The identity of indiscernibles is an ontological principle which states that two or more objects 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. This article is about ontology in philosophy. ... In philosophy, an object is a thing, an entity, or a being. ... This article is about the concept of an entity. ... The word property, in philosophy, mathematics, and logic, refers to an attribute of an object; thus a red object is said to have the property of redness. ... In linguistics and logic, a predicate is an expression that can be true of something. ...

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. Both are famously used in his arguments with Newton and Clarke in the Leibniz-Clarke correspondence. Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... The principle of sufficient reason states that anything that happens does so for a definite reason. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Samuel Clarke. ... The Leibniz-Clarke correspondence was a scientific, theological and philosophical debate conducted epistolary means, between the German thinker Gottfried Wilhelm Leibniz and Samuel Clarke an English supporter of Isaac Newton. ...

Philosophers have to come to recognise, 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), and purported predicates denoting what someone believes about an object (which some philosophers have considered to be valid predicates), may both be excluded. 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.^{[citation needed]}

Associated with this principle is also the question as to whether it is a logical principle, or merely an empirical principle. Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... 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 consequences that are observable by the senses. ...

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).^{[citation needed]} Note that these are all second-order expressions. Neither of these principles can be expressed in first-order logic. In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ... First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ...

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.

     

   • For any x and y, if x and y differ with respect to some property, then x is non-identical to y.

     

2. 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 is non-identical to y, then x and y differ with respect to some property.

     

Principle 1. is taken to be a logical truth and (for the most part^{[citation needed]}) uncontroversial. Principle 2. is controversial. Max Black famously argued against 2. (see Critique, below). Max Black (24 February 1909, Baku, Russian Empire [present-day Azerbaijan] – 27 August 1988, Ithaca, New York, United States) was a distinguished British-American philosopher, who was a leading influence in analytic philosophy in the first half of the twentieth century. ...

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: if x is non-identical to y, then there will always be a putative "property" which distinguishes them, namely "being identical to x". In propositional logic, a tautology (from the Greek word ταυτολογία) is a sentence that is true in every valuation (also called interpretation) of its propositional variables, independent of the truth values assigned to these variables. ...

On the other hand, it is incorrect to exclude all predicates which are materially equivalent (i.e. contingently 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. Modal logic, or (less commonly) intensional logic is the branch of logic that deals with sentences that are qualified by modalities such as can, could, might, may, must, possibly, and necessarily, and others. ...

Critique

Symmetric universe

Max Black 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 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. ^[1] This is of course asumptive in that given one symmetric universe, the properties of the object would be inverse but symetrical in their relativity to each other - one sphere would have an identical sphere to one side and the other would be on the opposite. The issue is the understanding of the properties of the objects required to fully describe the objects in space-time. Max Black (24 February 1909, Baku, Russian Empire [present-day Azerbaijan] – 27 August 1988, Ithaca, New York, United States) was a distinguished British-American philosopher, who was a leading influence in analytic philosophy in the first half of the twentieth century. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ...

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 in his Meditations on First Philosophy. Descartes concluded that he could not doubt the existence of himself (the famous cogito ergo sum 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. Descartes redirects here. ... The title page of the Meditations Meditations on First Philosophy (subtitled In which the existence of God and the real distinction of mind and body, are demonstrated) is a philosophical treatise written by René Descartes first published in Latin in 1641 . ... René Descartes (1596–1650) Cogito, ergo sum (Latin: I think, therefore I am) or Dubito, ergo cogito, ergo sum (Latin: I doubt, therefore I think, therefore I am) is a philosophical statement used by René Descartes, which became a foundational element of Western philosophy. ...

This argument is normally rejected by modern philosophers on the grounds that it 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.^{[citation needed]} Numerous counterexamples are given to debunk Descartes' reasoning via reductio ad absurdum, such as the following argument based on a secret identity: Reductio ad absurdum (Latin: reduction to the absurd) 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, derives an absurd or ridiculous outcome, and then concludes that the original assumption... For other uses of this term, please see Secret identity (disambiguation). ...

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 thinks that Clark Kent cannot fly.
4. Lois Lane thinks that Superman 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 else
   • A person's knowledge about x is not a predicate of x, thus undermining Descartes' argument.

For the Dutch girl group, see Loïs Lane. ... For other uses, see Clark Kent (disambiguation). ... Superman is a fictional character and comic book superhero , originally created by American writer Jerry Siegel and Canadian artist Joe Shuster and published by DC Comics. ...

Notes and references

• Lecture notes of Kevin Falvey / UCSB

See also

• Duck test
• Elephant test

The duck test is a specific form of inductive reasoning whereby one can infer the nature of an unknown based upon its outwardly visible traits. ... The term elephant test refers to the ability to recognise something while being unable to describe it. ...

External links

• http://www.consciousentities.com/tactics.htm
• Stanford Encyclopedia of Philosophy entry