# Précis of epistemology/The logic of identity and similarities

## Contents

- 1 Nothing new under the sun
- 2 Difference or identity of indiscernibles?
- 3 Science of the individual or science of the general?
- 4 Similarity
- 5 Reasoning by similarity and typology
- 6 Structures as properties
- 7 Symmetrical structures
- 8 The principle of equivalence of all observers
- 9 Does Nature really obey laws?
- 10 The diversity of names of the same being
- 11 The principle of indiscernibility of identicals
- 12 The identity of individuals in naturally possible worlds
- 13 The identity of properties and relations

## Nothing new under the sun[edit]

If everything was always new, we could never predict anything, because a predicted novelty is not really a novelty.

If we could never predict anything, we could not verify our observations, because to verify, we must anticipate what must be observed.

If we could not verify our observations, we could not develop knowledge about the observable reality.

In order for knowledge to be possible, everything must not always be new, rather, everything must always be the same.

The light that comes from distant stars is the same as that of the Sun, or the light we produce on Earth. It always behaves the same way. *« There is nothing new under the Sun. »* (Ecclesiastes) The laws of optics are among the best known and they are always verified, often with excellent precision. Throughout the Universe light is always the same and always obeys the same laws.

Light reveals the properties of matter. A natural substance can always be identified by spectroscopy, the analysis of the light absorbed or emitted.

Light reveals that matter is always the same everywhere in the Universe. We can know the chemical composition of distant stars by analyzing their light.

Natural substances always behave in the same way as soon as they are pure. Pure water always has the properties of pure water. It always obeys the same laws. For it too, there is nothing new under the Sun.

A natural substance is pure if it consists of molecules or atoms of the same species. Pure water contains only molecules.

Being of the same species does not exclude a priori having different properties. A species can be divided into subspecies. For a natural substance to be truly pure, the molecules of the same species must not be able to be separated into subspecies, otherwise the substance would be a mixture. The chemical species that characterize pure materials are ultimate species, which can not be further analyzed.

The existence of chemically pure substances suggests that atoms and molecules are naturally indiscernible, because two molecules of the same chemical species can not belong to different varieties. But the argument is not fully conclusive. Molecules could be individualized by different properties without these properties distinguishing subspecies.

That light always obeys the same laws suggests that its particles, the photons, are naturally indiscernible. Nothing in the behavior of a photon can distinguish it from another photon.

Two natural beings are naturally indiscernible when they have the same intrinsic natural properties. The intrinsic natural properties are the properties that they retain when the circumstances vary, when they are taken in various naturally possible arrangements.

The nature of a being is determined by its natural properties. A being is nothing without its properties. They make everything it is. Two beings with the same natural properties are essentially similar. One can always be replaced by the other. Everything that is naturally possible with one is naturally possible with the other.

The physics or elementary particles and thermodynamics require that particles, atoms, and molecules of the same species are naturally indiscernible.

## Difference or identity of indiscernibles?[edit]

The principle of identity of indiscernibles: if all that is true of one being is equally true of the other and vice versa, then they are the same being.

In a sense, the identity of indiscernibles is trivial. If x is different from y, there is at least one property of x that is not a property of y, to be identical to x.

But for natural indiscernibility the principle of identity of indiscernibles is false. The particles, atoms and molecules of the same species are naturally indiscernible, they have essentially the same natural properties.

The principle of identity of indiscernibles seems to be true if one walks in a garden in search of two naturally indiscernible leaves, because leaves are very complex natural beings. The diversity of naturally possible leaves far exceeds all the leaves that our universe could contain. There is a negligible probability that two leaves have exactly the same natural properties, but this possibility is not excluded by the laws of Nature. They do not forbid that natural beings have exactly the same constitution and the same natural properties. On the contrary, they show that all the elementary beings of the same natural species are naturally indiscernible.

Complex beings such as ourselves are always naturally discernible. There can not be two twins that are naturally indiscernible, because in the actual world the natural possibilities only occur once, as soon as they are sufficiently complex. The probability that two complex natural possibilities are indiscernible is always negligible.

