"To say of two things that they are equal is absurd,
and to say of a thing that it is equal to itself
is to say nothing at all" (Wittgenstein, Tractatus 5.5303).
"It is self-evident that identity is not a relation
between objects" (Wittgenstein, Tractatus 5.5301).
The Concepts
Identity
The identity −also called "equality"− is a relationship between two terms, a and b, which takes two possible forms:
a=a. In theory, identity is posed as a relation of an object to itself.
a=b. This form, if a and b are different, is apparently a contradiction and leads to a paradox, the so-called "identity paradox" or "identity problem". The paradox consists in the fact that two things that are different are being made equal. The solution to this paradox lies in Frege's distinction between the concepts of "sense" and "reference": a and b are two different forms, expressions or senses of the same reference. The example Frege gives is that of "the morning star" = "the evening star", where the terms of equality refer to the same object: Venus. [see Union of Opposites - The Union of Meaning and Denotation].
The first form is a necessarily true, a priori, synthetic statement, which does not provide information. The second form is a contingently true, a posteriori, analytic, information-providing statement.
Identity types
There are many points of view on the concept of identity and, therefore, different types of identity, among them: ontological, epistemological, logical, abstract, quantitative, quantitative, algebraic, necessary, contingent, contingent, qualitative, formal, generic, specific, absolute, relative, intrinsic, extrinsic, extrinsic, temporal, causal, psychological, theoretical, strict, etc. We highlight, among them, the following:
Ontological identity. When the essential properties of two objects are considered to be the same.
Epistemological identity. When we consider that two concepts or ideas have the same meaning.
Logical identity. It also takes two forms: 1) When a proposition implies itself: p → p (if p, then p), which is a tautology, and therefore p ↔ p (p if and only if p); 2) When two propositions imply each other, p ↔ q (p if and only if q), e.g., "x is father of y" ↔ "y is a child of x".
Physical identity. When we say, for example, that all electrons are identical. At the intuitive level, two identical physical objects are "copies" of each other.
Predicative identity. When two objects have the same properties.
Relative identity. When a and b are identical with respect to a set of properties. For example, two identical cars with respect to color.
Necessary identity. For example, a=a (reflexivity) and a+b = b+a (commutativity of the sum).
Contingent identity. For example, "Cervantes is the author of Don Quixote" (Cervantes might not have been the author of Don Quixote).
Temporal identity. When a and b have the same properties at an instant t.
Identity theory (or identity thesis) is the theory according to which there is an identity between mental and brain processes and states.
The law of identity states that one thing is identical to itself and different from another, which uses the symbols "=" and "≠", respectively. For example, 3=3 and 3≠4. It is discussed whether or not the law of identity is part of logic.
Identity vs. indistinguishability
From the point of view of properties, we must differentiate the concepts of identity and indistinguishability, the definitions of which are as follows:
Identity.
Two objects are identical if they have exactly the same properties (definition coming from Leibniz). It is an a priori characteristic, by definition, deep and synthetic.
Indistinguishability.
Two objects are indistinguishable if we do not find any differentiating property. It is an a posteriori, experiential, superficial and analytical characteristic.
In both cases it is assumed that the properties under consideration are partial or relative, since the two objects must always differ in some respect. Identity or total or absolute indistinguishability is absurd because we could never speak of two objects, but only of one, since to say "two" is already to differentiate them. That is why we speak of "relative identity" with respect to a set of properties. And also of "intrinsic properties" (or necessary) and "extrinsic properties" (or contingent). For example, two identical cars that differ only in color (which is an extrinsic property).
The concept of identity is of great importance, mainly at the logical and philosophical level. The same is not true for that of indistinguishability, which is considered of little interest and even false from the logical point of view.
The issues
On the concept of identity, several questions arise, such as the following:
What conditions must two terms fulfill in order to establish an identity between them?
When we say, for example, "Cervantes is the author of Don Quixote", are we establishing an identity? And if so, what kind?
