Space-time in logic

Space-Time Reasoning in Logic

Pavel B. Ivanov


Written: 11 Oct 1998

Every phenomenon of the physical world occurs in space and time, and these are the two fundamental forms of any motion. Though subjectivity does not obey any spatial or temporal limitations in any straightforward way, it has to be implemented in interacting material bodies, and every such implementation is bound to reflect the space-time properties of the material used. In particular, logical reasoning in humans must manifests both spatial and temporal features in every single act, and there are cases of domination of one mode of reasoning over another.

Since any object can be a part of different higher-level systems, it can manifest quite different features when viewed in respect to a specific external process. The opposition of space and time is only one possible aspect, and its representation in human reasoning can in no way exhaust its possible manifestations. Though some types of complementary descriptions (like geometric and dynamic methods in analytical mechanics, functional and operator forms of quantum mechanics etc.) could be related to the opposition of space and time, there are other, apparently quite different kinds of complementarity, such as, for example, the juxtaposition of configuration space and phase space pictures of dynamics. In this case, one will find a kind of space-time duality: motion in configuration space transforms into the geometry of phase space, so that a dynamic (time dependent) picture gets complemented with a static (space-like) description. A point in phase space may correspond to a straight trajectory in configuration space; inversely, two points in phase space (fixing a straight line) determine a point in the conjugate configuration space (the intersection of two trajectories).

One could also recall the two paradigms of statistical physics, where there are both time averages and ensemble averages, and one needs an ergodic system to make them equal.

In logic, there are two complementary aspects of any particular act of reasoning, one resembling space, and the other more like time. Considering the universe of established facts (for instance, a universe of true sentences of propositional logic, or a collection of axioms and theorems of a mathematical theory), one could extend it in two opposite ways, either via adding a new instance or via deriving a consequence of the already existing facts as a hypothesis to be converted into a new fact. The former (extensive) way is associated with spatial expansion, while the latter (serial) way has a serial organization similar to that of physical time. The two directions of logical development are relatively independent, since adding new true sentences or axioms does not require any inference, and the application of the pre-defined inference rules does not require any new axioms to proceed. This resembles the orthogonality of space and time coordinates in classical physics. However, many physical processes obey a requirement of contingence, which could be assimilated to the consistency of reasoning in logic: in a system with fixed dynamics, the possible trajectories cover the whole configuration space, and every point in the system's configuration space can only be achieved from the points lying on the same trajectory. This holds for either mechanical (both classical and quantum) or non-mechanical (thermodynamic and other) systems, with the appropriate distinction of space and time variables. In the same way, formal logical reasoning may only follow certain trajectories determined by the system of inference rules adopted. Two sentences are logically independent (within a given inference system) if they do not lie on the same logical trajectory, that is, they cannot be inferred from each other. This means that the addition of a new axiom or definition requires a consistency check against the already accepted ones: new space points have to be either dynamically achieved or unachievable under the current dynamics.

The problem with this analogy is that logical "trajectories" do not entirely resemble the trajectories of classical mechanics, since every conclusion is drawn from at least two statements rather than from a single one. However, one can demonstrate that the difference is only apparent, and the analogy between logical reasoning and physical processes is much closer and more deep rooted.

In the basis of all the traditional logic of most scientific theories (including mathematics), there is the fundamental figure of syllogism known under the name of modus ponendo ponens, or simply modus ponens. If one shows that modus ponens is constructed in a space-time manner of classical physics, it would mean that any scientific discourse is like physical motion, sharing its space-time attribution. Modus ponens binds three sentences (propositions) of special structure:

Minor premise: S is M.
Major premise: All M are P.
Conclusion: S is P.

It could be shown that all the other figures of syllogism are the unfolded forms of modus ponens, with additional statements implied, the most frequent being the assumption of completeness, or the existence of an exhausting class.

One can see that the major premise is of a different structure than the minor premise and the conclusion. This suggests the idea of an analogy between a universal statement (containing the universal quantifier "all") and an operator, or a transformation rule, which would not belong to the space of sentences it acts upon. Therefore, modus ponens can be considered as connecting one sentence (minor premise) with another (conclusion) applying an operator (major premise). This exactly corresponds to the situation in classical mechanics, where one point of a manifold gets connected to the adjacent point through an element of the tangent space in the fist point, which is known to be a differential operator corresponding to the vector of velocity. To put it plain, one obtains one spatial point from another using a difference operator:

x' = x + Δx,

which becomes differential in the infinitesimal case:

x' = x + dx.