## Science of the individual or science of the general?[edit]

Is science more about knowing individuals or laws?

Science must be the knowledge of the truth about reality. But only individuals are truly real. Laws are only theories. Science must be the science of individuals to really be science.

Science must be universal knowledge, the same for everyone. Concrete individuals have an ephemeral existence. They come from the dust and become dust again. How could the knowledge of an individual deserve the name of science? How could it be a knowledge that deserves to be shared by all?

A law is always about a domain of individuals. It may not mention any particular individual. Its truth is then general and depends only on the properties and relations it mentions. It is therefore a knowledge of the universal. It seems that to deserve to be shared by all, science must be the knowledge of the laws, so be a knowledge of the universal. Science is sometimes defined in this way: the universal knowledge of the universal (Aristotle, *Posterior Analytics*).

Is it necessary to conclude that science knows only laws and that it ignores individuals?

The knowledge of the laws is not separated from that of the individuals because we know the individuals from their properties and their relations and because we know the properties and relations from the laws that determine their place in the totality.

To know an individual well it is not enough to perceive it, or to know what others have perceived of it, it is also necessary to know how to reason about it, because one can thus deduce what has not yet been perceived. And to reason, one must know laws. Even if one is interested above all in concrete individuals, and not in laws, one needs laws, hence the science of the general, to make the science of individuals.

An individual is always known with the constellation of its properties and relations. Such a conjunction is generally sufficient to identify it among the other individuals, because it is sufficient to distinguish it from the others, but it does not completely determine its individuality, because it could be equally true of another individual, a kind of twin, naturally very similar. An individual is therefore never really known in its individuality, but only as an example of a natural possibility.

Some properties are true of only one individual: the property of being identical to x is true only of x. They are called haecceities. A haecceity identifies a being in its individuality. Now a concrete individual is always known from its observable properties and it can always in principle have a twin, which has the same intrinsic observable properties. To determine a haecceity one needs more than the intrinsic observable properties of the identified individual. A haecceity is determined by the relations between the identified individual and other individuals.

Permanence is an identification criterion that identifies a sustainable being. If we follow the trajectory of a mobile, and if it has not evaporated along the way, we can be pretty sure that the individual at the end is the same as at the beginning. This is how we recognize ourselves as individuals. I know that I am myself and that I will never be another because I am a permanent witness of myself. Permanence enables one to identify a sustainable being by its relations with its origins (Kripke 1972).

More generally, all beings are identified by their relations with other beings. In particular, we situate all concrete beings from ourselves and identify them by their relations with us.

## Similarity[edit]

Two beings are similar when a part of what is true of one is also true of the other. Two beings are different when a part of what is true of one is false of the other.

Assigning a property to an individual is a way of knowing its similarities and differences. As soon as the same property is true of several beings they are all similar, and all different from all beings which do not have the same property.

As individuals are always known from their properties (relations are properties of couples or of systems of individuals), to know is always to know similarities and differences.

Why are there some general truths? Events and individuals are always very different. Why should the same affirmation be true of numerous individual cases? Beings are all different, but they are also very similar. Laws state what they have in common.

When a law is true of many individuals, they are all alike, because they obey the same law, and all different from those who do not obey it. To know the laws, one must know the ways in which beings can be alike. The principles of science always state fundamental similarities between individuals and between systems.

## Reasoning by similarity and typology[edit]

A reasoning by similarity is to say that since a and b are similar, what was said to be true of a must be equally true of b. Used without restriction a reasoning by similarity is usually wrong, because two similar beings are also different, there are true statements about one that are not true about the other. A reasoning by similarity is not logically correct.

To justify a reasoning by similarity, we can use typology. a and b are not only said to be similar, they are said to belong to the same type, and that what has been said about a is true of all instances of this type. Thus we state a law. Then we can conclude logically that what has been said about a is also true of b.

A common property is shared by all individuals of the same type. If p is a common property of the type t, then the law 'for all x of type t, x is p' is true.

