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Relating Intensional Semantic Theories: Established Methods and Surprising Results

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Abstract

Formal semantics comprises a plethora of theories which interpret natural language through the use of different ontological primitives (e.g. individuals, possible worlds, situations, propositions, individual concepts). The ontological relations between these theories are, today, still largely unexplored. In particular, it remains an open question whether the primitives from some of these theories can be coded in terms of objects from other theories, or whether the ontologies of some theories can even be reduced to the ontologies of other, ontologically poorer, theories. This paper answers the above questions for a proper subset of formal semantic theories which are designed for the interpretation of doxastic attitude reports. The paper formalizes some ontological relations between these theories that are only suggested (but are not made explicit) in the literature, and identifies several new relations. The paper uses these relations to show that ‘the’ unifying theory for attitude reports is, in fact, a class of theories whose members are equivalent up to coding.

K. Liefke—I wish to thank two anonymous referees for LENLS 14 for their valuable comments and suggestions on an earlier version of this paper. The paper has profited from discussions with Chris Barker, Daisuke Bekki, Lucas Champollion, Ivano Ciardelli, Shalom Lappin, Roussanka Loukanova, Ed Zalta, and Ede Zimmermann. The research for this paper is supported by the German Research Foundation (via Kristina Liefke’s grant LI 2562/1-1 and Ede Zimmermann’s grant ZI 683/13-1).

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Notes

  1. 1.

    These are possible individuals, which exist in some possible world at some time.

  2. 2.

    We here use a simplified version of the computer science-notation for function types. Thus, ‘st’ abbreviates the type \((s \rightarrow t)\), which corresponds to Montague’s type \(\langle s, t\rangle \).

  3. 3.

    The equivalence of (3a) and (3b) presupposes the interpretation of kind terms as rigid designators (see [18]).

  4. 4.

    This difference is exemplified by the different truth-values of (4a) and (4c).

  5. 5.

    These include physically impossible objects (e.g. perpetuum mobiles) and logically impossible or self-contradictory objects (e.g. the round square).

  6. 6.

    Partial worlds are more commonly known as situations.

  7. 7.

    These are ordered pairs \(\langle i, x\rangle \) consisting of an index i and an individual x.

  8. 8.

    These additionally include the empty set of truth-values, \(\{\} =: \mathbf {N}\) (i.e. neither-true-nor-false), and/or the set \(\{\mathbf {T}, \mathbf {F}\} =: \mathbf {B}\) (i.e. both-true-and-false).

  9. 9.

    For example, Zalta [38, pp. 644–646] represents situations and impossible worlds by (abstract) objects in the domain of individuals.

  10. 10.

    For reasons of perspicuity, not all last-mentioned relations are visualized in Table 1.

  11. 11.

    In contrast to IL, TY\(_{2}\) validates full universal instantiation, beta conversion, and Leibniz’ Law and has the diamond property (see [29, pp. 23–24]; cf. [11, p. 323]).

  12. 12.

    For a refutation of this claim, see the discussion of [39] below.

  13. 13.

    More accurately, Kaplan shows “that Frege’s ontology [can be represented] within that part of it which constitutes Russell’s ontology” (see [15, p. 719]). Our description of Kaplan’s result is motivated by the identity of Frege’s ontology (when combined with a haecceitist position on trans-world identity; see [ibid., pp. 725–729]) with the ontology of IL, and by the identity of Russell’s ontology with the ontology of IL\(^{-}\).

  14. 14.

    Kaplan’s strategy can be significantly simplified by requiring that the type \(s \alpha \) be in \(^{\lozenge }\)-normal form (see [22, p. 11]). This form gathers occurrences of the index-type s immediately before t. Thus, the type s(et) has the \(^{\lozenge }\)-normal form e(st). Since e(st) is already an IL\(^{-}\)-type, objects of type s(et) will be represented in this type, rather than in the more complex type et(st).

  15. 15.

