Wittgenstein, Modeling and the Notion of Logical Space

Many attempts at understanding Wittgenstein climax in the wistful prayer that he had himself observed the closing proposition of his Tractatus logico-philosophicus: “Whereof one cannot speak, thereof one must be silent.” And indeed, he never published any other item in his lifetime. Deliberately speaking about logical space without the aid of formulaic language tempts one to reminisce about that mantra. In the terse, minimalistic language of the Tractatus, the facts in logical space are the world. In his ontology, the world is determined by the facts, and by these being all the facts. For the totality of facts determines both what is the case, and also all that is not the case. Mindful that “the world is all that is the case,” “the totality of facts, not of things,” Wittgenstein had entered upon the notion of logical space by reading Bertrand Russell, Georg Cantor and Gottlob Frege. It does not play a major role in his Tractatus but it does conceptually: logical space is, of course, a notion from set theory, an analogy to physical space modeled on Boltzmann’s and Hertz’ idea of a multi-dimensional space of physical possibility, comprising any system of relations that have the same or similar logical properties. The logical space of the Tractatus is nothing other than the set of all potential worlds, of which the real world is merely an element, being the only set of facts in logical space the elements of which are, without exception, facts. Hence logical space is infinite. Logical form, then, represents the possible within logical space. Wittgenstein’s method is that of logical atomism where all possible statements about a complex object consisting of parts can be reduced to the sum of statements about its parts.

Since all facts in logical space are logically independent of each other, objects forming part of facts cannot be complex, composed of simpler objects, but must be simple objects or atomic facts. Every statement about complexes is capable of being broken down analytically into a statement about their constituent parts, and into those propositions that completely describe the complexes. Inevitably, this happens as we make pictures of facts. The picture, then, presents the facts in logical space, the existence and non-existence of atomic facts.

Topologically speaking, logical space is a place for propositions that are subject to definitions. It limits other propositions in their actions on a given proposition and delimits any room for action a given proposition may take. Logical space is filled by topological places constituted by truth functions: our principal interest in the notion of logical space is due to its potential for clarifying connections between thought, meaningfulness of propositions, and truth. Each proposition partitions logical space into those possibilities of truth that agree and those that disagree with it. A place in logical space is filled if the state of affairs exists. A range is filled by possible states of affairs compatible with it being true. By definition, truth or falsity of each proposition is independent of that of others. Logical space may also define a field for possible change, namely for changing configurations of objects in facts.

Only propositions contain sense because only they, considered as a whole and as expression of a thought, can represent the world. Similarly to the object that is the substance of the world, the word as such is the simple constituent part of the proposition which can have only meaning. Significantly, Wittgenstein uses “substance” and “logical space” synonymously in reference to the set of all logically possible worlds. The world is only one of the possible worlds in logical space, but one wherein all that is the case are facts. Therefore, the substance of the world is the set of all logically possible worlds that have their respective objects in common - the object is the fixed, the existent; the configuration is the changing, the variable. Only if objects exist, then an extensional space of possible worlds exists. “Sense” is the truth value of the description of the atomic fact. “Meaning” is its correspondent in the logical space of the object. In this context, a name “means” an object. The object is its meaning (‘A‘ is the same sign as ‘A‘). Only propositions have sense. The propositional sign and the logical co-ordinates: that is the logical place. The geometrical and the logical place agree in that each is the possibility of an existence.

This logical space is a mathematical construct of conceptual functions among a set of properties and relations. If each element in it receives as many coordinates as there are dimensions relating to it, each element and its relationships will be represented by a matrix set of multiples of real numbers. Time can also be conceived as a logical space, forming a real line of relations between events and of conceptual causal and other functions between them. Indeed, all space is logical space, as all geography is logical geography. The categories of logical space and of logical topology are the most formal and abstract formulations of space conceived by the intellect as an ideal form of order. Structure is logic. Logic provides structure for experience. Structure integrates reason, reflection and experience.

Without the existence of logical space that comprises all logically possible worlds, and without the existence of corresponding propositions that have content, regardless if true or false, it would be impossible to form a picture of the world. The substance of the world is comprised of objects. They cannot be compound. If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true, then it would consequently be impossible to form a picture of the world, be it true or false.

Logical pictures can depict the world because the picture has the logical form of representation in common with what it pictures. Pictures depict reality by representing a possibility of the existence and non-existence of atomic facts. In any picture, the world and facts therein are the results of logical construction. They exist only in logical space characterized by propositions. Because propositions constitute thoughts, thoughts become the meanings of propositions. Eventually, meanings reach a logical structure, the general form of propositions. Logical atoms remain the same in all the different possible worlds. They represent “the fixed, the existent” within logical space. Possible worlds differ as to the configuration of objects forming different states of affairs and atomic facts. Since logical space is the set comprising all logical possibilities, it is the aggregate of all possible-and-existing states of affairs, but also of all possible-and-non-existing states of affairs.

The verbal aspect of theories presupposes descriptive logic relative to certain types of models. Descriptive features of various models mutually exclude and condition one another. Any potential description of a model must follow rules of description. Describing the model has normative implications (e.g., “anything positioned on one side of this coordinate cannot be identified as being on its other side”). Tautologies can express the structure of models. They leave the entirety of logical space to reality because tautologies are either true or false regardless of the facts. Now, any model may be said to form a logical space. Modeling the world, as Wittgenstein addresses “pictures,” is one form of map-making. Like a fashion model, one can follow a model, utilize it, trust it, but no such thing exists as a confirmation or assessment of a model. A model does not represent anything that can later be confirmed. It is merely a means of orientation in the world, one possible way to trust it. Thus, mathematical models become "representative” - they describe something that can possibly happen or be the case. Elaboration of models is purely a matter of deriving sense, sometimes by means of exemplars.

Facts do not compose the world randomly: there is a structure of logical relations amongst them because each thought contains the possibility of the particular state of affairs which it thinks. What is thinkable is also always possible. We cannot think anything illogical, for otherwise we should have to think illogically. We could not truly say of an "illogical" world how it would look because the “real” world imaginable to us consists of facts.  To present in language anything that "contradicts logic" is as impossible as in geometry to present by its coordinates a figure which contradicts the laws of space; or to give the coordinates of a point which does not exist.

Decades after Tractatus, Wittgenstein’s notion of a single logical space gradually gave way to a vision of a plurality of grammars and syntaxes. Starting from the conclusion that we cannot describe language, value or purpose fully within its means, and arriving, after his famous, almost ideologically impassioned critique of Kurt Gödel’s ground-breaking incompleteness theorems, at an increasingly less hesitant allowance for metaphysical elements, Wittgenstein later vastly complicated his once clear tenets in the Tractatus of how the conceptual structure in question should be characterized.