In
modal logic, the necessity of identity is the thesis that for every object x and object y, if x and y are the same object, it is
necessary that x and y are the same object. The thesis is best known for its association with
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 em ...
, who published it in 1971, although it was first derived by the logician
Ruth Barcan Marcus
Ruth Barcan Marcus (; born Ruth Charlotte Barcan; 2 August 1921 – 19 February 2012) was an American academic philosopher and logician best known for her work in modal and philosophical logic. She developed the first formal systems of quant ...
in 1947, and later, in simplified form, by
W. V. O. Quine W. may refer to:
* SoHo (Australian TV channel) (previously W.), an Australian pay television channel
* ''W.'' (film), a 2008 American biographical drama film based on the life of George W. Bush
* "W.", the fifth track from Codeine's 1992 EP ''Bar ...
in 1953.
Kripke's derivation
The derivation in Kripke's 'Identity and Necessity' is in three steps:
:(1)
.
:(2)
.
:(3)
The first premise is simply postulated: every object is identical to itself. The second is an application of the
principle of substitutivity: if a = b, then a has all the properties b has, thus from Fa, infer Fb, where F is
. The third follows by elementary predicate logic.
Rigid designation
In the later ''
Naming and Necessity
''Naming and Necessity'' is a 1980 book with the transcript of three lectures, given by the philosopher Saul Kripke, at Princeton University in 1970, in which he dealt with the debates of proper names in the philosophy of language. The transcript ...
'', Kripke suggested that the principle could be derived directly, assuming what he called
rigid designation
In modal logic and the philosophy of language, a term is said to be a rigid designator or absolute substantial term when it designates (picks out, denotes, refers to) the same thing in ''all possible worlds'' in which that thing exists. A designat ...
. A term is a rigid designator when it designates the same object in every
possible world
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their me ...
in which that object exists. When a name's referent is fixed by the original act of naming, it becomes a rigid designator. Some examples of rigid designators include proper names (i.e. ‘Richard Nixon’), natural kind terms ( i.e. ‘gold’ or ‘H2O’) and some descriptions.
Proper names are typically rigid designators, but
definite description
In formal semantics and philosophy of language, a definite description is a denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun. The definite description is ''proper'' if X applies to a unique individual or o ...
s are typically not. So we can speak of "Richard Nixon" referring to the same person in all possible worlds, but the description "the man who won the 1968 election" could refer to many different people. According to Kripke, the proper name "Richard Nixon" can only be used rigidly, but the description "the man who won the 1968 election" can be used non-rigidly. Kripke argues, that if names are rigid designators, then identity must be necessary, because the names ‘a’ and ‘b’ will be rigid designators of an object x if a is identical to b, and so in every possible world, ‘a’ and ‘b’ will both refer to this same object x, and no other, and there could be no situation in which a might not have been b, otherwise x would not have been identical with itself.
:Waiving fussy considerations deriving from the fact that x need not have necessary existence, it was clear from
and Leibniz’s law that identity is an ‘internal’ relation:
. (What pairs (x, y) could be counterexamples? Not pairs of distinct objects, for then the antecedent is false; nor any pair of an object and itself, for then the consequent is true.) If ‘a’ and ‘b’ are rigid designators, it follows that ‘a = b’, if true, is a necessary truth. If ‘a’ and ‘b’ are not rigid designators, no such conclusion follows about the statement ‘a = b’ (though the objects designated by ‘a’ and ‘b’ will be necessarily identical).
[Naming and Necessity p.3]
This does not mean that we have knowledge of this necessity. Before the discovery that Hesperus (the evening star) and Phosphorus (the morning star) were the same planet, this fact was not known, and could not have been inferred from
first principles
In philosophy and science, a first principle is a basic proposition or assumption that cannot be deduced from any other proposition or assumption.
First principles in philosophy are from First Cause attitudes and taught by Aristotelians, and nua ...
. Thus there can be
a posteriori necessity
''A posteriori'' necessity is a thesis in metaphysics and the philosophy of language, that some statements of which we must acquire knowledge a posteriori are also necessarily true. It challenges previously widespread belief that only ''a priori ...
.
The principle can also be applied to
natural kind
"Natural kind" is an intellectual grouping, or categorizing of things, in a manner that is reflective of the actual world and not just human interests. Some treat it as a classification identifying some structure of truth and reality that exists wh ...
s. If
water
Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...
is H
2O, then water is ''necessarily'' H
2O. Since the terms 'water' and 'H
2O' pick out the same object in every possible world, there is no possible world in which 'water' picks out something different from 'H
2O'. Therefore, water is necessarily H
2O. It is possible, of course, that we are mistaken about the chemical composition of water, but that does not affect the necessity of identities. What is not being claimed is that water is necessarily H
2O, but ''conditionally'', ''if'' water is H
2O (though we may not know this, it does not change the fact if it is true), then water is necessarily H
2O.
See also
*
A posteriori necessity
''A posteriori'' necessity is a thesis in metaphysics and the philosophy of language, that some statements of which we must acquire knowledge a posteriori are also necessarily true. It challenges previously widespread belief that only ''a priori ...
*
Rigid designator
In modal logic and the philosophy of language, a term is said to be a rigid designator or absolute substantial term when it designates (picks out, denotes, refers to) the same thing in ''all possible worlds'' in which that thing exists. A designat ...
*''
Naming and Necessity
''Naming and Necessity'' is a 1980 book with the transcript of three lectures, given by the philosopher Saul Kripke, at Princeton University in 1970, in which he dealt with the debates of proper names in the philosophy of language. The transcript ...
''
Notes
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Philosophical logic