Identity (Sameness)

In philosophy, identity (also called sameness) is whatever makes an entity definable and recognizable, in terms of possessing a set of qualities or characteristics that distinguish it from other entities. Or, in layman's terms, identity is whatever makes something the same or different..

In logic, the identity relation (also called "equality") is normally defined as the binary relation that holds only between a thing and itself. That is, identity is the two-place predicate, "=", such that for all x and y, "x = y" is true if x is the same thing as y. Identity is transitive, symmetric, and reflexive. It is an axiom of most normal modal logics that for all x and y, if x = y then necessarily x = y. That is, identity does not hold contingently, but of necessity.

Subsumption is a primitive relation between an object and an idea (an object is subsumed under an idea if that idea represents it, e.g. Socrates is subsumed under the idea “philosopher”).

Subordination is a relation between ideas defined in terms of subsumption (an idea is subordinate to another idea if all objects subsumed under the former idea are subsumed under the latter but not conversely, e.g. “Greek” is subordinate to “European”),

Part relation between ideas (an idea is a part of another idea, e.g. “rational” is a part of the idea “man”), intersection (denoted as ∩) of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

The negative cases give rise to three kinds of incompatibility: exclusion, contradiction, opposition and contrariety (incompatibility without contradiction). Exclusion differs from incompatibility only in comparing three or more ideas or collections of ideas: the ideas A, B, C, … exclude each other if they are incompatible and if not even two of them are compatible with each other. To define contradiction, Bolzano needs also the universal class which is the extension of the concept “something in general”. As all these relations are derived from compatibility and its negation, it is possible to represent both the relations between ideas and those between propositions in the form of a genealogical tree.

In classical logic, a contradiction (contradictoriness) consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle’s law of non-contradiction states that “One cannot say of something that it is and that it is not in the same respect and at the same time.”

The relationship between opposites is known as opposition. Opposites are words that lie in an inherently incompatible binary relationship as in the opposite pairs male - female, long - short, up - down, and precede - follow. The notion of incompatibility here refers to fact that one word in an opposite pair entails that it is not the other pair member. For example, something that is long entails that it is not short. It is referred to as a 'binary' relationship because there are two members in a set of opposites. A member of a pair of opposites can generally be determined by the question What is the opposite of X ?

Contrariety is the relation between contraries. The sentences 'X is blue all over' and 'X is red all over' are contraries since both cannot be simultaneously true. On the Aristotelian square of opposition, the A and E type propositions ('All As are Bs' and 'No As are Bs', respectively) are contraries of each other. Propositions that cannot be simultaneously false (e.g. 'Something is red' and 'Something is not red') are said to be subcontraries.

The logical operations with concepts

The basic logical operation is definition. A definition is a passage that explains the meaning of a term (a word, phrase or other set of symbols), or a type of thing. The term to be defined is the definiendum (plural definienda). A term may have many different senses or meanings. For each such specific sense, a definiens (plural definientia) is a cluster of words that defines it.

A chief difficulty in managing definition is the need to use other terms that are already understood or whose definitions are easily obtainable. The use of the term in a simple example may suffice. By contrast, a dictionary definition has additional details, typically including an etymology showing snapshots of the earlier meanings and the parent language.

Like other words, the term definition has subtly different meanings in different contexts. A definition may be descriptive of the general use meaning, or stipulative of the speaker's immediate intentional meaning. For example, in formal languages like mathematics, a 'stipulative' definition guides a specific discussion. A descriptive definition can be shown to be "right" or "wrong" by comparison to general usage, but a stipulative definition can only be disproved by showing a logical contradiction.

A precising definition extends the descriptive dictionary definition (lexical definition) of a term for a specific purpose by including additional criteria that narrow down the set of things meeting the definition.

An intensional definition, also called a coactive definition, specifies the necessary and sufficient conditions for a thing being a member of a specific set. Any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition.

An extensional definition, also called a denotative definition, of a concept or term specifies its extension. It is a list naming every object that is a member of a specific set. So, for example, an intensional definition of “prime minister” might be the most senior minister of a cabinet in the executive branch of government in a parliamentary system. An extensional definition would be a list of all past, present and future prime ministers.

One important form of the extensional definition is ostensive definition. This gives the meaning of a term by pointing, in the case of an individual, to the thing itself, or in the case of a class, to examples of the right kind. So, you can explain who Alice (an individual) is by pointing her out to me; or what a rabbit (a class) is by pointing at several and expecting me to 'catch on'.

An enumerative definition of a concept or term is an extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible for finite sets and only practical for relatively small sets.

A new definition can be composed of two parts:

1. A genus (or family): An existing definition that serves as a portion of the new definition; all definitions with the same genus are considered members of that genus, and a definition can be composed of multiple genera (more than one genus).

2. The differentia: The portion of the new definition that is not provided by the genera.

For example, consider these two definitions: “a triangle is a plane figure bounded by 3 straight sides”; “a quadrilateral is a plane figure bounded by 4 straight sides”. Those definitions can be expressed as a genus and 2 differentiae:

2.1. A genus: A plane figure.

2.2. Differentiae:

- the differentia for a triangle: bounded by 3 straight sides.

- the differentia for a quadrilateral: bounded by 4 straight sides.

Continuing the process of differentiation:

- a rectangle: a quadrilateral with 4 right angles.

- a rhombus: a quadrilateral with all 4 sides having the same length.

Importantly, differentiae can include genera. For instance, consider the following: a square - a rectangle where all 4 sides are the same length. This definition could be recast as follows:

• a square: a rectangle that is a rhombus.

• a square: a rhombus that is a rectangle.

• a square: a quadrilateral that is both a rectangle and a rhombus.

• a square: both a rectangle and a rhombus.

The rules for definition by genus and differentia

Certain rules have traditionally been given for this particular type of definition:

1. A definition must set out the essential attributes of the thing defined.

2. Definitions should avoid circularity. To define “a priori” as “transcendental” would convey no information whatsoever. For this reason, a definition of the term must not comprise the terms which are synonymous with it. This would be a circular definition, a “circulus in definiendo”. Note, however, that it is acceptable to define two relative terms in respect of each other. Clearly, we cannot define 'antecedent' without using the term 'consequent', nor conversely.

3. The definition must not be too wide or too narrow. It must be applicable to everything to which the defined term applies (i.e. not miss anything out), and to nothing else (i.e. not include any things to which the defined term would not truly apply).

4. The definition must not be obscure. The purpose of a definition is to explain the meaning of a term which may be obscure or difficult, by the use of terms that are commonly understood and whose meaning is clear. The violation of this rule is known by the Latin term obscurum per obscurius. However, sometimes scientific and philosophical terms are difficult to define without obscurity.

5. A definition should not be negative where it can be positive. We should not define 'wisdom' as the absence of folly, or a healthy thing as whatever is not sick. Sometimes this is unavoidable, however. We cannot define a point except as 'something with no parts', nor blindness except as 'the absence of sight in a creature that is normally sighted'.