Identity of indiscernibles
In logical systems identity is introduced via indistinguishability: The principle of identity (also law of identity ) states that an object A is then identical with an object B if they can find no difference between A and B. The method that is detected by the identity, the comparison. The identity principle is often attributed to Gottfried Wilhelm Leibniz and therefore also called Leibniz 's law ( Leibniz's law).
- 2.1 Different formulations of the principle of identity
- 2.2 Explanation
- 2.3 Identity in the computer science
Intuitive justification
The principle of identity can be split into two claims:
- The identity of indistinguishable things
- The indistinguishability of identical things
The identity of indistinguishable things
The identity of indistinguishable things states that when things are indistinguishable, they are also identical, or equivalent: If they are not identical, so there must be a difference between them give. For example, have two different coins, even if they look absolutely identical, differ in some respects, such as their position in space.
The indistinguishability Identical things
The indistinguishability of identical things states that identical things are indistinguishable: Is there a difference between them, so they can not be identical. If it is found that a coin made entirely of copper and is one with the same value all of gold, it can not be of the same coin, because this coin would be both entirely of copper as well as all of gold, which is obviously contradictory. Indistinguishable, however, are the values of both coins as you can see on an account after the Deposit: The account will contain the values of both coins, but it is impossible to determine which part of the account balance to which the paid coins as a.
Historical consideration
The philosophical formulation of a principle of " identity of indiscernibles " goes way back, and can already be found in considerations of the Stoic, the modern view of the identity goes back to considerations of Leibniz. The historical discussion of the intuitive properties of Ununterscheidbarem found mostly by his Latin tag as principium identitatis indiscernibilium.
Discussion
Various formulations of the principle of identity
From the principle of identity, there are different formulations. The first is the allgemeinverständlichste but unpräziseste; the third, most precise formulation goes back to Leibniz:
Explanation
The relationship between the first two formulations arises from the fact that a difference between two things is always accompanied by a property which belongs to one thing and the other not. Thus, for example, a color - difference are that the one thing the property " red " plays, the other not.
Number three is a version of Leibniz 's famous formulation, " sunt quae sibi Eadem ubique substitui possunt, salva veritate " ( " The same, which may replace everywhere, with preservation of truth "). In the explanation we first go out of two expressions for the same object, eg
- The highest mountain in the world
- Mount Everest
We now replace in the statement Mount Everest is located in the Himalayas " Mount Everest " by " the highest mountain in the world ", we obtain:
- The highest mountain in the world is located in the Himalayas
The identity principle now states that this substitution is given the truth value, that is, if the first sentence is true, this must also be the second set and vice versa. Indeed, this must apply to all sentences in which the expression occurs a. If we assume, however, of expressions that do not designate the same object, such as
- The Matterhorn
- Mount Everest
So there must be, according to a principle of identity set in which an appropriate replacement will not get the truth value. Such a set is, for example:
- The Matterhorn is 8000 meters high.
This sentence is false, but we will replace him in " Matterhorn " with " Mount Everest ", we obtain the true sentence:
- Mount Everest is about 8000 meters high.
The identity principle applies fully only in so-called "extensional " languages such as the language of mathematics. In " intensional " languages ( such as the German vernacular ), it is only with restrictions. This issue affects only the principle of indistinguishability Identical, not the identity of indistinguishable. Consider the sentences:
- Frank believes that Mount Everest is located in the Himalayas.
- Frank believes that the highest mountain in the world is located in the Himalayas.
Assuming that Frank does not know that the highest mountain, Mount Everest, is now the first sentence could be true and the second false. But precisely this is likely, according to principle of identity, expressions that designate the same object, not the case. The solution of this difficulty is that the principle of identity with so-called intensional expressions (which also " believe " part ) is overridden. That is, the statements in which the change is made, such expressions can not contain (see also opaque context).
Assuming one of the first two formulations of the identity principle, so one would say: Properties like " by Frank for the Himalayas kept lying to be " are not actual properties of things ( but of Frank) and may therefore not to distinguish between Mount Everest and the highest mountain in the world are used.
Identity in the computer science
In computer science, the difference between identical store and same memory values is easily recognizable: Does the implementation of a variable in the form of a memory address to the same memory cell, the contents of a second reference to the same memory cell is identical, is in a different memory cell only may the same value.
Properties of the identity
The identity is a binary relation, that is, a relationship between two things. More particularly, it is an equivalence relation, i.e., they have the following properties:
- Reflexivity: Everything is identical with itself.
- Symmetry: If A is identical with B, then also B, where A
- Transitivity: If A is identical with B, B with C, then A with C
More specifically, can the identity relation be defined as the " finest-grained " equivalence relation in a language. This means that when a = b, for each equivalence relation R: a R b. Another equivalence relation example, would be " equal weight ". It is therefore important that when a with b is identical to a well weigh the same as b. The same is true ( same size, same color, etc. ) for all other equivalence relations.
It can be shown that this last property the maximum fine granularity characterizes the relation of identity in an unambiguous way. That is, there is in addition to the identity relation is a further equivalence relation R * with this property as: a = b if and only if a R * b
Introduction of the identity relation in formal systems
There are several ways to introduce the identity relation in a predicate logic based formal system.
In the case of predicate logic second level (or higher), the identity can be defined directly and in general:
Where F is to be a predicate variable.
This definition is a straightforward implementation of Leibniz's principle of identity.
In the case of first-order predicate logic can a definition be given if a formal theory contains a finite number of non- defined predicates. Let us consider the case of set theory with the element stem - predicate as the only undefined predicate. Then the identity is defined as follows:
For more predicates would have to be added yet for this appropriate clauses.
In the case of first order predicate logic, there is no general definition, which would be independent of the predicates used. But there is the possibility of a general introduction of either rules or axioms.
By controlling the identity can be introduced as follows
Identity elimination
Off and follows
( wherein A is the formula, have been replaced in some or all occurrences of "a" by "b ").
Identity introduction
The following applies:
The intuition behind these rules is that if it has been shown that a = b, can ( in some or all points ) in each set in a proof of A can be replaced by b. Furthermore, you can put in a proof always a = a, since this is apparently never wrong.
In the axiomatic introduction to sets the following axiom schema (called Hao -Wang formula ):
,
Read: is only true if b for all the fact that b is identical to a, implies that. The Axiom immediately implies the identity elimination of it can, however, also very easy the introduction of identity, a = a, are derived.