Observation Topologies for Finite Propositional Semantics:
Distinguishability, Distance, and Inference
Brian Theory
September 2026
Abstract
Fix a propositional language over finitely many variables and regard its truth assignments
as points. A selected finite family of formulas acts as a family of positive-certification tests,
and the topology generated by their truth regions records the distinctions supported by those
tests. In this finite setting, observation signatures give an explicit coordinate description of
the resulting Alexandrov topology: topological indistinguishability is equality of signatures, the
specialization preorder is inclusion of positive observations, and a weighted directed set-difference
quasi-pseudometric generates the same topology. Symmetrizing that directed distance gives
weighted Hamming distance. We distinguish positive observation from a complete two-sided
record, and observation refinement from premise restriction. Classical inference itself remains
ordinary inclusion of the common premise region in the conclusion region. The paper is a
self-contained expository synthesis of standard finite Boolean, order-topological, observational,
and quasi-metric constructions; its aim is to make their interfaces and presentation-dependence
explicit in one propositional model rather than to claim a new class of topological spaces.
Keywords: finite topology; Alexandrov space; propositional logic; observation; specialization
preorder; quasi-pseudometric; Hamming distance.
Contents
1 Introduction 2
2 Valuations, truth regions, and partitions 3
2.1 Formula tuples, truth vectors, and truth regions . . . . . . . . . . . . . . . . . . . . . 4
3 Observation-generated spaces 5
3.1 Partitions induced by observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Distinguishability and separation 8
5 Asymmetric separation and the specialization preorder 9
6 Distances on observation signatures 10
7 Interior, closure, and observational boundary 11
7.1 Open truth regions and Heyting implication . . . . . . . . . . . . . . . . . . . . . . . 13
18 Information refinement and logical update 14
8.1 Refining the observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
8.2 Updating the possible valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
9 Inference by truth-region restriction 15
9.1 A remark on minimal truth spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
9.2 Modus Ponens and Modus Tollens . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9.3 Hypothetical Syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
9.4 Pullback formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
10 Continuous maps and preservation of information 16
11 Scope, limitations, and deferred directions 17
11.1 A presentation-dependent nerve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
11.2 Focused directions beyond the synthesis . . . . . . . . . . . . . . . . . . . . . . . . . 17
12 Conclusion 17
1 Introduction
Propositional formulas can be represented by the truth assignments on which they hold. If every
such subset of a finite valuation space is declared open, the resulting topology is discrete and contains
no structure beyond classical truth-set semantics. A nontrivial topological treatment begins instead
with a restricted family of formulas that can be used as observations. Two truth assignments may
then be logically different but observationally indistinguishable.
This paper develops five linked ideas:
(i) complete truth assignments give the atomic partition for finite Boolean semantics;
(ii) selected observable formulas determine distinguishability and indistinguishability;
(iii) the specialization preorder records asymmetric logical separation;
(iv) XOR of observation signatures gives symmetric disagreement, while a source mask gives
directed loss; and
(v) adding observations refines the topology, while adding premises restricts the current space of
possibilities.
The intended reader knows propositional truth tables and basic set notation but may be meeting
finite topology, specialization order, or topological semantics for the first time. The expository aim
is to use one finite observation table to show which constructions share data and which do not: in
particular, truth-region intersection versus intersection of topologies, complete semantic profiles
versus partial certificates, and an indexed observation presentation versus the topology it generates.
The numbered propositions and theorems below are included to make the synthesis self-contained;
they are not priority claims for the underlying finite-topology, order, Hamming, or quasi-metric
facts. The contribution is organizational: the same finite signature data are used consistently to
compare positive evidence, two-sided observation, refinement, update, distance, and inference.
The qualitative construction can be summarized as
Φ 7− →oΦ 7− →τΦ ← →⪯Φ .
2Given positive weights a = (a1, . . . , am), the same presentation also gives
(Φ, a) 7− →d+
Φ,a 7− →(⪯Φ, τΦ).
Here Φ is an indexed family of observable formulas, oΦ is the observation signature, τΦ is the
generated topology, ⪯Φ is its specialization preorder, and d+
Φ is a directed logical distance. The last
arrow is not reversible: the topology determines its specialization preorder on this finite carrier, but
it does not determine the chosen weights or observation presentation.
The Boolean representation underlying the truth-region map is classical and is closely related
to the representation of Boolean algebras by fields of sets [ 15]. Topological semantics for logical
operations has a long history, notably in the work of McKinsey and Tarski [11]. Finite topologies and
their order-theoretic structure are classical as well [ 16, pp. 325–327]; see also [ 3]. Sierpi´ nski-valued
predicates provide a standard description of opens and positive observation [ 17, Ch. 1]; see also [ 18,
Example 1.12(3)]. Observation frames provide a related abstract separation between observable
predicates and all predicates [ 4]. Abramsky and Vickers place observation and testing in a broader
algebraic semantics of observational logic [1]. More recently, Achilleos and Kyriakou study topologies
generated by finite behavioural observations and characterize open sets as properties verifiable
by those observations [2]. Their process-semantic setting is different from the finite propositional
carrier used here, but it is a close conceptual precedent for the observation-to-topology viewpoint.
The purpose here is to assemble these ideas around partitioned propositional states, asymmetric
distinguishability, signature distance, refinement, and inference. No new duality, completeness
theorem, or general existence theorem for observation topologies is claimed.
Two further precedents use essentially the same object–attribute data. In Pawlak’s information
systems, equality of attribute descriptions induces an indiscernibility partition, whose lower and
upper approximations are interior and closure in the associated partition topology [ 12, Sec. 2.2,
pp. 343–344]. That construction agrees here with two-sided observation; positive observation may
instead retain strict specialization comparisons. In formal concept analysis, the incidence v I i
defined by v |= φi is a formal context, and a finite conjunction of attributes has the corresponding
extent [8, Sec. 1.1]. Topological constructions have also been built directly from formal contexts;
Pei et al. construct a topology on the attribute set rather than on the valuation/object carrier used
here [14]. Our topology also admits arbitrary unions of such extents, which need not themselves be
extents, so its open-set lattice is not in general the concept lattice. For Φ = ( X, Y ), for instance,
the three-element set of valuations {(1, 0), (1, 1), (0, 1)} is open but is not an extent of the context
with just the attributes X and Y .
Table 1 records the distinctions that organize the paper. In particular, changing what can be
observed and changing which valuations remain possible are different operations.
2 Valuations, truth regions, and partitions
Let P = {p1, . . . , pn} be a finite set of propositional variables and let
ΩP = {0, 1}P
be the set of valuations. For v ∈ ΩP , the expression Vv(L) is the value of a formula L at v, under
the usual classical truth tables. We identify T with 1 and F with 0.
3Operation Observation data Carrier Effect
Positive observation Φ Ω P Generates τΦ from affirmative
truth regions.
Two-sided
observation
Φ± ΩP Makes complete signature fibers
clopen; yields a partition
topology.
Observation
refinement
enlarge Φ Ω P Refines distinguishability and can
only make the topology finer.
Premise update keep Φ SΓ ⊆ ΩP Restricts possible valuations and
uses the induced subspace
topology.
Table 1: Four operations that share truth-table data but should not be conflated.
2.1 Formula tuples, truth vectors, and truth regions
Definition 2.1 (Formula tuple and truth vector). A formula tuple of arity m is an ordered tuple of
formulas
L = (L1, . . . ,Lm).
At a valuation v ∈ ΩP , its truth vector is the Boolean vector
Vv(L) :=

