General Law of Identity


Overcoming the Problem of Self-reference

Published in the Philosophical Review, 2019, 

 Winner of the 2019 Boethius Prize


If we go by how “Self-reference” is defined, then the Law of Identity (A = A) is its evident example:

“Self-reference occurs in natural or formal languages when a sentence idea or formula refers to itself”. (1)

Identifying “A” by itself, carries no useful information.  In logical terms, the Law of Identity is a tautology (a useless repetition, as for example: 1 =1).  While this is the case, one may feel puzzled indeed that a tautology has been considered as the first law of logic.  

Among those who objected to this concept of the Law of Identity was Friedrich Hegel (1770 – 1841)

 “Likewise, Hegel attacked the Law of Identity and claimed that “the Law of Identity says very little in itself”. The fact that A equals A is no more than a tautology and has little meaning – it tells us almost nothing about the identity of a thing.  The only way a thing truly takes on identity is through what Hegel called its“otherness” or what is not.  What a thing ‘is not’ is as necessary to its identity as what ‘it is’ since what ‘is not’ is what gives a thing boundaries, definition and meaning. Thus a thing’s otherness must be contained within the very identity of a thing”. (2)

In other word, Hegel was saying that: we need to mention “thing’s otherness” - for example a ‘contrast’ between an object and what-is-not that object, to be able to distinguish it.  This is a convincing argument, but for some reason it did not attract further attempts to examine the root problem of self-reference in the Law of Identity.  Instead, some philosophers tried to find at least one field where A = A can be meaningful.  As Ilya Satvinsky explains in his article “The essence of formal logic” - the only validity of the Law of Identity is when we treat “A” just as a symbol without physical content - as is the case in mathematics:

“In mathematical logic the expression “A = A” has a well-defined meaning, which no one disputes but the expression loses its meaning outside of mathematical logic”. (3)

It can be argued, however, that mathematics speaks about equality of quantities, not about their identity.  For example, two lines in a plane can be “equal” in length, but each line has its own coordinates, which are differently situated; the lines are equal, but they are not the same:

Plato notes that the sense of “the same” as applied to mathematical objects and to the ideas is different: properly speaking, sameness (identity) applies only to ideas while in mathematics sameness means equality….” (4)

Identity based on Uniqueness

In his work on the concept of individuation, Leibnitz (1646 – 1716) formulated a statement – suggesting that an individual object “A” needs nothing else but itself (to be the individual object “A”):

Every singular substance does not need as individuating principle anything more than its entity” (5)

Don’t we perceive a spirit of self-reference hidden within Leibnitz' Principle of Individuation?  If by “A” we refer to a certain “singular substance”, then the ‘Principle of Individuation’ says that the entity of this “singular substance” is just “A” – but this is no more than a circular reference of “A” to itself. 

We miss Hegel when we read Leibnitz.  Hegel was enlightening us to the importance of “others” – in general terms - to identify an object or an individual, while Leibnitz made “the others” completely transparent in his focus on object’s self-sufficiency.  

The necessity to include “the others” in the process of defining the existence of an entity is an essential principle in Eastern philosophical teachings.  From the perspective of Eastern philosophical concepts, the starting point in observation of an object is where does it belong: in what group of other objects it would fit.  This perspective is derived from the essential truth of “Interconnectedness of beings”.  Nothing exists on its own.  The origin of an individual object cannot be itself; it is dependent on other objects:

“[Dependent Origination] teaches that no beings or phenomena exist on their own; they exist or occur because of their relationship with other beings and phenomena. Everything in the world comes into existence in response to causes and conditions. That is, nothing can exist independent of other things or arise in isolation”. (6)

Could it be that Leibnitz meant that an object must be unique to distinguish it as an individual object?  This was not clearly stated in his principle, but the concept of individuation seems to resonate well with “uniqueness” of discerned object.

On the other hand, everyone is unique.  To be distinguished as being a unique individual, one must belong to a group of individuals among whom one is distinguished.  A person can be unique among peers, among other people - but not among trees or fish.  This means that identifying a unique entity necessarily requires referencing it to a general group of belonging, in which the unique entity is a specific member.

Identity through the fusion of the two categories: the “General” and the “Specific”

Any object under observation must be contained within a pattern, a category, or a general set of objects.   The following example offers an explanation about how to identify a specific object.  Suppose that an accident involving a certain car took place and that the police is trying to identify that car.  As it is common in such circumstances, the description of the sought car would be usually given first through its general properties (such as manufacturer, model and year), followed by its specific properties (such as registration, colour, etc.).  

In this example, identifying a certain car “A” would involve a reference to two components: one related to general properties (A general), and the other related to the specific properties (A specific), The sought car “A” is then fully identified by its specific properties (A specific) of registration, colour, etc, as a unit belonging to its general group (A general) of manufacturer, and model.  Proper identification must include both the general and the specific.  An individual, for example is identified through a specific name and a family of belonging.    

An object “A” can be easily defined by its specific properties {A specific} - or simply {As}, and by its general set of belonging{A general}- or simply {Ag}.  Using the symbol ∈, which stands for “belonging to”, we have a full expression for identifying “A”:

A ≡ {As} ∈ {Ag}

This expression of identity can be referred to as the General Law of Identity, in which the problem of self-reference is dissolved by regarding object “A” as a unique object within a general group of reference (to which “A” belongs).  

