A theory of bondage

Philosophical Review

Volumn 115, Issue 4, 2006, Pages 415-448

  Salmon, Nathan ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

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EID: 38349144261     PISSN: 00318108     EISSN: 15581470     Source Type: Journal    
DOI: 10.1215/00318108-2006-009     Document Type: Article

References (61)

1. Gottlob Frege, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, trans. J. L. Austin (Evanston, IL.: Northwestern University Press, 1968), x, 71, 73.
2. Hence I reject Donald Davidson's appeal to "pre-Fregean semantic innocence", which I believe is based largely (though not entirely) on a confusion bet ween expressionbased and occurrence-based semantics. See the closing paragraph of his "On Saying That", Synthese 19(1968-69):130-46. (See note 20 below.)
3. See by way of comparison Richard Montague's treatment of this sentence in his "The Proper Treatment of Quantification in Ordinary English", in Approaches to Natural Language: Proceedings of the 1970 Stanford Workshop on Grammar and Semantics, ed. J. Hintikka, J. Moravcsik, and P. Suppes (Dordrecht: D. Reidel, 1973), 221-42, at 240.
4. An expression-occurrence standing within a formula must not be confused with a token of the expression, such as an inscription or an utterance. A token is a physical embodiment, physical event, or other physical manifestation of the expression (type). An occurrence of an expression is, like the expression itself, an abstract entity. For most purposes, an expression occurrence may be regarded as the expression together with a position that the expression occupies within a larger sequence of expressions. In contemporary philosophy of language, it has become a common practice to attribute semantic values neither to expressions themselves nor to their occurrences but to expression-utterances. I regard t his speech-act-centered conception of semantics a giant leap backward, lamentable in the extreme. See my "Two Conceptions of Semantics", in Semantics versus Pragmatics, ed. Zoltán Gendler Szabó (Oxford: Clarendon, 2005), 317-28, reprinted in my forthcoming Content, Cognition, and Communication (Oxford University Press).
5. The letters in 'Nathan' are four: 'A', 'H', 'N', and 'T'. Two of these occur twice, making six letter-occurrences in all. With some trepidation, I follow the common vernacular in speaking of "bound variables" in a sentence where what are mentioned are actually bound occurrences, or of "the initial quantifier of", or "the 'he' in" a sentence, and so on, where what is mentioned is actually an occurrence of a quantifier or the pronoun. I have taken care to see that my usage unambiguously decides each case. For example, there is only one lowercase, italic letter 'x' and only one English word 'he', but there are infinitely many occurrences of either, so that any talk of "the bound variables" (plural) of a formula containing no variable other than 'x ', or of "the 'he'" (with definite article) in a sentence, cannot sensibly concern expressions.
6. Positions within quotation marks and similar devices, including 'believes that', are not extensional. An occurrence of a well-formed expression ζ, is said to be within the scope of an occurrence of a variable-binding- operator phrase ⌈ (Bα) ⌈, where B is a variable-binding operator and α is a variable, if the latter occurrence is the initial part of an occurrence of a well-formed expression of the form ⌈ (Bα) φ⌈, where φ is a formula and the former occurrence stands within that occurrence of ⌈ (Bα) φ⌈. (It may be assumed that the universal quantifier is '∀'-and as a notational convenience is routinely deleted-so that a universal-quantifier phrase written ⌈ (α) ⌈ is of the form ⌈ (Bα) ⌈.)
7. Alonzo Church, "The Need for Abstract Entities in Semantic Analysis", American Academy of Arts and Sciences Proceedings 80 (1951):100-112. Church does not follow Frege's Context Principle. Church's semantics is on expressions, not on their occurrences. He therefore does not distinguish between designatum and customary designatum, or between sense and customary sense, and has no notion of indirect designatum or indirect sense.
8. Peter Geach, "Ryle on Namely-Riders", Analysis 21, no. 3 (1960-61), reprinted in his Logic Matters (Oxford: Basil Blackwell, 1972), 88-92, at 92.
9. Peter Geach, Reference and Generality (Ithaca, NY: Cornell University Press, 1962), at 125-26, and passim.
10. 2 = 9'. See, by way of comparison, Donald Kalish, Richard Montague, and Gary Mar, Logic: Techniques of Formal Reasoning (1964; 2nd ed., New York: Harcourt Brace Jovanovich, 1980), at 206, 311-12.
11. This function is the customary extension of ⌈ (λα) [ζ]⌈ under s. Compare: The indirect extension of 'Snow is white' is the customary sense-the proposition that snow is white-which is the customary extension of 'that snow is white'. (Frege: "In indirect discourse the words have their indirect designata [ungerade Bedeutungen], which coincide with what are customarily their senses. In this case then the clause has as its designatum a thought [Gedanke], not a truth-value; its sense is not a thought but is the [customary] sense of the words 'the thought that...'" [Über Sinn und Bedeutung].)
12. i. This is the constant function to the customary designatum, s ('y'), defined over the range of 'x'. For most pur poses, this may be replaced with s ('y') itself.
13. This function is the customary extension of ⌈ (λα) (λβ) [ζ]⌈ under s.
14. The notion of double bondage extension, and the distinction between it and customary designation, is relevant to resolving Kit Fine's development of Russell's antinomy of the variable, in "The Role of Variables", Journal of Philosophy 100 (2003):605-31. The problem, as Fine poses it, is this: "How is it that any two variables ranging over a given universe have the same semantic role and yet have a different semantic role?" As he develops the problem, the question becomes "How is it that there is no crosscontextual difference in semantic role between the variables 'x 'and 'y', and yet there is a cross-contextual difference in semantic role between the pair <'x', 'y'> and the pair <'x', 'x'>?" (6 0 8). Our theory of bondage provides one possible response to Fine's question. The extension of each of the occurrences of 'x' in '(∃x) (x loves x)' is t he bondage extension of 'x' with respect to itself: the identity function on the universe over which 'x' ranges. This is equally the extension of the occurrences of 'y' in '(∃y) (y loves y) '. Here is one sense in which there is no "cross-contextual difference in semantic role" between 'x' and 'y '. By contrast, the occurrences of 'x' and 'y' in '(x) (∃y) (x loves y)', though they range over the same universe, differ in extension. The double bondage extensions of 'x' and 'y' with respect to <'x', 'y'> are neither of them the same as the single bondage extension of 'x' with respect to itself, or that of 'y' with respect to itself. Here is one sense in which there is a "cross-contextual difference in semantic role" between the variables in' (∃x) (∃y) (x loves y) ', on the one hand, and those in' (∃x) (x loves x)' or in '(∃y) (y loves y)', on the other. The apparent dichotomy here is illusory. The single bondage extension of 'x' with respect to 'x' is not the same as that of 'y' with respect to 'x'. (See note 12.) And although the double bond age extension of 'x' with respect to <'x', 'y'> is not the same as that of 'y' with respect to the same pair <'x', 'y'>, it is the same as that of 'y' with respect to the converse pair <'y', 'x'>. Furthermore, the double bondage extension of 'y' with respect to <'x', 'y'> is the same as that of 'x' with respect to the reflexive pair <'x', 'x'>.
15. (n+1), might be taken as axioms. This construal may seem more appropriate for the latter principle than the former, which is plausibly construed instead as a definition of 'customary extension'. (See note 6.) What entity the customary extension of an expression is can be determined by invoking the classical characterization of extension simpliciter.
16. α under s is nonempty.
17. See notes 11 and 12 above. By contrast, on Frege's hierarchies of multiply indirect extensions, the (n + 1)-fold indirect extension is the n-fold indirect sense-which, as Russell noted in his infamous "Gray's Elegy" argument, is a new entity, entirely distinct from the n-fold indirect extension.
18. This might be the concept of being a nonempty class, or the second-order propositional function Σ that maps any first-order propositional function F to the proposition that F is "sometimes true", that is, that F yields a true proposition for at least one argument, or the corresponding concept, or something similar.
19. An actual proof that a modestly restricted principle of strong compositionality is satisfied (or falsified) awaits a suitable theory of concepts analogous to Zermelo-Frankel set theory.
20. I argue this in "On Indirect Sense and Designation" (unpublished). My attitude resonates to some extent with Rudolf Carnap's in Meaning and Necessity: A Study in Semantics and Modal Logic (1947; 2nd ed., Chicago: University of Chicago Press, 1956), chap. 3, especially secs. 29-32, pp. 124-44.
21. (But see note 2 above.) Carnap calls expressionbased semantics the method of extension and intension, and Frege's occurrence-based semantics the method of the name-relation. Carnap saw Frege's occurrence-based semantics as flowing naturally from his assimilation of semantic extension to "the name-relation" between a singular term and its designatum.
22. See ibid., sec. 28, especially at page 123. (Occurrence-based semantics per se does not require this assimilation. I believe the Context Principle also flows fairly naturally from a "truth-conditional" semantics that does not assimilate extension to designation. I have set out occurrence-based semantics without assuming the assimilation.) A resolute advocate of the expression-based semantic method over Frege's occurrence-based semantics, Carnap points out that the expression-semantic notion of extension and Frege's notion of designation ("nominatum", Bedeutung), though they are very similar, are not to be identified; and likewise the expression-semantic notion of content ("intension") and Frege's notion of sense, though very similar, are not to be identified. "A decisive difference between our method and Frege's consists in the fact that our concepts, in distinction to Frege's, are independent of the context" (125).
23. Still, Carnap noted, the expression-semantic notions of extension and content coincide, respectively, with Frege's notions of customary designatum and sense. (See Carnap's principles 29-1 and 29-2, pp. 125-26.)
24. Carnap advises against doing semantics both ways simultaneously (128-29) and complains that Frege's method led him to postulate an insufficiently explained notion of indirect sense (129) and leads ultimately to Frege's infinite hierarchies (131-32).
25. Russell had previously blamed the Fregean hierarchy not on occurrence-based semantics, but on the expression-semantic thesis that definite descriptions are singular terms. See note 17. My own view is that the hierarchy discredits neither the Context Principle nor the thesis that definite descriptions are singular terms and is to be traced instead to the union of two fundamental principles of Fregean theory: that any expression-occurrence that has a designatum also has a sense, which is a concept of the designatum; and that the indirect designatum of an expression is the customary sense. See my "On Designating", Mind 114 (2005):1069-1133,
26. reprinted in my Metaphysics, Mathematics, and Meaning (Oxford: Clarendon, 2005), 286-334;
27. and also "On Indirect Sense and Designation." There is an analogue to the Fregean hierarchy in Alonzo Church's elegant "Logic of Sense and Denotation" ("LSD"), in Structure, Method, and Meaning: Essays in Honor of Henry M. Sheffer, ed. Paul Henle, Horace M. Kallen, and Susanne K. Langer (New York: Liberal Arts, 1951), 3-24;
28. Noûs (1973):24-33,
29. 135-56. As Carnap recognizes (132, 137-38), however, the hierarchies in LSD are not semantic values of single expressions. They are the senses of infinitely many different expressions.
30. Since 1905 it has been illegitimate to presume without argument that definite descriptions are singular terms and not restricted quantifiers-even if it is at least as illegitimate, based largely on intuitions concerning what is mentioned, to presume without argument that definite descriptions are quantifiers and not singular terms. Some of the arguments of Russell and his followers have shaken confidence in the orthodox view that definite descriptions are singular terms. (See my "On Designating.") By contrast, the thesis that demonstratives and 'that'-clauses are singular terms remains quite plausible, also based largely on intuitions concerning what is mentioned, while the rival thesis that they are quantifiers remains enormously implausible. Many of the arguments of Kripke and others that names are not descriptions transfer easily to demonstratives and 'that'-clauses. In particular, that demonstratives are singular terms is common sense, and no persuasive evidence has been adduced that they are quantifiers. Specifically, as will be seen, the general argument presently to be considered provides no evidence whatever concerning demonstratives or 'that'-clauses. (I thank Zoltán Szabó for pressing me to address this. It should not be assumed that he agrees with my assessment.)
