Identity of indiscernibles
Encyclopedia
The identity of indiscernibles is an ontological
principle which states that two or more object
s or entities
are identical (are one and the same entity) if they have all their properties
in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa. A related principle is the indiscernibility of identicals, discussed below.
A form of the principle 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
. Both are famously used in his arguments with Newton
and Clarke
in the Leibniz–Clarke correspondence. Because of its association with Leibniz, the principle is sometimes known as Leibniz's law. (However, the name 'Leibniz's Law' is also commonly used for the converse of the principle, the indiscernibility of identicals (described below), which is logically distinct and not to be confused with the identity of indiscernibles.)
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 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..
Willard Van Orman Quine
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
holds that this failure may be the result of the use of the disquotational principle
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
principle.
expressions. Neither of these principles can be expressed in first-order logic
.
Principle 1 is taken to be a logical truth
and (for the most part) uncontroversial. Principle 2 is controversial. Max Black
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:
If all such predicates are included, then the second principle as formulated above can be trivially and uncontroversially shown to be a logical tautology
: 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
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.
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 provides a model
in which there are two distinct (non-identical) things that have all the same properties. He claimed that in a symmetric universe wherein only two symmetrical spheres exist, the two spheres are two distinct objects, even though they have all the properties in common.
in his Meditations on First Philosophy
. Descartes concluded that he could not doubt the existence of himself (the famous cogito
argument), but that he could doubt the existence of his body.
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
, such as the following argument based on a secret identity
:
A response may be that the argument in the Meditations on First Philosophy
isn't that Descartes cannot doubt the existence of his mind, but rather that it is beyond doubt, such that no being with understanding could doubt it. This much stronger claim doesn't resort to relational properties, but rather presents monadic properties, as the foundation for the use of Leibniz's law. One could expound an infinite list of relational properties that may appear to undermine Leibniz's law(i.e. Lois Lane loves Clark Kent, but not Superman. etc.) but nonetheless any approach focused on monadic properties will always produce accurate results in support of Descarte's claim.
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
principle which states that two or more object
Object (philosophy)
An object in philosophy is a technical term often used in contrast to the term subject. Consciousness is a state of cognition that includes the subject, which can never be doubted as only it can be the one who doubts, and some object or objects that may or may not have real existence without...
s or entities
Entity
An entity is something that has a distinct, separate existence, although it need not be a material existence. In particular, abstractions and legal fictions are usually regarded as entities. In general, there is also no presumption that an entity is animate.An entity could be viewed as a set...
are identical (are one and the same entity) if they have all their properties
Property (philosophy)
In modern philosophy, logic, and mathematics a property is an attribute of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property however differs from individual objects in that...
in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa. A related principle is the indiscernibility of identicals, discussed below.
A form of the principle 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 reason: no state of affairs can obtain, and no statement can be true unless there is sufficient reason why it should not be otherwise...
. Both are famously used in his arguments with Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
and Clarke
Samuel Clarke
thumb|right|200px|Samuel ClarkeSamuel Clarke was an English philosopher and Anglican clergyman.-Early life and studies:...
in the Leibniz–Clarke correspondence. Because of its association with Leibniz, the principle is sometimes known as Leibniz's law. (However, the name 'Leibniz's Law' is also commonly used for the converse of the principle, the indiscernibility of identicals (described below), which is logically distinct and not to be confused with the identity of indiscernibles.)
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 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..
Willard Van Orman Quine
Willard Van Orman Quine
Willard Van Orman Quine was an American philosopher and logician in the analytic tradition...
thought that the failure of substitutivity
Substitution (logic)
Substitution is a fundamental concept in logic. Substitution is a syntactic transformation on strings of symbols of a formal language.In propositional logic, a substitution instance of a propositional formula is a second formula obtained by replacing symbols of the original formula by other formulas...
in intensional contexts (e.g., "Sally believes that p," "It is necessarily the case that q") shows that modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
is an impossible project. Saul Kripke
Saul Kripke
Saul Aaron Kripke is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the 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 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 proposition...
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 or experimentation. Empirical data are data produced by an experiment or observation....
principle.
Identity and indiscernibility
There are two principles here that must be distinguished (equivalent versions of each are given in the language of the predicate calculus). Note that these are all second-orderSecond-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 logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
.
- The indiscernibility of identicals
- For any x and y, if x is identical to y, then x and y have all the same properties.
- For any x and y, if x is identical to y, then x and y have all the same properties.
- The identity of indiscernibles
- For any x and y, if x and y have all the same properties, then x is identical to y.