Universality of the General Law of Identity

While the conventional Law of Identity (A = A) is valid only for mathematical objects,

the general Law of Identity A ≡ {As} ∈ {Ag} applies to all objects, abstract and physical alike.  

Take for example a number, say A = 5.  According to the General Law of Identity, the general property of number 5 is its belonging to the general set of positive integers, while its specific property is that it is the only element in that general set that is greater than 4 and smaller than 6.  Obviously, there is no number 5, which can exist without having also other numbers like 4, 6 and in fact all others.  On the other hand, to identify number 5 by just as being number 5 (according to the Conventional Law of Identity 5 = 5 ) this becomes a tautology with no information about the nature and belonging of that number.

As for the application of the General Law of Identity (A ≡ As ∈ Ag) in the physical field of objects, then the previously mentioned example of identifying a car, can be extended – for example – to identifying a person. A person is a unique individual, described by his/her specific properties {As} and by belonging to a certain general group being a family or race {Ag}.  Various features of individual can be developed in many ways, and this means that the properties {As} are not fixed or stationary but allow for change.  

There is, however, one concrete element of uniqueness, which never changes: the individuals' event of birth within a family to whom the individual belongs.  We may find many individuals in the world who share most of their specific personal properties, including the one-off event of date and place of birth, but when birth is referenced to a family of belonging, there can be no two individuals who can share the date, time, place and origin.  

Defining the identity of an individual requires not only the individual’s self-entity, but a reference of others (here being the family).  Uniqueness of the event of birth in a certain family is carried along unchanged over the passage of time - and remains as such.  Even that most (or all) specific properties of an individual can change - nonetheless uniqueness of identity is maintained over the passage of time.

Identity over time

The reason why individuals undergo drastic changes in their identity is that their specific properties include the potential of change.  For example, in identifying a certain child “A”, the child’s specific properties include the potential of becoming an adult, an essential property of being a child.  A child and an adult are different and separate states, but a child ‘possessing the potential to be an adult’ combines both states.  The actual state (of being a child) contains within its properties the potential (of becoming an adult).

Identity over time is a difficult philosophical problem.  For example, the following passage (from an article published by “Hume Society”) reflects the difficulty and limitations in explaining identity over changing phases of time:

“What is identity?….  Hume thinks that our idea of identity involves confusion, first of all because it is based on vacillating between viewing (A) as two things and viewing it as one, and more fundamentally because it mistakenly applies passage of time, or duration, to the unchanging (A)”. (7)

Where does this confusion about identity over time come from?  Regarding a dynamic entity “A” as fixed (A = A) does not allow “A” to express its potentials of change. To overcome this problem, we can employ the perspective of identifying “A” through the set of specific properties (As), which are a function of time.  This perspective allows for the property of growth (or change over time) to exist within the identity of A ≡ As ∈ Ag.

Despite changes a person undergoes, the uniqueness of the person (being the event of birth within a family of belonging) - this uniqueness is an unchanging event of existence – and thus can be maintained over time.

Identity as a dynamic entity

The static formula of A = A does not help in defining a dynamic entity.  According to Aristotle, an object possesses its potential to change, and to refer to this property of change he used the word “dumanis”.  However, potentiality implies various scenarios of the future state of observed object, and not a fixed or concrete description.  This lack of concreteness about the future state of an object made potentiality undefinable:

A dunamis in this sense is not a thing’s power to produce a change but rather its capacity to be in a different and more completed state.  Aristotle thinks that potentiality so understood is indefinable, claiming that the general idea can be grasped from a consideration of cases”. (8)

Applying the terminology o the General Law of Identity A ≡ {As} ∈ {Ag}, the potentials of change of object “A” are part of the description of the specific properties {As}.  This mean that scenarios of potentiality of future states are contained within the actual state of object (although undefined in concrete terms).

The concept of potentials (or latent states of identity) is also discussed in Eastern philosophies, and it emerged through the search for a solution to the problem of sufferings.  One’s actual state can be that of hardships, but the same actual state of current hardships contains within itself a potential (or a latent state) of change - allowing for transformation of one state into another.  

The concept of ‘potentiality contained within the actual state’ described by the specific properties {As} - applies also to any physical phenomenon or natural occurrence under observation:     

“An understanding of [latency], therefore, helps us to see that, despite how we may see them, things–people, situations, relationships, our own lives–are not fixed, but dynamic, constantly changing and evolving. They are filled with latent potential which can become manifest at any time”. (9)

Differently from the static structure of the Conventional Law of Identity A = A,  the General Law of Identity: 

A ≡ {As} ∈ {Ag}

can explain the dynamism of a developing identity, maintaining its uniqueness over time.



(1)  Definitions for Self-Reference 

(2)  Jesus After Modernity, p. 84, James Danaher

(3)  The Development of Laws of Formal Logic of Aristotle

(4)  Phiolosophia Sceintiae. Studies in the history of Sciences and Philosophy

(5)  Leibnitz on Individualisation

(6)  Dependent Origination, Nichiren Buddhism Dictionary

(7)  Constancy, Coherence and Causality, Ira Schnall, Hume Studies vol.30  

(8)  Aristotle – metaphysics 

(9)  The concept of Emptiness in SGI Buddhism