Equality, disjointness, inclusion and partial overlap, with worked examples and reasoning practice Read the bilingual lesson for Four Relations Between Universal Concepts and test your understanding with a focused quiz.
Lesson
1. From classification to relations
Previous lessons introduced universals such as human, animal, and body. The new task is to compare the ranges of two universals: Is one broader? Are the ranges separate? Do they partly overlap? This is the textbook’s discussion of relations between two universal concepts.
2. What it means to apply to individuals
The textbook repeatedly uses “applies to” (ṣādiq ānā). Here it means that a universal correctly describes an individual, not merely that a sentence is true. Zayd falls under both human and animal; a horse falls under animal but not human. Relations arise by comparing these ranges.
3. The four possible patterns
The four relations are equality, disjointness, strict inclusion, and partial overlap. Equality: same extension. Disjointness: no shared members. Absolute generality/specificity: one class lies wholly within a broader class. Partial generality/specificity: some overlap, with additional members exclusive to each.
4. Definition of equality
Two universals are equal when every individual of each is an individual of the other. They are mutasāwiyān. Both universal statements must hold: all A are B, and all B are A. Only one direction is not enough to establish equality.
5. Textbook example: human and rational
The source pairs “human” with nāṭiq (rational) as equal universals. The traditional term means endowed with rational capacity, not simply able to speak aloud. In that stipulated classification, every human is rational and every rational being is human. Equality depends on the technical sense.
6. Equal extensions need not mean identical definitions
Equality compares which individuals fall under each concept. It does not require identical wording or the same role in a definition. In the book, “human” names a species while “rational” expresses an essential differentiating feature, yet their extensions coincide in the classical framework.
7. Definition of disjointness
Disjoint universals have no shared individual. Neither class applies to any member of the other. The textbook uses human and horse: in the intended classification, no individual is both a human and a horse. These are mutabāyinān.
8. Work through human and horse
In the source example, faras means horse. No horse is human and no human is horse. The fact that both fall within a third, wider genus “animal” does not make the two species overlap. Sharing a broader class and sharing individual members are different.
9. Shared properties do not erase disjointness
Humans and horses share characteristics such as breathing and eating, yet remain disjoint species: no individual belongs to both. Human versus animal and horse versus animal, by contrast, are inclusion relations. The comparison depends on the exact pair of concepts.
10. Definition of strict inclusion
Absolute generality/specificity is strict inclusion: every member of one class belongs to a broader class, but not every member of the broader class belongs to the narrower one. The broad class is ʿāmm muṭlaq; the narrow class is khāṣṣ muṭlaq.
11. Textbook example: animal and human
The book uses animal and human. Every human is an animal, but not every animal is human: a horse is a counterexample to the reverse claim. Thus animal is absolutely more general and human absolutely more specific, relative to this pair.
12. The two directions of inclusion
Test both directions: all humans are animals; all animals are not humans. Some animals are human. The narrower class is fully contained within the broader class, and the broader class includes further members outside it.
13. A term can be broad in one pair and narrow in another
Generality is relative to the compared concepts. Animal is broader than human but narrower than growing body in the textbook’s hierarchy. Growing body is narrower than body. Always ask “more general than what?” instead of attaching broad or narrow permanently to a term.
14. Definition of partial overlap
Partial overlap requires three nonempty regions: individuals in both classes, individuals in only the first, and individuals in only the second. Neither class contains the other entirely, and they are not disjoint. Each is more general in one respect and more specific in another.
15. Textbook example: animal and white
The source compares animal with abyaḍ, “white.” A white horse belongs to both; a black horse belongs only to animal; a white wall belongs only to white things. These three witnesses show that the concepts overlap partially rather than one containing the other.
16. Why one common example is insufficient
A white horse shows overlap, but that fact alone does not distinguish equality, inclusion, and partial overlap. You also need an animal that is not white and a white object that is not an animal. These exclusive examples establish partial overlap.