- 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
Logical truth
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement.Logical...
and (for the most part) uncontroversial. Principle 2 is controversial. Max Black
Max Black
Max Black was a British-American philosopher, who was a leading influential figure in analytic philosophy in the first half of the twentieth century. He made contributions to the philosophy of language, the philosophy of mathematics and science, and the philosophy of art, also publishing studies...
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:
- "is identical to x"
- "is identical to y"
- "is not identical to x"
- "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 logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
: 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 neither true under every possible valuation nor false under every possible valuation . A contingent proposition is neither necessarily true nor necessarily false...
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.
Symmetric universe
Max BlackMax Black
Max Black was a British-American philosopher, who was a leading influential figure in analytic philosophy in the first half of the twentieth century. He made contributions to the philosophy of language, the philosophy of mathematics and science, and the philosophy of art, also publishing studies...
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 provides a model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
in which there are two distinct (non-identical) things that have all the same properties. He claimed that in a symmetric universe wherein only two symmetrical spheres exist, the two spheres are two distinct objects, even though they have all the properties in common.
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 DescartesRené Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
in his Meditations on First Philosophy
Meditations on First Philosophy
Meditations on First Philosophy is a philosophical treatise written by René Descartes and first published in 1641 . The French translation was published in 1647 as Méditations Metaphysiques...
. Descartes concluded that he could not doubt the existence of himself (the famous cogito
Cogito
Cogito is a philosophical magazine published by Nefeli publications and a member of the Eurozine network. It was founded in 2004 with the intention of making philosophy accessible to the lay reader without compromising it through oversimplification. Cogito is therefore an attempt to approach...
argument), but that he could doubt the existence of his body.
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
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
, such as the following argument based on a secret identity
Secret identity
A secret identity is an element of fiction wherein a character develops a separate persona , while keeping their true identity hidden. The character also may wear a disguise...
:
- Entities x and y are identical if and only if any predicate possessed by x is also possessed by y and vice versa.
- Clark Kent is Superman's secret identity; that is, they're the same person (identical) but people don't know this fact.
- Lois LaneLois LaneLois Lane is a fictional character, the primary love interest of Superman in the comic books of DC Comics. Created by writer Jerry Siegel and artist Joe Shuster, she first appeared in Action Comics #1 ....
thinks that Clark KentClark KentClark Kent is a fictional character created by Jerry Siegel and Joe Shuster. Appearing regularly in stories published by DC Comics, he debuted in Action Comics #1 and serves as the civilian and secret identity of the superhero Superman....
cannot fly. - Lois Lane thinks that SupermanSupermanSuperman is a fictional comic book superhero appearing in publications by DC Comics, widely considered to be an American cultural icon. Created by American writer Jerry Siegel and Canadian-born American artist Joe Shuster in 1932 while both were living in Cleveland, Ohio, and sold to Detective...
can fly. - Therefore Superman has a property that Clark Kent does not have, namely that Lois Lane thinks that he can fly.
- Therefore, Superman is not identical to Clark Kent.
- 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
- The application of Leibniz's law is erroneous; the law is only applicable in cases of monadic, not polyadic, properties; or
- What people think about are not the actual objects themselves; or
- A person is capable of holding conflicting beliefs.
-
-
- Any of which will undermine Descartes' argument.
-
A response may be that the argument in the Meditations on First Philosophy
Meditations on First Philosophy
Meditations on First Philosophy is a philosophical treatise written by René Descartes and first published in 1641 . The French translation was published in 1647 as Méditations Metaphysiques...
isn't that Descartes cannot doubt the existence of his mind, but rather that it is beyond doubt, such that no being with understanding could doubt it. This much stronger claim doesn't resort to relational properties, but rather presents monadic properties, as the foundation for the use of Leibniz's law. One could expound an infinite list of relational properties that may appear to undermine Leibniz's law(i.e. Lois Lane loves Clark Kent, but not Superman. etc.) but nonetheless any approach focused on monadic properties will always produce accurate results in support of Descarte's claim.
See also
- Disquotational principleDisquotational principleThe disquotational principle is a philosophical theorem 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 proposition...
- Identical particles, a similar idea in quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
- Intensional fallacyIntensional fallacyIn philosophical logic, the intensional fallacy is committed when one makes an illicit use of Leibniz's law in an argument. Leibniz's law states that, if one object has a certain property, while another object does not have the same property, the two objects cannot be identical. For example, if...
External links
- http://www.consciousentities.com/tactics.htm
- Stanford Encyclopedia of Philosophy entry