Abstract
In this paper I develop a dynamic semantics for a first-order fragment, which incorporates insights from work on anaphora in logic, and the trivalent approach to presupposition projection. The resulting system—EDS—has interesting features which set it apart, both conceptually and empirically, from earlier iterations of dynamic semantics. Conceptually, the meanings of the logical connectives are derived by systematically generalizing the Strong Kleene connectives into a dynamic setting—the system is thereby predictive, drawing a tight connection between the logic of presupposition projection and patterns of anaphoric accessibility. On the empirical side, EDS diverges sharply from earlier proposals. In this paper, I focus mainly on disjunction, arguing that EDS provides a simple and elegant account of the dynamics of disjunction, including traditionally problematic cases such as Partee disjunctions and program disjunctions.
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Notes
- 1.
This will later prove useful when embedding EDS in a concrete discourse pragmatics.
- 2.
The definition of relational composition (which is totally standard) is given below:
$$\begin{aligned} R \circ S := \{(g,i)|\exists h[(g,h) \in R \wedge (h,i) \in S]\} \end{aligned}$$This operation plays a central role in both DPL and EDS.
- 3.
G&S importantly assume that a multi-sentence discourse is translated into DPL as a conjunctive sentence.
- 4.
N.b. the universal quantifier is defined as the dual of the existential.
- 5.
As emphasized by Mandelkern and Rothschild [10] the kind of situation-based e-type approach to anaphora developed in [11] and refined in [12, 13] does not (in its current state, at least) consitute a viable alternative. E-type theories have not addressed in detail how to capture notions of anaphoric accessibility in complex sentences, beyond donkey sentences, and as shown in [14], were they to do so, they would require entries for the logical connectives which manipulate minimal situations in an apparently arbitrary fashion.
- 6.
If \(\phi \) and \(\lnot \lnot \phi \) are equivalent, then choosing to use a sentence of the form \(\lnot \lnot \phi \) is naturally expected to trigger a Manner implicature. I leave the interesting question of the pragmatics of doubly-negated sentences to future work.
- 7.
- 8.
This observation is often attributed to the later [18]. The intuition behind the analysis is already implied by the fact that the two indefinites are annotates with the same variable.
- 9.
For an excellent recent overview of DPL, which expands on many issues which I don’t have the space to discuss here, see [21].
- 10.
Since I’m exclusively concerned with anaphoric presuppositions here, I make the simplifying assumption that all predicates are bivalent, i.e., if all of the values of the variables are known then an atomic sentence is always either true or false. One way of extending the logical language in order to model (non-anaphoric) presuppositions while maintaining bivalent predicates would be to incorporate Beaver’s unary presupposition operator [23].
- 11.
Note that there are different ways in which to interpret the third truth value in a trivalent setting, e.g., as standing in for undefinedness. Here, it is explicitly referred to as “unknown”, since this framing is a natural fit for the Strong Kleene logic of indeterminacy, which is exploited extensively later in the paper. Undefinedness typically goes together with Weak Kleene logic. I’m grateful to an anonymous reviewer for pressing me to clarify this point.
- 12.
- 13.
The generalization of Strong Kleene trivalent semantics to a dynamic setting will out of necessity remain rather impressionistic in this paper. The procedure of lifting truth-functional operators into a dynamic setting has however been made precise in important work by Charlow [26]. Simon Charlow (p.c.) points out that the recipe for lifting the Strong Kleene semantics used here can be formalized as a lifting of the Strong Kleene connectives into the State.Set applicative, following [26]. See [20] for more details.
- 14.
In order to keep the definitions relatively terse, I take advantage of the convention that
is understood as
. - 15.
I’ll simply note here that the validity of de Morgan’s in the presence of anaphoric dependencies seems independently desirable given our intuitions about natural language. The following sentences are all arguably truth-conditionally equivalent.
- 16.
Assuming a Strong Kleene semantics for material implication. Something interesting to note here is that EDS predicts existential truth conditions for donkey sentences, unlike, e.g., [3]. Egli’s corrolary therefore doesn’t hold.
This is by no means a bad prediction—it has been widely reported that such existential readings are attested for donkey sentences, as alluded to in fn. 7 (see also [17, 27]). The following example, for example, is clearly true if Gabor owns two credit cards but only pays with one of them.
The empirical picture is however much more complicated, and donkey sentences do often have stronger, universal readings. Relatedly, [15] reports that Partee disjunctions have universal readings—unlike donkey sentences, to my knowledge very little work has been done examining the distribution of existential and universal readings of Partee disjunctions. A detailed discussion of universal readings will have to wait for another occasion, but see [28] for one recent approach.
- 17.
This is what von Fintel calls ‘Stalnaker’s bridge’ [30], in the context of a dynamic setting.
- 18.
Various pragmatic justifications can be given for the formal contingency requirement stated in (43). What is important for my purposes is that if a disjunctive sentence is asserted by a speaker \(s\) in a context which trivializes one of the disjuncts, the assertion is judged to be ‘odd’.
- 19.
There seem to be information-structural constraints on program disjunctions in natural language which are beyond the remit of EDS. For example, it seems that some degree of parallelism is required to hold between the disjuncts. Singular anaphora, by my reckoning, is extremely difficult in the following example:
I speculate that this is related to constraints on co-indexing. I leave this interesting issue to future work on program disjunctions.
- 20.
This kind of suggestion raises the issue of how exactly natural language sentences can be mapped compositionally to EDS Logical Forms. After all, one doesn’t want to allow for too much flexibility, otherwise the resulting grammar won’t be sufficiently constrained.
is understood as
.
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Acknowledgments
Aspects of this work have been presented in various venues, including at Rutgers, NYU, MIT, ENS, and most recently at the Third Tsinghua Interdisciplinary Workshop on Logic, Language, and Meaning. I’m grateful to participants on all such occasions for insightful and challenging feedback on this material, which has shaped the current form. The logic outlined in this paper was developed for the Spring 2022 Topics in Semantics seminar at MIT, and I’m especially grateful to Filipe Hisao Kobayashi and Enrico Flor for their input. I remain solely responsible for any mistakes.
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Elliott, P.D. (2023). Disjunction in a Predictive Theory of Anaphora. In: Deng, D., Liu, M., Westerståhl, D., Xie, K. (eds) Dynamics in Logic and Language. TLLM 2022. Lecture Notes in Computer Science, vol 13524. Springer, Cham. https://doi.org/10.1007/978-3-031-25894-7_4
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