The use of typologies is fundamental to all sciences. One can think of mathematical structures (as types of mathematical objects), species of elementary particles, atoms and molecules, living species...

## Structures as properties[edit]

When we speak of similarity between two individuals, we mean that some of the properties that are attributed to one can be attributed to the other. When we speak of similarity between two systems, the expression 'what is true of one is equally true of the other' can receive a more subtle meaning. We mean that there exists a projection f which makes it possible to replace the individuals x of the first system by individuals f(x) of the second system, in such a way that true statements about the first system are replaced by true statements about the second system. Such a projection is called in mathematics a morphism, or an isomorphism if it is bijective, to say that the two systems have the same form, or the same structure.

The current use of the concept of structure is ambiguous. The structure designates sometimes the object, the system, sometimes its property. Structures have a structure. From a logical point of view, a structure as an object is a logically possible world or part of such a world. A structure as a property can be defined from the equivalence relation x has the same structure as y. This equivalence relation can be defined with the concept of isomorphism:

**Two structures** (or two systems)** have the same structure if and only if they are isomorphic.**

An isomorphism between two structures E and F is a bijective function f which replaces the individuals of E by individuals of F so that all fundamental properties and relations are retained. Formally:

If P is a fundamental property, for all x in E, x has the property P if and only if f(x) has the property P.

If R is a fundamental binary relation, for all x and all y in E, xRy if and only if f(x)Rf(y)

The same goes for the fundamental relations between more terms.

(A relation between the elements of E and the elements of F defines an application of E into F when each element of E is connected to a single element of F. An application of E into F is bijective when each element of F is connected to a single element of E. In other words, a bijective function is an application whose inverse is also an application.)

An isomorphism between two structures makes it possible to transform all true statements about one into true statements about the other, by replacing everywhere x by f(x). When two structures are isomorphic, they are models of the same theories. Any system of axioms of true one is necessarily true of the other.

A complex natural being is a natural structure, defined with natural properties and relations. Two isomorphic complex natural beings are essentially similar, naturally indiscernible. They have the same natural properties. Everything that is naturally possible with one is naturally possible with the other. The nature of a complex natural being is its structure. Two isomorphic complex natural beings have the same nature.

The concept of isomorphism is often defined in a more general way. The bijective function f is allowed to replace not only individuals but also properties and relations, always in such a way that true statements on one system are replaced by true statements on another system. When the similarity between systems is defined in this way, it is commonly said that similar systems are analogous and that the projection f is an analogy. An isomorphism can be defined as a bijective analogy.

We can also define the concept of structure in a more general way:

**Two structures have the same structure if and only if they are models of the same theory.**

With this second definition, a structure as a property is determined by the axioms of a theory. More precisely, systems of different axioms define the same structure when they have the same models, when any model of one is a model of the other.

A theory is categorical when all its models are isomorphic. The fundamental structures of mathematics, the set of natural numbers and the set of real numbers in particular, are determined with categorical theories. A categorical theory forbids any contingency. There is essentially one logically possible world that obeys its principles. The laws of Nature do not determine a categorical theory of Nature. They leave room for contingency.

When a theory is not categorical, different, non-isomorphic, structures or systems may have the same structure, as defined by the theory. For example, we can say of all vector spaces that they have a vector space structure.

## Symmetrical structures[edit]

An automorphism of a structure E is an internal isomorphism, an isomorphism from E to E.

Every structure has a trivial automorphism, the identity-function defined by id(x)=x.

A structure is symmetrical when it has at least one non-trivial automorphism.

A non-trivial automorphism is a symmetry of a structure.

The automorphisms of a structure form a group, in the algebraic sense, because the inverse of an automorphism is an automorphism and because the composite of two automorphisms is also an automorphism.

The group of all the automorphisms of a structure is also called the group of its symmetries. For example, the group of symmetries of a circle, or a disc, is the group of rotations around their center and reflections with respect to a diameter.

When there is an automorphism g such that y=g(x), x and y are essentially indistinguishable within the structure, in the sense that any truth on one can be transformed into an equivalent truth on the other. .