Vv(L1), . . . ,Vv(Lm)


∈ {0, 1}m.
A formula tuple is therefore syntactic, whereas a truth vector is one of its possible semantic
values. Neither should be confused with a valuation v ∈ ΩP , which assigns truth values to every
variable in the fixed language and serves as a point of the topological carrier.
Definition 2.2 (Truth set or truth region). For a formula L, define
ς(L) := {v ∈ ΩP : Vv(L) = T}.
The notation VT
v (L) abbreviates v ∈ ς(L), and VF
v (L) abbreviates v /∈ ς(L).
The terms truth set and truth region refer to this same subset of Ω P . We use “region” when
emphasizing its role inside a topological space. The superscripts T and F above denote truth
assertions; they do not define additional numerical evaluation maps.
Using a fixed carrier ensures that ς depends only on the logical equivalence class of L. It obeys
ς(L1 ∧ L2) = ς(L1) ∩ ς(L2),
ς(L1 ∨ L2) = ς(L1) ∪ ς(L2),
ς(¬L) = ΩP \ ς(L).
Write L1 ⇕ L2 for Boolean exclusive OR. For two formulas, their union separates into an overlap
and an exclusive remainder:
ς(L1 ∨ L2) = ς(L1 ∧ L2) ˙∪ ς(L1 ⇕ L2).
Equivalently,
L1 ∨ L2 ≡ (L1 ∧ L2) ⇕ (L1 ⇕ L2),
4because the overlap and exclusive remainder are disjoint. The complete two-formula truth-pattern
partition is
L1 ∧ L2, L1 ∧ ¬L2,
¬L1 ∧ L2, ¬L1 ∧ ¬L2,
after empty cells are removed. Thus XOR identifies the two exclusive cells; it is not a conversion of
the conjunction into a disjunction.
For every v ∈ ΩP , define its complete minterm
mv :=
n^
i=1
ℓi, ℓ i =
(
pi, v (pi) = 1,
¬pi, v (pi) = 0.
The family {ς(mv) : v ∈ ΩP } is the singleton partition of Ω P . Every formula has the canonical
complete disjunctive form
L ≡
_
v∈ς(L)
mv,
where the empty disjunction is F and the empty conjunction is T. Thus the complete minterms
give the atoms of the finite Boolean algebra P(ΩP ). They form a topological basis only when the
topology is discrete.
For two variables use the cell labels
A = X ∧ ¬Y, B = X ∧ Y,
C = ¬X ∧ Y, D = ¬X ∧ ¬Y, ΩX,Y = {A, B, C, D}.
Each letter denotes the unique valuation satisfying the displayed minterm; the same letter may also
label that singleton truth cell when the type is clear. For example,
ς(X) = {A, B}, ς (Y ) = {B, C}, ς (X ⇒ Y ) = {B, C, D}.
3 Observation-generated spaces
The topology should record a limited observational vocabulary rather than all definable subsets.
Definition 3.1 (Observation family and signature). Let I := {1, . . . , m} and let the observations
be the indexed finite family
Φ = (φi)i∈I .
Put
bi(v) := 1[v ∈ ς(φi)]
and define the observation signature
oΦ(v) := (bi(v))i∈I ∈ {0, 1}I .
When subset notation is used, we identify this vector with its support
oΦ(v) = {i ∈ I : v |= φi} ⊆ I.
Different indices may carry logically equivalent formulas, or even literal repetitions, and may receive
distinct weights.
5The truth vector of the indexed formula tuple Φ at v is exactly the observation signature:
Vv(Φ) = oΦ(v).
The signature notation emphasizes that these coordinates are the observations available to the
topology.
The evidence convention is asymmetric: a subbasic event records positive certification that φi
is true. Failure to certify φi is not itself a positive certificate for ¬φi. If both outcomes of every
Boolean test are operationally available, the appropriate family is Φ ± from Proposition 3.5 below,
and the resulting topology is the two-sided partition topology. Here “positive” is relative to the
selected tests; a selected formula may itself contain a negation.
The signature is a complete semantic profile: bi(v) = 0 asserts that φi is false at v. It does
not mean that a certificate has merely failed to arrive. The topology describes the affirmative
distinctions allowed by the selected interface; it does not by itself model time, partial transcripts, or
delivery of certificates. Likewise, the distances in Section 6 are defined on complete profiles and are
not automatically measurable from an unfinished positive-only transcript.
For a family G of subsets of Ω P , let Top(G) denote the smallest topology on Ω P containing G.
Definition 3.2 (Observation-generated topology). The topology generated by Φ is
τΦ := Top{ς(φi) : i ∈ I}
on the fixed carrier Ω P . The displayed truth regions are a subbasis; their finite intersections form a
basis. We call (Ω P , τΦ) the observation space generated by Φ.
Its points are valuations and its designated subbasic opens are truth sets. The carrier is always
displayed because a family of subsets is not a topology until its underlying set is fixed. The indexed
family Φ and any assigned weights are presentation data; they are not part of the topology τΦ and
need not be recoverable from it.
Because complete minterms define every subset of the finite carrier, every topology on Ω P can
be presented by choosing formulas for its open sets. Thus the propositional language supplies a
finite presentation and interpretation; observation-generated spaces are not a new restricted class of
finite topological spaces.
The empty intersection of subbasic opens is Ω P . Hence a convenient basis is
BΦ =
(\
i∈J
ς(φi) : J ⊆ I
)
.
Equivalently, give S = {0, 1} the Sierpi´ nski topology{∅, {1}, S}. Then τΦ is the initial topology
of the coordinate maps bi : Ω P → S, or of their product oΦ : Ω P → SI. This is the standard
topological formulation of positive observable predicates [18, Example 1.12(3)].
Order the Boolean cube {0, 1}I coordinatewise, or equivalently order its subsets by inclusion.
Theorem 3.3 (Signature representation). The opens of τΦ are precisely the inverse images of
upward-closed sets of realized observation signatures:
τΦ = {o−1
Φ (U) : U ⊆ oΦ(ΩP ) is upward closed }.
Proof. For a realized signature s, its principal upward set is
↑s = {t : s ⊆ t}.
6Its inverse image is the basic open
o−1
Φ (↑s) =
\
i∈s
ς(φi).
Every upward-closed set in a finite poset is the union of the principal upward sets of its members,
so its inverse image is open. Conversely, a basic open indexed by any J ⊆ I is the inverse image of
the upward set
{s ∈ oΦ(ΩP ) : J ⊆ s}.
Unions of basic opens give the reverse inclusion.
3.1 Partitions induced by observations
Define
v ∼Φ w ⇐ ⇒ oΦ(v) = oΦ(w).
This equivalence relation partitions Ω P into observational states.
