The identity of indiscernibles is an
ontological
In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality.
Ontology addresses questions like how entities are grouped into categories and which of these entities exis ...
principle that states that there cannot be separate
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
s or
entities
An entity is something that exists as itself, as a subject or as an object, actually or potentially, concretely or abstractly, physically or not. It need not be of material existence. In particular, abstractions and legal fictions are usually ...
that have all their
properties
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy and ...
in common. That is, entities ''x'' and ''y'' are identical if every
predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...
possessed by ''x'' is also possessed by ''y'' and vice versa. It states that no two distinct things (such as
snowflake
A snowflake is a single ice crystal that has achieved a sufficient size, and may have amalgamated with others, which falls through the Earth's atmosphere as snow.Knight, C.; Knight, N. (1973). Snow crystals. Scientific American, vol. 228, no. ...
s) can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the
indiscernibility
The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities ''x'' and ''y'' are identical if every predicate possessed by ''x'' ...
of identicals, discussed below.
A form of the principle is attributed to the German philosopher
Gottfried Wilhelm Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
. While some think that Leibniz's version of the principle is meant to be only the indiscernibility of identicals, others have interpreted it as the conjunction of the identity of indiscernibles and the indiscernibility of identicals (the converse principle). Because of its association with Leibniz, the indiscernibility of identicals is sometimes known as Leibniz's law. It is considered to be one of his great metaphysical principles, the other being the
principle of noncontradiction and the
principle of sufficient reason
The principle of sufficient reason states that everything must have a reason or a cause. The principle was articulated and made prominent by Gottfried Wilhelm Leibniz, with many antecedents, and was further used and developed by Arthur Schopenhau ...
(famously been used in his disputes with
Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
and
Clarke
Clarke is a surname which means "clerk". The surname is of English and Irish origin and comes from the Latin . Variants include Clerk and Clark. Clarke is also uncommonly chosen as a given name.
Irish surname origin
Clarke is a popular surname i ...
in the
Leibniz–Clarke correspondence
The Leibniz–Clarke correspondence was a scientific, theological and philosophical debate conducted in an exchange of letters between the German thinker Gottfried Wilhelm Leibniz and Samuel Clarke, an English supporter of Isaac Newton during the ...
).
Some philosophers have decided, however, that it is important to exclude certain predicates (or purported predicates) from the principle in order to avoid either triviality or contradiction. An example (detailed below) is the predicate that 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.
Willard Van Orman Quine
Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". ...
thought that the failure of
substitution in intensional contexts (e.g., "Sally believes that ''p''" or "It is necessarily the case that ''q''") shows that
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
is an impossible project.
Saul Kripke
Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American philosopher and logician in the analytic tradition. He was a Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emerit ...
holds that this failure may be the result of the use of the
disquotational principle
The disquotational principle is a philosophical principle which holds that a rational speaker will accept "''p''" if and only if he or she believes ''p''. The quotes indicate that the statement ''p'' is being treated as a sentence, and not as a ...
implicit in these proofs, and not a failure of substitutivity as such.
[Kripke, Saul. "A Puzzle about Belief". First appeared in, ''Meaning and Use''. ed., A. Margalit. Dordrecht: D. Reidel, 1979. pp. 239–283]
The identity of indiscernibles has been used to motivate notions of
noncontextuality within quantum mechanics.
Associated with this principle is also the question as to whether it is a
logical
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
principle, or merely an
empirical
Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
principle.
Identity and indiscernibility
Both identity and indiscernibility are expressed by the word "same".
''Identity'' is about ''numerical sameness'', it is expressed by the equality sign ("="). It is the relation each object bears only to itself. ''Indiscernibility'', on the other hand, concerns ''qualitative sameness'': two objects are indiscernible if they have all their properties in common.
Formally, this can be expressed as "
". The two senses of ''sameness'' are linked by two principles: the principle of ''indiscernibility of identicals'' and the principle of ''identity of indiscernibles''. The principle of ''indiscernibility of identicals'' is uncontroversial and states that if two entities are identical with each other then they have the same properties.
The principle of ''identity of indiscernibles'', on the other hand, is more controversial in making the converse claim that if two entities have the same properties then they must be identical.
This entails that "no two distinct things exactly resemble each other".
Note that these are all
second-order
Second-order may refer to:
Mathematics
* Second order approximation, an approximation that includes quadratic terms
* Second-order arithmetic, an axiomatization allowing quantification of sets of numbers
* Second-order differential equation, a di ...
expressions. Neither of these principles can be expressed in
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
(are
nonfirstorderizable In formal logic, nonfirstorderizability is the inability of a natural-language statement to be adequately captured by a formula of first-order logic. Specifically, a statement is nonfirstorderizable if there is no formula of first-order logic which ...
). Taken together, they are sometimes referred to as ''Leibniz's law''. Formally, the two principles can be expressed in the following way:
#The indiscernibility of identicals: