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The identity of indiscernibles is an ontological
Ontology

Ontology in philosophy is the study of the nature of being, existence or reality in general, as well as of the basic category of being and their relations....
 principle which states that two or more object
Object (philosophy)

In philosophy, an object is a thing, an entity, or a being. This may be taken in several senses.In its weakest sense, the word object is the most all-purpose of nouns, and can replace a noun in any sentence at all....
s or entities
Entity

An entity is something that has a distinct, separate existence, though it need not be a material existence. In particular, abstractions and legal fictions are usually regarded as entities....
 are identical (are one and the same entity), if they have all their properties
Property (philosophy)

In modern philosophy, mathematics, and logic, a property is an attribute of an Object ; thus a red object is said to have the property of redness....
 in common. That is, entities x and y are identical if any predicate
Predicate (logic)

Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common....
 possessed by x is also possessed by y and vice versa. A related principle is the indiscernibility of identicals, discussed below.

The principle is also known as Leibniz's law since a form of it is attributed to the German philosopher Gottfried Wilhelm Leibniz.






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The identity of indiscernibles is an ontological
Ontology

Ontology in philosophy is the study of the nature of being, existence or reality in general, as well as of the basic category of being and their relations....
 principle which states that two or more object
Object (philosophy)

In philosophy, an object is a thing, an entity, or a being. This may be taken in several senses.In its weakest sense, the word object is the most all-purpose of nouns, and can replace a noun in any sentence at all....
s or entities
Entity

An entity is something that has a distinct, separate existence, though it need not be a material existence. In particular, abstractions and legal fictions are usually regarded as entities....
 are identical (are one and the same entity), if they have all their properties
Property (philosophy)

In modern philosophy, mathematics, and logic, a property is an attribute of an Object ; thus a red object is said to have the property of redness....
 in common. That is, entities x and y are identical if any predicate
Predicate (logic)

Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common....
 possessed by x is also possessed by y and vice versa. A related principle is the indiscernibility of identicals, discussed below.

The principle is also known as Leibniz's law since a form of it is attributed to the German philosopher Gottfried Wilhelm Leibniz. It is one of his two great metaphysical principles, the other being the principle of sufficient reason
Principle of sufficient reason

The principle of sufficient reason states that anything that happens does so for a definite reason. In virtue of which no fact can be real or no statement true unless it has sufficient reason why it should not be otherwise....
. Both are famously used in his arguments with Newton
Isaac Newton

Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history....
 and Clarke
Samuel Clarke

Samuel Clarke was an English philosopher.The son of Edward Clarke, an alderman who represented the city of Norwich, England in parliament, was educated at the free school of Norwich and at Caius College, Cambridge....
 in the Leibniz-Clarke correspondence
Leibniz-Clarke correspondence

The Leibniz-Clarke correspondence was a scientific, theological and philosophical debate conducted by epistolary means, between the Germany thinker Gottfried Wilhelm Leibniz and Samuel Clarke, an England supporter of Isaac Newton between 1715 and 1716....
.

Some philosophers have decided, however, that it is important to exclude certain predicates - or purported predicates - from this principle. This is necessary to avoid either triviality or contradiction. For example (as detailed below), the predicate which denotes whether an object is equal to x (often considered a valid predicate). As a consequence, there are a few different versions of the principle in the philosophical literature, of varying logical strength - and some of them are termed "the strong principle" or "the weak principle" by particular authors, in order to distinguish between them..

Quine
Quine

In computing, a quine is a computer program which produces a copy of its own source code as its only output.A quine is a fixed point of an execution environment, when the execution environment is viewed as a function....
 thought that the failure of substitutivity in intensional contexts (e.g., "Sally believes that p," "It is necessarily the case that q") shows that modal logic is an impossible project. Saul Kripke
Saul Kripke

Saul Aaron Kripke is an American philosophy and logician, now emeritus from Princeton University. He teaches as distinguished professor of philosophy at CUNY Graduate Center....
 holds that this failure may be the result of the use of the disquotational principle
Disquotational principle

The disquotational principle is a philosophical theorem which holds that a speaker will accept 'p' if he or she believes p. The scare quotes indicate that the statement p Use?mention distinction....
 implicit in these proofs, and not a failure of substitutivity as such.

Associated with this principle is also the question as to whether it is a logical principle, or merely an empirical
Empirical

The word empirical denotes information gained by means of observation, experience, or experiment, as opposed to theory. A central concept in science and the scientific method is that all evidence must be empirical, or empirically based, that is, dependent on evidence or Logical consequence that are observable by the senses....
 principle.

Identity and indiscernibility

There are two principles here that must be distinguished (two equivalent versions of each are given in the language of the predicate calculus). Note that these are all second-order
Second-order logic

In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
 expressions. Neither of these principles can be expressed in first-order logic
First-order logic

First-order logic is a formal deductive system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus , the lower predicate calculus, the language of first-order logic or predicate logic....
.

  1. The indiscernibility of identicals
    • For any x and y, if x is identical to y, then x and y have all the same properties.
  2. The identity of indiscernables
    • For any x and y, if x and y have all the same properties, then x is identical to y.


Principle 1. is taken to be a logical truth and (for the most part) uncontroversial. Principle 2. is controversial. Max Black
Max Black

Max Black was a distinguished United Kingdom-United States philosopher, who was a leading influence in analytic philosophy in the first half of the twentieth century....
 famously argued against 2. (see Critique, below).

The above formulations are not satisfactory, however: the second principle should be read as having an implicit side-condition excluding any predicates which are equivalent (in some sense) to any of the following:
  1. "is identical to x"
  2. "is identical to y"
  3. "is not identical to x"
  4. "is not identical to y"
If all such predicates are included, then the second principle as formulated above can be trivially and uncontroversially shown to be a logical tautology
Tautology (logic)

In propositional logic, a tautology is a propositional formula that is true under any possible Valuation of its propositional variables. For example, the propositional formula is a tautology, because the statement is true for any valuation of A....
: if x is non-identical to y, then there will always be a putative "property" which distinguishes them, namely "being identical to x".

On the other hand, it is incorrect to exclude all predicates which are materially equivalent (i.e. contingent
Contingency (philosophy)

In philosophy and logic, contingency is the status of propositions that are not necessarily true or necessarily false. Here are four classes of propositions, some of which overlap:...
ly equivalent) to one or more of the four given above. If this is done, the principle says that in a universe consisting of two non-identical objects, because all distinguishing predicates are materially equivalent to at least one of the four given above (in fact, they are each materially equivalent to two of them), the two non-identical objects are identical - which is a contradiction.

Critique


Symmetric universe

Max Black
Max Black

Max Black was a distinguished United Kingdom-United States philosopher, who was a leading influence in analytic philosophy in the first half of the twentieth century....
 has argued against the identity of indiscernibles by counterexample. Notice that to show that the identity of indiscernibles is false, it is sufficient that one provide a model
Model theory

In mathematics, model theory is the study of mathematical Structure such as Group , fields, graph , or even models of set theory, using tools from mathematical logic....
 in which there are two distinct (non-identical) things that have all the same properties. He claimed that in the symmetric universe where only two symmetrical spheres exist, the two spheres are two distinct objects, even though they have all the properties in common.

Black's argument is significant because it shows that even relational properties (properties specifying distances between objects in space-time) fail to distinguish two identical objects in a symmetrical universe. The two objects are, and will remain, equidistant from the universe's line of symmetry.

Hacking objects to Black's claim that Leibniz's principle is violated, however, by citing in modern physics the possibility that one object can appear to be two objects in a non-Euclidian space. Any journey, theoretical or otherwise, between objects in a symmetrical universe is subject to the possibility that one has gone not from object a to b, but from object a back to a. It is thus vain, claims Hacking, to use possible spatiotemporal worlds to refute or support Leibniz's principle.

Indiscernibility of identicals

As stated above, the principle of indiscernibility of identicals - that if two objects are in fact one and the same, they have all the same properties - is mostly uncontroversial. However, one famous application of the indiscernibility of identicals was by René Descartes
René Descartes

Ren? Descartes , , also known as Renatus Cartesius , was a French philosophy, mathematician, scientist, and writer who spent most of his adult life in the Dutch Republic....
 in his Meditations on First Philosophy
Meditations on First Philosophy

Meditations on First Philosophy is a philosophy treatise written by Ren? Descartes first published in Latin language in 1641. The French language translation was made by the Duke of Luynes with the supervision of Descartes and was published in 1647 with the title M?ditations Metaphysiques....
. Descartes concluded that he could not doubt the existence of himself (the famous cogito ergo sum
Cogito ergo sum

"'" , sometimes misquoted as ' , is a philosophy statement in Latin used by Ren? Descartes, which became a foundational element of Western philosophy....
 argument), but that he could doubt the existence of his body. From this he inferred that the person Descartes must not be identical to his body, since one possessed a characteristic that the other did not: namely, it could be known to exist.

This argument is rejected by many modern philosophers on the grounds that it allegedly derives a conclusion about what is true from a premise about what people know. What people know or believe about an entity, they argue, is not really a characteristic of that entity. Numerous counterexamples are given to debunk Descartes' reasoning via reductio ad absurdum
Reductio ad absurdum

Reductio ad absurdum , also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument and derives an absurd or ridiculous outcome, and then concludes that the original claim must have been wrong as it led to an abs...
, such as the following argument based on a secret identity
Secret identity

A secret identity is an Fiction#Elements of fiction wherein a character develops a separate persona , while keeping their true identity hidden. The character also may wear a disguise ....
:

  1. Entities x and y are identical if and only if any predicate possessed by x is also possessed by y and vice versa.
  2. Clark Kent is Superman's secret identity; that is, they're the same person (identical) but people don't know this fact.
  3. Lois Lane
    Lois Lane

    Lois Joanne Lane-Kent is the primary love interest of Superman in the DC Comics? Superman stories. Created by writer Jerry Siegel and artist Joe Shuster, she First appearance in Action Comics #1 ....
     thinks that Clark Kent
    Clark Kent

    Clark Joseph Kent is a fictional character created by Joe Shuster and Jerry Siegel. He serves as the civilian and secret identity of the superhero Superman....
     cannot fly.
  4. Lois Lane thinks that Superman
    Superman

    Superman is a Character , a comic book superhero widely considered to be an American cultural icon. Created by American writer Jerry Siegel and Canadian-born artist Joe Shuster in 1932 while both were living in Cleveland, Ohio, Ohio, and sold to DC Comics in 1938, the character first appeared in Action Comics Action Comics 1 and subseque...
     can fly.
  5. Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly.
  6. Therefore, Superman is not identical to Clark Kent.
  7. Since in proposition 6 we come to a contradiction with proposition 2, we conclude that at least one of the premises is wrong. Either:
    • Leibniz's law is wrong; or
    • A person's knowledge about x is not a predicate of x; or
    • A person is capable of holding conflicting beliefs.
Either of which will undermine Descartes' argument.

However, this depends on the assumption that modal properties are reducible to beliefs or knowledge; if they are not, then this argument fails. A defender of Descartes will argue that his argument is not about what we know or believe, but about what is logically possible; our knowledge or belief is a way of knowing possibility, rather than possibility itself.

See also

  • Disquotational principle
    Disquotational principle

    The disquotational principle is a philosophical theorem which holds that a speaker will accept 'p' if he or she believes p. The scare quotes indicate that the statement p Use?mention distinction....


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