Abstract
The source, status, and significance of the derivation of the necessity of identity at the beginning of Kripke’s lecture “Identity and Necessity” is discussed from a logical, philosophical, and historical point of view.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Remark (A) comes in the second sentence after Kripke’s statement of the conclusion of the argument as his displayed formula (4), and remarks (B)–(D) come in the second paragraph thereafter. As the preceding sentence illustrates, when citing passages in works more readily available in reprints than in the original, I will generally locate them by internal divisions, such as section and formula numbers, since unlike pagination these remain the same from printing to printing.
Except in direct quotations I will use the modern notation \(\rightarrow ,\Rightarrow ,\leftrightarrow ,\Leftrightarrow \) in place of the older horseshoe and fishhook and triple bar and quadruple bar symbols for the truth-functional or material and modal or strict versions of the conditional or implication and the biconditional or equivalence. As the preceding sentence illustrates, when mentioning expressions of formal languages, I will generally let them name themselves, eschewing quotation or quasi-quotation marks.
Here \(\Phi \) may be any formula having free occurrences of a variable \(z\) and no quantification on the variables \(x\) and \(y\), while \(x/z\) and \(y\)/z indicate substitution of \(x\) and of \(y\), respectively, for all free occurrences of \(z\). (2) is the instance of (4) where \(\Phi \) is \(\square x=z\).
In the sense that (1) and (2) \(\rightarrow \) (3) is a substitution instance of a thesis of classical predicate:\( (\forall xPx \& \forall x\forall y(Qxy\rightarrow (Px\rightarrow Rxy)))\!\rightarrow \! \forall x\forall y(Qxy\rightarrow Rxy)\).
The significance of the letter is discussed by Fetzer and Humphreys (1998) in the editorial introduction (p. viii) to their collection. Marcus declined their invitation to have the letter included in their volume, but she had already by then circulated it widely, the present writer being one of its many recipients. There is a briefer remark tending in the same direction as the passage I am about to quote in the introduction (p. viii) to the collection (Marcus 1993).
For a more detailed analysis see footnote 4 of Soames (1995).
See displayed formulas (25) and (26) in Marcus (1961).
Kripke does not mention Marcus, and may not at the time have been aware of her role, since the particular formulation of Quine’s that he cites is the one that happens not to mention her. In any case, Kripke really needed Quine’s and not Marcus’s version, since he was working in a first-order and not a second-order context, with identity as a primitive and not a defined notion.
Not everyone who studied with Quine realized that he was the originator of the derivation, since one published source misattributing the derivation (1)–(3) to Marcus is Føllesdal (2004), a reprinting of (a 1963 revision of) Føllesdal’s 1961 doctoral dissertation (later published as Føllesdal 2004), written under Quine’s direction. There seems to have been a miscommunication between the doctoral candidate and his supervisor, since Føllesdal, when giving (on pp. 40–41) a derivation similar to (1)–(3), describes it as a Quinean reformulation of the argument of “Miss Barcan”, when he ought to call it a Quinean replacement for that argument. He then goes on, in describing what is and isn’t done in the derivation, to speak of what is and isn’t done by “Miss Barcan” rather than by Quine.
At the beginning of their chapter 11. Since this is the presentation mentioned in the Marcus letter quoted earlier, it may be mentioned that Hughes and Creswell cite Marcus for the conclusion (3) and give no separate citation for their derivation. This might leave the careless reader with the impression that the derivation is also to be attributed to Marcus; but the careful reader will note that this is not something Hughes and Creswell explicitly say.
The formulations Kripke presents actually require a slight amendment as per Fine (1983), but that matter does not affect any issue under discussion here.
Prior (1956) showed that (6), which unlike (5) is not derivable even in the Barcan system based on S4, becomes derivable if the underlying sentential modal logic is strengthened to S5. Indeed, it is enough to have available the “Brouwerian” formula \(\diamond \square \Phi \rightarrow \Phi \). Having this formula and (3), one can further derive the law of the necessity of distinctness and its contrapositive: \(\forall x\forall y(x\ne y\rightarrow \square x\ne y)\quad \quad \forall x\forall y(\diamond x=y\rightarrow x=y)\).