Definition 3.4 (Partition topology). For a partition Π of a set S, define
τΠ :=
n[
A : A ⊆ Π
o
.
Its blocks are both open and closed.
An arbitrary observation topology need not be a partition topology. The difference is essential:
partition topologies describe symmetric indistinguishability classes, whereas general finite topologies
can also record one-way observational dependence.
Proposition 3.5 (Positive and two-sided observation). Let Φ± be indexed by I × {+, −}, with
φ(i,+) = φi and φ(i,−) = ¬φi. Then τΦ± is the partition topology whose blocks are the ∼Φ-classes.
Proof. For every realized signature s, its fiber is
o−1
Φ (s) =
\
i∈s
ς(φi) ∩
\
i∈I\s
ς(¬φi).
It is therefore open in τΦ±. Every open generated by Φ ± is saturated under equality of signatures,
so it is a union of these fibers.
Example 3.6 (Two positive observations). Take Φ = (X, Y ) on Ω X,Y . Then
τΦ = {∅, {B}, {A, B}, {B, C}, {A, B, C}, ΩX,Y }.
Adding ¬X and ¬Y makes each of A, B, C, D open and gives the discrete topology.
For later reference, Table 2 collects the complete signatures and smallest positive-observation
neighborhoods for this running example. The last two columns anticipate the implication and XOR
examples without changing their classical truth values.
7Cell o(X,Y ) N(X,Y ) X ⇒ Y X ⇕ Y
A (1, 0) {A, B} F T
B (1, 1) {B} T F
C (0, 1) {B, C} T T
D (0, 0) Ω X,Y T F
Table 2: The four-cell example used throughout the paper.
4 Distinguishability and separation
Three questions should not be conflated:
1. Formulas L1, L2 are semantically different when ς(L1) ̸= ς(L2).
2. Valuations v, w are observationally distinguishable by Φ when oΦ(v) ̸= oΦ(w).
3. Points are topologically separated according to the neighborhoods available in τΦ.
Overlap between two truth regions does not by itself establish topological indistinguishability.
Proposition 4.1 (Semantic XOR criterion). For formulas L1, L2,
ς(L1) ̸= ς(L2) ⇐ ⇒ ς(L1 ⇕ L2) ̸= ∅.
At a particular valuation v, their truth values are distinct exactly when v ∈ ς(L1 ⇕ L2).
Proof. The XOR truth region is the symmetric difference
ς(L1) △ ς(L2).
It is nonempty exactly when the two truth regions are unequal.
Observing only L1 ∧ L2 gives one positive bit and groups together all three ways in which the
conjunction can be false. Replacing that observation by the pair Φ = ( L1, L2) refines those states
according to the realized truth vectors (1 , 1), (1, 0), (0, 1), (0, 0). Adding both negations makes the
nonempty truth-pattern cells clopen. This is the precise observational sense in which the conjuncts
become distinguishable.
Definition 4.2 (Topological indistinguishability). Points v, w ∈ ΩP are topologically indistinguish-
able when they belong to exactly the same open sets of τΦ.
Theorem 4.3 (Indistinguishability theorem). For v, w ∈ ΩP , the following are equivalent:
(i) v and w are topologically indistinguishable in τΦ;
(ii) v and w have the same subbasic open memberships;
(iii) oΦ(v) = oΦ(w);
(iv) v ∼Φ w.
Proof. Equality of all open memberships implies equality of subbasic memberships. Subbasic
memberships are exactly the coordinates of oΦ. Conversely, equality of signatures gives equality
of membership in every finite intersection of subbasic opens and hence in every union of such
intersections.
8Corollary 4.4 (T0 quotient). The space (ΩP , τΦ) is T0 exactly when oΦ is injective. Its Kolmogorov
quotient is naturally identified with the set oΦ(ΩP ) of realized signatures, equipped with the subspace
upper-set topology inherited from SI.
The next separation level requires distinguishability in both directions.
Proposition 4.5 (T1 and complete finite separation). The observation space is T1 exactly when,
for every distinct v, w,
oΦ(v) ⊈ oΦ(w) and oΦ(w) ⊈ oΦ(v).
Equivalently, oΦ is injective and its image is an antichain. A finite T1 space is discrete; therefore
finite Hausdorff observation spaces are discrete as well.
Proof. There is an open containing v but not w exactly when some observation is true at v and false
at w, which is equivalent to oΦ(v) ⊈ oΦ(w). The T1 condition requires this in both directions. In a
finite T1 space, each singleton is the finite intersection of open complements of the other singletons,
so every singleton is open.
Example 4.6 (Sierpi´ nski observation space). Let P = {X} and Φ = (X). With valuations 0 , 1, the
topology is
τΦ = {∅, {1}, {0, 1}}.
It is T0 but not T1. The positive observation X distinguishes 1 from 0, but no positive observation
in Φ distinguishes 0 from 1. This is the canonical two-point example of asymmetric observation.
5 Asymmetric separation and the specialization preorder
We use the specialization convention
v ⪯Φ w ⇐ ⇒ every open neighborhood of v contains w.
Some authors use the reverse convention; the direction is fixed here by the display above.
Because ΩP is finite, every point has a smallest open neighborhood
NΦ(v) :=
\
{U ∈ τΦ : v ∈ U }.
Theorem 5.1 (Positive-information order). For all v, w ∈ ΩP ,
v ⪯Φ w ⇐ ⇒ oΦ(v) ⊆ oΦ(w).
Moreover,
NΦ(v) =
\
i∈oΦ(v)
ς(φi) = {w : oΦ(v) ⊆ oΦ(w)}.
Proof. If every open containing v contains w, this holds for every subbasic open, so every observation
true at v is true at w. Conversely, if oΦ(v) ⊆ oΦ(w), then every basic open containing v contains w,
and therefore every open containing v contains w. The neighborhood formula follows.
Thus w preserves all positive information observable at v and may satisfy additional observations.
Failure of symmetry is not a defect: it records that positive evidence can separate one direction
without separating the reverse. After quotienting by ∼Φ, the preorder becomes the inclusion order
on realized signatures.
96 Distances on observation signatures
Coordinatewise XOR naturally produces a difference vector. Let a = (ai)i∈I be positive real weights.
As noted in Section 3, these formulas use complete semantic profiles, even when the topology is read
as an interface for affirmative certificates. The distances are deliberately presentation-dependent:
changing weights, duplicating coordinates, or replacing Φ by a different family that generates the
same topology can change their numerical values.
Definition 6.1 (Observable XOR difference). Define
∆Φ(v, w) := oΦ(v) ⇕ oΦ(w) :=