31. Benson Mates, "Descriptions and Reference", Foundations of Language 10 (19 73):409-18, at 415. The general form of argument has been employed or endorsed by several others during the past three decades. The following is a chronological partial bibliography: Gareth Evans, "Reference and Contingency", Monist 62 (1979):161-89, at 169-70;
32. Stephen Neale, Descriptions (Cambridge, MA: MIT Press, 1990), at 56n28;
33. Neale, "Term Limits", in Logic and Language, vol. 7 of Philosophical Perspectives, ed. James E. Tomberlin (Atascadero, CA: Ridgeview, 1993), 89-123, at 107;
34. Jeffrey King, "Are Complex 'That' Phrases Devices of Direct Reference?" Noûs 33 (1999):155-82, at 157-58, 161-62;
35. Ernest Lepore and Kirk Ludwig, "The Semantics and Pragmatics of Complex Demonstratives", Mind 109 (2000):200-241, at 205-206, 210-22, and passim;
36. King, Complex Demonstratives (Cambridge, MA: MIT Press, 2001), at xi-xii, 1, 10-11, 20-22;
37. Kent Johnson and Ernest Lepore, "Does Syntax Reveal Semantics? A Case Study of Complex Demonstratives", in Language and Mind, vol. 16 of Philosophical Perspectives, ed. James E. Tomberlin (Atascadero, CA: Ridgeview, 2002), 17-41, at 31;
38. Jason Stanley, "Review of Jeffrey King, Complex Demonstratives, Philosophical Review 111 (2002):605-9.
39. See, by way of comparison, my "Being of Two Minds: Belief with Doubt", Noûs 29 (1995):1-20, at 18n26, and "Demonstrating and Necessity", Philosophical Review 111 (2002):497-537, at 534-35n47; both reprinted in my Content, Cognition, and Communication.
40. Before Mates, Geach had drawn a somewhat different conclusion from the same data: that the occurrence of the definite description in (3), since it does not designate, does not "have the role of a definite description." See his "Ryle on Namely-Riders", in Logic Matters, 91-92; also, "Referring Expressions Again", Analysis 24 (1963-64), reprinted in Geach's Logic Matters, 97-102, at 99-100.
41. See note 6. The bracketed material represents variations or restrictions that Mates might have in mind. The restricted class C excludes such problematic sentences as ⌈β does not exist⌈ and things that entail it.
42. α. (See the appendix.)
43. α is the function that assigns truth to everything in the range of the variable α, and is falsehood otherwise.
44. α. Each of these corrected versions effectively blocks the objection.
45. n, i) = truth, if there is a unique such i, and is undefined otherwise.
46. Saul Kripke, "Is There a Problem about Substitutional Quantification?" in Truth and Meaning: Essays in Semantics, ed. Gareth Evans and John Henry McDowell (Oxford: Clarendon, 1976), 325-419, at 408.
47. Geach, Reference and Generality, at 129ff; and "Quine's Syntactical Insights", in Logic Matters, at 118-19.
48. See, by way of comparison, Gareth Evans, "Pronouns, Quantifiers, and Relative Clauses (I)", Canadian Journal of Philosophy 7 (1977):777-97; "Pronouns", Linguistic Inquiry 11 (1980):337-62. The analogous discourse fragment-Just two actors starred in City Lights. They were both multitalented-is not equivalent to the quantified generalization 'Just two actors both: starred in City Lights and were multitalented'. (The latter allows, while the former does not, that a third, nonmultitalented actor also starred in City Lights.) Many, including several critics, have followed Evans in concluding that the pronoun 'they' in the discourse fragment is an occurrence of a closed expression; hence too, by analogy, the pronoun in (4).
49. In the vernacular of theoretical linguistics, the term 'E-type pronoun' is used for an anaphoric pronoun-occurrence whose grammatical antecedent is a quantifieroccurrence that does not c-command that pronoun-occurrence. Linguists and linguisticsoriented philosophers almost invariably phrase this in terms of a "pronoun" and its antecedent "quantifier", where what are at issue are actually occurrences. (See note 5 above and recall again the cautionary note to which it is appended.)
50. Insofar as the modal truth-conditions for (4) yield this result, the 'he' does not function in (4) as a demonstrative. By contrast with (4ii), the sentence 'Dthat[the comedian who composed the musical score for City Lights] was multitalented' is true with respect to a context c and a possible world w if and only if the comedian who in the possible world of c (rather than w) composed the musical score for City Lights, was multitalented in w.
51. This argument for the pronoun's nonrigidity is McKinsey's, in "Mental Anaphora", Synthese 66 (1986):159-75, at 161.
52. It is echoed by Scott Soames, in his review of Gareth Evans's Collected Papers, Journal of Philosophy 86 (1989):141-56, at 145.
