Comparison Across Domains in Delineation Semantics


This paper presents a new logical analysis of quantity comparatives (i.e. More linguists than philosophers came to the party.) within the Delineation Semantics approach to gradability and comparison (McConnell-Ginet in Comparison constructions in English. PhD thesis, University of Rochester, Rochester, 1973; Kamp in Formal semantics of natural language. Cambridge University Press, Cambridge, 1975; Klein in Linguist Philos 4:1–45, 1980) among many others. Along with the Degree Semantics framework (Cresswell in Montague grammar. Academic Press, New York, 1976; von Stechow in J Semant 3:1–77, 1984; Kennedy in Projecting the adjective. PhD thesis. University of California, Santa Cruz, 1997, among many others) Delineation Semantics is one of the dominant logical frameworks for analyzing the meaning of gradable constituents of the adjectival syntactic category; however, there has been very little work done investigating the application of this framework to the analysis of gradability outside the adjectival domain. This state of affairs distinguishes the Delineation Semantics framework from its Degree Semantics counterpart, where such questions have been investigated in great deal since the beginning of the twenty-first century. Nevertheless, it has been observed [for example, by Doetjes (Seventeenth amsterdam colloquium. University of Amsterdam, Amsterdam, 2011); van Rooij (The vagueness handbook. Springer, Dordrecht, 2011c)] that there is nothing inherently adjectival about the way that the interpretations of scalar predicates are calculated in Delineation Semantics, and therefore that there is enormous potential for this approach to shed light on the nature of gradability and comparison in the nominal and verbal domains. This paper is a first contribution to realizing this potential within a Mereological extension of a simple version of the DelS system.

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  1. 1.

    Along with many and few, much and little are called Q-adjectives because they can appear with degree modifiers (such as very and so) that otherwise only co-occur with expressions of the adjectival syntactic category.

    1. (i)

         a.    Very many linguists came to the party.

         b.    So few philosophers stayed home.

         c.    This much wine was drunk.

         d.    Too little beer is left in the fridge.

  2. 2.

    In this work, I assume that the predicates in comparison classes are simple adjectives (like tall), but, in principle, they could be more complex like tall for a basketball player.

  3. 3.

    This semantic analysis follows rather closely van Rooij (2011b), who calls this function Lots, not much. This analysis is also very similar to Wellwood (2014)’s analysis set within the Degree Semantics framework. Wellwood explicitly integrates her semantics with Bresnan’s syntax, and therefore refers to this function as much. It is not clear from van Rooij’s paper whether he has Bresnan’s syntactic proposals in mind; therefore, I stress that this aspect of the proposal is my own (inspired by Wellwood).

  4. 4.

    For readability considerations, I will often omit the model notation, writing only \([\![\cdot ]\!]_X\) for \([\![\cdot ]\!]_{X,M}\).

  5. 5.

    Note that van Rooij proposes that this axiom set governs the context-sensitivity of relative adjectives like tall, beautiful, expensive etc., not Q-adjectives like much.

  6. 6.

    A relation \(R\) is semi-transitive just in case, for all \(a_1, a_2, a_3, a_4\), if \(a_1Ra_2\) and \(a_2 R a_3\), then \(a_1 Ra_4\) or \(a_4 R a_3\).

  7. 7.

    A relation \(R\) satisfies the interval order property just in case, for all \(a_1, a_2, a_3, a_4\), if \(a_1Ra_2\) and \(a_3 R a_4\), then \(a_1 Ra_4\) or \(a_2 R a_3\).

  8. 8.

    In the literature following Kennedy, we saw that  prohibits crisp judgments.

  9. 9.

    A relation \(R\) is almost connected just in case, for all \(a_1, a_2\), if \(a_1 R a_2\), then for all \(a_3\), \(a_1 R a_3\) or \(a_3 R a_2\).

  10. 10.

    See (Bartsch and Vennemann 1972; Klein 1980; Bierwisch 1989; Klein 1991; Kennedy 1997; Bale 2008; Doetjes 2011, among many others).

  11. 11.

    See Morzycki (in press) for a recent review.

  12. 12.

    Büring also makes the empirical proposal that cross-polar comparatives of the form A \(^+\) er than \(A^-\) are impossible; however, I consider this to be more of a preference rather than a grammatical constraint, since, for example,  features a relative comparative with a positive (total) adjective (silent) paired with a negative (partial) adjective (bent).

  13. 13.

    In this paper, I assume for convenience that nouns are not gradable. This is clearly false and, indeed, I believe that the Delineation framework and the methods developed here would allow for an interesting application to the analysis of gradable nouns like idiot, heap and disaster (see Morzycki 2009, for a recent DegS proposal). However, I leave this extension to future work.