bi(v) ⇕ bi(w)


i∈I .
This vector identifies exactly which observations disagree.
Definition 6.2 (Symmetric signature distance). Define the weighted Hamming distance between
signatures by
dΦ(v, w) :=
X
i∈I
ai|bi(v) − bi(w)|.
This is a standard weighted form of Hamming distance [9].
Proposition 6.3. The function dΦ is a pseudometric on ΩP , and
dΦ(v, w) = 0 ⇐ ⇒ v ∼Φ w.
Consequently, dΦ induces a metric on the underlying set ΩP /∼Φ. If τdΦ denotes the pseudometric
topology on ΩP , then
τdΦ = τΦ±.
The induced metric topology on the finite quotient set is discrete and need not equal its Kolmogorov
quotient topology.
Proof. Each coordinate contributes the weighted discrete metric on {0, 1}; their sum is a pseudo-
metric. It vanishes exactly when every observation coordinate agrees. If I ̸= ∅, every sufficiently
small ball at v is exactly the signature fiber [ v]Φ, while every ball is a union of such fibers. Hence
the pseudometric topology is the partition topology of Proposition 3.5. For I = ∅ there is one fiber
and both topologies are indiscrete. A genuine metric on a finite set has discrete topology, giving the
final qualification.
For example, the Sierpi´ nski space of Example 4.6 is alreadyT0, so its Kolmogorov quotient has
the same nondiscrete topology. Its Hamming metric is the ordinary discrete two-point metric. Thus
“metric on the quotient set” does not mean “metric inducing the quotient topology.”
Symmetric distance measures disagreement but cannot by itself encode the direction of the
specialization preorder. For that purpose define the cost of losing observations.
Definition 6.4 (Directed loss distance). Define
d+
Φ(v, w) :=
X
i∈I
ai 1[bi(v) = 1 and bi(w) = 0].
Equivalently, this is weighted set difference from the support of v to the support of w. In XOR
notation the direction is supplied by the source mask:
d+
Φ(v, w) =
X
i∈I
aibi(v)

bi(v) ⇕ bi(w)