Since the elements of the phase space (momentum vectors) are proportional to velocity (or, in a different formalism, are linear combinations of the elements of the tangent space of the manifold), the correspondence between the dynamics of a mechanical system and logical inference is established, with singular sentences treated as the points of a configuration space and general sentences forming the corresponding phase space. Like in mechanics, there may be different displacements of the same point represented by different differential operators. There may be two possibilities with direct analogs in logic: longitudinal and transverse displacements. For instance, in a two-dimensional space, one can distinguish the case of

(x1, y) = (x, y) + (Δx1, 0) and (x2, y) = (x, y) + (Δx2, 0)

from the case of

(x', y) = (x, y) + (Δx, 0) and (x, y') = (x, y) + (0, Δy).

The first case corresponds to a chain of logical inference:

S is M → (All M are M') → S is M' → (All M' are P) → S is P.

One can construct inferences of different mediation depth in this way, always following the serial paradigm. Another dimension would be introduced in a parallel structure like:

S is M1 → (All M1 are P1) → S is P1
S is M2 → (All M2 are P2) → S is P2

These are the two paradigms (spatial and temporal, extensive and serial), which are combined in different proportions in any logical discourse.

Since the application of any inference rule is local, the manifold of logic reasoning (the inference space) may have rather complex topology and geometry, like the manifolds of analytical mechanics. Also, one could consider various non-traditional ways of reasoning, extending the analogy to quantum mechanics and relativism.

There may be problems with studying the global structure of inference space in formal theories, if some local properties become extrapolated to the whole inference space. For instance, the introduction of negation is based on the assumption of a simple nearly Euclidean global geometry, without torsion, cusps, lacunas and other singularities. In such an "almost plain" space individual trajectories do not intersect, they do not form loops, and the order of trajectories in any dimension remains the same along the trajectory. This may be not so for many sciences, and especially for non-scientific modes of reasoning.

The problem of completeness is close to that of negation. Thus, the presence of a pole singularity in the inference space of a theory will result in that no finite trajectory can ever reach the point of singularity, though this point may be well included in the inference space embedded in some other space with a simpler topology. There may also be singularities in phase space, which may result in that a non-singular point of the inference space could not be reached in a finite time (via a finite inference) from certain regions of the inference space, though there may be definitely finite trajectories leading to that point. The only way of restoring "completeness" in such cases would be to embed the inference space to another space, thus making transition to a more general theory. There is no way to establish the completeness of an inference space from within it, without recourse to global information obtained from a wider view.

One more related problem is that of logical loops. If the geometry of the inference space is nearly plain, any detection of a logical loop would indicate the presence of a logical error in the discourse. However, in a more complex theory, there may appear circular trajectories, associated either with the presence of singular points or with an essentially non-Euclidean global topology. In the former case, following a circular trajectory around the singularity point would lead to opening new "leaves" of the same inference space (like the leaves of logarithmic function in the complex plane). The sequence of transfinite ordinals may be an example of that kind of behavior. The examples of the latter possibility can be provided by every defining dictionary, where one word is defined trough another, which is, in its turn defined through the first one. This kind of logic is also characteristic of the categorial schemes of philosophy, and in particular, various methodological works.

Finally, one could touch the problem of infinity. For the extensive (spatial) paradigm, infinity is a datum, and it can only be postulated as a peculiarity of the inference space geometry. For the serial (temporal) paradigm, infinity is a process that can never end. The two kinds of infinity are called actual and potential, respectively. There may be "ergodic" theories, in which there is a correspondence between the two types of infinity, and a method of mutual conversion. For instance, an almost plain dynamics is ergodic; also, there may be ergodic motion of the chaotic type, and various quasi-ergodic spaces with singularities. In non-plain manifolds, one has also to make distinction between potential infinity and openness of the manifold. If a point (or the border of the manifold) cannot be reached in a finite process, this point (or border) does not need to be infinitely remote, it may just be singular.

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