53. It is also endorsed by Stephen Neale in "Descriptive Pronouns and Donkey Anaphora", Journal of Philosophy 87 (1990):113-50, at 130,
54. and again in Descriptions (Cambridge, MA: MIT Press, 1990), 186.
55. See, by way of comparison, my "Demonstrating and Necessity", at 536-37 n 52.
56. My critique has benefited from discussion with Alan Berger, who realized independently that the arguments of Evans and McKinsey are incorrect.
57. See his Terms and Truth (Cambridge, MA: MIT Press, 2002), 171-78. Though the pronoun 'he' is rigid, so-called laziness occurrences (in addition to bound occurrences) may be nonrigid. The occurrence in (4ii) is not a laziness occurrence.
58. By contrast with (4), the two "E-type" pronoun-occurrences in 'If a man has a home, it is his castle' are more naturally taken as variable-occurrences bound by implicit universal-quantifier occurrences. Compare the account of Berger, Terms and Truth, 159-89, 203-27. The analysis Berger provides for discourse-fragments like (4) looks to be a notational variant of (4'). (Berger has informed me that he is inclined to think it is.) The anonymous referee for the Philosophical Review worries that although the two E-type pronouns in the following discourse are anaphorically linked to each other, on the analysis proposed here they are not co-bound by the same quantifier-occurrence: (5) (i) I spoke to a philosopher yesterday. (ii) He sides with Geach against Evans. (iii) He lives in California. Imagine the referee spoke with only two male philosophers yesterday, one of whom sides with Geach against Evans but does not live in California, the other lives in California but does not side with Geach against Evans. Then (5ii) and (5iii) are not both true. The worry is misplaced. The underlying logical form of (5) is arguably given by: (5') (i) [a x: philosopher (x)] (I spoke to x yesterday). (ii) [a y: philosopher (y); I spoke to y yesterday] (y sides with Geach against Evans). (iii) [a z: philsopher (z); I spoke to z yesterday; z side s with Geach against Evans)] (z lives in California). On this analysis each of the E-type pronouns is a bound variable. Whereas (5'ii) is true in the envisaged circumstance, (5'iii) is false-evidently in conformity with the English sentences they represent. The final occurrence of 'z' in (5'iii) is indeed cobound with the second-to-last, as it should be, by the initial quantifier phrase '[a z]'. It is not co-bound with the occurrences of 'y' in (5'ii). Nor should it be. If the two E-type pronouns in (5) were co-bound variables, (5ii) would be an open sentence and, as such, would not have truth-value. (The conjunction corresponding to (5') is equivalent to the conjunct (5'iii) a lone.)
59. The discourse fragment mentioned in note 32 is plausibly regarded as having an underlying logical form given, nearly enough, by: (i) [just two x: actor (x)] (x starred in City Lights). (ii) [every y: actor (y); y starred in City Lights] (y was multitalented). The occurrence of 'they' corresponds to the final occurrence of 'y'. See the previous note. Consider, in contrast, the discourse fragment: (i) A man and a woman starred in City Lights. (ii) The man was multitalented. If this does not entail that only one man starred in City Lights (... 'Another man who also starred in City Lights was not multitalented'), its logical form is arguably given by, (i) [a x: man (x)] (x star red in City Lights) and [a x: woman (x)] (x starred in City Lights). (ii) [a y: man (y); y starred in City Lights] ([the z: man (z)] (z = y) was multitalented). It is an interesting question under what circumstances a so-called E-type pronoun or similar occurrence is bound by an implicit (typically restricted) universal-quantifier occurrence and under what circumstances it is bound instead by an implicit existentialquantifier occurrence. In many cases, the issue might not be settled unambiguouslyfor example, 'Some senators are liars, but they have redeeming qualities'. It is possible that some E-type pronouns (occurrences) are pronouns of laziness rather than bound.
60. See by way of comparison also my Frege's Puzzle (Atascadero, CA: Ridgeview, 1986), at 144-47.
61. See note 28. There is a similarly corrected version of King's (K2):(K2') If a singular term β is directly referential, then every free occurrence in a sentence of β not within the scope of any nonextensional operator (other than classical variable-binding operators) is a directly referential occurrence. As with the replacement of (M) by (M"), and (M+) by (M+'), correcting (K2) effectively blocks King's argument.