17. Strict inclusion versus partial overlap
Both inclusion and partial overlap have shared members. The decisive distinction is whether all members of either class lie in the other. Humans are fully within animals, giving absolute inclusion. White and animal each have members outside the other, giving partial overlap.
18. A four-question diagnostic
Use a systematic test. First check both-way inclusion (equality); then absence of overlap (disjointness); then one-way inclusion (absolute inclusion); finally overlap with members exclusive to each (partial overlap). A correct classification rests on the exact range of both concepts.
19. Worked set: four textbook pairs
Worked source pairs: human/rational = equality; human/horse = disjointness; animal/human = absolute inclusion; animal/white = partial overlap. The criterion matters more than the label: same individuals, none shared, one-way inclusion, or three regions respectively.
20. Practice with broader and narrower classes
From the textbook exercise: growing body/tree is inclusion, because all trees are growing bodies but not vice versa. Stone/body is inclusion, because stones are bodies but not all bodies are stones. Human/stone is disjointness, because there is no shared member.
21. More pairs from the source questions
Further exercise pairs: Roman/human gives inclusion; sheep-or-goat/donkey gives disjointness; donkey/animal gives inclusion. The goal is to identify the range of each universal, even when unfamiliar Arabic terms appear.
22. Color and distinctive-property examples
The exercise includes animal/black: a black horse is in both, a white horse in animal only, and a black stone in black only, so the relation is partial overlap. It also includes horse/ṣāhil (neighing), treated as an identifying property coextensive with horse in the conventional textbook example. Such judgments depend on the stipulated meanings.
23. A new worked example: student and athlete
Consider student and athlete in a setting where there is a student athlete, a student who is not an athlete, and an athlete who is not a student. All three regions exist, so the relation is partial overlap. This additional everyday example applies the book’s three-region criterion.
24. Common errors and corrections
Three misconceptions: human and horse are not equal merely because both are animals; animal/human is not partial overlap merely because they share some individuals; animal/white is not equality merely because a white horse fits both. You must compare all members and seek counterexamples to each universal claim.
25. Final review: one decisive test each
Equality: both classes include exactly the same individuals. Disjointness: no individual is shared. Absolute inclusion: the narrower class is entirely inside the broader class, with additional members of the broader class. Partial overlap: shared members plus members unique to each class. The four textbook anchors are human/rational, human/horse, animal/human, and animal/white.
Practice questions
Question 1
What do we compare to determine the relation between two universals?
- Number of letters
- Which individuals fall under each universal
- Their page positions
- Which word is pronounced first
Question 2
If each universal applies to every individual of the other, the relation is:
- Disjointness
- Partial overlap
- Equality
- Absolute inclusion
Question 3
Which is the textbook’s primary equality example?
- Human and rational
- Human and horse
- Animal and human
- Animal and white
Question 4
What does nāṭiq mean in the textbook example?
- Only speaking loudly
- Possessing rational capacity
- Only a writer
- Only an Arabic speaker
Question 5
What is the relation between human and horse?
- Equality
- Disjointness
- Absolute inclusion
- Partial overlap
Question 6
What is decisive evidence for disjointness?
- No shared individual exists
- One is larger
- One shared individual appears
- Some members overlap
Question 7
Why can human and horse be disjoint despite both being animals?
- Because they have different spellings
- Because animal means nothing
- Because no individual belongs to both
- Because every horse is human
Question 8
Which pair is an example of absolute inclusion?
- Human/horse
- Human/rational
- Animal/human
- Animal/white
Question 9
Which is absolutely more general: animal or human?
- Human
- Animal
- Both equal
- Neither overlaps
Question 10
Which is absolutely more specific in animal versus human?
Question 11
What additional fact establishes strict inclusion for human and animal?
- Every animal is human
- Not every animal is human
- No human is animal
- They are disjoint
Question 12
Why is animal broader than human but narrower than growing body?
- Its meaning changes daily
- Generality is relative to the comparator
- Animal is always the broadest
- Human is always narrowest
Question 13
How many regions are checked for partial overlap?
