The donkey strikes back: Extending the dynamic interpretation “constructively”

The donkey strikes back Extending the dynamic interpretation "constructively" Tim Fernando fernando@cwi, nl Centre for Mathematics and Computer Science P.O. Box 4079, 1009 AB Amsterdam, The Netherlands Abstract The dynamic interpretation of a formula as a binary relation (inducing transitions) on states is extended by alternative treat- ments of implication, universal quantifi- cation, negation and disjunction that are more "dynamic" (in a precise sense) than the usual reductions to tests from quanti- fied dynamic logic (which, nonetheless, can be recovered from the new connectives). An analysis of the "donkey" sentence followed by the assertion "It will kick back" is pro- vided. 1 Introduction The line If a farmer owns a donkey he beats it (1) from Geach [6] is often cited as one of the success sto- ries of the so-called "dynamic" approach to natural language semantics (by which is meant Kamp [12], Heim [9], Sarwise [1], and Groenendijk and Stokhof [7], among others). But add the note It will kick back (2) and the picture turns sour: processing (1) may leave no beaten donkey active. Accordingly, providing a referent for the pronoun it in (2) would appear to call for some non-compositional surgery (that may upset many a squeamish linguist). The present pa- per offers, as a preventive, a "dynamic" form of im- plication =~ applied to (1). Based on a "construc- tive" conception of discourse analysis, an overhaul of Groenendijk and Stokhof [7]'s Dynamic Predicate Logic (DPI.) is suggested, although :=~ can also be introduced less destructively so as to extend DPL conservatively. Thus, the reader who prefers the old "static" interpretation of (1) can still make that choice, and declare the continuation (2) to be "se- mantically ill-formed." On the other hand, Groe- nendijk and Stokhof [7] themselves concede that "at least in certain contexts, we need alternative exter- nally dynamic interpretations of universal quantifi- cation, implication and negation; a both internally and externally dynamic treatment of disjunction." A proposal for such connectives is made below, extend- ing the dynamic interpretation in a manner analo- gous to the extension of classical logic by constructive logic (with its richer collection of primitive connec- tives), through a certain conjunctive notion of par- allelism. To put the problem in a somewhat general per- spective, let us step back a bit and note that in as- signing a natural language utterance a meaning, it is convenient to isolate an intermediate notion of (say) a formula. By taking for granted a translation of the utterance to a formula, certain complexities in natu- ral language can be abstracted away, and semantics can be understood rigorously as a map from formu- las to meanings. Characteristic of the dynamic ap- proach mentioned above is the identification of the meaning of a formula A with a binary relation on states (or contexts) describing transitions A induces, rather than with a set of states validating A. In the present paper, formulas are given by first-order for- mulas, and the target binary relations given by pro- grams. To …

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