From Context to Concepts: When Are Two Things Similar?

1. Similarity Depends on Context

Are a dolphin and a cat similar? Are a dolphin and a fish similar?

If we care about how animals feed their young, a dolphin is similar to a cat: both feed their young with milk. If we care about where animals live, a dolphin is similar to a fish: both live underwater. The animals have not changed, but the criterion by which we compare them has.

Similarity depends on context.

This raises a simple question: how does a context determine which entities are similar, and how can concepts emerge from these similarities?

2. A Running Example: Five Animals and Three Propositions

Consider five animals: a cat, a dolphin, a fish, a lizard, and a bird. Suppose that, for the moment, our knowledge about them is limited to three questions: Does the animal feed its young with milk? Does it live underwater? Does it have four legs?

We denote the corresponding propositions by $P_1$, $P_2$, and $P_3$:

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

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

2.1. First Context: Milk and Water

Suppose that we care only about whether an animal feeds its young with milk and whether it lives underwater. Under these two questions, the cat, dolphin, and fish can all be distinguished from one another. The lizard and the bird, however, cannot: neither feeds its young with milk, and neither lives underwater.

We call the propositions currently used to distinguish entities a context. In this case,

$$c_1=\{P_1,P_2\}.$$

The context therefore separates the five animals into the groups

$$\{\text{cat}\}\mid \{\text{dolphin}\}\mid \{\text{fish}\}\mid \{\text{lizard,bird}\}.$$

In particular, the lizard and the bird are indistinguishable under this context. We may write

$$\text{lizard}\sim_{c_1}\text{bird}.$$

2.2. Second Context: Milk and Four Legs

Now replace the question about living underwater with the question about having four legs. The lizard and the bird are immediately distinguished because the lizard has four legs whereas the bird does not. But a new pair becomes indistinguishable: the fish and the bird neither feed their young with milk nor have four legs.

Only after observing this change do we write the second context as

$$c_2=\{P_1,P_3\}.$$

It produces the grouping

$$\{\text{cat}\}\mid \{\text{dolphin}\}\mid \{\text{lizard}\}\mid \{\text{fish,bird}\}.$$

Thus the same five entities give two different groupings. Under the first context, the lizard and the bird are similar; under the second, the fish and the bird are similar.

The entities did not change; the context did. Changing context changes which distinctions matter.

The move from $c_1$ to $c_2$ is a change of viewpoint rather than simply an addition or removal of information. We will call this a context shift.

2.3. Keeping Only What the Two Contexts Share: Meet

The two contexts share one question: whether an animal feeds its young with milk. Suppose that we forget the other questions and retain only this common one.

The cat and the dolphin can no longer be distinguished because both feed their young with milk. Similarly, the fish, lizard, and bird become indistinguishable because none of them does. We therefore obtain only two groups:

$$\{\text{cat,dolphin}\}\mid \{\text{fish,lizard,bird}\}.$$

Only now do we identify this common context as the meet of the two contexts:

$$c_1\wedge c_2=\{P_1\}.$$

Keeping less contextual information removes distinctions and produces a coarser grouping.

2.4. Combining the Two Contexts: Join

Instead of forgetting information, suppose that we combine all the questions used by the two contexts. We now ask whether each animal feeds its young with milk, whether it lives underwater, and whether it has four legs.

With all three questions available, every animal can be distinguished from every other animal. The resulting groups are

$$\{\text{cat}\}\mid \{\text{dolphin}\}\mid \{\text{fish}\}\mid \{\text{lizard}\}\mid \{\text{bird}\}.$$

The combined context is the join:

$$c_1\vee c_2=\{P_1,P_2,P_3\}.$$

Combining contextual information can create more distinctions and therefore a finer grouping.

