The necessity of identityKripke versus Gibbard

1. Kripke on the necessity of identity• Kripke gives the short proof of the necessity of identity at the

beginning of his seminal article (1971/1979, 136)• Kripke gives credit to Wiggins (1965)• However, the conclusion was derived earlier by Ruth Barcan Marcus -

in her “Identity of individuals in a strict functional calculus of second order” (series of three papers)• In the present form was first given by Quine (Burgess 2014)

The argument

1. ∀x y (x = y → (Fx → Fy))∀ - indiscernibility of identicals

2. ∀x □x = x - universal necessary self-identity

3. ∀x y (x = y → (□x = x →□x = y))∀ - instance of (1)

4. ∀x y (x = y →□x = y)∀ - necessity of identity

Derivation

5. ∀x □x = x Premiss

6. ∀x∀y (x = y → (□x = x →□x = y)) Premiss (instance of Lebniz’s Law)

7. ∀x∀y (□x = x → (x=y →□x = y)) (6) Equi.

8. ∀x y (x = y →□x = y)∀ (5), (7) MPP

Presuppositions

2. ∀x □x = x - universal necessary self-identity• The most important premiss, postulated by Kripke

• Weak modal reading - metaphysical modality:• “Let us interpret necessity here weakly. We can count statements as

necessary if whenever the objects mentioned therein exist, the statement is true.” (Kripke 1971, 480)

• Kripke: acceptance of the weak reading of (2) commits one to the conclusion (4)

Conclusion – formula (4)

4. x y (x = y →□x = y)∀ ∀ - necessity of identity

• Kripke says that formula (4):• “does not say anything about statements at all. It says for every object x and

object y, if x and y are the same object, then it is necessary that x and y are the same object.” (Kripke 1979, 480)

Modality de re versus de dicto

• If the modality were construed de dicto the argument would be invalid• De dicto modality – Quine’s example (Quine 1961)

9. ‘number 8 is greater than 7’ - necessarily true10. ‘the number of planet is greater than 7’ – contingently true

• 9. and 10. are extensionally equivalent – pick out the same number• truth of the statement can depend on how the referent is picked out• If picked out by the proper name, such as ‘number 8’ then the statement can

express a necessary truth• If picked out by the description ‘the number of planets’ then it does not

express a necessary truth.

Why Kripke does not prove premiss 2.?• Informally it seems straightforward to derive (2)

11. ∀x x = x 11a. a=a.11b. □a=a2. ∀x □x = x Universal generalization from 11b. since a is arbitrary

Problem: the derivation presupposes Barcan formulae

• Barcan schema: Converse Barcan Schema:• ∀x □Fx → □ x Fx∀ □ x Fx → x □Fx∀ ∀

( x ∃ ◊x=x → ◊ x∃ ◊ x=x) (◊ x∃ x=x → x ∃ ◊ x=x)

11. ∀x x = x Premiss12. □ x x = x ∀ 11. Rule of necessitation13. □ ∀x x = x → ∀x □x = x CBS 2. ∀x □x = x 12., 13. → elim. (Burgess 2014, 1573)

Problem: the necessity of existence• Barcan Schemas – allows the proof according to which• everything that could exist does exist, and everything exists necessarily

(Hayaki 2006)• In particular Timothy Williamson (2002) has argued that he exists

necessarily• BS x ◊x=x → ◊ x x=x is equivalent to □ x Fx → x □Fx∃ ∃ ∀ ∀• CBS ◊ x x=x → x ◊ x=x is equivalent to x □Fx → □ x Fx∃ ∃ ∀ ∀

2. Defense of the necessity of identity• Kripke examines three different kinds of identity claims and explains

in what way they could be contingent and why if they are contingent do not count against the conclusion of the argument that all things are necessarily identical to themselves

Three types of identity statements

A) Identity statements that relate individuating descriptions: such as “the inventor of bifocals is identical with the first Postmaster general of United States”.B) Identity statements that relate proper names: such as “Cicero is Tully”.C) Identity statements that pertain to theoretical reductions in science: such as “pain is identical to firing of C-fibers” or “Heat is mean kinetic energy”, etc. (Burnyeat 1979)

• A) type of identity statements can be contingent – this is consistent with 4)• Example – Russell’ theory of descriptions (scope of descriptions)

• „Just one thing x was the first Postmaster General of the United States and just one thing y was the inventor of bifocals and it is necessary that x = y” (Burnyeat 1979, 472)

• Formally: 14. [ὶx Px] & [ὶy Iy] & □x=y

• B) and C) denote necessary identity claims• Proponents of contingent identity make the mistake of confusing

metaphysical and epistemological notions• Hesperus=Phosphorus – discovered a posteriori (epistemology) - necessary identical (metaphysics)

Proper names – rigid designators

• Proper names refer rigidly:• “a term that designates the same object in all possible worlds.”

(Kripke, 1979, 488) • The function of proper names is simply to refer to objects no matter

how they are described - their function is to pick out the same object (under our current usage of words) in all possible worlds where that object exists• Similar considerations apply to theoretical terms, namely they also

refer rigidly

3. Contingent identity

• Allan Gibbard (1975) – counterexample to 4)• Lumpl (piece of clay) and Goliath (statue) • „I make a clay statue of the infant Goliath in two pieces, one the part above

the waist and the other the part below the waist. Once I finish the two halves, I stick them together thereby bringing in to existence simultaneously a new piece of clay and a new statue. A day later I smash the statue, thereby bringing to an end both statue and piece of clay. The statue and the piece of clay persisted during exactly the same period of time.” (Gibbard 1975, 191)

• Lumpl=Goliath

• „suppose I had brought Lumpl in to existence as Goliath,just as I actually did, but before the clay had a chance to dry, I squeezed it into a ball. At that point, according to the persistence criteria I have given, the statue Goliath would have ceased to exist, but the piece of clay Lumpl would still exist in a new shape. Hence Lumpl would not be Goliath, even though both existed” (Gibbard 1975, 191)

• Lumpl = Goliath & ◊(Lumpl exists & Goliath exists & Lumpl≠Goliath).

