role in the formation of logical concepts. Epistemology recapitulates ontology.[3]

[1] See my “Logic and Space”.

[2] See my “Space, Time, and Logic”.

[3] I keep thinking of Kant: he sensed the absolutely central role of space in the construction of reality, ontologically and epistemologically. Human sensibility is deeply spatial, but so is objective reality; indeed, the former is true because of the latter. It is strange that we ever stray from strict spatiality—why do we even seem to conceive of non-spatial things? Do other animals entertain such fantasies—aren’t they strict Kantians? Told of non-spatial entities, they respond, “Humbug, away with such nonsense!”—or words to that effect. Is it language that creates such strange ideas is us? Do we have a misguided propensity towards abstraction, suggested by words? I have the feeling that Mr. Spock is a strict spatialist; perhaps he instinctively thinks of the mind as a spatial thing, paceDescartes (but then he does have a very sophisticated conception of space). As to logic, he sees the link between logic and space quite clearly; only human emotion could obstruct such perception. In his mind, logic has reality just because space does: both are staring us in the face. He is not sentimental about logic. Here he sides with terrestrial beasts (and Immanuel Kant, the original Space Man).