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On the Fundamental Role of ‘Means That’ in Semantic Theorizing

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Abstract

Our aim is to illuminate the interconnected notions of meaning and truth. For this purpose, we investigate the relationship between meaning theories based on commonsensical ‘means that’ and interpretive truth theories. The latter are Tarski–Davidson-style truth theories serving as meaning theories. We consider analytically true semantic principles containing ‘means’ and ‘means that’ side to side with ‘denotes’, ‘satisfies’, and ‘true’, which constitute the extensional semantic constants of interpretive truth theories. We show that these semantic constants are definable in terms of ‘means’ and ‘means that’ operators. The definitions themselves are semantic principles in the role of meaning postulates. We extend a meaning theory based on ‘means that’ by adjoining semantic principles to the axioms of the theory. Then, all axioms, hence all theorems, of a corresponding interpretive truth theory are provable in the extension of the meaning theory. Furthermore, every interpretive truth theory can be included in the extension of a corresponding meaning theory. Therefore, the extension of a meaning theory resulting from the adjunction of semantic principles constitutes a unified meaning-and-truth theory since it includes both a meaning theory and an interpretive truth theory.

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Notes

  1. Concerning this distinction see Lewis (1972/1983, pp.163, 164) and Davies (1981, pp. 6ff).

  2. See Tarski (1956). However, as mentioned by Kripke, “Tarski’s metalanguages contain no [primitive] semantical terms” (Kripke, 1976, p. 348).

  3. We use ‘sentence’, unless otherwise specified, exclusively in the sense of ‘declarative closed sentence’. On the other hand, we shall use the phrase ‘(closed or open) sentence’ in place of the more usual term ‘formula’.

  4. This term is introduced in Kaplan (1989, p. 522). Kaplan defines a context as “a package of whatsoever parameters are needed to determine content.” (Kaplan, 1989, p. 591).

  5. We deal exclusively with languages free of paradoxical or pathological sentences such as the Liar.

  6. Concerning expressions-in-a-context and, especially, sentences-in-a-context: (i) Frege writes that “the time of utterance [of a time dependent sentence type] is part of the expression of the thought (Gedankenausdruck) (See Frege (1918/1997, p. 322). (ii) Kripke represents Frege’s expression of a thought by an ordered pair of a sentence type and a time (Kripke, 2008, p. 201 and p. 203 n.61). (iii) Kaplan calls “the combination of an expression [type] and a context” an occurrence [of the expression in the context]”. See Kaplan (1989, p. 524 and p. 547).

  7. In our use of substitutional quantification, we follow Kripke (1976, esp. sec. 1 pp. 328–331 and sec 5(a) pp. 365–368).

  8. We adopt the term ‘referential’ from Kripke (1976, p. 329). Note that, following Quine, Soames uses ‘objectual’ in place of Kripke’s ‘referential’. See Soames (1999, p. 42).

  9. We adopt the symbol ‘Q’ from Grover (1973, p. 101 and 1992, p. 235) and the term ‘quotation-functor’ from Tarski (1956, p. 161). Most often the quotation functor is expressed merely by a pair of quotation marks rather than by a prefix such as ‘Q.’ For example, see Tarski (1956, pp. 160–1), Curry (1963, pp. 42–3), and Belnap and Grover (1973, p. 19).

  10. We adopt the term ‘quotatum’ (as used in the sense of quoted expression) from Saka (2013, p. 940).

  11. We adopt the term ‘meaning specification’ from Davies (1981, p. 3).

  12. We adopt the term ‘M-form sentence’ from Ray (2014, p. 83 n. 8).

  13. Cf. Evans and McDowell (1976, p. viii). The importance of the distinction between translation and meaning specification expressed by ‘means that’ is emphasized by Lewis (1972/1983, pp. 169–170), Evans and McDowell (1976, pp. vii–xi) and Davies (1981, p. 4).

  14. This test has been introduced by Langford in the following way: “A word that is being used is to be translated, while a word that is talked about [i.e. mentioned] must not be” (Langford, 1937, p. 53).

