Four Relations Between Universals and Relative Particularity

1. Comparing two universal concepts

The preceding lesson defined a universal as a concept whose very meaning does not prevent its application to many individuals. We now ask how the instances of two universal concepts can relate to each other.

For example, both “human” and “animal” are universals. Every human is an animal, but not every animal is human. The textbook's pair “human” and “rational” has a different relationship.

The governing test is which individuals each concept can truly be predicated of. This yields four relations, followed by the distinction between real and relative particularity.

2. The fourfold source classification

فصل في النسبة بين الكليين: اعلم أن النسبة بين الكليين تتصور على أنحاء أربعة؛
Section on the relation between two universals: know that four kinds of relation are possible between two universal concepts.

“Relation” here refers to the range of things to which each meaning applies, not to how similar the words sound or look. The words “human” and “animal” differ in form; that is not the question. We ask which individuals fall under each meaning.

The four relations are:
1. Equality: exactly the same instances.
2. Disjointness: no common instances.
3. Absolute inclusion: everything under one concept falls under the other, but not conversely.
4. Overlap in one respect: some common and some exclusive instances on each side.

3. Predication, not verbal resemblance

In logic, to say that an individual falls under a universal is to say that the universal can truly be predicated of that individual. Zayd falls under both “human” and “animal.”

Three questions guide the classification:
1. Is there any shared instance?
2. Does every instance of the first fall under the second?
3. Does every instance of the second fall under the first?

One shared example cannot establish equality, and one different example cannot establish disjointness. Each relation concerns the whole range of application.

4. Equality: the source definition

فإما أن يصدق كل منهما على ما يصدق عليه الآخر فهما متساويان، كالإنسان والناطق؛
If each universal is predicable of everything of which the other is predicable, they are equal, as with human and rational.

Equality requires predication in both directions:
Every human is rational.
Every rational being, in the textbook's intended classification, is human.

Here rational (nāṭiq) signifies possession of the rational faculty, not merely the ability to utter sounds. A silent human remains rational, whereas a parrot that imitates words is not thereby rational in this technical sense.

5. Same extension does not mean identical concepts

“Human” and “rational” may apply to exactly the same individuals while presenting their meaning under different aspects. “Human” names the species; “rational” marks its intellectual distinguishing feature.

Thus equality of predication does not require synonymy or exactly matching definitions. It concerns extension, the set of individuals falling under each concept.

Zayd is human and rational.
Amr is human and rational.
No individual is human without being rational in this technical sense, and none is rational without being human within the intended classification.

Only when both universal directions hold is the relation equality.

6. Equality versus sharing one instance

The fact that two concepts apply to Zayd does not establish equality. Both “human” and “animal” apply to him, yet it does not follow that every animal is human.

Equality requires that neither concept have even one instance outside the other.

If one counterexample appears in either direction, equality fails. We must then determine whether the difference exists only on one side or on both sides to identify the correct remaining relation.

7. Disjointness: the source wording

أو لا يصدق شيء منهما على شيء مما يصدق عليه الآخر، فهما متباينان، كالإنسان والفرس،
Or if neither applies to any individual to which the other applies, they are disjoint, like human and horse.

Disjointness means that the two extensions never intersect:
No human is a horse.
No horse is a human.

Both nevertheless fall under “animal.” Having a common genus does not create a common individual belonging simultaneously to both the human and horse species.

8. Common genus does not defeat disjointness

“Human” and “horse” are different species concepts. One individual cannot be a human and a horse in these same specific respects, so they are disjoint.

Yet both share animality:
Every human is an animal.
Every horse is an animal.
Nevertheless, no human is a horse.

Being contained under a wider third concept does not make the original two equal or inclusive. The decisive question is whether one individual can fall under both at once.

9. Two negative directions in disjointness

Disjointness can be expressed by two universal negative propositions:
No human is a horse.
No horse is a human.

If no instance of the first lies in the second, there is equally no instance of the second in the first. Disjointness is therefore symmetric.

A single statement such as “Zayd is not a horse” is not enough to prove the universal relation. The classification concerns all instances of both concepts, not one isolated example.