  14. 14.

    Where similarity is defined as in Definition 6.

  15. 15.

    Proof: Suppose \([\![\textsc {er}^+(a_1, P_1\circ N_1, a_2, P1\circ N_1)]\!]_{X\circ N_1} = 1\). Then \(\langle a_1, P_1\circ N_1 \rangle > \langle a_2, P_1 \circ N_1\rangle \). So there is some \(\langle a_3, P_1\circ N_1\rangle \) such that \(\langle a_1, P_1\circ N_1\rangle \sim \langle a_3, P_1\circ N_1\rangle \) and \(\langle a_3, P_1\circ N_1 \rangle \succ \langle a_2, P_1 \circ N_1 \rangle \) or \(\langle a_2, P_1\circ N_1\rangle \sim \langle a_3, P_1\circ N_1\rangle \) and \(\langle a_1, P_1\circ N_1 \rangle \succ \langle a_3, P_1 \circ N_1 \rangle \). Without loss of generality, suppose \(\langle a_1, P_1\circ N_1\rangle \sim \langle a_3, P_1\circ N_1\rangle \) and \(\langle a_3, P_1\circ N_1 \rangle \succ \langle a_2, P_1 \circ N_1 \rangle \). Then there is some \(X\) such that \(\langle a_3, P_1\circ N_1\rangle \in [\![(\textsc {much}P_1)\circ N_1]\!]_{X\circ N_1}\) and \(\langle a_2, P_1\circ N_1\rangle \in [\![(\textsc {little}P_1)\circ N_1]\!]_{X\circ N_1}\). Since \(\langle a_2, P_1\circ N_1\rangle \) and \(\langle a_3, P_1\circ N_1\rangle \) are in \(X\circ N_1\), by Definition 9, \(a_2, a_3 \in [\![N_1]\!]\). Likewise, since \(\langle a_1, P_1\circ N_1\rangle \sim \langle a_3, P_1\circ N_1\rangle \), there is some comparison class \(X'\circ N_1\) such that \(\langle a_1, P_1\circ N_1\rangle , \langle a_3, P_1\circ N_1\rangle \in X'\circ N_1\). So, by Definition 9, \(a_1 \in [\![N_1]\!]\).\(\square \)

  16. 16.

    Proof: Suppose \([\![\textsc {er}^+(a_1, P_1\circ N_1, a_2, P_1\circ N_1)]\!]_{X\circ N_1} = 1\). Then \(\langle a_1, P_1\circ N_1 \rangle > \langle a_2, P_1 \circ N_1\rangle \). So there is some \(\langle a_3, P_1\circ N_1\rangle \) such that \(\langle a_1, P_1\circ N_1\rangle \sim \langle a_3, P_1\circ N_1\rangle \) and \(\langle a_3, P_1\circ N_1 \rangle \succ \langle a_2, P_1 \circ N_1 \rangle \) or \(\langle a_2, P_1\circ N_1\rangle \sim \langle a_3, P_1\circ N_1\rangle \) and \(\langle a_1, P_1\circ N_1 \rangle \succ \langle a_3, P_1 \circ N_1 \rangle \). Without loss of generality, suppose \(\langle a_1, P_1\circ N_1\rangle \sim \langle a_3, P_1\circ N_1\rangle \) and \(\langle a_3, P_1\circ N_1 \rangle \succ \langle a_2, P_1 \circ N_1 \rangle \). Since \(\langle a_3, P_1\circ N_1 \rangle \succ \langle a_2, P_1 \circ N_1 \rangle \), there is some comparison class \(X\) such that \(\langle a_3, P_1\circ N_1\rangle \in [\![(\textsc {much}P_1)\circ N_1]\!]_{X\circ N_1}\) and \(\langle a_2, P_1\circ N_1\rangle \in [\![(\textsc {little}P_1)\circ N_1]\!]_{X\circ N_1}\). By Definition 10, \(\langle a_3, P_1\rangle \in [\![\textsc {much}]\!]_X\) and \(\langle a_2, P_1\rangle \in [\![\textsc {little}]\!]_X.\) Since \(\langle a_1, P_1\circ N_1\rangle \sim \langle a_3, P_1\circ N_1\rangle \), \(\langle a_1\rangle \sim \langle a_3, P_1\rangle \). So \(\langle a_1, P_1\rangle > \langle a_2, P_2\rangle \). \(\square \)

  17. 17.

    In this work, I will only discuss comparatives in existential DP subjects. The extension to other argument positions [such as the direct object position (i)] is straightforward.

    1. (i)

         Mary saw a taller man than John.

  18. 18.

