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BulletinoftheSectionofLogic
Volume11:1/2(1982),pp.84{87
reedition2009[originaledition,pp.84{88]
Boguslaw Wolniewicz
ONLOGICALSPACE
1. Consider a classic propositional language L. By the logical space of
L we mean with Wittgenstein a metaphysical construction SP comprising
all possibilities expressible in that language. These are situations, and
S
0
is
to be their totality. Thus, in a sense to be dened, every possible situation
is comprised in logical space:
^
S SP:
(1)
S2
S
0
The situation presented by a proposition is S(). With Meinong
we call it the objective of . Objectives are equal i the corresponding
propositions are strictly equivalent:
S() = S() == : (2)
If, however, is a contradiction, then it has no objective in S
0
. To provide it
with one, the set S
0
is argument with the impossible situation
V
: S = S
0
[
f
V
g. Thus S : L ! S is a function mapping propositions into situations.
2. To dene our terms we start with a universe SE" of elementary
situations. These correspond to conjunctions of atomic propositions; if
is such a conjunction, then for some x 2 SE" : S() = fxg. The
universe SE" consists of two parts: of the set SE of proper (=contingent)
elementary situations, and of the two improper ones: the empty one o, and
the impossible one . I.e., SE" = SE [fo;g.
One elementary situation may obtain in another: x6y. This is a
partial ordering such that o6x6, for any x 2 SE". Under it, SE" is
a lattice. (Cf. (1)). The join x; y = supfx;yg corresponds to conjunction;
the meet x!y = inffx;yg has no clear-cut counterpart in language.
 On Logical Space
85
fg =
V
. For any w
i
2 SP the set R
i
= fx 2 SE" : x6w
i
g is a maximal
ideal of SE". With Los we call it a realization, and
R
is to be the totality
of them.
3. Situations are sets of elementary situations:
S
P(SE"). There are
several ways to determine
S
exactly, but the following is the simplest. Two
sets of elementary situations are said to be V -equivalent i they intersect
the same realizations:
A
v
B ==
^
R2
R
(A\R = ;, B\R = ;):
(3)
Situations are then the minima of the unions of V -equivalence-classes:
S
= fS SE" :
_
ASE"
S = Min(
[
jAj
v
)g:
(4)
its zero and the objective of tautology;
V
is the unit). For any A;B SE",
we dene now:
A B ,
^
x2A
_
x6y:
(5)
y2B
Clearly, is a preordering, SE SP, and (1) is a theorem. This relation,
however, should not be confused with that of involvement { dened with
\6" reversed { which is the counterpart of entailment, and the ordering of
the Boolean algebra of situations.
Taking the simplest case as an example, set SE = ;. Thus SE" =
fo;g, P(SE") = fSE";Q
o
;
V
;;g, and
R
= fQ
o
g. Moreover SE"
v
Consequently,
S
= fQ
o
;
V
g, i.e. there are in that case just two situations.
This is Frege, as interpreted by Lukasiewicz and Suszko. And, following a
suggestion by K. E. Pledger, the equation SP = Q
o
might be then taken
to mean that the logical space of L is void of empirical content.
4. Stipulating further conditions for SE, or for SP, gives rise to a
typology of logical spaces.
The minimal elements of SE, if any, are logical atoms (or states of
aairs). The maximal ones are logical points (or possible worlds), and
logical space is the totality of them. I.e.: SA = fog = Q
o
, and SA =
Under suitably dened operations (cf. [2]),
S
is a Boolean algebra. (Q
o
is
Q
o
, and
V
v
;. Hence Min(
S
jSE"j) = Q
o
, and Min(
S
j;j) =
V
.
86
Boguslaw Wolniewicz
If every elementary situation is a join of atoms, i.e., if we have:
^
_
x = supA;
(6)
x2SE"
ASA
then SP is said to be atomistic; otherwise { non-atomistic. An atomistic
SP may be either dimensionally determinate, or indeterminate. It is de-
terminate i either SE = ;, or the following holds: there is a partition of
logical atoms into logical dimensions such that in each possible world there
obtains exactly one atom of every dimension. I.e., if
_
^
^
1
_
x6w;
(7)
D
2Part(SA)
w2SP
D2
D
x2D
where \
1
W
" is the singular quantier.
x
A dimensionally determinate SP is either zero-dimensional (if SE = ;),
or one-dimensional (if SE = SA = SP), or multi-dimensional (if SE 6= ;,
and SA 6= SP). The zero-dimensional case is represented historically by the
ontologies of Logical Monism. The situation Q
o
corresponds then to \the
One" of Parmenides, to \the Substance" of Spinoza, to \the Absolute" of
Hegel, and to \the True" of Frege. One-dimensional cases are represented
by Leibniz, by Laplace, and by modern possible-worlds semantics. The
multi-dimensional case leads to further subdivisions.
The logical dimensions of SP are said to be orthogonal i the atoms of
dierent dimensions are always composible; i.e., i we have:
^
[
^
x;x
0
2A
(x 6= x
0
) D(x) 6= D(x
0
)) ) supA 6= ]: (8)
ASA
This is Logical Atomism, as propounded by Hume and Russell.
Now observe that each logical dimension consists of at least two atoms:
cardD
i
>2, for any D
i
2 D. Thus a special case of an orthogonal, multi-
dimensional logical space is one with binary dimensions only: cardD
i
= 2,
for any D
i
2 D. Marking then in each dimension one atom as \positve",
one as \negative", we have: SA = SA
+
[SA
. The dimensions being or-
thogonal, we have here also a one-to-one correspondence between SP and
P(SA
+
). Hence SA
may be dropped, and so we arrive at the variant
of Logical Atomism represented by Wittgenstein's Tractatus (1922). The
On Logical Space
87
elements of S4
+
are his \Sachverhalte", and those of SP are his \Wahre-
heitsmoglichkeiten der Elementarsatze". Observe also that the former are
the \basic particular situations" of Cresswell's [3], and the latter are the
\L-states" of Carnap's [4].
Finally, the number of logical dimensions may be nite or innite. The
nite case has been investigated in [1] and [2], for logical spaces with or-
thogonal dimensions of an arbitrary cardinality.
References
[1] B. Wolniewicz, On the Lattice of Elementary Situations, thisBul-
letin, vol. 9 no. 3 (1980), pp. 115{121.
[2] B. Wolniewicz, The Boolean Algebra of Objectives, thisBulletin,
vol. 10 no. 1 (1981), pp. 17{23.
[3] M. J. Cresswell,LogicsandLanguage, London 1973.
[4] R. Carnap,IntroductiontoSemantics, Cambridge Mass. 1946.
Institute of Philosophy
Warsaw Univeristy
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