Abstract
Kaplan (1989a) insists that natural languages do not contain displacing devices that operate on character—such displacing devices are called monsters. This thesis has recently faced various empirical challenges (e.g., Schlenker 2003; Anand and Nevins 2004). In this note, the thesis is challenged on grounds of a more theoretical nature. It is argued that the standard compositional semantics of variable binding employs monstrous operations. As a dramatic first example, Kaplan’s formal language, the Logic of Demonstratives, is shown to contain monsters. For similar reasons, the orthodox lambda-calculus-based semantics for variable binding is argued to be monstrous. This technical point promises to provide some far-reaching implications for our understanding of semantic theory and content. The theoretical upshot of the discussion is at least threefold: (i) the Kaplanian thesis that “directly referential” terms are not shiftable/bindable is unmotivated, (ii) since monsters operate on something distinct from the assertoric content of their operands, we must distinguish ingredient sense from assertoric content (cf. Dummett 1973; Evans 1979; Stanley 1997), and (iii) since the case of variable binding provides a paradigm of semantic shift that differs from the other types, it is plausible to think that indexicals—which are standardly treated by means of the assignment function—might undergo the same kind of shift.
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Notes
Note that this is not to say that the ban on monsters is incompatible with the semantics being compositional at the level of character. After all, if the semantics is compositional at the level of content, then it is thereby compositional at the level of character. One could in principle provide composition rules that are defined over characters and then supply a context to get the contents (and extensions) of the complex expressions. (See Westerståhl forthcoming for a detailed analysis of how compositionality at different semantic levels relate to each other.) The question rather is this: assuming that the language in question is compositional at the level of character, is it also compositional at the level of content? In this way the monster prohibition and the compositionality of character and content are connected via the following biconditional: a semantics is monstrous iff (i) it is compositional at the level of character and (ii) it fails to be compositional at the level of content – or as Westerståhl forthcoming puts it “Monsters destroy the compositionality of content”.
Although Kaplan does not provide an explicit argument against the existence of monsters, I think a fair rational reconstruction of his reasoning proceeds as follows: (i) the semantic composition rules are defined over the information contents associated with expressions, (ii) the information (or assertoric) content of an expression is never equal to the character-level value of an expression, (iii) thus, the language fails to contain monstrous operations.
Again we should more precisely say that the monster ban requires that there not be any operators in the language that must take characters as argument. The qualification of “must” should be included because many non-character operators can be transformed into “equivalent” ones that takes characters as argument, e.g., any truth-functional connective can be given a semantics in terms of functions on characters.
The monster prohibition as stated rules out all hyperintensional operators, which is well-motivated since Kaplan glosses his monster prohibition as the thesis that “all operators that can be given an English reading are at most intensional" (Kaplan 1989a, p. 502 footnote 27). The definition would need to be modified if one wanted to allow for purported non-monstrous hyperintensional operators (e.g., quotational operators). The definitions are also simplified by limiting the focus to monstrous sentential operators. But in full generality a monster could be of any syntactic category. In general the monster prohibition is the prohibition of the following composition rule (in the style of Heim and Kratzer 1998): Monstrous functional application. If α is a branching node and {β, γ} the set of its daughters, then for any context c and circumstance i: if \({[\!\![\beta]\!\!]^{{\text{c}},{\text{i}}}}\) is a function whose domain contains λc,i.\({[\!\![\gamma]\!\!]^{{\text{c}},{\text{i}}}}\), then \({[\!\![\alpha]\!\!]^{{\text{c}},{\text{i}}}}\) = \({[\!\![\beta]\!\!]^{{\text{c}},{\text{i}}}}\)(λc,i.\({[\!\![\gamma]\!\!]^{{\text{c}},{\text{i}}}}\)).
Actually, Tarski (1936) formulated it in terms of functions from sequences to individuals. Assignments are functions from variables to individuals, whereas Tarski's sequences were just sequences of individuals – and variables were indexed to positions in sequences. There is clearly no essential difference here. I use the formulation in terms of assignments for continuity with Kaplan (1989a) and contemporary semantic frameworks, e.g. Heim and Kratzer (1998).
This actually gives Tarski's definition of “satisfaction by a sequence”, Tarski reserves the term “truth” for formulae that are satisfied by all sequences.