    Proof: \(\Rightarrow \) Suppose \([\![\exists (\textsc {much}P_1)\circ N_1(V_1)]\!]_{\exists X\circ N_1} = 1\). Then, by Definition 16, \(\langle [\![V_1 ]\!], \exists P_1\circ N_1\rangle \in [\![\exists (\textsc {much}P_1)\circ N_1 ]\!]_{\exists X\circ N_1}.\) So, by Definition 15, . Therefore, by Definition 13, \([\![V_1]\!]\in [\![\exists (\textsc {much}P_1)\circ N_1]\!]_{X\circ N_1}\), and so \( [\![\exists (\textsc {much}P_1)\circ N_1(V_1)]\!]_{X\circ N_1} = 1\). \(\Rightarrow \) Suppose \([\![\exists (\textsc {much}P_1)\circ N_1(V_1)]\!]_{X\circ N_1} = 1.\) So \([\![V_1]\!]\in [\![\exists (\textsc {much}P_1)\circ N_1]\!]_{X\circ N_1}\). By Definition 13, . So there is some \(a_1\in [\![V_1]\!]\) such that \(\langle a_1, P_1\circ N_1\rangle \in [\![(\textsc {much} P_1)\circ N_1]\!]_{X\circ N_1}.\) Since \(\{a_1\} \subseteq [\![V_1]\!]\) and \([\![V_1]\!]\in \bigvee \{a_1\}\). So, by Definition 14, \(\langle [\![V_1]\!], \exists P_1\circ N_1\rangle \in \exists X\circ N\), and \(\langle [\![V_1]\!], \exists P_1\circ N_1\rangle \in [\![\exists (\textsc {much}P_1)\circ N_1 ]\!]_{\exists X\circ N_1}\). Therefore, by Definition 16, \([\![\exists (\textsc {much}P_1)\circ N_1(V_1)]\!]_{\exists X\circ N_1} = 1\). \(\square \)

  19. 19.

    Note this \(\preceq \) symbol should not be confused with the \(\succ \) symbols that notate the semi-order ‘implicit scale’ relations. \(\preceq \) notates invariant, predicate independent relations that are part of the model structure. I apologize if this notation is confusing.

  20. 20.

    This particular axiomatization is taken from Hovda (2008) (p.81). The version of fusion used here is what Hovda calls ‘type 1 fusion’.

  21. 21.

    Where identity and proper part are defined as follows:

    Definition 22 Identical (\(=\)). For all \(a_1, a_2 \in D\), \(a_1 = a_2\) iff \(a_1 \preceq a_2\) and \(a_2 \preceq a_1\).

    Definition 23 Proper part (\(\prec \)). For all \(a_1, a_2 \in D\), \(a_1 \prec a_2\) iff \(a_1 \mathbf {\preceq } a_2\) and \(a_1 \ne a_2\).

  22. 22.

    In real Distributive Morphology, roots do not even have a syntactic category; however, for convenience in our small logical language, we will suppose that there is a category distinction between \(N\) predicates and \(V\) predicates.

  23. 23.

    Above we stipulated that the domain was finite, and therefore every predicate denotation (mass or count) must have some minimal elements. A common proposal (since at least Quine 1960) is that the denotation of mass nouns does not have minimal elements. That is, in a mereological framework, this would boil down to saying that the denotation of mass predicates is a continuous join semi-lattice. The ‘no minimal parts’ hypothesis has notorious difficulty treating examples of mass predicates such as footwear or furniture, which clearly have minimal parts. I therefore assume that at least some mass predicates have these elements in their denotation. This being said, the proposals that I make here for mass quantity comparatives are compatible with both discrete and continuous lattices, so I leave it open whether we want to have mass predicates denote discrete or continuous lattices (or both).

  24. 24.

    See Bale and Barner (2009) for pictorial representations of the differences between individuated and non-individuated lattices.

  25. 25.

    For example, an English mass term denoting a non-individuated lattice might be water; whereas, a mass term denoting an individuated semi-lattice might be furniture.

  26. 26.

    Actually, we might suppose that (50) holds also for adjectival much/little; however, since the denotations of singular adjectival predicates do not have any mereological structure in them, its application is vacuous.

  27. 27.

    Strangely, such sentences do not sound so good with much and mass terms, possibly because of competition with the expression a lot in English.

    1. (i)

         a.    #This beer is much.

         b.    This beer is a lot (to drink in one sitting).

  28. 28.

    Note that we give an analysis only for the quantity interpretation of (60-a) (there are more linguists and philosophers within this set of women). We set aside the ‘metalinguistic interpretation’ that we see with Sara is more linguist than philosopher.

    Note also that, again, in these constructions, we are faced with questions of (in)commensurability: while (60-a) is fine contrasting linguists and philosophers, its minimal pair (i) would require a very unusual context to be felicitous.