The equivalence class, or orbit, of an element x of a symmetric structure is the set of y such that y=g(x) where g is an automorphism of the structure.

An equivalence class is a set of elements that are essentially indistinguishable within the structure. For example, all the points of a circle are in the same equivalence class because there is nothing on the circle to distinguish them. All the points of a disc at the same distance from the center are also in the same equivalence class, but different concentric circles are different equivalence classes, because the points are distinguished by their distance to the center.

A structure is symmetrical when it contains distinct but essentially indistinguishable elements, because their properties and their relations within the structure determine distinct but equivalent places.

A natural structure is perfectly symmetrical when it contains naturally indiscernible elements such that their relations within the structure give them equivalent places.

A natural structure is imperfectly symmetrical when it contains naturally very similar elements such that their relations within the structure give them equivalent or almost equivalent places.

## The principle of equivalence of all observers[edit]

*« It is not possible that the divinity be envious. »*

(Aristotle, *Metaphysics*, Book A, 983a)

To develop empirical science, we must assume that all experimenters are equivalent in the sense that any observation made by one can be made again by another. Experiments must be reproducible. If an experiment is not so, then it is not well controlled. For experiments to be reproducible, it is necessary in particular that their results do not depend on the place or time. Experimental conditions can be reproduced always and everywhere, and have to lead always to the same result. By postulating the principle of equivalence of observers, we thus assume at the same time that the laws of physics are true always and everywhere. This leads to define the symmetry group of space-time. All points of space-time are necessarily similar, they are all in the same equivalence class. When we know one, we know them all. It is the same for all directions in space and more generally for all frames of reference. There is no center of space-time, no preferred direction in space (isotropy, no up and down) and no absolute state of rest (Galileo-Einstein relativity principle). Like the Knights of the Round Table, but in a much larger space, space-time observers never have a privileged position. The group of symmetries of space-time (Poincaré's group) is a mathematical translation of the principle of equivalence of all observers, just as the group of symmetries of a round table is a mathematical translation of the principle of equivalence of all knights.

The principle of equivalence of all observers is not only a foundation of theoretical physics but also of all sciences, because reason requires that knowledge be universal, that all that is known by one can be known by all others.

When we understand that we are all equally capable of knowing, we understand at the same time the great principle on which we can base all rational knowledge. Everything happens just like if the truth was a generous divinity, which gives wisdom to those who really want to know it. The first truth about truth is precisely that it is generous. It is not jealous, it does not deprive us of the best. It would not be the best if it deprived any of us of the best. Knowing that knowledge can be shared by all, we have the fundamental knowledge which enables us to understand all rational knowledge.

## Does Nature really obey laws?[edit]

We believe in the conclusions of our reasoning because we believe in the truth of the laws with which we reason. We believe that we are capable of developing science because we believe in the principle of equivalence of all observers.

It is in the nature of the mind to reason and therefore to postulate laws with which to reason. A mind can not develop its mind without thinking of laws. It seems, then, that the existence of laws results from the nature of the mind. But, in general, matter seems naturally without spirit, why should it obey laws?

To justify our knowledge and the principle of equivalence of all observers, we need to postulate that Nature obeys laws, but is it really a justified belief? Is not it rather taking one's desire for a reality? It may be that all the laws of Nature to which we now believe are all refuted by future observations. And could not Nature be without law?

Matter would not be matter if it did not obey laws. Matter is necessarily detectable, so it must obey laws of detection, which result from fundamental laws of interaction. A matter which obeys no law would not be detectable, and there would be no reason to call it matter. We do not know what it might be, it seems inconceivable.

It is as if matter and spirit had been made for each other, because the nature of matter is to obey laws, and the nature of mind is to know the laws.

Neither matter, nor a fortiori life and consciousness, could exist and develop if Nature did not obey laws. We would not be here to talk about it.

