Bulletin of the Section of Logic
Volume 11:1/2 (1982), pp. 84–87

reedition 2009 [original edition, pp. 84–88]

Bogus̷law Wolniewicz

ON LOGICAL SPACE

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

′

is

to be their totality. Thus, in a sense to be defined, every possible situation
is comprised in logical space:

⋀

𝑆∈𝑆

′

𝑆 ⊂⋅ 𝑆𝑃.

(1)

The situation presented by a proposition 𝛼 is 𝑆(𝛼). With Meinong

we call it the objective of 𝛼. Objectives are equal iff the corresponding
propositions are strictly equivalent:

𝑆(𝛼) = 𝑆(𝛽) == 𝛼 ≡ 𝛽.

(2)

If, however, 𝛼 is a contradiction, then it has no objective in 𝑆

′

. To provide it

with one, the set 𝑆

′

is argument with the impossible situation

⋀ : 𝑆 = 𝑆

′

∪

{

⋀}. Thus 𝑆 : 𝐿 → 𝑆 is a function mapping propositions into situations.

2.

To define our terms we start with a universe 𝑆𝐸” of elementary

situations. These correspond to conjunctions of atomic propositions; if
𝛼 is such a conjunction, then for some 𝑥 ∈ 𝑆𝐸” : 𝑆(𝛼) = {𝑥}.

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 𝜆. I.e., 𝑆𝐸” = 𝑆𝐸 ∪ {𝑜, 𝜆}.

One elementary situation may obtain in another: 𝑥 ⩽ 𝑦. This is a

partial ordering such that 𝑜 ⩽ 𝑥 ⩽ 𝜆, for any 𝑥 ∈ 𝑆𝐸”. Under it, 𝑆𝐸” is
a lattice. (Cf. (1)). The join 𝑥; 𝑦 = 𝑠𝑢𝑝{𝑥, 𝑦} corresponds to conjunction;
the meet 𝑥!𝑦 = 𝑖𝑛𝑓 {𝑥, 𝑦} has no clear-cut counterpart in language.

On Logical Space

85

The minimal elements of 𝑆𝐸, if any, are logical atoms (or states of

affairs).

The maximal ones are logical points (or possible worlds), and

logical space is the totality of them. I.e.: 𝑆𝐴 = {𝑜} = 𝑄

𝑜

, and 𝑆𝐴 =

{𝜆} =

⋀. For any 𝑤

𝑖

∈ 𝑆𝑃 the set 𝑅

𝑖

= {𝑥 ∈ 𝑆𝐸” : 𝑥 ⩽ 𝑤

𝑖

} is a maximal

ideal of 𝑆𝐸”. With ̷Lo´

s 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 iff they intersect
the same realizations:

𝐴 ∼

𝑣

𝐵 ==

⋀

𝑅∈𝑅

(𝐴 ∩ 𝑅 = ∅ ⇔ 𝐵 ∩ 𝑅 = ∅).

(3)

Situations are then the minima of the unions of 𝑉 -equivalence-classes:

𝑆 = {𝑆 ⊂ 𝑆𝐸” :

⋁

𝐴⊂𝑆𝐸”

𝑆 = 𝑀 𝑖𝑛(

∪

∣𝐴∣

∼

𝑣

)}.

(4)

Under suitably defined operations (cf. [2]), 𝑆 is a Boolean algebra. (𝑄

𝑜

is

its zero and the objective of tautology;

⋀ is the unit). For any 𝐴, 𝐵 ⊂ 𝑆𝐸”,

we define now:

𝐴 ⊂⋅𝐵 ⇔

⋀

𝑥∈𝐴

⋁

𝑦∈𝐵

𝑥 ⩽ 𝑦.

(5)

Clearly, ⊂⋅ is a preordering, 𝑆𝐸 ⊂⋅ 𝑆𝑃 , and (1) is a theorem. This relation,
however, should not be confused with that of involvement – defined with
“⩽” 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” =

{𝑜, 𝜆}, 𝑃 (𝑆𝐸”) = {𝑆𝐸”, 𝑄

𝑜

,

⋀, ∅}, and 𝑅 = {𝑄

𝑜

}. Moreover 𝑆𝐸” ∼

𝑣

𝑄

𝑜

, and

⋀

∼

𝑣

∅. Hence 𝑀 𝑖𝑛(

∪ ∣𝑆𝐸”∣) = 𝑄

𝑜

, and 𝑀 𝑖𝑛(

∪ ∣∅∣) = ⋀.

Consequently, 𝑆 = {𝑄

𝑜

,

⋀}, 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.

86

Bogus̷law Wolniewicz

If every elementary situation is a join of atoms, i.e., if we have:

⋀

𝑥∈𝑆𝐸”

⋁

𝐴⊂𝑆𝐴

𝑥 = 𝑠𝑢𝑝𝐴,

(6)

then 𝑆𝑃 is said to be atomistic; otherwise – non-atomistic. An atomistic
𝑆𝑃 may be either dimensionally determinate, or indeterminate. It is de-
terminate iff 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

⋁

𝐷∈𝑃 𝑎𝑟𝑡(𝑆𝐴)

⋀

𝑤∈𝑆𝑃

⋀

𝐷∈𝐷

1

⋁

𝑥∈𝐷

𝑥 ⩽ 𝑤,

(7)

where “

1

⋁

𝑥

” is the singular quantifier.

A dimensionally determinate 𝑆𝑃 is either zero-dimensional (if 𝑆𝐸 = ∅),

or one-dimensional (if 𝑆𝐸 = 𝑆𝐴 = 𝑆𝑃 ), or multi-dimensional (if 𝑆𝐸 ∕= ∅,
and 𝑆𝐴 ∕= 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 iff the atoms of

different dimensions are always composible; i.e., iff we have:

⋀

𝐴⊂𝑆𝐴

[

⋀

𝑥,𝑥

′

∈𝐴

(𝑥 ∕= 𝑥

′

⇒ 𝐷(𝑥) ∕= 𝐷(𝑥

′

)) ⇒ 𝑠𝑢𝑝𝐴 ∕= 𝜆].

(8)

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

∈ 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

∈ 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-

heitsm¨

oglichkeiten der Elementars¨

atze”. 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 finite or infinite. The

finite 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, this Bul-

letin, vol. 9 no. 3 (1980), pp. 115–121.

[2] B. Wolniewicz, The Boolean Algebra of Objectives, this Bulletin,

vol. 10 no. 1 (1981), pp. 17–23.

[3] M. J. Cresswell, Logics and Language, London 1973.
[4] R. Carnap, Introduction to Semantics, Cambridge Mass. 1946.

Institute of Philosophy
Warsaw Univeristy