  15. Our use—at least in the present context—of the notion of invariance under translation seems to be unaffected by Davidson’s indeterminacy of translation. Indeed, Davidson himself remarks that “[i]n determinacy of meaning or translation does not represent a failure to capture significant distinctions.” It marks the fact that certain apparent distinctions are not significant” (Davidson, 1974, p. 322). Moreover, Church remarks that “[t]he existence of more than one language is not usually required as fundamental ground of the conclusion …” (Church, 1954, p. 72, n. 21).

  16. We adopt the term ‘M-sentence’ from Lepore and Ludwig (2005, p. 42).

  17. Cf. Church’s remark that “an item of factual information” is contained in the fact “that ‘man is a rational animal’ means in English that man is a rational animal” (Church, 1949, p. 98).

  18. We use ‘proposition’ with a lower-case ‘p’ as the meaning of a sentence where meaning is construed as an entity. On the other hand, we shall use ‘Proposition’ with a capital ‘P’ in the sense of an asserted sentence in accordance with mathematicians’ usage.

  19. We agree here with Davidson who remarks that: “[p]aradoxically, the one thing meanings [as entities] do not seem to do is oil the wheels of a theory of meaning [which] non-trivially give[s] the meaning of every sentence in the language” (Davidson, 1967/2001, pp. 20–1).

  20. Cf. Grice’s following remark:

    It remains to inquire whether there is any reasonable alternative program for the problems about meaning other than of the provision of a reductive analysis of the concept of meaning. The only alternative which I can think of would be that of treating “meaning” as a theoretical concept (Grice (1987/1989, pp. 358–9).

  21. Kripke (1982, esp. pp. 13-14, 77–79).

  22. Cf. Boghossian who formulates and critically discusses the thesis that “judgments about meaning are factual, irreducible, and judgment-independent” (Boghossian, 2002, pp. 185–7).

  23. Davidson’s radical interpretation, or cognitive science in general, may constitute a procedure for testing M-form sentences.

  24. For example, Carnap writes “S1 contains descriptive constants (that is nonlogical constants) …” See Carnap (1947, p. 3).

  25. Carnap characterizes the relationship of the meaning postulates to the non-logical constants (hence the semantic constants too) as follows: The [semanticists] are free to choose their [meaning] postulates, guided not by their beliefs concerning facts of the world but by their intention with respect to the meanings, i.e. the ways of use of the [semantic] constants (Carnap, 1952, p. 66 and p. 69 (6)).

  26. Given the distinction between meaning specifications and translation rules (pointed out in sec. 0.6) we abstain from including the translational tradition in our Historical Background.

  27. We will critically evaluate Ray’s theory in sec. 8.3. below.

  28. Indeed Lepore and Ludwig characterize the theorems of a truth theory meeting Convention T as follows:

    If we know that a sentence of the form (T) [i.e. σ is T iff p] is one of these theorems, then we can replace ‘is T iff’ with ‘means that’ to yield a true M-sentence(Lepore and Ludwig, 2007, p. 28).

  29. Davies (1981) constructs a meaning theory for a language with sentential connectives (pp. 41–44), for a subject-predicate language (pp. 93–94), and for a quantificational language with substitutional quantifiers (pp. 122–123).

  30. See the definition of ‘interpretational’ in Davies (1981, p. 34).

  31. This definition is a simplified version of Grice and Strawson’s following definition: “Two sentences are synonymous if and only if any true answer to the question “What does it mean?” asked of one of them is a true answer to the same question, asked of the others” (Grice and Strawson, 1956/1989, p. 201).

  32. This definition is a simplified version of Soames’ following definition: “two expressions are synonymous when the answer to … the question ‘What does it mean?’… is the same for both” (Soames, 2010, p. 4).

  33. Cf. Church (1941, esp. pp. 208–9).

  34. Cf. Klement (2002, pp. 102–103).

  35. Concerning such a use/display distinction see Jespersen (2019, sec. 3.3) and Duží (2019, sec. 2).

  36. We consider exclusively meaning theories based on the meaning-operators. Davies calls such theories “meaning theories strictly so-called” in order to distinguish them from Davidsonian truth theories that serve as meaning theories. See Davies (1984, p. 86).