    The latter coding is justified w.r.t. to the \(^{\lozenge }\)-normal form of s((se)t), i.e. (se)(st).

  16. 16.

    The relevant literature uses 0 and \(0^{*}\) as the types for natural numbers, respectively for coded sequences of natural numbers. To ensure the transferability of results, we identify 0 with e.

  17. 17.

    In contrast to the theory’s original relational formulation, this version has a primitive type for truth-combinations.

  18. 18.

    Since the truth-combination B includes both T and F on this ordering, it will be represented by the set \(\{\mathbf {T}, \mathbf {F}\}\). Since the truth-combination N includes neither T nor F (or B), it will be represented by the empty set. Since each of T and F only includes N and itself, they will be represented by their singleton sets. The relation \(\sqsubseteq \) is discussed in detail in [2].

  19. 19.

    Possible worlds will then be represented by singleton sets containing these worlds. Situations/impossible worlds will be represented by sets of worlds whose members extend the information of the situation, resp. whose members capture a total consistent part of the world’s information. The relation \(\le \) is introduced in [29, pp. 69–74].

  20. 20.

    This excludes OT and IS, which assign more fine-grained meanings to attitude complements (see Sect. 2.2).

  21. 21.

    The impossibility of representing primitive propositions by type-st (or type-(st)(tt)) objects motivates the distinction between generalized and property theories. This distinction will be discussed in more detail in Sect. 2.4.

  22. 22.

    These contexts allow the truth-preserving substitution of equivalent expressions.

  23. 23.

    To ensure the finer grain of hyperintensions (than intensions), it is commonly assumed that this homomorphism is not an isomorphism (see [30, 36]).

  24. 24.

    This follows from the analysis of proper names as rigid designators (which have the same referent at all indices) (see [18]; cf. [31, p. 260]) and from the assumption that primitive individual concepts are more fine-grained than functions from indices to individuals (see [31, pp. 273, 277–278]).

  25. 25.

    In virtue of the possibility of representing possible worlds by ultrafilters of propositions, the ontologies of PT and IntL further enable the adequate interpretation of intensional constructions like (7).

  26. 26.

    This variant reflects the possibility that neither a proposition nor its complement are true at a situation and that both a proposition and its complement are true at an impossible world or situation. To capture this possibility, we represent worlds and situations by (type-pt, or -p(tt)) functions from primitive propositions to truth-combinations.

  27. 27.

    We assume that this is a designated language, whose constants are associated with the lexical elements of the target natural language (here: English).

  28. 28.

    This is a language whose constants are associated with the lexical elements of (1a) and (1b).

  29. 29.

    This is due to the fact that \(=\) is totally defined, s.t. the interpretation of \(\lambda {{\varvec{i}}} \lambda \vartheta [\textit{plus}(1)\) \((1) = 2]\) at a particular situation-argument is independent of this argument.

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Authors and Affiliations

  1. Institute for Linguistics, Goethe University Frankfurt, 60323, Frankfurt am Main, Germany

    Kristina Liefke

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  1. Kristina Liefke
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Editors and Affiliations

  1. Chiba University, Chiba, Japan

    Sachiyo Arai

  2. National Institute of Advanced Industrial Science and Technology, Ibaraki, Japan

    Kazuhiro Kojima

  3. Ochanomizu University, Tokyo, Japan

    Koji Mineshima

  4. Ochanomizu University, Tokyo, Japan

    Daisuke Bekki

  5. National Institute of Informatics, Tokyo, Japan

    Ken Satoh

  6. Fujitsu Laboratories Limited, Kanagawa, Japan

    Yuiko Ohta

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Liefke, K. (2018). Relating Intensional Semantic Theories: Established Methods and Surprising Results. In: Arai, S., Kojima, K., Mineshima, K., Bekki, D., Satoh, K., Ohta, Y. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2017. Lecture Notes in Computer Science(), vol 10838. Springer, Cham. https://doi.org/10.1007/978-3-319-93794-6_12

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