Is the identity part of the logic?
Is identity the same as equivalence?
What is the relationship between identity and consciousness?
Philosophical principles
In philosophy there are two opposite or dual principles:
The Principle of Indiscernibility of Identics (Pin).
If two objects are identical, then they have the same properties.
In logical notation:
(x)(y)(x=y) → (P)(Px ↔ Py)
The Principle of Identity of Indiscernibles (Pid).
If two objects have the same properties, then they are identical.
In logical notation:
(x)(y)(P)(Px ↔ Py) → (x=y)
Pid is often referred to as "Leibniz's law" (although it is sometimes attributed to Pin). But really, Leibniz was referring to the impossibility of the existence of exactly identical objects, an application of his "principle of sufficient reason" (everything must have a reason, cause, or ground).
In principle, we can suppose that Pin implies Pid, i.e., that two identical objects are indiscernible (since they have the same properties), since one advances from the deep to the superficial. But Pind does not imply Pin, i.e., two indiscernible objects might not be identical, since one walks from the superficial to the deep.
The Pin is considered correct and valid, as a fundamental principle of reason, a truth by definition and a priori. In contrast, Pid has been questioned to the point of being considered even as a false and meaningless principle, in parallel with the concept of indistinguishability.
These two principles are supposed to apply to particulars, not universals.
The principle of substitution of identicals. The problem of substitution
According to Leibniz, if there exists a statement of identity between two terms, then either of the two terms, can substitute for the other in any statement without changing the truth value of the latter. This is the "principle of substitution of identicals" or "principle of substitutability of identity". And conversely: two terms are the same if one can be substituted for the other in any sentence salva veritate (i.e., as long as truth is preserved).
For example: 1) "Cervantes is the author of Don Quixote". The identity between "Cervantes" and "the author of Don Quixote" is established in principle; 2) "Cervantes is Spanish". From this it is inferred (by substitution): 3) "The author of Don Quixote is Spanish".
The problem arises when this principle is not applicable in certain contexts. For example: "Juan wants to know if Cervantes is the author of Don Quixote". Substituting "author of Don Quixote" for "Cervantes", the result is "Pepe wishes to know if Cervantes is Cervantes", which alters its original meaning.
To account for this type of cases in which the principle of substitutability of identicals is not applicable, Quine [1968] proposed to distinguish between transparent contexts (in which the principle is applicable) and opaque contexts (in which the principle is not applicable, since the meaning of the utterance is changed). For Quine, substitutability is one of the fundamental principles governing identity.
The Different Conceptions of Identity
Aristóteles
Aristotle, in his Metaphysics, states that the notion of identity occurs in 3 forms: 1) as unity of being; 2) as unity of a multiplicity of beings; 3) as unity of a single being treated as multiple.
Hegel
For Hegel, identity expresses a relation of a higher type, by uniting particulars.
Frege
For Frege, identity is a deep relation, between senses, a and b being two different ways or senses of referring to the same object. Identity refers to the same object, not to different objects. There is no sense of identity between objects. Frege conceives identity as a primitive logical notion, i.e. indefinable from others.
Russell
In Principia Mathematica (PM), Russell defines identity from the point of view of predicate logic:
(x=y) = (φ)(φx → φy)
This is the generalized version of Leibniz's law: two objects are identical if every property that one has also has the other. And conversely: if two objects have the same properties they are identical. For Russell, then, the two principles (Pid and Pin) imply each other, i.e., they are logically equivalent.
According to this definition, in principle, identity is a derived concept, not a primitive one. But actually this definition is circular, since it uses the notion of identity to define identity.
Russell treats identity mainly in two contexts:
Theory of Descriptions.
According to Russell, the notion of identity is of no logical interest when proper names are used, but it becomes useful when definite descriptions are used. For example, a sentence like "Scott is the author of Waverley" is an identity statement between two definite descriptions: "the proper-named entity 'Scott'" and "the author of Waverley", which in turn become logical names. In effect, the sentence becomes "there exists a single x such that x is named Scott and x is the author of Waverley".