The four contexts and their induced groupings can now be viewed together:

Context	Propositions used	Equivalence classes
$c_1\wedge c_2$	$\{P_1\}$	$\{\text{cat,dolphin}\}\mid\{\text{fish,lizard,bird}\}$
$c_1$	$\{P_1,P_2\}$	$\{\text{cat}\}\mid\{\text{dolphin}\}\mid\{\text{fish}\}\mid\{\text{lizard,bird}\}$
$c_2$	$\{P_1,P_3\}$	$\{\text{cat}\}\mid\{\text{dolphin}\}\mid\{\text{lizard}\}\mid\{\text{fish,bird}\}$
$c_1\vee c_2$	$\{P_1,P_2,P_3\}$	$\{\text{cat}\}\mid\{\text{dolphin}\}\mid\{\text{fish}\}\mid\{\text{lizard}\}\mid\{\text{bird}\}$

These four cases are not unrelated. They form a natural order according to which propositions are retained:

$$\begin{array}{ccccc} &&c_1\vee c_2&&\\ &\nearrow&&\nwarrow&\\ c_1&&&&c_2\\ &\nwarrow&&\nearrow&\\ &&c_1\wedge c_2&& \end{array}$$

This is a small lattice of contexts. Moving upward adds available distinctions. Moving downward forgets distinctions. Moving from $c_1$ to $c_2$, by contrast, changes which distinctions are being made.

Context controls not only how many distinctions we can make, but also which distinctions we make.

3. From Propositions to Concepts

The animal example suggests a more general viewpoint. The lizard and the bird belong to the same group under the first context because every proposition considered in that context gives the same answer for both animals. Likewise, the fish and the bird belong to the same group under the second context because that context cannot distinguish them.

This observation gives a natural notion of contextual equivalence. For entities $x$ and $y$ and a context $c$, we say that they are equivalent when every proposition selected by the context gives the same truth value for both entities:

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

Thus a context does more than provide background information. By selecting which propositions matter, it induces an equivalence relation among entities.

Propositions selected by a context induce equivalence.

An equivalence relation partitions the entities into equivalence classes. In our example, under the meet context, the class containing the cat is $\{\text{cat,dolphin}\}$, while the other class is $\{\text{fish,lizard,bird}\}$. These classes represent the distinctions that remain visible at that context.

We can therefore take one further conceptual step: treat each equivalence class as a concept generated at the resolution provided by the context. Formally, if $E$ is the set of entities, the context $c$ produces the quotient

$$E/{\sim_c}.$$

Its elements are the concepts induced by that context.

Context selects propositions; propositions induce equivalence; equivalence classes give rise to concepts.

The construction can be summarized as

$$\text{Propositions} \xrightarrow{\text{select a context}} \text{Equivalence relation} \longrightarrow \text{Concepts}.$$

This also gives a useful interpretation of the familiar statement

$$\text{Knowledge}=\text{Concepts}+\text{Propositions}.$$

Concepts and propositions need not be completely independent ingredients. Propositions can themselves generate conceptual structure once a context determines which propositions are relevant.

3.1. Connection to Rough Set Theory

Return once more to the first context. The lizard and the bird are grouped together not because they are identical animals, but because the available questions cannot distinguish them. This idea of indistinguishability relative to available information is central to Rough Set Theory.

In the classical rough-set setting, objects are described by attributes. Selecting a collection of attributes determines an indiscernibility relation: two objects are indiscernible when they agree on every selected attribute. The resulting equivalence classes are often viewed as information granules.

Our running example has exactly this form. The propositions play the role of observable attributes, a context selects which of them are relevant, and contextual equivalence plays the role of indiscernibility:

$$\text{context}\longleftrightarrow\text{selected propositions},$$

$$\sim_c\longleftrightarrow\text{indiscernibility relation}.$$

Rough sets therefore provide a natural mathematical home for the crisp version of the phenomenon developed from our example.

But the example also suggests a next question. Must contextual similarity always be crisp? A dolphin may be closer to a fish when we think about habitat and closer to a cat when we think about reproduction, even when neither comparison should be reduced to exact equality of a few propositions.

This motivates replacing contextual equivalence by contextual geometry:

$$\sim_c\quad\longrightarrow\quad d_c(x,y).$$

The next step is to ask whether context can determine not only equivalence, but also the geometry in which entities and concepts become near or far.