.
10This finite weighted sum is an elementary directed set-difference construction. Pavlovic uses the
cardinality |y \ x|, with the opposite argument orientation from our loss convention [ 13, Sec. 2],
while de Brecht uses a supremum of decaying weights on a countable powerset [5, Sec. 4].
The theorem below is therefore an explicit coordinate representation for the chosen observation
presentation, not a claim that quasi-metrizability of finite Alexandrov spaces is new. In particular,
Dovgoshey and Shanin prove that every T0 Alexandrov space is quasi-metrizable by an equidistant
quasi-metric [ 7]. Here d+
Φ works already before the T0 quotient, at the price of being a quasi-
pseudometric when distinct points have the same signature.
Theorem 6.5 (Quasi-pseudometric representation). The function d+
Φ is a quasi-pseudometric: it is
nonnegative, d+
Φ(v, v) = 0, and
d+
Φ(v, u) ≤ d+
Φ(v, w) + d+
Φ(w, u).
It also satisfies
d+
Φ(v, w) = 0 ⇐ ⇒ v ⪯Φ w,
and
dΦ(v, w) = d+
Φ(v, w) + d+
Φ(w, v).
The topology generated by the forward balls, for r > 0,
B+(v, r) := {w : d+
Φ(v, w) < r }
is exactly τΦ.
Proof. For each coordinate, an observation lost from v to u must be lost either from v to w or
from w to u. Summing the resulting coordinatewise inequality proves the triangle inequality.
Vanishing means that no observation true at v is false at w, which is exactly oΦ(v) ⊆ oΦ(w). The
symmetrization identity follows because every disagreement is a loss in exactly one direction.
Every forward ball is upward closed in ⪯Φ: if w belongs to the ball and w ⪯Φ u, then u loses
no more observations from v than w does. Hence every forward ball is τΦ-open. If I ̸= ∅ and
0 < r ≤ mini∈I ai, then
B+(v, r) = NΦ(v).
The smallest neighborhoods generate τΦ, so the forward-ball topology is exactly τΦ. If I = ∅, the
directed distance is identically zero and every positive-radius forward ball is Ω P , again giving the
correct indiscrete topology.
The zero-distance relation of d+
Φ recovers the preorder and its forward balls recover the topology.
Its positive numerical values contain additional presentation data: changing weights, or adding
a topologically redundant coordinate, can change distances without changing ⪯Φ or τΦ. This is
compatible with the general role of nonsymmetric distance in generalized metric theory [10].
7 Interior, closure, and observational boundary
Topological operators describe what can be verified or ruled out using the chosen observations.
Definition 7.1. For an event E ⊆ ΩP , let
IntΦ(E) := IntτΦ(E), ClΦ(E) := ClτΦ(E), ∂ ΦE := ClΦ(E) \ IntΦ(E).
11The interior IntΦ(E) is the largest observable open event contained in E. If the actual valuation
lies in this interior, a finite conjunction of available positive observations certifies membership in
E. The closure ClΦ(E) consists of valuations at which the available observations cannot exclude
contact with E. The boundary ∂ΦE is therefore an observational ambiguity region. Equivalently,
∂ΦE = ClΦ(E) ∩ ClΦ(ΩP \ E).
It contains precisely those points whose available neighborhoods cannot settle membership in E one
way or the other.
Proposition 7.2 (Finite neighborhood formulas). For every E ⊆ ΩP ,
IntΦ(E) = {v : NΦ(v) ⊆ E},
ClΦ(E) = {v : NΦ(v) ∩ E ̸= ∅}.
Consequently,
v ∈ ∂ΦE ⇐ ⇒ NΦ(v) ∩ E ̸= ∅ and NΦ(v) ∩ (ΩP \ E) ̸= ∅.
Proof. The point v is interior to E exactly when it has an open neighborhood contained in E.
Since NΦ(v) is its smallest neighborhood, this is equivalent to NΦ(v) ⊆ E. A point belongs to the
closure of E exactly when each of its neighborhoods meets E, which in a finite space is equivalent
to NΦ(v) ∩ E ̸= ∅.
Remark 7.3 (Boundary versus XOR) . The symbol ∂Φ denotes the topological boundary operator
above. It is not the Boolean symmetric-difference operation that removes the overlap of two formulas.
That Boolean operation is the exclusive remainder
ς(L1 ∨ L2) \ ς(L1 ∧ L2) = ς(L1 ⇕ L2).
The two constructions can differ even in the simplest cases: in a discrete topology every topological
boundary is empty, while an XOR truth region may be nonempty.
Example 7.4 (Boundary of implication under positive observations) . For Φ = (X, Y ),
E = ς(X ⇒ Y ) = {B, C, D}.
Using the topology in Example 3.6,
IntΦ(E) = {B, C}, ClΦ(E) = ΩX,Y , ∂ ΦE = {A, D}.
Thus X ⇒ Y is neither open nor closed in this observation topology. The point D satisfies the
implication but is not positively certifiable as such using only X and Y , while A lies observationally
adjacent to satisfying states despite falsifying the implication.
Example 7.5 (XOR region under coarse and discrete observation) . Retain the partition labels
A = X ∧ ¬Y , B = X ∧ Y , C = ¬X ∧ Y , and D = ¬X ∧ ¬Y . The exclusive truth region is
E⊕ := ς(X ⇕ Y ) = {A, C}.
First use only the positive observations Φ = ( X, Y ). The smallest neighborhoods are
NΦ(A) = {A, B}, N Φ(B) = {B}, N Φ(C) = {B, C}, N Φ(D) = ΩX,Y .
12Hence no nonempty τΦ-open set is contained in E⊕, while the neighborhoods of A, C, and D each
meet it. Thus
IntΦ(E⊕) = ∅, ClΦ(E⊕) = {A, C, D}, ∂ ΦE⊕ = {A, C, D}.
The points A and C belong to the XOR region but are not positively certifiable using only X and
Y , because their smallest neighborhoods also contain B. The point D is not in the XOR region,
but it remains on its boundary because every positive-observation neighborhood of D contains both
XOR and non-XOR valuations.
Now take the two-sided observation family
Φ± = (X, ¬X, Y, ¬Y ).
Its topology is discrete. Therefore E⊕ is both open and closed, and
IntΦ±(E⊕) = ClΦ±(E⊕) = {A, C}, ∂ Φ±E⊕ = ∅.
The XOR truth set is unchanged by the topology; only its observable boundary changes. This is
the distinction between Boolean symmetric difference and topological ambiguity. Figure 1 displays
the same calculation on the four truth-pattern cells.
X ¬X
Y B = X ∧ Y
non-XOR
C = ¬X ∧ Y
XOR
¬Y A = X ∧ ¬Y
XOR
D = ¬X ∧ ¬Y
non-XOR
Figure 1: The four truth-pattern cells for X and Y . The bold cells form E⊕ = ς(X ⇕ Y ) = {A, C}.
Under positive observation Φ = ( X, Y ), the observable boundary is ∂ΦE⊕ = {A, C, D}; under Φ ± it
is empty.
7.1 Open truth regions and Heyting implication
The opens of any topology form a Heyting algebra. For U, V ∈ τΦ, define
U ⇒H V := IntΦ