    1. (i)

         # These women are more linguists than red-heads.

  29. 29.

    Proof: Suppose \(a_1 \preceq a_2\) and suppose for a contradiction that \(\langle a_1, N^*_M\rangle > \langle a_2, N^*_M\rangle \). Since \(\langle a_1, N^*_M\rangle > \langle a_2, N^*_M\rangle \), there is some \(\langle a_3, N^*_M\rangle \) such that \(\langle a_1, N^*_M\rangle \sim \langle a_3, N^*_M\rangle \) and \(\langle a_3, N^*_M\rangle \succ \langle a_2, N^*_M\rangle \) or \(\langle a_3, N^*_M\rangle \sim \langle a_2, N^*_M\rangle \) and \(\langle a_1, N^*_M\rangle > \langle a_3, N^*_M\rangle \). Without loss of generality, suppose \(\langle a_1, N^*_M\rangle \sim \langle a_3, N^*_M\rangle \) and \(\langle a_3, N^*_M\rangle \succ \langle a_2, N^*_M\rangle \). Then there is some comparison class \(X\) such that \(\langle a_3, N^*_M\rangle \in [\![\textsc {much}]\!]_X\) and \(\langle a_2, N^*_M \rangle \in [\![\textsc {little}]\!]_X\). So by Downward Difference (54), \(\langle a_3, N^*_M\rangle \in [\![\textsc {much}]\!]_{\{\langle a_2, N^*_M\rangle , \langle a_3, N^*_M\rangle \}}\) and \(\langle a_2, N^*_M \rangle \in [\![\textsc {little}]\!]_{\{\langle a_2, N^*_M\rangle , \langle a_3, N^*_M\rangle \}}\). Now consider the comparison class \(\{\langle a_2, N^*_M\rangle , \langle a_3, N^*_M\rangle , \langle a_1, N^*_M\rangle \}\). Since \(\langle a_3, N^*_M\rangle \sim \langle a_1, N^*_M\rangle \), by applications of Upward Difference (53) and Downward Difference (54), \( \langle a_1, N^*_M\rangle \in [\![\textsc {much}]\!]_{\{\langle a_2, N^*_M\rangle , \langle a_1, N^*_M\rangle \}}\) and \(\langle a_2, N^*_M\rangle \in [\![\textsc {little}]\!]_{\{\langle a_2, N^*_M\rangle , \langle a_1, N^*_M\rangle \}}\). Since \(a_1\preceq a_2\), \(a_1 \vee a_2 = a_2\). Therefore, by (50), \(a_2 \in [\![\textsc {much}]\!]_{\{\langle a_2, N^*_M\rangle , \langle a_1, N^*_M\rangle \}}\). But by Contraries (51), \(a_2 \notin [\![\textsc {much}]\!]_{\{\langle a_2, N^*_M\rangle , \langle a_1, N^*_M\rangle \}}\) \(\bot \). \(\square \)

  30. 30.

    A natural analysis for a subset comparative (which would capture the entailment that the Ling 100 class came to the party) would be using a (pseudo) formula such as \(\textsc {er}^+(\text {came},\exists \text {linguist}^*_C,\text { came,} \textsc {ling100})\); however, we would need to say something about what it means for a pair like \(\langle \)came, ling100 \(\rangle \) to be in \([\![\textsc {little}]\!]_{\exists X}\). Perhaps just plays a role in allowing for the ‘scalar’ interpretation of the Ling100 class. I therefore leave the analysis of subset comparatives to future work.

  31. 31.

    This is not to say that the grammar only provides a single way of interpreting subject DPs. For example, Rett (2014) shows that DPs containing many (like other DPs that do not contain Q-adjectives) can have what she calls a ‘degree interpretation’ (i-b), in addition to an ‘individual’ interpretation (i-a).


    In her 2014 paper, Rett proposes that the ‘degree interpretation’ of many guests is given by a null measure operator that can apply to all kinds of DPs (not just ones containing Q-adjectives). Such an operator could be easily integrated into my proposal [which concerns only the individual interpretation (i-a)] to extend this framework to account for examples like (i-b).

  32. 32.

    Additionally, there is a large body of work on the expression most which is generally analyzed as the superlative of either many (Bresnan 1973; Hackl 2009) or more (Bobaljik 2012) that shows that the acquisition of the meaning of this expression is independent of counting ability (Halberda and Feigenson 2008) and that both children and adults (in certain experimental settings) use their ANS system to evaluate sentences with most (Pietrosky et al. 2009).


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Burnett, H. Comparison Across Domains in Delineation Semantics. J of Log Lang and Inf 24, 233–265 (2015).

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  • Delineation semantics
  • Mereology
  • Comparatives
  • Plurality