Perhaps the first to see that a linguistic conception of necessity admits at most de dicto and not de re modality was J. S. Mill, the well-known advocate of the principle that the only necessity is verbal necessity, who also held that there are no individual essences. For him “Philosophers are rational” can be necessary because “rational” is part of the connotation of “philosopher”, but “Auguste Comte is rational” cannot be necessary because “Auguste Comte” has no connotation for “rational” to be part of. Of course, Mill here assumes his doctrine that the signification of a proper name consists solely of its denotation, that a proper name has no connotation. Quine’s ambition was to make the case without any special assumption about proper names or other singular terms. It is one of the ironies of the debate that many of Quine’s opponents thought they could answer his critique by positing Millian proper names, without explicitly arguing, as Kripke was to do, against Mill’s assumption that all necessity is verbal. Unless that is done, the assumption of Millian names succeeds in reducing de re to de dicto modality only at the cost of making de dicto modality vacuous or a mystery for dicta involving proper names.
Always, that is, after he had arrived at the doctrines expressed in “Identity and Necessity” and Naming and Necessity. We will see later that he had not yet arrived at them at the time he did his famous technical work on model theory for modal logic.
For another instance, in systems of so-called free logic (whose use is nowadays often advocated in connection with quantified modal logic) there may be constants, but the instantiation rule (26) is rejected in favor of the following weaker rule, with an additional premise asserting existence for \(t\):
from \(\forall x\Phi \left( x \right) \) and \(\exists zz=t\) to infer \(\Phi \left( t \right) \)
The second premise may alternatively be expressed as E!\(t\), using a special existence or “ecce” predicate. But to pursue this matter further would take us back to distracting issues about possible nonexistence, which I wish to set aside here.
Even experts, however, sometimes forget this. Marcus (1961) in her Boston talk claims that if \(a=b\) (her formula (13)) is true, then just like \(a = a\) (her formula (14)) it is necessarily true. She further claims that this first claim “is precisely the import of” her theorem on the necessity of identity (her formula (18), our formula (15)). But the claim about the necessary truth of true identity statements does not follow from Marcus’s theorem without the instantiation rule (29).
Quine, who did not put much weight on the distinction even after 1970, early and late tends to mention Føllesdal and Kripke in the same breath, as when he says, “Circumstances of this kind have led Professors Føllesdal and Kripke to posit a special kind of singular term called a genuine singular term or rigid designator...” in Quine (2008), p. 339.
I have been ignoring, as I earlier said I would, the distracting issue of possible non-existence, but it is worth mentioning that if one is only taking the box in a weak sense, the argument for (27) requires only that the denotation of \(a\) and likewise that of \(b\) should remain invariant in all worlds in which \(a\) and \(b\) exists. A main reason Føllesdal undertook the 1963 revision of his 1961 thesis was to take account of this fact. If one seeks not merely a sufficient but a necessary and sufficient condition, the obvious requirement is that the denotations of \(a\) and \(b\) should be actually the same, and should covary across the whole range of whatever kind of “possibilities” are under discussion, so that if the one changes, the other changes in the same way. A case where we would presumably have covariance without invariance, regardless of what kind of “possibilities” are under discussion, would be that of synonymous definite descriptions such as “the largest asteroid” and “the largest planetoid”.