We do not have to expect from our experiences that they definitely prove that Nature obeys laws, which they can not do, since any law verified today could be refuted tomorrow, but only that they would help to find the laws of Nature. We know beforehand that Nature obeys laws but we do not know which ones. Since Nature does not seem to be mischievous, but rather generous, it seems that honest work and well-controlled experiments are enough to find and prove the laws to which it obeys. If a law is verified by a well-controlled experiment, or if it is a logical consequence of already well-established premises, it can be regarded as proven, as long as it is not refuted.

## The diversity of names of the same being[edit]

**x = y** means that x and y are names of the same being. We need the relation of identity when we can not conclude from the diversity of names to the diversity of beings because the same being can be named in many ways.

Knowing the diversity of the names of the same being can teach us a great deal about it when nouns are compound expressions. Aristotle is the best student of Plato means Aristotle = The best student of Plato. "The best student of Plato" is one of the many names of Aristotle.

"The best student of" is the name of a function that associates a teacher with his or her best student. In a general way, we name all beings by giving them simple names and names composed with functions.

A function **f(x)** with one argument **x** defines a binary relation **y = f(x)**.

A function **f(x,y)** with two arguments **x** and **y** defines a ternary relation **z = f(x,y)** .

The same goes, of course, for functions that have more arguments.

Functions are also called operators.

By replacing the functions with the relations they define, one can always associate with a structure defined with functions an equivalent structure defined only with relations. This is why it is not necessary to mention functions in the definition of logically possible worlds.

## The principle of indiscernibility of identicals[edit]

Knowing that **x=y** means that x and y are names of the same being, the principles of reflexivity of identity **x=x**, of symmetry, **if x=y then y=x**, and of transitivity, **if x=y and y=z then x=z** are true by definition, like the principle of indiscernibility of identicals:

If **x=y**, all that is true of x is also true of y.

**If E(x) and x=y then E(y)**

for any statement **E(x)** about x.

The principle of indiscernibility of identicals makes it possible to prove the principle of transitivity. By replacing **E(z)** with **w=z** we get:

**If w=x and x=y then w=y**

It can also be used to deduce the principle of symmetry from the principle of reflexivity, by replacing **E (z)** by **z=x**:

**If x=x and x=y then y=x**

Or **x=x**

So :

**If x=y then y=x**

**x=x** can be understood in two ways: a being is always identical to itself, or a name x must always name the same being.

## The identity of individuals in naturally possible worlds[edit]

When we reason about the possibilities that are available to us, we reason about the naturally possible arrangements of actual beings, including ourselves. We therefore reason on different possible worlds that contain the same beings. The same individuals exist virtually in many possible worlds.

When one argues about absolute possibilities, there is not much sense in identifying the same individual in different worlds. For example, if one reason on two different possible material universes, there is no sense in saying that a point or a particle of one is identical to a point or a particle of the other. And although I imagine that I could have other destinies, the other versions of me are never really me. I am not responsible for their virtual acts.

A natural being exists in only one naturally possible world. For us, this world is the actual world. But the nature of a natural being is determined by its natural properties, and the nature of natural properties is determined by their place in all naturally possible worlds. That is why the nature of a natural being is determined by its place in all naturally possible worlds even if a natural being exists in a single naturally possible world.

A reasoning on the same individual in several naturally possible worlds can always be replaced by reasoning on different but naturally indiscernible individuals (Lewis 1986, but his theory of possible worlds is different).

## The identity of properties and relations[edit]

A property or a natural relation is determined by its place in all naturally possible worlds, therefore by its place in a system of axioms which defines the laws of Nature.

More generally, a property or a theoretical relation is determined by its place in a system of axioms which defines a theory.

Two natural properties that are true of the same beings in all naturally possible worlds occupy the same place. They are therefore essentially the same property. The same goes for natural relations. We have therefore justified the principle of extensionality of properties and natural relations:

**Two properties or natural relations are identical if and only if they are true of the same beings in all naturally possible worlds.**

The same principle of extensionality is obtained for theoretical properties and relations:

**Two properties or theoretical relations are identical if and only if they are true of the same beings in all the models of the theory, that is to say in all the logically possible worlds such that its axioms are true.**