  37. We follow Church in adopting the notation ‘(κ1κ2)’. Cf. Benzmüller and Andrews (2019, esp. pp. 3–6 and pp. 9–19).

  38. We adopt the examples ‘Theaetetus flies’ and ‘the father of Annette’ from Davidson (1967/2001, p. 17).

  39. See Davidson (1967/2001, p. 18).

  40. We introduce ‘M-form theorem’ in analogy to Lepore and Ludwig’s ‘T-form theorem’ (2005, p. 145).

  41. We adopt ‘M-theorem’ from Lepore and Ludwig (2005, p. 120).

  42. We use the term ‘sample language’ in analogy to Lepore and Ludwig’s use of “sample theory” (Lepore and Ludwig, 2007, p. 29).

  43. This definition of ‘meansNm’ is analogous to that given in Davies (1984, p. 87(v)). Note that ‘R(L0, t, a)’ is equivalent to ‘the name t meansNm (in L0) a.’.

  44. This proposition is based on Davies’ axioms for predicate terms in Davies (1981, pp. 93–94).

  45. We adopt “structure-reflecting” from Evans (1985, pp. 323, 325).

  46. Davies’ meaning theory for L0 is given in Davies (1981, pp. 93–4).

  47. In Davies’ notation (AxD.ɣ1) and (AxD.Φ1) are formulated respectively as follows (Davies, 1981):

    1. (1)

      MRef (‘m1’, n1)

    2. (2)

      \(\forall\)ɣ \(\forall \)v [ MRef (ɣ, v) → (⌜\({P}_{1}\)ɣ⌝ means that Q1v)]

  48. See Davies (1981, p. 87) and Taylor (1982, p. 197). \({\theta }_{{L}_{0}}^{{M}_{DT}}\) is Davies’ axiom system corresponding to Taylor’s axiom system for the Gricean analysis of the meanings of L0’s sentences. See Taylor (1982, esp. pp. 188, 191, 197).

  49. In our own framework ‘P’ should stand for ‘[λx.x is a planet].’

  50. Concerning sentence (1) see sec. 7.2 (iv).

  51. See Davies (1981, p. 42 (M2)) and Davies (1984, p. 87).

  52. We adopt ‘T-form theorem’ from Lepore and Ludwig (2005, p. 145).

  53. According to Lepore and Ludwig a T-sentence “is a T-form sentence in which the right-hand side translates or interprets the sentence mentioned on the left-hand side” (Lepore and Ludwig, 2007, p. 33 n. 15).

  54. We adopt the term ‘meets Convention T’ from Lepore and Ludwig (2007, p. 28).

  55. We adopt the term ‘truth-from-meaning principle’ from Collins (2002, p. 506).

  56. Horwich (2010, p. 164) qualifies (TM) as being “explanatorily fundamental.”

  57. Axioms (Ax. ɣi), (Ax. Φj), and (Ax.comp) correspond to Evans’ axioms as formulated in Evans (1985, p. 327).

  58. See also Ludwig (2017, p. 26).

  59. The main clause of such a meaning theory, viz. [1] is formulated in sec. 0.9. above.

  60. See Davidson (1967/2001, pp. 22–4).

  61. Indeed, given that ‘σ is true (in L) \(\leftrightarrow \) p’ is a T-sentence provable in \({\theta }_{L}^{T}\), and ‘q’ stands for a logically true sentence in LML, the sentence ‘σ is true (in L) \(\leftrightarrow \) (p \(\wedge \) q)’ is provable in \({\theta }_{L}^{T}\) although it is not a T-sentence. Cf. Lepore and Ludwig (2005, pp. 109–110).