See (25)–(27) above. This is a kind of derivation that, to reiterate, played no role in establishing (27) for rigid designators by the argument (30)–(33). Now given (28) for non-modal \(\Phi \), Becker’s rule, which is available even in the minimal modal logic K, gives us
$$\begin{aligned} \square a=b\rightarrow \left( {\square \Phi \left( {a/z} \right) \rightarrow \Phi \left( {b/z} \right) } \right) \end{aligned}$$while
$$\begin{aligned} \square \left( {\Phi \left( {a/z} \right) \rightarrow \Phi \left( {b/z} \right) } \right) \rightarrow \left( {\square \Phi \left( {a/z} \right) \rightarrow \square \left( {b/z} \right) } \right) \end{aligned}$$is an instance of the sole axiom of K. With (27) for the metaphysical reading of \(\square \) we then get
$$\begin{aligned} a=b\rightarrow \left( {\square \Phi \left( {a/z} \right) \rightarrow \square \left( {b/z} \right) } \right) \end{aligned}$$and this is the key step in a proof (by induction on logical complexity of \(\Phi )\) of the indiscernibility law (28) for \(\Phi \) containing metaphysical modalities (but no other non-classical logical operators). In this sense, indiscernibility follows from necessity of identity in a Kripkean context, rather than the other way around.
Some recent defenders of Russellian logically proper names (nowadays often called Kaplanian directly referential names, from the terminology used in Kaplan (1989) and related earlier work), such as Scott Soames, have rejected (39) while accepting (38), very roughly on the grounds that in contrast to “it is apriori that...”, the context “it is analytic that...”, though it does not overtly involve quotation, is covertly a context of mention rather than use. But it will not be needful to go into this issue for present purposes, since none of the early workers on quantified modal logic with whose views I will be concerned anticipated taking such a line.
Russell’s distinctive position is discussed by Kripke in the “Naming and Necessity” lectures, note 4, and by Føllesdal in his dissertation, passim. It is safe to say that all participants in early debates about quantified modal logic were well enough aware of Russell’s views.
Prior’s formulations illustrate Quine’s complaint that use/mention confusions were endemic in the modal logic of his day. By Quine’s lights, one really should not be speaking about \(x\) and \(y\) as “naming” or “describing” objects, since variables range over a domain of objects but do not have specific objects as denotations; one should be speaking instead about instantiating the variables with “names” or “descriptions”.
The expressions in question are called “tags” in (1961) and “names in an ideal sense” in (1962). The original version of (1963), dating from before the Naming and Necessity lectures, is more revealing than the rewrite, Marcus (1974), dating from after. In particular, the passages I will be quoting shortly disappear in the rewrite.
While Quine urged that names should be replaced by descriptions or iota-terms and then eliminated by Russell’s method, he never held a “descriptivist” theory of names as that label is usually understood. As Delia Graff Fara has reminded the present writer, in §36 of his Methods of Logic (1950) Quine explicitly declares himself neutral. If “Pegasus” or “Socrates” is not synonymous with any existing description in the language, we can simply introduce a new predicate “Pegasizes” or “Socratizes”, amounting to “is-Pegasus” or “is-Socrates” with the ”is” of predication rather than of identity, and replace the names by “the Pegasizer” or “the Socratizer”. Kripke in Naming and Necessity, footnote 5, explicitly exempts Quine’s proposal from his critique of “descriptivism” or “the Frege-Russell theory”. He does insist that Quine is only trading the question “How is the reference of ‘Socrates’ determined” by the question “How is the extension of ‘Socratizes’ determined?” but adds “Of course I do not suggest that Quine has ever claimed the contrary.”
The verbatim transcript of the audiotape of the discussion—though as is usual in such transcripts it contains many false starts, incomplete sentences, and the like—is in some ways even more revealing than the heavily-edited published version, Marcus et al. (1962). One noteworthy change between the verbatim transcript and the edited, published version of the discussion is that the word “analytic” in Kripke’s question is changed to “necessary”. The switch misled Soames (1995), note 26 into thinking that “Marcus responds to a straightforward question about the necessity of identities involving names...with remarks about epistemology and synonymy.” In fact, the question to which Marcus was responding was phrased in terms of analyticity to begin with. Fetzer and Humphreys (1998) lament their inability to include a verbatim transcript—they had a new one prepared from the original audiotapes—in their collection, but the partial catalogue seems to indicate that Quine’s copy of the 1962 transcript is now available to scholars in his Nachlaß (folder 673) at the Houghton. Of the two Boston quotations below, the first is from the unedited transcript, the second from the published version, of Marcus’s last speech.