10. Absolute generality and specificity

أو يصدق أحدهما على كل ما يصدق عليه الآخر، ولا يصدق الآخر على جميع أفراد أحدهما، فبينهما عموم وخصوص مطلقا، كالحيوان والإنسان،
Or if one universal applies to everything to which the other applies, but the other does not apply to every instance of the first, they stand in absolute generality and specificity, as with animal and human.

Here animal is absolutely more general and human absolutely more specific:
Every human is an animal.
Not every animal is human: a horse is an animal but not a human.

“More general” and “more specific” describe a relationship between the chosen concepts, not permanent labels independent of comparison.

11. Direction matters in absolute inclusion

In the pair animal → human, “animal” covers more individuals and “human” covers only a part of that extension. Moving from the general to the specific narrows the range.

Valid judgments:
Every human is an animal.
Some animals are humans.
Some animals are not humans, such as horses.

“Every animal is human” is invalid. Inclusion of all specific instances within the general does not reverse the containment. This one-way inclusion is precisely what distinguishes the relation from equality.

12. Generality depends on the comparison

“Human” is more specific than “animal,” yet more general than the particular individual Zayd. Zayd is one individual; “human” applies to him and many others.

The ordered comparison is:
Animal — more general than human.
Human — more specific than animal, but wider than Zayd.
Zayd — a specified individual.

Our main classification still concerns two universals. Zayd is introduced to illustrate the relativity of the words general and specific, preparing for the technical term “relative particular.”

13. Partial overlap: animal and white

أو يصدق بعض كل واحد منهما على بعض ما يصدق عليه الآخر، فبينهما عموم وخصوص من وجه، كالأبيض والحيوان،
Or if some instances of each universal fall under the other, the relation is generality and specificity in one respect, as with white and animal.

Partial overlap means that some instances are shared while neither universal is wholly contained in the other:
A white duck — both white and an animal.
A dark horse — an animal but not white.
Snow — white but not an animal.

These three cases together establish the source relation of overlap in one respect.

14. Three regions of partial overlap

Partial overlap requires three kinds of instances:
1. Intersection: both predicates are true, as in a white duck.
2. First exclusive region: animal but not white, as in a dark horse.
3. Second exclusive region: white but not animal, as in snow.

Showing only a shared instance leaves equality or inclusion possible. Showing an exclusive instance on only one side also fails to settle the relation. A shared case plus a distinct case on each side establishes partial overlap.

15. Testing the textbook cases

The original text elaborates the case as follows:
ففي البط يصدق كل منهما، وفي الفرس يصدق الحيوان فقط، وفي الثلج والعاج يصدق الأبيض فقط،
Both are true of the duck; only “animal” is true of the horse; only “white” is true of snow and ivory.

The intended duck is a white duck, and the horse is one that is not white. Ducks are not invariably white and horses are not invariably non-white. The logical point is that appropriately described examples exist.

Accordingly, the extensions meet in some instances and diverge in others: neither equality, disjointness nor one-way inclusion applies.

16. Why overlap is not absolute inclusion

If “animal” and “white” stood in absolute inclusion, all instances of one would belong to the other. But:
A dark horse is an animal, not white.
Snow is white, not an animal.

The first defeats “every animal is white”; the second defeats “everything white is an animal.” Yet a white duck falls under both, so the concepts are not disjoint either.

The three observations together establish partial overlap. A single particular example cannot establish the full universal relationship.

17. A reliable four-way decision procedure

Given two universals, test their instances:
Equality: each applies to all instances of the other — human and rational.
Disjointness: no shared instance — human and horse.
Absolute inclusion: one, but not both, contains all instances of the other — animal and human.
Partial overlap: at least one shared instance and exclusive instances on both sides — animal and white.

The labels general and specific are conclusions from examining predication and extension, not substitutes for that examination.

18. Identity of instances and distinct definitions

Calling “human” and “rational” equal does not make their meanings identical. Here we examine the individuals of which they are predicated.

“Human” and “animal” are also conceptually connected, but their extensions differ: humans belong to the animal class, while horses belong to the animal class without belonging to the human class.

We therefore distinguish closeness of meaning from relation of instances. The distinction will also matter when studying the five predicables.

19. The commentary on kinds of disjointness

The commentary notes that disjointness need not involve two positive species names. Two universal predicates may also be disjoint because no individual can possess both in the same respect.