Arithmetic.
For Russell, identity is indispensable also from the arithmetical point of view. In "An Inquiry into Meaning and Truth" he states that it is theoretically impossible to count objects if identity is not considered. He gives the following example. Suppose we wish to count a collection of 5 objects (A, B, C, D, E) and that B and C are identical. If we do not consider the identity, we will count 5 objects. And if we consider it, then, when counting B, we are also counting C and, therefore, we will count only 4 objects.
Wittgenstein
Wittgenstein, in the Tractatus, subjects the philosophical notion of identity to a thorough critique:
"To say of two things that they are equal is absurd, and to say of a thing that it is equal to itself is to say nothing" (Tractatus 5.5303).
The notion of identity is irreproachable, but to express identity between objects it is not necessary to define identity formally or to use the sign of identity at all. The difference between objects is expressed by the signs themselves. If the signs are equal, there is identity. Identity is not a relation between objects but between equal signs. "Frege says that these expressions have the same meaning but different senses. But the gist of the equation is that it is not necessary to show that two expressions which are connected by the sign of equality have the same meaning: for this can be perceived from the two expressions themselves" (Tractatus 6.232). "Therefore, the sign of identity is not an essential component of logical notation" (Tractatus 5.533).
Difference between the concepts of identity and equality: "If two expressions are joined by the sign of equality, it means that they are substitutable for each other" (Tractatus 6.23).
If "being identical with itself" is a genuine predicate, then "being different from" is also. But there is no such property as "being different from" in the abstract. No two objects differ from each other just because they are different. There are neither the monadic predicates of identity nor its opposite, difference. Identity and difference are shown through signs, according to whether they are the same or different.
The notion of identity in philosophy does not correspond to the normal notion. To speak of identity in normal language is to speak of partial or relative identity.
Pin is a meaningful and true proposition. Pid is nothing but a poorly expressed tautology. Anyway, when talking about the "same properties, we must differentiate between:
Non-shared properties. For example, two people, A and B, have identical cars, but each his own.
Shared properties. For example, two siblings, A and B, have a mother in common. It makes no sense to say that A's mother is identical to B's mother, since they are the same person.
If Pid is interpreted to mean that there are no two different objects that have all their properties in common, then the principle is untestable, but trivial.
When one speaks of necessary properties of an object one is not strictly speaking of properties (one is not predicating anything) but of definitions of the objects themselves in the context of a grammar.
Russell's definition of identity is circular. It is not a formal definition, but merely a notational device to facilitate the expression of ideas. And it does not even correspond to the normal interpretation of identity in natural language.
Counting does not require the concept of identity, as Russell argues.
Russell makes a double use of the sign of identity:
As a definition or substitution, which is a mechanism for abbreviating expressions. For example, 7+5+8 = 20, i.e. we can specify 20 instead of 7+5+8. And the form a=a is a pseudo-expression, since interpreted as a substitution it makes no sense.
In the conversion of defined descriptions into logical names ("the x such that..."). But in the sentences where logical names appear there is no allusion to identity.
In short −Wittgenstein concludes− the problematic of identity is nothing but a consequence of conceptual confusions. In philosophy there are no genuine problems but mere conceptual entanglements.
Ramsey
Frank Plumton Ramsey, in his famous article "The Foundations of Mathematics" [2001], reflects on the logical concept of identity used by Russell in Principia Mathematica (PM):
The notion of identity used in PM is the result of a misinterpretation of mathematics in general. A consequence of this misinterpretation is the "axiom of infinity" that Russell needs for the logical foundation of mathematics. If the identity is correctly conceived, this axiom is either a tautology or a false proposition. [The axiom of infinity −criticized because it does not appear to be a logical truth − states that there exists an infinite set, the set of natural numbers].
Moreover, in PM, the identity depends on the "axiom of reducibility". Rejecting this axiom implies automatically rejecting the notion of identity.