To be sure, a quarter century later Marcus was to write as follows of the above quoted statement about “having recourse to a dictionary”:
I had in mind here a biographical “dictionary” which as linguists take it are not lexical dictionaries but encyclopedias. Such a “dictionary” might not resolve the question. Encyclopedias can be mistaken.
(The quoted formulation is from the letter cited earlier, but there are similar remarks in the preface and notes to the reprintings of Marcus (1961) and Marcus et al. (1962) in the collection Marcus (1993).) And this amounts, not merely to a retraction of the endorsement of (39) and even (38), but to a virtual denial that she ever meant to make such an endorsement. But what we have in this autobiographical reminiscence is clearly another instance of an unsuccessful attempt to reconstruct decades-old thoughts without consulting the full range of contemporary documents. Specifically, Marcus seems not to have consulted the verbatim transcript as opposed to the edited, published version of Marcus et al. (1962), or the original Marcus (1963) as opposed to the rewrite Marcus (1974). For those sources, from which I have just quoted in the body of the text, clearly show that Marcus meant by “dictionary”, as she used the term in Boston, a book that would tell us when two words have the same meaning—what she in Helsinki called a “lexicon”, and not an “encyclopedia”—and that she was mentioning the “dictionary” in order to explain in what sense an identity statement would be analytic. No one, of course, explains the notion of analytic truth in terms of what non-lexical information or misinformation is to be found in an encyclopedia that may be mistaken.
Soames (1995) suggests that Marcus would have liked to be able to claim that ordinary proper nouns are Russellian logically proper names, but was aware of some difficulties or objections, and so wavered between bolder formulations that clearly seem to be about “names” in the ordinary sense and more guarded formulations that almost equally clearly seem to be about “names” only in some idealized sense. The “dictionary” passage seems of the latter kind.
The philosophical issues can hardly be satisfactorily sorted out if doxographical issues remain in confusion. Nothing has been more damaging to philosophical understanding of the issues in this area than sensationalistic “revisionist” attempts, of the kind objected to in Soames (1995), to read doctrines expressed in later writings by Kripke back into earlier writings of Marcus and others. For this sort of thing can be accomplished only by willfully conflating notions and theses — such as various of (35)–(41) above — that crucially need to be distinguished.
References
Church, A. (1950). Review of Fitch (1949). Journal of Symbolic Logic, 15, 63.
Donnellan, K. (1966). Reference and definite descriptions. Philosophical Review, 75, 281–304.
Fetzer, J. H., & Humphreys, P. W. (Eds.). (1998). The new theory of reference: Kripke, Marcus, and its origins. Dordrecht: Kluwer.
Fine, K. (1983). The permutation principle in quantificational logic. Journal of Philosophical Logic, 12, 33–37.
Fitch, F. B. (1949). The problem of the morning star and the evening star. Philosophy of Science, 16, 137–141.
Fitch, F. B. (1950). Attribute and class. In M. Farber (Ed.), Philosophic thought in France and the United States (pp. 640–647). Buffalo, NY: University of Buffalo Press.
Føllesdal, D. (2004). Referential opacity and modal logic. London: Routledge.
Hughes, G. E., & Creswell, M. J. (1968). An introduction to modal logic. London: Methuen.
Kaplan, D. (1989). Demonstratives: An essay on the semantics, logic, metaphysics, and epistemology of demonstratives and other indexicals. In J. Almog, J. Perry, & H. Wettstein (Eds.), Themes from Kaplan. Oxford: Oxford University Press.
Kripke, S. (1959). A completeness theorem in modal logic. Journal of Symbolic Logic, 24, 1–14.
Kripke, S. (1963). Semantical considerations on modal logic. Acta Philosophica Fennica, 16, 83–94.
Kripke, S. (1971). Identity and necessity. In M. K. Munitz (Ed.), Identity and individuation (pp. 161–191). New York: New York University Press. Reprinted in Kripke 2011, pp. 1–26.