For example:
Human and non-human cannot both apply to the same individual in the same respect.
Stone and non-stone are similarly exclusive.

The notes discuss related contrasts involving negation and affirmation. The essential point remains: in disjointness there is no common instance, whether the terms are both positive or one is negative.

20. Changing one concept changes the relation

The relation cannot be determined merely because some words or ideas recur. Compare:
Animal and human — absolute inclusion.
Animal and white — partial overlap.
Human and horse — disjointness.

Different second concepts change the outcome, even when some concepts are related under a common genus.

It is therefore incorrect to brand the term “animal” permanently as general or disjoint. The relation belongs to a specified pair of concepts.

21. Relative particularity: the new definition

Immediately after the four relations, the source introduces another use of “particular”:
فصل: وقد يقال للجزئي معنى آخر، وهو ما كان أخص تحت الأعم؛
Section: “particular” may also be used in another sense, namely what is more specific under a more general concept.

This is relative particularity (juzʾī iḍāfī). “Human” remains a universal because it applies to many humans, yet it is narrower than “animal,” and thus a relative particular under animal.

The word relative signals that particularity is determined by comparison with a wider concept, not solely by how many individuals the concept can apply to.

22. Real particularity is intrinsic

As explained earlier, a real particular (juzʾī ḥaqīqī) is a concept whose own representation precludes application to many distinct individuals in the same sense. “Zayd,” as the name of a specified individual, is a straightforward case.

A relative particular need not be restricted to one individual. “Human” applies to many, yet remains relatively particular beneath “animal.”

The criteria therefore differ:
Real particular — restricted by its own conceived identity.
Relative particular — narrower than some other concept in a comparison.

23. Human can be universal and relatively particular

Both statements about human can hold:
1. Human is a universal: Zayd, Amr, Khalid and many others fall under it.
2. Human is a relative particular: it is narrower than animal and falls under that more general concept.

There is no contradiction. The first concerns application to individual humans; the second concerns comparison with animal.

Whenever “particular” is used, we must ask whether real or relative particularity is meant rather than assuming it always refers to a single person.

24. Zayd: both kinds of particularity

The source further clarifies the relation:
والنسبة بين الجزئي الحقيقي وبين هذا الجزئي المسمى بالجزئي الإضافي عموم وخصوص مطلقا؛
Between the real particular and the particular called relative there is absolute generality and specificity.

Zayd is a real particular, because he is a specified individual. He is also relatively particular, since the individual Zayd falls under the wider concept human.

Both descriptions therefore apply to Zayd, while “human” exemplifies relative particularity without being a real particular. This illustrates one-way inclusion between the classifications.

25. The source explains one-way inclusion

لاجتماعهما في «زيد» مثلاً، وصدق الإضافي بدون الحقيقي في «الإنسان» فإنه جزئي إضافي وليس بجزئي حقيقي؛ لأن صدقه على كثيرين غير ممتنع.
They coincide in Zayd, for example, while relative particularity applies without real particularity to “human”: it is relative, not real, since it can apply to many individuals.

The argument has two parts:
Zayd — both real and relative particular.
Human — relative but not real particular.

Accordingly, the category of real particulars is included within relative particulars, but not everything relatively particular is a real particular.

26. A diagnostic test for both senses

For any term, ask two different questions:
Question 1: Does its precise conceived identity exclude application to many distinct individuals? If so, it may be a real particular.
Question 2: Is it more specific than some wider concept under which it falls? If so, it is a relative particular in that comparison.

Zayd: a specified individual falling under human — both.
Human: applicable to many and falling under animal — relative, not real.

The distinction separates the identity of an individual from the position of a species within a wider genus.

27. Integrated practice and final synthesis

Review the complete classifications:
Human / rational: equality; identical range of instances.
Human / horse: disjointness; no common individual.
Animal / human: absolute inclusion; every human is an animal but not vice versa.
Animal / white: partial overlap; a white duck is shared, a dark horse and snow lie on opposite exclusive sides.

A real particular excludes multiple distinct instances by its own conceived identity. A relative particular is narrower than some wider concept. Zayd exemplifies both; human is a relative particular.

These distinctions prepare for the discussion of the five predicables and, later, rigorous definition.