# Identity (logic)

In logical systems, identity is introduced through indistinguishability: The identity principle (also the principle of identity ) states that an object A is identical to an object B if and only if no difference can be found between A and B. The method by which identity is recognized is through comparison . The principle of identity is often Gottfried Wilhelm Leibniz attributed and therefore law Leibniz ( english Leibniz's law called).

## Intuitive justification

The identity principle can be split into two requirements:

• the identity of indistinguishable things
• the indistinguishability of identical things

### The identity of indistinguishable things

The identity of indistinguishable things means that if things are indistinguishable, they are also identical, or equivalent: if they are not identical, there must be a difference between them. For example, two different coins, even if they look absolutely alike, must differ in some way, such as their position in space.

### The indistinguishability of identical things

The indistinguishability of identical things means that identical things are indistinguishable: if there is a difference between them, they cannot be identical. If one finds that a coin is made entirely of copper and one with the same value is entirely made of gold, then it cannot be the same coin, because this coin would then be made entirely of copper as well as entirely of gold, which is obviously contradictory . On the other hand, the values ​​of both coins are indistinguishable , as you can see after they have been deposited into an account: the account then contains the values ​​of both coins, but it is impossible to determine which part of the account balance belongs to which of the deposited coins.

### Historical consideration

The philosophical formulation of a principle of the identity of the indistinguishable goes back a long way and can already be found in the considerations of the Stoics, the modern view of identity goes back to considerations by Leibniz. The historical discussion of the intuitive properties of the indistinguishable can usually be found under its Latin catchphrase as principium identitatis indiscernibilium .

## discussion

### Different formulations of the identity principle

There are different formulations of the identity principle. The first is the most easily understandable but the most imprecise; the third, most precise formulation goes back to Leibniz:

1. An object A is identical to an object B if and only if there is no difference between A and B.
2. An object A is identical to an object B if and only if all properties that A belong to also belong to B and vice versa.
3. A and B denote the same object if and only if A can be substituted for B in all statements while maintaining the truth value.

### Explanation

The connection between the first two formulations results from the fact that a difference between two things is always associated with a property that belongs to one thing and not to the other. For example, a color difference could be that one thing has the property red and the other not.

Number three is a version of Leibniz's famous formulation Eadem sunt quae sibi ubique substiti possunt, salva veritate ("The same ones are who can replace each other everywhere, while maintaining the truth"). In the explanation we start with two expressions for the same object, e.g. B. from

• the highest mountain on earth
• the mount everest

If you replace the term Mount Everest in the statement that Mount Everest is in the Himalayas with the highest mountain on earth you get:

• The highest mountain on earth is in the Himalayas

The identity principle now states that this substitution receives the truth value. If the first sentence is true, so must the second sentence and vice versa. In fact, this must apply to all sentences in which the one expression occurs. If, on the other hand, we start from expressions that do not designate the same object as

• the Matterhorn
• the mount everest

so, according to the principle of identity, there must be a sentence in which a corresponding substitution does not receive the truth value. Such a sentence is for example:

• The Matterhorn is over 8000 meters high.

This sentence is wrong; but if you replace the Matterhorn in it with Mount Everest, we get the true sentence:

• Mount Everest is over 8000 meters high.

The principle of identity applies without restriction only in extensional languages ​​such as the language of mathematics. In intensional languages ​​such as colloquial German it only applies with restrictions. This problem only concerns the principle of the indistinguishability of identical, not that of the identity of indistinguishable. Consider the following sentences.

• Frank believes that Mount Everest is in the Himalayas.
• Frank believes that the highest mountain on earth is in the Himalayas.

Given that Frank does not know that the highest mountain is Mount Everest, the first sentence could be true and the second false. However, according to the principle of identity, this is precisely what should not be the case with expressions that designate the same object. The solution to this problem is that the identity principle is overridden in the case of intensional expressions (which also include believing that belongs). The statements in which the substitution is made must not contain such expressions (see also opaque context ).