Where ∏ is the integer product of the sequence of truth-values (i.e., the sequence of 0s and 1s) and p is a function from assignments to truth-values, i.e., of type 〈γ, t〉.
This is also evident in the algebraization of the semantics of predicate logic in terms of cylindrical algebra.
The formal system LD is presented in Kaplan (1989a), sect. XVIII, pp. 541–553. In what follows I make a few notational changes to ease the exposition.
If we added the first person pronoun ‘I’, we would add the clause: \({[\!\![{{\text{I}}]\!\!]^{c,g,t,w}}}\) = the agent of c. If we added the modal operator ‘□’ we would add the clause: \({[\!\![{\Box\phi]\!\!]^{c,g,t,w}}}\) = 1 iff for all w′ ∈ W, \({[\!\![{\phi]\!\!]^{c,g,t,w^\prime}}}\) = 1 etc.
To my knowledge the fact that Kaplan himself employs monsters in LD has never been argued for before, although a related issue in terms of bound pronouns is discussed in Zimmerman (1991) (see especially sect. 4.1).
I have never seen a lexical entry given for the lambda binders but it seems fairly obvious and uncontroversial that this is the way to do it. And I hope that some will find this explicit rendering of the semantics of lambda enlightening. An alternative way to see the situation is as follows: A lambda binder λα turns a sentence into a predicate by taking the set of assignments at which the sentence is true and outputting a set of individuals, namely the set of individuals that are in the α-slots of the assignments (i.e., the individuals who satisfy the relevant predicate). We could say that the semantic value of a lambda binder is the powerset transform of a projection function.
Importantly, one could make the same point with other examples of natural language variable binding (see Partee (1989) for various cases). The argument here does not essentially rely on the use of lambda binders nor on a syntactic story about quantifier raising. For example, consider the type of binding that is at issue in so-called binding arguments in Stanley (2000): an utterance of “Every bottle is green” in context might express the proposition that every bottle in this room is green. But when “Every bottle is green” is embedded, e.g., in “In every room, every bottle is green” the quantifier domain variable is bound such that the utterance expresses the proposition that in every room x, every bottle in x is green. So the variable binding operator is monstrous.
One might insist that compositional semantics should not concern itself with a notion of “what is said” or assertoric content. If so, then so much the worse for Kaplan's monster prohibition, since it is fundamentally entangled with such a notion.
There is an analogous maneuver in Salmon (1986), where he re-defines “character” as the function that maps a context to a function from times to contents.
See Ninan (2010) and Rabern (forthcoming) for some recent critical discussion of the dogma that compositional semantic values are to be identified with the objects of assertion (of course historically there has been an undercurrent of theorists who have gone against the dogma, most notably Dummett 1973; Evans 1979; Stanley 2002).
Predicate abstraction rule: Let α be a branching node with daughters β and γ, where β dominates only a lambda binder λx. Then, for any variables assignment g, \({[\!\![{\alpha}]\!\!]^g}\) = λz.\({[\!\![{\gamma}]\!\!]^{g[x:=z]}}\) (Heim and Kratzer 1998, p. 186).
See Pagin and Westerståhl (2010) for a detailed analysis of when a rule is genuinely compositional.
It’s unclear whether the motivation for Heim and Kratzer's syncategorematic treatment of variable binding was done for merely pedagogical reasons or for some unstated theoretical reason. But I suspect it was the former, since they are theoretically guided by Frege's Conjecture (i.e., the claim that semantic evaluation proceeds via functional application). This seems especially likely since although \({[\!\![{.}]\!\!]^g}\) is not compositional \({[\!\![{.}]\!\!]}\) itself clearly is compositional.
Kaplan (1989a), p. 489 says: “[Pronouns] have uses other than those in which I am interested (or, perhaps, depending on how you individuate words, we should say that they have homonyms in which I am not interested)”.
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Acknowledgment
For helpful comments on earlier drafts thanks to David Chalmers, John Cusbert, Karen Lewis, Daniel Nolan, Jim Pryor, Landon Rabern, Paolo Santorio, Wolfgang Schwarz, Clas Weber, and an anonymous referee.
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Rabern, B. Monsters in Kaplan’s logic of demonstratives. Philos Stud 164, 393–404 (2013). https://doi.org/10.1007/s11098-012-9855-1
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DOI: https://doi.org/10.1007/s11098-012-9855-1