- Only shared members
- Shared, first-only, second-only
- Only first and second
- None
Question 14
Which textbook pair illustrates partial overlap?
- Human/rational
- Human/horse
- Animal/human
- Animal/white
Question 15
Which is in both animal and white?
- White horse
- Black horse
- White wall
- Red stone
Question 16
Which lies in animal but not white?
- White wall
- Black horse
- White horse
- White paper
Question 17
Which lies in white but not animal?
- White wall
- White horse
- Black horse
- Black cow
Question 18
What does finding one white horse establish?
- They are disjoint
- They share at least some members
- They are equal
- Every animal is white
Question 19
Which condition distinguishes absolute inclusion from partial overlap?
- One class lies entirely inside the other
- Both words are Arabic
- They share no members
- Both consist of one word
Question 20
What is the relation between growing body and tree?
- Equality
- Disjointness
- Absolute inclusion
- Partial overlap
Question 21
What is the relation between stone and body?
- Disjointness
- Equality
- Partial overlap
- Absolute inclusion
Question 22
What is the relation between human and stone?
- Disjointness
- Equality
- Absolute inclusion
- Partial overlap
Question 23
What is the relation between Roman and human?
- Equality
- Absolute inclusion
- Disjointness
- Partial overlap
Question 24
What is the relation between sheep/goat and donkey?
- Equality
- Partial overlap
- Disjointness
- Absolute inclusion
Question 25
Which is broader: donkey or animal?
- Donkey
- Animal
- Both equal
- Neither
Question 26
How are animal and black related?
- Partial overlap
- Equality
- Disjointness
- Absolute inclusion
Question 27
In the source’s conventional usage, horse and neighing are treated as:
- Disjointness
- Equality
- Absolute inclusion
- Partial overlap
Question 28
In the traditional classification, sentient and animal are:
- Equality
- Disjointness
- Absolute inclusion
- Partial overlap
Question 29
If all A are B and all B are A, which relation holds?
- Disjointness
- Absolute inclusion
- Equality
- Partial overlap
Question 30
If all A are B but not all B are A, which relation holds?
- Absolute inclusion
- Disjointness
- Equality
- Partial overlap
Question 31
If A and B have no shared members, which relation holds?
- Equality
- Disjointness
- Absolute inclusion
- Partial overlap
Question 32
If some A are B, some A are not B, and some B are not A, the relation is:
- Equality
- Absolute inclusion
- Disjointness
- Partial overlap
Question 33
If student-athletes, nonathlete students, and nonstudent athletes exist, the relation is:
- Disjointness
- Equality
- Partial overlap
- Absolute inclusion
Question 34
What is wrong with “Human and horse are both animals, so they are equal”?
- Confusing shared genus with shared individuals
- Animal has no meaning
- Every human is horse
- Every horse is human
Question 35
What follows from saying only that animal and human share some members?
- Partial overlap follows immediately
- Equality follows immediately
- The complete relation still requires further checking
- They are disjoint
Question 36
Why may animal be broader in one comparison and narrower in another?
- Generality is relative to the other class
- Because pronunciation changes
- Because broad classes have no members
- All universals are always equal
Question 37
Which fact refutes equality and establishes absolute inclusion?
- All A are B, with one B outside A
- No overlap
- Same members both ways
- Partial overlap with members unique to both
Question 38
What does “applies to” mean here?
- Every sentence being true
- A universal correctly describing an individual
- Reading aloud
- Calling two words synonyms
Question 39
How many relations between two universals does this lesson enumerate?
Question 40
To establish equality, knowing only that all humans are animals is sufficient.
Question 41
Human and horse remain disjoint even though both are animals.
Question 42
Partial overlap has shared and exclusive members on both sides.
Question 43
Every animal is human.
Question 44
The relation between universals is determined by the number of words.
Question 45
A white horse, black horse, and white wall jointly demonstrate animal/white partial overlap.
Question 46
A class broader than human must be broader than every other class.
Question 47
The textbook treats human and rational as an example of equality.
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