If one proceeds from one of the first two formulations of the identity principle, one would say that properties like those of Frank to be held in the Himalayas are not actual properties of things (but of Frank) and are therefore not allowed to distinguish between Mount Everest and highest mountain on earth.

### Identity in computer science

In computer science , the difference between identical memories and the same memory values ​​is easier to recognize: If the implementation of a variable in the form of a memory address refers to the same memory cell , the content of a second reference to the same memory cell is identical; in another memory cell there may only be the same value.

## Properties of identity

Identity is a two-digit relationship , a relationship between two things. More precisely, it is an equivalence relation with the following properties:

• Reflexivity : Everything is identical to itself.
• Symmetry : If A is identical with B, so is B with A.
• Transitivity : If A is identical to B, and B to C, then A to C as well

The identity relation can be determined even more precisely than the finest-grained equivalence relation in a language. That means that also applies to for every equivalence relation . Another equivalence relation would be equally difficult , for example . So if with is identical, that is also as difficult as is. The same applies to all other equivalence relations (same size, same color, etc.). ${\ displaystyle a = b}$ ${\ displaystyle \ sim}$ ${\ displaystyle a \ sim b}$ ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle a}$ ${\ displaystyle b}$ It can be shown that this last property of maximum fineness characterizes the identity relation in a unique way. If there is a further equivalence relation with this property besides the identity relation , then the following applies if and only if${\ displaystyle \ sim ^ {*}}$ ${\ displaystyle a = b}$ ${\ displaystyle a \ sim ^ {*} b}$ ## Introduction of the identity relation in formal systems

There are different ways of introducing the identity relation in a formal system based on predicate logic .

In the second level (or higher) predicate logic, the identity can be defined directly and generally with a predicate variable : ${\ displaystyle F}$ ${\ displaystyle a = b \ equiv _ {def} \ forall F (F (a) \ equiv F (b))}$ This definition is a straightforward implementation of the Leibnizian identity principle.

In first order predicate logic, a definition can be given when a formal theory contains a finite number of undefined predicates . Let us consider the case of a set theory with the element predicate as the only undefined predicate. Then the identity is to be defined as follows: ${\ displaystyle \ in}$ ${\ displaystyle a = b \ equiv _ {def} \ forall x (a \ in x \ equiv b \ in x) \ wedge \ forall x (x \ in a \ equiv x \ in b)}$ If there are several predicates, corresponding clauses would have to be added for them.

However, in first order predicate logic there is no general definition that is independent of the predicates used. But there is the possibility of a general introduction either through rules or axioms .

The identity can be introduced through rules as follows

Identity Elimination

Out and follows${\ displaystyle a = b}$ ${\ displaystyle \ phi}$ ${\ displaystyle \ phi [a \ leftarrowtail b]}$ (where is a formula in which some or all of the occurrences of have been replaced by). ${\ displaystyle \ phi [a \ leftarrowtail b]}$ ${\ displaystyle a}$ ${\ displaystyle b}$ Identity introduction

The following applies: ${\ displaystyle a = a}$ The intuition behind these rules is that once you have shown that a = b, you can replace a (in some or all of the places) with b in any theorem in a proof. Furthermore, one can always set a = a in a proof, since this is obviously never wrong.

The axiomatic introduction uses the following axiom scheme (the so-called Hao-Wang formula):

${\ displaystyle \ phi \ equiv \ forall b (a = b \ supset \ phi [a \ leftarrowtail b])}$ ,

Read: is true if and only if for all the fact that is identical with , conditions that . The axiom immediately implies the elimination of identity; however, it is also very easy to derive the identity introduction ,, from it. ${\ displaystyle \ phi}$ ${\ displaystyle b}$ ${\ displaystyle b}$ ${\ displaystyle a}$ ${\ displaystyle \ phi [a \ leftarrowtail b]}$ ${\ displaystyle a = a}$ 