  62. See Davies (1981, p. 33) and Lepore and Ludwig (2005, p. 111).

  63. Concerning the empty name ‘Vulcan’ of an alleged planet see Sainsbury (2005, pp. 86–89).

  64. Cf. Sainsbury (2004, p. 370(1)) and Sainsbury (2005, p. 46, [9]).

  65. Sentence (11) is literally mentioned in McDowell (1977, p. 161).

  66. Cf. Sainsbury (2005, pp. 15 and 35–7).

  67. The definition of denotation (Df.Den) corresponds to the one formulated by Wallace. Indeed, he remarks that “we can use” ∃z [x = z and Trans ‘z’ = y]” as a definition of “y denotes x” (Wallace, 1975, p. 189). We see that in Wallace’s definition of ‘denotes’ the symbols ‘x’, ‘y’, and ‘z’ stand respectively for our ‘x’, ‘ɣ’, and ‘a’, so that ‘Trans ‘z’ = y’ stands for ‘Trans ‘a’ = ɣ’. The terms ‘Trans’ is introduced by Camp by the example that “Trans ‘Able’ can be thought … as the object language translation of ‘Able’” (Camp, 1975, p. 174). Presumably ‘Trans ‘z’ = y’ can be read as ‘y means z’, hence in our notation, as ‘ɣ means a’ (In Camp’s terminology it is read as ‘ɣ is the object language translation of a’. Kripke criticizes Wallace’s definition of denotation by making two objections. The first one is to the effect that the quantifiers ‘∃z’ should be replaced by the substitutional quantifiers ‘Σz’. The second, and more important, objection consists in the remark that “the use of the symbol ‘ = ’ is … problematic unless a denotation is already provided for all the terms [of the purported definition]” (Kripke, 1976, p. 417). Our own definition (Df.Den) is not subject to either of Kripke’s objections. Indeed, on the one hand the existential quantifier is explicitly a substitutional one, and on the other hand our recourse to free logic enables us to use non-denoting terms in the definition.

  68. Horwich (2010, p. 131).

  69. Horwich (2010, p. 132).

  70. The informal definition of ‘true’ is stated by Rabinowicz (2010, p. 308) as follows: “For any statement S, S is true iff for some p, S means that p and p”.

  71. The notion of M-correlate is used by Davies (1981, p. 34) and by Lepore and Ludwig (2007, p. 28). Lepore and Ludwig (2011, p. 267) call the inference rule leading from a canonically proved T-form theorem of an interpretive truth theory to its M-correlate “Transference (T)”.

  72. The notion of T-correlate is introduced by Davidson (1967/2001, p. 23).

  73. Our Definition 7 is equivalent to Lepore and Ludwig’s following definition: “call any truth theory that meets Convention A interpretive” (Lepore and Ludwig, 2011, p. 268).

  74. Proposition 3 is analogous, though not equivalent, to Lepore and Ludwig’s following statement: “… if the axioms meet Convention A …, the [truth] theory is true [i.e. sound]” (Lepore and Ludwig, 2011, p. 273). It is not equivalent since \({\theta }_{{L}_{1}}^{M}\) is not involved in that statement.

  75. All of these properties, except (ii) have been already established by Lepore and Ludwig, though in a quite different framework.

  76. That is how Tarski characterizes a desired definition of truth. See Tarski (1944, p. 341).

  77. Davidson (1967/2001, p. 22) and Foster (1976, p. 6) raises strong objections to meaning theories strictly so called, whereas Davies (1981, sec. II.3) argues in their favor and eventually himself constructs such theories.

  78. We adopt the abbreviation ‘L&L’ from Hoeltje (2013, p. 832).

  79. Note that Ludwig (2017, p. 29 n.1 [pp. 24–25]) agrees with Hoeltje’s objection to Ray’s means-that theory. Ray’s means-that theory is also criticized by Kirk-Giannini and Lepore (2017).

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    Teo Grünberg, David Grünberg & Oğuz Akçelik

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Grünberg, T., Grünberg, D. & Akçelik, O. On the Fundamental Role of ‘Means That’ in Semantic Theorizing. J of Log Lang and Inf 32, 601–656 (2023). https://doi.org/10.1007/s10849-022-09393-8

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