[According to Russell, the axiom of reducibility is a generalized version of the Leibnizian principle of identity of indiscernibles. This axiom was introduced by Russell as part of his branching theory of types, in an attempt to solve the problem of paradoxes. It has been criticized for being too complex and for not being clearly justified at the logical level. An axiom must be somewhat self-evident, and this axiom is not].
A symbolism is needed to express that two things A and B have the same properties and yet are different.
The identity is not a genuine propositional function, but a mere symbolic notation.
Kripke
For Kripke, there are necessary and contingent identities. He defines "rigid designator" as a name that is the same in all possible worlds in which that entity exists, and does not designate anything else in those possible worlds in which that entity does not exist. Proper names are rigid designators, such as "water" and H2O", since both designate the same kind of matter in every possible world.
A necessary identity is that both terms are rigid designators. If the expression a=b is true, and if a and b are rigid designators, then it is a metaphysically true identity, i.e., true in all possible worlds. For example, "water=H2O" is metaphysically necessary, even though it cannot be known a priori that water is H2O.
A contingent identity is one that contains a term that is not a rigid designator. For example, "Franklin is the inventor of bifocals" is contingent, since Franklin might not have been the inventor of bifocals. And "the inventor of bifocals" is not a rigid designator.
Is identity part of logic?
It is justified to ask whether or not identity should form as part of logic, that is, whether logic can formalize the concept of identity. In this connection, we should take into account the following aspects:
Logical identity is nothing more than two expressions that refer, not to an object, but to a relation (e.g., the relation parent−child) between propositions.
Identity theory is a basic, neutral, universal theory, like the logical theory of quantification.
Identity is sometimes included among the logical primitives of the predicate calculus as a binary predicate. In this case it is called " first-order predicate logic with identity".
The language of the predicate calculus can be used to characterize identity using predicates and quantifiers, but there are limitations associated with such a language. Even substitutability is guaranteed without recourse to any ontological principle.
Gödel proved in 1930 that first-order predicate logic with identity is a complete axiomatic system. Gödel did not attempt to define the identity, but to characterize its operation at the logical level.
If logical truths are so by reason of their structure, then truths of identity theory (such as x≡x) would not be logical truths. To accept identity as part of logic would be to question its formal character. For example, the expressions (∀x)(x=x) and (∃ y)(x=y) are unacceptable as logical truths because they are falsifiable when we replace "=" by another predicate. However, the identity laws can be considered as abbreviated forms of logical truths of the theory of quantification, i.e., as purely formal truths that do not refer to any particular predicate.
Frege −as the logician he was− was clear: identity is part of logic. Willard van Orman Quine [1970], after studying the pros and cons, also decided to include identity in the realm of logic.
MENTAL: Equivalence and Substitution
Identity is a very important concept because it facilitates and simplifies the description of reality. It is the tendency of reason and consciousness to convert multiplicity into classes, where the elements of each class are identical to each other.
The problem of identity (and all its associated context), as well as its formalization, has a very simple solution in MENTAL by means of two essential concepts perfectly differentiated (although closely related), which are primary archetypes that cannot be dispensed with: equivalence and substitution, represented respectively by the symbols "≡" and "=".
Equivalence
Identity between two different objects is meaningless, as Frege and Wittgenstein claimed. It only makes sense when it refers to different ways or expressions of referring to the same object. Therefore, it is more appropriate to speak of "equivalence", to avoid also the phantom of the identity paradox arising, because the paradox arises fundamentally from the very denomination of "identity" or "equality". When we consider two equivalent expressions, we are connecting them, which implies an elevation of consciousness, by ascending to a higher position from which they are contemplated as equivalent. It is consciousness that unites them. The equivalence is, at the same time, ontological and epistemological.
Two expressions a and b are equivalent if they evaluate or represent the same expression. Examples:
The expressions 2^3, 4*2 and 5+3 are not the same, they are equivalent, since they all refer to the number 8.