Contingent identity and proper names• sortal theory of proper names• it does not make a sense to ask “what that thing would be, (…), a part from

the way it is designated”, because “(p)roper names like 'Goliath or 'Lumpl refer to a thing as a thing of a certain kind:' Goliath' refers to something as a statue; 'Lumpl as a lump.” (Gibbard 1975, 194-195)

• Goliath refers to a thing qua statue• Lumpl refers qua lump of clay• Questions of identity through possible worlds only make sense when

we relativize them to a sortal

Rigid designators in the sortal theory• „A designator may be rigid with respect to a sortal: it maybe statue-rigid, as

'Goliath' is, or it maybe lump-rigid as 'Lumpl' is. A designator for instance is statue-rigid if it designates the same statue in every possible world in which that statue exists and designates nothing in any other possible world. What is special about proper names like 'Goliath' and ‘Lumpl’ is not that they are rigid designators it is rather that each is rigid with respect to the sortal it invokes. ‘Goliath' refers to its bearer as a statue and is statue rigid; 'Lumpl' refers to its bearer as a lump and is lump-rigid.” (Gibbard 1975, 195)

Persistence criteria

• What explains the possibility of contingent identity is the fact that the sortals through which things are picked out invoke persistence criteria• Lumpl qua lump of clay and Goliath qua statue have different

persistence criteria• Reference of a term is determined by the origin of the object qua

something and by the persistence criteria

Persistence criteria• Persistence criteria for pieces of clay:“(a) The domain of P is an interval of time T.(b) For any instant t in T, P(t) is a portion of clay the parts of which, at t, are both stuck to each other and not stuck to any clay particles which are not part o f P(t).(c) The portions of clay P(t) change with t only slowly, if at all. (d) No function P * which satisfies (a), (b), and (c) extends P, in the sense that the domain of P* properly includes the domain of P and the function P is P* with its domain restricted.” (Gibbard 1975, 189)• a statue (e.g. Goliath) is identical to a lump of clay with a certain

shape

3.1. Contingent Identity and Leibniz’s law• The problem with the sortal account of proper names is that it

seemingly violates Leibniz’s law of identity15. □(Lumpl exists → Lumpl=Lumpl) – necessary truth16. Goliath=Lumpl contingent identity17. □ (Lumpl exists → Goliath=Lumpl) false

• However, by Leibniz’s Law 17. follows from 15. and 16.

Gibbard’s solution

• Leibniz’s law applies to properties and relations (ibid., 201). Therefore, if we are going to apply the law in the present context one needs to argue that modal contexts attribute properties to things• Gibbard follows Quine in claiming that concrete things do not have modal

properties:• „Modal expressions do not apply to concrete things independently of the way they

are designated. Lumpl, for instance, is the same thing as Goliath: it is a clay statue of the infant Goliath which I put together and then broke. Necessary identity to Lumpl, though, is not a property which that thing has or lacks, for it makes no sense to ask whether that thing, as such, is necessarily identical with Lumpl. Modal contexts, then, do not attribute properties or relations to concrete things-so the proponent of contingent identity can respond to Leibniz's Law.” (Gibbard 1975, 201-202)

Individual concepts• Frege and Carnap• In modal contexts names and variables shift their reference to their normal sense

18. □(Lumpl exists → Goliath=Lumpl)• In 18) ‘Goliath’ and ‘Lumpl’ to the concept of Goliath and Lumpl

19. (Lumpl exists and x≠Lumpl)• x does not range over concrete objects; rather it ranges over individual concepts.• Individual concepts are functions from possible worlds whose values are objects in

those possible worlds.

References

• Burgess, John P. 2014. "On a derivation of the necessity of identity." Synthese 191: 1567–1585.• Hayaki, Reina. 2006. "Contingent objects and the Barcan formula." Erkenntnis 64: 75-83.• Burnyeat, Myles. 1979. "Saul Kripke: Identity and Individuation." In Philosophy as it is, edited by Ted Honderich

and Myles Burnyeat, 467-477. London: Penguin books.• Gibbard, Allan. 1975. "Contingent Identity." Journal of Philosophical Logic 4: 187-221.• Kripke, Saul. 1979. "Identity and necessity." In Philosophy as it is, edited by Ted Honderich and Myles Burnyeat,

476-513. London: Penguin Books ltd.• Kripke, Saul. 1971. "Identity and Necessity." In Identity and Individuation, edited by Milton K. Munitz, 135-164.

New York: New York University Press.• Marcus, Ruth Barcan. 1946. “A functional calculus of first order based on strict implication.” Journal of Symbolic

Logic 11: 1-16.• Quine, Willard V. O. 1961. "Reference and Modality." In From a Logical Point of View, by Willard V. O. Quine,

139-59. New York: Harper and Row.• Wiggins, David. 1965. "Identity statements." In Analytic philosophy: Second series, edited by R. J. Butler, 40–71.

Oxford: Basil Blackwell.• Williamson, Timothy. 2002. “Necessary Existents.” In Logic, Thought, and Language, edited by Anthony O’Hear,

233-251. Cambridge: Cambridge University Press.