Definition according to AristotlePhilip Pennance1 -Version: July 5, 2014

1. Aristotle (384-322 BC) introduced theconcept of definition. According toAristotle, the definition of a speciesconsists of genus proximum and differ-entia specifica. The differentia specificais that part of the definition not pro-vided by the genus.

2. The genus expresses an is–a relation-ship. Two species with the same genusare said to be members of that genus.The differentia is the attribute by whichone species is distinguished from allothers of the same genus. Two mem-bers of a genus are distinct if and onlyif their differentiae are distinct. Defi-nition of species within a fixed genus is(not surprisingly) called differentiation.

3. Differentiation

(a) An animal is a human if it has thecapacity for reason.

(b) An animal is an elephant if it hasa trunk.

Example (a) defines, per genus proxi-mum et differentia specifica, the specieshuman by presenting the genus (ani-mal) and the differentia specifica (ca-pacity for reason). Example (b) definesanother member of the same genus bydifferentiation.

4. Definition in Chemistry:

A reaction is exothermic if it liberatesheat.

The species of chemical reaction calledexothermic is defined by specifying thegenus (chemical reaction) and the dif-ferentia specifica (“liberates heat”).

5. Mathematical definition is also ofgenus-differentia type. For example:

An integer n is even if there exists an

integer k such that n = 2k.

In the definition of even number the setof integers plays the role of genus. Thedifferentia specifica, ”twice an integer”,is expressed by the predicate, “there ex-ists an integer k such that n = 2k”.

6. It is convention in mathematical writ-ing, and elsewhere, to bold, italicize, orunderline, the species.

7. Relations

The differentia is often a compoundstatement. In the definition of relation,the genus is triple. The differentia is aconjunction of the three predicates (a)–(c) below. Specifically:

A triple (A,B,G) is a relation if

(a) A is a set,

(b) B is a set,

(c) G is a subset of A×B.

8. Example

A relation (A,B,G) is a function if forevery element a ∈ A, there is exactlyone element b ∈ B such that (a, b) ∈ G.

We can use all or part of an existing def-inition to define a new genus, a processknown as abstraction. The definition offunction above takes the previously de-fined concept of relation as the genus.Thus relation is both a species of thegenus triple and the genus of the speciesfunction.

9. Example

A function (A,B,G) is an endomor-phism if A = B.

In this example, the concept of functionis abstracted to provide the genus of thethe species endomorphsm.

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Substance(summum genus)

Man (dead or alive)(the individual substance

of a rational nature)

Body(material)

Spirit(immaterial)

Prof. PennanceLiving

(animate)Nonliving

(inanimate)

Animal(sensitive)

Plant(insensitive)

Tree of Porphyry - Improved

The processes of differentiation and ab-straction can be illustrated by a lattice, eachvertex of which represents a species. Ver-tices are labelled with the name of the speciesand possibly also the corresponding differen-tia. The lowest leaves represent the lowestspecies in the chains of abstraction. Lowestspecies remain undefined.

Porphyry (234 - 305 AD) presented Aris-totle’s classification of categories as a directedtree (drawn above). In this example differen-tiae are enclosed by parentheses.

The lattice below shows some species ob-tained by differentiation and abstraction ofthe summum (highest) genus triple.

Triple (Summum Genus)(A,B,G)

Relation

Partial FunctionFunction Complete Relation

(A,B,A×B)

Endomorphism Bijection TrigonometricFunction

EvenFunction

(IR, IR, G)where

G = {(x, 2x + 1) : x ∈ IR}

LogarithmicFunction

CosineFunction

Natural Logarithm (IR, IR+, {(x, x2) : x ∈ IR})

Some Important Triples

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Exercises

1. Give definitions per genus proximus et differentia specifica of the following concepts:

(a) Triangle.

(b) Maximum of anordered set.

(c) Sequence.

(d) Odd integer.

(e) Prime integer.

(f) Binary relation.

(g) Valid argument.

(h) Rhombus.

(i) Implication.

2. Explain what (if anything) is wrong with each of the following “definitions”:

(a) A circle is a figure all of whose points are equidistant from a given point.

(b) An elephant is an animal with four feet.

(c) A man is a dentist if he practices dentistry.

(d) A thief is a man who steals money.

3. Look up the concept of virtue in Aristotle’s works on Ethics. Identify the genus anddifferentia.

4. In Plato’s Meno how does Socrates explain what is wrong with Meno’s definition,“virtue is the capacity to rule”?

5. In the Mathematics Standards of Puerto Rico, the term pattern is mentioned multipletimes, but the notion of definition nowhere. Provide a clear definition of the wordpattern as used in these standards.

References

Philip Pennance, Mathematics Standards of the Puerto Rico Department of Education:Analysis and Recommendations

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