The expressions {a b c} and {c b a} are equivalent, because they represent the same set.
The expressions ( 1...4 ) and (1 2 3 4) are equivalent because they both represent 1234.
The expressions (∞ =: 1+∞) and (∞ =: ∞+1) are equivalent because both represent infinity.
An equivalence expression is a horizontal relationship between expressions. The paradigmatic example is
⟨( x+y ≡ y+x )⟩
(commutativity of the sum of expressions).
To make two expressions equivalent, two conditions are needed: 1) The expressions on both sides of the equivalence symbol must refer to the same thing; 2) The two expressions must have the same level of significance. You cannot make something superficial and something deep equivalent. Equivalence between expressions must be horizontal, not vertical. And the substitution of an expression for its equivalent makes sense when both expressions have the same level of significance. It is not, then, a problem of opaque or transparent contexts, as Quine said, but a problem of the equivalence expression itself.
For example, it makes no sense to say that "Cervantes" is equivalent to "author of Don Quixote", i.e. (Cervantes ≡ Author(Quixote)), since the first expression we have only a name, and the second expression supposes a higher level of meaning. Neither can we make equivalent the terms Madrid and "The capital of Spain", i.e. . In both cases we have vertical relationships.
On the other hand, a horizontal relation is the already mentioned that expresses the commutativity of the sum. And also
(author(Don Quixote) ≡ author("Exemplary novels")
We can also specify equivalences between names. For example,
(London ≡ Londres).
Equivalence is not a primitive logical notion, as Frege, Russell and Quine claimed. Equivalence transcends logic, and is different from and independent of logic. The only logical notion that exists is "condition", which is totally different and independent of equivalence. Equivalence is an archetype, a semantic dimension that is indefinable, like the rest of the primitives, but everything is built from them.
The term "not equivalent" appears only as a descriptive expression or as a condition:
(x ≡' y) (x is not equivalent to y)
(z ← (x ≡' y))
(z if x and y are not equivalent)
To describe the set of natural numbers does not require the concept of identity −as Russell− claims, but that of potential substitution. For example, simply, {1...}.
In any expression, all its equivalents are always considered, the one with the greatest simplicity being automatically chosen (simplification principle). For example:
{a b c}+{c b a} // ev. 2*{a b c}
In addition to absolute equivalence (independent of context), there is relative equivalence. For example, the 4 solutions of the equation x^4 = 1 are relative equivalents:
Logical equivalence is different from normal equivalence. It is two expressions that imply each other. For example:
⟨((parent(x) = y) ↔ (child(y) = x))⟩
Logical equivalence is an equivalence relation, since the reflexive, symmetric and transitive properties are fulfilled, like normal equivalence, creating equivalence classes formed by expressions that mutually imply each other:
Reflexive:
⟨(( (p ↔ p) )⟩
Symmetric:
⟨( ((p ↔ q) → (q ↔ p)) )⟩
Transitive:
⟨( ((p ↔ q) → (q ↔ r) → (p ↔ r) )⟩
Substitution
Just as equivalence is a horizontal relationship, substitution is a vertical relationship between two expressions: a = b (a is evaluated as b). There are two forms or models of substitution:
The immediate or current substitution (x = y), which is a vertical downward relation.
Potential, deferred or representation substitution (x =: y), which is a vertical upward relationship.
With both forms, definitions (particular or general) can be made. There is additionally the "initial substitution" (x := y).