Kripke, S. (1977). Speaker’s reference and semantic reference. In P. A. French, T. E. Uehling, & H. K. Wettstein (Eds.), Studies in the philosophy of language (pp. 255–276). Minneapolis, MN: University of Minnesota Press. reprinted in Kripke 2011, pp. 99–124.
Kripke, S. (1979). A puzzle about belief. In A. Margalit (Ed.), Meaning and use. Dordrecht: Reidel. Reprinted in Kripke 2011, pp. 125–161.
Kripke, S. (1980). Naming and necessity. Cambridge, MA: Harvard University Press.
Kripke, S. (2011). Philosophical troubles: Collected papers (Vol. 1). Oxford: Oxford University Press.
Marcus, R. B. [Ruth. C. Barcan] (1946a). A functional calculus of first order based on strict implication. Journal of Symbolic Logic, 11, 1–16.
Marcus, R. B. [Ruth. C. Barcan] (1946b). A deduction theorem in a functional calculus of first order based on strict implication. Journal of Symbolic Logic, 11, 115–118.
Marcus, R. B. [Ruth. C. Barcan] (1947). Identity of individuals in a strict functional calculus of second order. Journal of Symbolic Logic, 12, 12–15.
Marcus, R. B. (1960). Extensionality. Mind, 69, 55–62.
Marcus, R. B. (1961). Modalities and intensional languages. Synthese, 13, 303–322.
Marcus, R. B. (1963). Attribute and class in extended modal systems. Acta Philosophical Fennica, 16, 123–136.
Marcus, R. B. (1974). Classes collections, and individuals. American Philosophical Quarterly, 11, 227–232.
Marcus, R. B. (1993). Modalities: Philosophical essays. Oxford: Oxford University Press.
Marcus, R. B., Quine, W. V., Kripke, S., & Føllesdal, D. (1962). Discussion on the paper of Ruth B. Marcus. Synthese, 14, 132–143.
Prior, A. N. (1956). Modality and quantification in S5. Journal of Symbolic Logic, 21, 60–62.
Prior, A. N. (1963). Is the concept of referential opacity really necessary? Acta Philosophical Fennica, 15, 189–199.
Prior, A. N. (1967). Logic, modal. In P. Weiss (Ed.), Encyclopedia of philosophy. New York: Macmillan.
Quine, W. V. (1947a). The problem of interpreting modal logic. Journal of Symbolic Logic, 12, 43–48.
Quine, W. V. (1947b). Review of Marcus (1947). Journal of Symbolic Logic, 12, 95–96.
Quine, W. V. (1950). Methods of logic. Cambridge, MA: Harvard University Press.
Quine, W. V. (1953a). Reference and modality. In: From a logical point of view (pp. 139–159). Cambridge, MA: Harvard University Press.
Quine, W. V. (1953b) Three grades of modal involvement. In: Proceedings of the 11th International Congress of Philosophy (vol. 14, pp. 65–81). Amsterdam: North-Holland.
Quine, W. V. (1958). Correction to Quine (1947b). Journal of Symbolic Logic, 23, 342.
Quine, W. V. (1960). Word and object. Cambridge, MA: MIT Press.
Quine, W. V. (1961). Reply to professor Marcus. Synthese, 12, 323–330.
Quine, W. V. (2008). Panel on reference. In: D. Føllesdal & D. Quine (Eds.). Confessions of a confirmed extensionalist and other essays. Cambridge: Harvard University Press.
Smullyan, A. (1947). Review of Quine (1947a). Journal of Symbolic Logic, 12, 139–141.
Smullyan, A. (1948). Modality and description. Journal of Symbolic Logic, 13, 31–37.
Soames, S. (1995). Revisionism about reference: A reply to Smith. Synthese, 104, 191–216.
Wiggins, D. (1965). Identity statements. In R. J. Butler (Ed.), Analytic philosophy: Second series (pp. 40–71). Oxford: Basil Blackwell.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Burgess, J.P. On a derivation of the necessity of identity. Synthese 191, 1567–1585 (2014). https://doi.org/10.1007/s11229-013-0351-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-013-0351-8