(ΩP \ U) ∪ V


, ¬H U := IntΦ(ΩP \ U).
Then
U ∩ (U ⇒H V ) ⊆ V
is topological Modus Ponens, and
(U ⇒H V ) ∩ ¬H V ⊆ ¬ H U
is topological Modus Tollens. In a discrete topology the interiors do nothing, and these reduce to
their classical truth-set forms. In a nondiscrete observation topology, the interior records the loss
imposed by requiring the result to remain observable.
Indeed, for every open W ,
W ⊆ U ⇒H V ⇐ ⇒ W ∩ U ⊆ V,
13which is the defining Heyting adjunction. The two displayed inference inequalities follow from
it. These open-set operations belong to standard topological semantics [ 11, 18]; they should be
distinguished from the classical truth-region operations used earlier. A full intuitionistic interpreta-
tion would assign open sets to atoms and evaluate compound formulas recursively in this Heyting
algebra.
For example, in the Sierpi´ nski space of Example 4.6, interpret X by the open {1}. Then
¬H {1} = ∅ and ¬H ¬H {1} = ΩX, whereas the interior of the classical truth region ς(¬¬X) = {1} is
still {1}. Recursive Heyting evaluation therefore differs from taking one final interior of a classical
truth region.
8 Information refinement and logical update
The phrase “acquiring information” can describe two mathematically different operations.
8.1 Refining the observations
Let Φ = ( φi)i∈I and Ψ = ( ψj)j∈J, where I ⊆ J and ψi = φi for every i ∈ I. Thus Ψ extends Φ
while retaining the identities of its old coordinates.
Theorem 8.1 (Refinement theorem). For such an indexed extension Φ ,→ Ψ:
(i) τΦ ⊆ τΨ;
(ii) v ∼Ψ w implies v ∼Φ w;
(iii) ⪯Ψ ⊆ ⪯Φ as relations;
(iv) if Ψ retains the weights ai on i ∈ I, then d+
Ψ(v, w) ≥ d+
Φ(v, w) and dΨ(v, w) ≥ dΦ(v, w).
Proof. The subbasis for τΦ is contained in the subbasis for τΨ, proving (i). Equality of the longer
signatures implies equality after deleting the new coordinates, proving (ii). Inclusion of Ψ-signatures
implies inclusion after deleting those coordinates, proving (iii). The added coordinates make
nonnegative contributions to both distances, proving (iv).
Adding an observable can split an indistinguishability block, remove a one-way comparison, and
increase measured separation. A proposed observation ψ is topologically redundant relative to Φ
exactly when
ς(ψ) ∈ τΦ,
because adding an already open truth region does not refine the topology. Topological redundancy
need not mean numerical redundancy: an added coordinate can add a further term to dΦ and d+
Φ
while leaving the generated topology unchanged. For example, adding X ∧ Y to Φ = ( X, Y ) is
topologically redundant because ς(X ∧ Y ) is already open. With unit weights it nevertheless changes
d+(B, A) from 1 to 2.
8.2 Updating the possible valuations
Let S ⊆ ΩP be the current possibility region. Learning a premise γ produces
S 7− →S ∩ ς(γ).
14For premises Γ = {γ1, . . . , γr}, define
S[Γ] := S ∩
r\
j=1
ς(γj), S Γ := ΩP [Γ].
For an arbitrary current carrier S, the updated observation space is
XT
S,Γ :=

