Three Worlds of Knowledge: Possibility, Abstraction, and Realization

1. Possibility Knowledge

Suppose that we are looking at a collection of animals: a cat, a dolphin, a fish, a lizard, and a bird. There are many things that we could potentially say about these animals. Some feed their young with milk, some live underwater, and some have four legs. Before deciding which distinctions are important, let us simply regard all these animals and all these available propositions as forming a world of possibilities.

Let $C$ denote the collection of possible entities and let $P$ denote the collection of propositions that can be evaluated on them. We write

$$W=(C,P).$$

We call such a pair a propositional world. For example, we may take

$$C=\{\text{cat},\text{dolphin},\text{fish},\text{lizard},\text{bird}\},$$

and

$$P=\{P_1,P_2,P_3\},$$

where

$$P_1=\text{“feeds its young with milk”},\qquad P_2=\text{“lives underwater”},\qquad P_3=\text{“has four legs”}.$$

The propositions take the following values:

Entity	$P_1$: milk	$P_2$: underwater	$P_3$: four legs
cat	1	0	1
dolphin	1	1	0
fish	0	1	0
lizard	0	0	1
bird	0	0	0

Each entity can therefore be viewed through the answers it gives to all propositions in $P$. The cat has the profile $(1,0,1)$, while the dolphin has the profile $(1,1,0)$. In this example, all five animals have different profiles, so the available propositions distinguish all of them.

The important point, however, is that $C$ contains the possibilities themselves, not merely the distinctions that we happen to make between them. Two entities may be genuinely different elements of $C$ even when the available propositions cannot distinguish them.

We therefore call

$$W_{\mathrm{pos}}=(C,P)$$

the possibility knowledge.

Possibility knowledge describes what may exist and what may be said about it.

This immediately raises another question. If two possibilities cannot be distinguished by any proposition available to us, should our knowledge still treat them as different concepts?

2. Abstract Knowledge

To see what happens, suppose that our available knowledge is more limited. Imagine that we retain only the propositions concerning milk and underwater habitat:

$$P'=\{P_1,P_2\}.$$

The corresponding profiles are now

$$\text{cat}\mapsto(1,0),\qquad \text{dolphin}\mapsto(1,1),\qquad \text{fish}\mapsto(0,1),\qquad \text{lizard}\mapsto(0,0),\qquad \text{bird}\mapsto(0,0).$$

The lizard and the bird are different possibilities, but the propositions in $P'$ can no longer distinguish them. At this level of knowledge, they appear identical.

This suggests that propositions naturally induce an equivalence relation. For $x,y\in C$, define

$$x\sim_P y\quad\Longleftrightarrow\quad p(x)=p(y)\ \text{for every }p\in P.$$

Two possibilities are therefore equivalent whenever the entire propositional system gives exactly the same answers for both.

The resulting equivalence classes form a new world:

$$\bar C=C/{\sim_P}.$$

Instead of keeping every possible entity separately, $\bar C$ keeps only the distinctions visible through $P$. There is a natural map from the possibility world to this new world:

$$\mathcal A:C\longrightarrow\bar C,\qquad \mathcal A(x)=[x]_P.$$

We call $\mathcal A$ the abstraction map.

The propositions themselves also descend naturally to $\bar C$. For each $p\in P$, define

$$\bar p([x]_P)=p(x).$$

This is well defined because all members of the same equivalence class have the same value under every proposition in $P$. Let

$$\bar P=\{\bar p:p\in P\}.$$

We therefore obtain another propositional world,

$$W_{\mathrm{abs}}=(\bar C,\bar P),$$

which we call the abstract knowledge.

Abstract knowledge retains the distinctions visible to the propositions while forgetting differences that the propositions cannot detect.

Abstraction becomes especially clear when we reduce the available propositions. Suppose that

$$P'\subseteq P.$$

If two possibilities are indistinguishable under $P$, they must also be indistinguishable under $P'$. But the converse need not hold. Removing propositions may cause previously distinct classes to merge. Consequently, there is a natural surjective map

$$[C]_P\longrightarrow[C]_{P'}.$$

Thus abstraction has a simple structural interpretation:

$$\text{forget propositions}\longrightarrow\text{lose distinctions}\longrightarrow\text{merge concepts}.$$

Fewer propositions produce coarser concepts.

The abstraction world, however, no longer tells us which particular possibility should represent an abstract concept. An equivalence class may contain many possible realizations. This leads to the opposite question: how can an abstract concept become concrete again?

3. Realization Knowledge

Consider an abstract concept

$$\bar x=[x]_P\in\bar C.$$

The class $\bar x$ may contain several elements of the possibility world. Abstraction deliberately forgot the differences among them. To return to the possibility world, we must choose one concrete representative.

Let

$$\mathcal R:\bar C\longrightarrow C$$

be a map satisfying

$$\mathcal R([x]_P)\in[x]_P.$$

We call $\mathcal R$ a realization map. By construction,

$$\mathcal A(\mathcal R(\bar x))=\bar x$$

for every $\bar x\in\bar C$. Hence

$$\mathcal A\circ\mathcal R=I_{\bar C}.$$

The realization map selects one possible representative for every abstract concept. Its image

$$\widetilde C=\mathcal R(\bar C)$$

forms a new world inside the original possibility world.

The propositions on the abstract world can also be transported to this realization. For each $\bar p\in\bar P$, define a proposition $\widetilde p$ on $\widetilde C$ by

$$\widetilde p(\mathcal R(\bar x))=\bar p(\bar x).$$

Let

$$\widetilde P=\{\widetilde p:\bar p\in\bar P\}.$$

We then obtain the third propositional world,

$$W_{\mathrm{real}}=(\widetilde C,\widetilde P),$$

which we call the realization knowledge.

Realization knowledge gives a concrete representative of each abstract possibility while preserving its propositional structure.

The three worlds can now be viewed together:

$$(C,P)\xrightarrow{\mathcal A}(\bar C,\bar P)\xrightarrow{\mathcal R}(\widetilde C,\widetilde P).$$

The first world contains possibilities. The second identifies possibilities that cannot be distinguished propositionally. The third chooses one realization for each resulting abstract concept.

There is an important asymmetry between the two maps. Since $\mathcal R$ chooses a representative from the correct equivalence class,

$$\mathcal A\mathcal R=I_{\bar C}.$$

But generally,

$$\mathcal R\mathcal A\neq I_C.$$

Starting from a possibility $x$, abstracting it, and realizing the resulting class may return another member of the same class rather than $x$ itself. Nevertheless, repeating the process produces no further change:

$$(\mathcal R\mathcal A)^2=\mathcal R\mathcal A.$$

Thus abstraction followed by realization acts as an idempotent projection of the possibility world onto the realization world.

We can finally summarize the three forms of knowledge as

$$\boxed{\text{Possibility Knowledge}\xrightarrow{\mathcal A}\text{Abstract Knowledge}\xrightarrow{\mathcal R}\text{Realization Knowledge}.}$$

Possibility knowledge describes what could be. Abstract knowledge retains only the distinctions that matter propositionally. Realization knowledge chooses how those abstractions become concrete again.

Knowledge can move from possibilities to abstractions, and from abstractions back to realizations.