The terms "equality", "inequality" appear as conditions. Additionally inequality can be used as a description:
(z ← x=y) (z if x and y are equal)
(z ← x≠y) (z if x and y are different)
(x ≠ y) (describes that x is not equal to y)
As Wittgenstein argued, equality and difference is not a monadic predicate, but it can be expressed as dyadic, in the following alternative form:
((a b)/= = α) ↔ a=b) (a and b are equal, as condition)
((a b)/≠ = α) ↔ a≠b) (a and b are different, as a condition)
These expressions can be generalized for more than two expressions. For example,
((a b c)/= = α) ↔ (a=b ← a=c))) (a, b and c are equal)
Subject-predicate expressions vs. substitution
When we say, for example, "Cervantes is the author of Don Quixote", the "is", in general, does not indicate equivalence but predicate:
Cervantes/(author(Don Quixote))
But the fact that Cervantes is the author of Don Quixote does not indicate that he is the only author, since this work could have been written by several authors. But if we say "Cervantes is the author of Don Quixote", "is the" indicates uniqueness. This can be expressed in two ways:
(Cervantes =: author(Quixote)) (deferred substitution or representation)
In the first case, the author is identified through a particular function (author) and an argument (Quixote). In the second case, Cervantes is a tag to refer to the author of Don Quixote.
In general, "is" indicates attribute or property. And "de" indicates function argument.
We can also use declarative expressions. For example:
(satellite(Earth) = Moon) (The Moon is a satellite of the Earth)
(satellite(Earth) ≠ Ganymede) (Ganymede is not a satellite of the Earth)
Properties
Reflexive: ⟨ x≡x ⟩
Symmetrical: ⟨( x≡y → y≡x )⟩
Transitive: ⟨( x≡y → y≡z → x≡z)⟩
These first three properties determine an equivalence relation, giving rise to "equivalence classes", where a class is the set of expressions equivalent to each other.
Substitution implies equivalence, but not the other way around:
⟨( x=y → x≡y )⟩.
If two expressions x and y are equivalent, then any property p relative to x is equivalent to the same property p relative to y:
⟨( x≡y → (x/p≡ y/p) )⟩
If two expressions, x and y, evaluate as the same expression, then they are equivalent:
⟨( x=z → y=z → (x ≡ y) )⟩
The principle of identical substitution is correct. If we have an expression z containing another expression x, and if x≡y, then z is equivalent to the same expression by replacing x with y:
⟨( (z/(x = θ) ≠ z))→ (x ≡ y) → (z ≡ z/(x = y)) )⟩
The expression (z/(x = θ) ≠ z/(x = θ)° ) indicates the condition that the expression z contains the expression x. For example:
{a b c}/(b=θ) (ev. {a c}, since it contains b)
{a b c}/(u=θ) (ev. {a b c}, since it does not contain u)
If two sets have the same elements, they are equivalent:
⟨(x∈A ↔ x∈B)⟩ → (A ≡ B) )⟩
Bibliography
Fogelin, Robert J. Wittgenstein (Arguments of the Philosophers). Routledge, 1995.
Frege, Gottlob. Sobre sentido y referencia. En Estudios sobre Semántica, Ediciones Folio, 2003. Y en [Valdés Villanueva, Luis Manuel, 2012]. Disponible en Internet.
Kripke, Saul. El nombrar y la necesidad. UNAM, México, 1995.
Quine, Willard van Orman. Palabra y objeto. Labor, 1968.
Quine, Willard van Orman. Desde un punto de vista lógico. Paidós Ibérica, 2002.
Quine, Willard van Orman. Filosofía de la lógica. Alianza, 1998.
Ramsey, Frank Plumton. The Foundations of Mathematics and other Logical Essays. Routledge, 2001.
Ramsey, Frank Plumton. Obra filosófica completa. Comares, 2005.
Russell, Bertrand; Whitehead, Alfred North. Principia Mathematica. Paraninfo, 1981.
Russell, Bertrand. An Inquiry into Meaning and Truth. Routledge, 1996.
Russell, Bertrand. Los problemas de la filosofía. Labor, 1991. Disponible en Internet.
Russell, Bertrand. Introducción a la filosofía matemática. Paidós Ibérica, 2002. Disponible en Internet el original en inglés.
Russell, Bertrand. Los principios de la matemática. Círculo de Lectores, 1996. Disponible en Internet el original en inglés.
SEP (Stanford Encyclopedia of Philosophy). Relative Identity, 2007.