S[Γ], τΦ|S[Γ]


, τ Φ|S := {U ∩ S : U ∈ τΦ}.
When S = Ω P , write XT
Γ := XT
ΩP ,Γ. Sequential updates are associative, commutative, and
idempotent because set intersection has those properties. Inconsistent premises produce the empty
space. Smallest neighborhoods restrict accordingly:
NΦ|S[Γ](v) := S[Γ] ∩ NΦ(v) ( v ∈ S[Γ]).
Observation refinement and premise update need not have the same effect. Refinement changes
which distinctions are expressible; update changes which worlds remain possible.
9 Inference by truth-region restriction
Classical inference is a relationship between the common truth region of the premises and the truth
region of the conclusion.
Theorem 9.1 (Semantic consequence). For a finite set of premises Γ and a conclusion ψ,
Γ |= ψ ⇐ ⇒ SΓ ⊆ ς(ψ).
When this holds, the inclusion
XT
Γ ,→

ς(ψ), τΦ|ς(ψ)


is continuous.
Proof. The semantic statement says exactly that every valuation satisfying all premises satisfies ψ.
The inclusion between two subspaces of the same ambient space is continuous.
The topology contributes a structured representation of this consequence: the common premise
region carries an induced observation topology, and the valid inclusion is a continuous map into
the conclusion subspace. It does not alter which consequences are classically valid. In particular,
premise conjunction uses intersection of truth regions (or the pullback of their induced subspaces),
not intersection of two families of open sets. The latter is a different operation: a meet in the lattice
of topologies on one fixed carrier.
9.1 A remark on minimal truth spaces
For a truth region S, one may form the indiscrete space T T
S = (S, {∅, S}). Intersecting carriers and
then assigning this indiscrete topology transports the ordinary meet R ∩ S of the power set into new
notation; it supplies no additional topological inference principle. Moreover, T T
S generally differs
from the induced observation subspace ( S, τΦ|S), and its inclusion into the ambient observation
space need not be continuous. It therefore cannot replace the induced subspaces in the pullback
formulation below. The two spaces coincide exactly when the induced topology on S is indiscrete,
as for a singleton or a region whose points all have the same observation signature.
159.2 Modus Ponens and Modus Tollens
Using A, B, C, D,
ς(X ⇒ Y ) ∩ ς(X) = {B} = ς(X ∧ Y ) ⊆ ς(Y ),
ς(X ⇒ Y ) ∩ ς(¬Y ) = {D} = ς(¬X ∧ ¬Y ) ⊆ ς(¬X).
9.3 Hypothetical Syllogism
Write a three-variable valuation as the triple ( X, Y, Z). Then
ς(X ⇒ Y ) ∩ ς(Y ⇒ Z) = {(0, 0, 0), (0, 0, 1), (0, 1, 1), (1, 1, 1)} ⊊ ς(X ⇒ Z).
The inclusion is proper: (1 , 0, 1) satisfies X ⇒ Z but not X ⇒ Y . The two premises therefore carry
more information than the conclusion; validity requires inclusion, not equality.
9.4 Pullback formulation
For a single formula L, write
XT
L :=

ς(L), τΦ|ς(L)


.
The inclusions of two truth subspaces into (Ω P , τΦ) have pullback
XT
L1 ×(ΩP ,τΦ) XT
L2
∼= XT
L1∧L2.
For Modus Ponens,
XT
X⇒Y ×(ΩX,Y ,τΦ) XT
X ∼= XT
X∧Y ,→ XT
Y .
Because both arrows are subspace inclusions, this pullback is just the set-theoretic intersection
equipped with its subspace topology, via the diagonal identification v 7→ (v, v). Its universal property
is the standard one for intersections of subspaces: compatible maps into the two premise regions
factor uniquely and continuously through their common region. Thus the categorical notation
repackages premise conjunction; it does not add a new inference rule.
10 Continuous maps and preservation of information
Let Φ = (φi)i∈I and Ψ = (ψj)j∈J generate the observation spaces (Ω P , τΦ) and (Ω Q, τΨ). A map
f : ΩP − →ΩQ
is continuous when the inverse image of every Ψ-observable open event is Φ-observable. It suffices
to check the subbasic truth regions:
f −1
ς(ψj)


