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.
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