∈ τΦ (j ∈ J).
Proposition 10.1 (Monotonicity of continuous maps). If f is continuous and v ⪯Φ w, then
f(v) ⪯Ψ f(w).
Proof. Let U ∈ τΨ contain f(v). Then f −1(U) is a Φ-open set containing v, so it contains w. Hence
U contains f(w).
Continuity therefore preserves one-way observational dependence. This gives a rigorous starting
point for continuous valuation maps. Conversely, because these finite spaces have their upper-set
topologies, every monotone valuation map is continuous. Continuity may collapse distinguishable
points and does not by itself imply preservation of either weighted distance.
1611 Scope, limitations, and deferred directions
The results above form a finite expository theory of observation, distinguishability, distance,
refinement, and inference. They do not establish new homological or application-specific results.
The most direct extensions are questions about the design and use of observations in reasoning.
11.1 A presentation-dependent nerve
A standard relation-nerve construction can be applied to a family of truth regions. For an indexed
observation family Φ = (φi)i∈I, define a simplicial complex
KΦ :=
(
J ⊆ I :
\
i∈J
ς(φi) ̸= ∅
)
.
Its vertices are the indices of individually satisfiable observations, and its simplices are jointly
satisfiable indexed subfamilies. This is the observation-side Dowker complex of the satisfaction
relation [6]. It can record higher-order compatibility not visible from pairwise intersections, but it is
not an invariant of τΦ. For example, Φ = ( X, Y, ¬(X ∧ Y )) gives the boundary of a triangle. Its
observation regions are {A, B}, {B, C}, and {A, C, D}: their pairwise intersections are respectively
{B}, {A}, and {C}, while their triple intersection is empty. Thus the nerve is connected, whereas
the observation space has connected components {B} and {A, C, D}. Adjoining a tautology leaves
the generated topology and both signature distances unchanged, but makes the nerve a cone and
kills its reduced homology. Any homological use must therefore state the presentation and the
intended invariance separately: the homology of KΦ need not agree with the singular homology of
the observation space and is not determined by τΦ.
11.2 Focused directions beyond the synthesis
The most natural next step is an observation-design problem rather than a new name for the finite
topology itself: find a smallest or least-cost observation family that separates designated valuations,
certifies a target conclusion, or distinguishes competing premise sets. Such questions could support
genuinely new algorithmic or complexity results. A second direction is to study sequences in which
observation refinement and premise restriction alternate, or to compare the classical truth-region
semantics used here with the Heyting semantics of the open-set algebra. These are deliberately left
outside the present expository scope.
12 Conclusion
An observation-generated topology can be nontrivial when it records a selected family of formulas
rather than every definable truth set. Observation signatures simultaneously determine a partition
into indistinguishable states, a finite topology, and its specialization preorder. Coordinatewise XOR
records which observations differ; weighted Hamming distance measures symmetric disagreement;
and the directed distance d+
Φ measures loss of positive information while generating the selected
observation topology. Adding tests refines distinguishability, while adding premises restricts the
possibility space. Inference is the inclusion of that restricted region in the truth region of a conclusion.
The framework gives a common notation for partitioning, distinguishability, asymmetric sepa-
ration, logical distance, and refinement. Its individual ingredients are classical. Its value here is
expository: it makes explicit which structures are determined by the observations, which require
17extra weights or presentation choices, and which logical claims remain ordinary model-set inclusions.
Natural research extensions would add an algorithmic or complexity result, a new invariant, or a
validated application rather than more renamings of the same finite structures.
References
[1] S. Abramsky and S. Vickers, “Quantales, observational logic and process semantics,” Mathematical
Structures in Computer Science 3, no. 2 (1993), 161–227. doi:10.1017/S0960129500000189.
[2] A. Achilleos and V. Kyriakou, “A topological framework for finite behavioural observations and
verification,” arXiv:2606.23975 (2026). arXiv:2606.23975.
[3] P. Alexandroff, “Diskrete R¨ aume,”Recueil Math´ ematique, Nouvelle S´ erie2(44), no. 3 (1937), 501–519.
[4] M. M. Bonsangue, B. P. F. Jacobs, and J. N. Kok, “Duality beyond sober spaces: Topological spaces
and observation frames,” Theoretical Computer Science 151, no. 1 (1995), 79–124. doi:10.1016/0304-
3975(95)00048-2.
[5] M. de Brecht, “Quasi-Polish spaces,” Annals of Pure and Applied Logic 164, no. 3 (2013), 356–381.
doi:10.1016/j.apal.2012.11.001.
[6] C. H. Dowker, “Homology groups of relations,” Annals of Mathematics , second series, 56, no. 1 (1952),
84–95. doi:10.2307/1969768.
[7] O. Dovgoshey and R. Shanin, “On the closure of one point sets in T0-spaces,” Topology and its
Applications 382 (2026), 109756. doi:10.1016/j.topol.2026.109756.
[8] B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundations , Springer, Berlin, 1999.
doi:10.1007/978-3-642-59830-2.
[9] R. W. Hamming, “Error detecting and error correcting codes,” Bell System Technical Journal 29, no. 2
(1950), 147–160. doi:10.1002/j.1538-7305.1950.tb00463.x.
[10] F. W. Lawvere, “Metric spaces, generalized logic, and closed categories,” Rendiconti del Seminario
Matematico e Fisico di Milano 43 (1973), 135–166. doi:10.1007/BF02924844.
[11] J. C. C. McKinsey and A. Tarski, “The algebra of topology,” Annals of Mathematics , second series, 45,
no. 1 (1944), 141–191. doi:10.2307/1969080.
[12] Z. Pawlak, “Rough sets,” International Journal of Computer and Information Sciences 11 (1982),
341–356. doi:10.1007/BF01001956.
[13] D. Pavlovic, “Quantitative concept analysis,” in Formal Concept Analysis, Lecture Notes in Computer
Science 7278 (2012), 260–277. doi:10.1007/978-3-642-29892-9 24.
[14] Z. Pei, D. Ruan, D. Meng, and Z. Liu, “Formal concept analysis based on the topology for attributes of
a formal context,” Information Sciences 236 (2013), 66–82. doi:10.1016/j.ins.2013.02.027.
[15] M. H. Stone, “The theory of representations for Boolean algebras,” Transactions of the American
Mathematical Society 40, no. 1 (1936), 37–111. doi:10.1090/S0002-9947-1936-1501865-8.
[16] R. E. Stong, “Finite topological spaces,” Transactions of the American Mathematical Society 123, no. 2
(1966), 325–340. doi:10.1090/S0002-9947-1966-0195042-2.
[17] S. Vickers, Topology via Logic , Cambridge Tracts in Theoretical Computer Science 5, Cambridge
University Press, 1989. doi:10.1017/CBO9780511625900.
[18] S. Vickers, “Locales and toposes as spaces,” in Handbook of Spatial Logics, Springer, 2007, 429–496.
Author preprint.
18