GSV are interested in developing and establishing a reasonable theory
of discourse update. One way of looking at this paper is like this:
- GSV = GS + V, where
+ GSV = GS + V, where
- GS = Dynamic theories of binding of Groenendijk and Stokhof, e.g.,
- Dynamic Predicate Logic L&P 1991: dynamic binding, donkey anaphora
- Dynamic Montague Grammar 1990: generalized quantifiers, discourse referents
+ GS = Dynamic theories of binding of Groenendijk and Stokhof, e.g.,
+ Dynamic Predicate Logic L&P 1991: dynamic binding, donkey anaphora
+ Dynamic Montague Grammar 1990: generalized quantifiers, discourse referents
- V = a dynamic theory of epistemic modality, e.g.,
- Veltman, Frank. "Data semantics."
- In Truth, Interpretation and Information, Foris, Dordrecht (1984): 43-63.
+ V = a dynamic theory of epistemic modality, e.g.,
+ Veltman, Frank. "Data semantics."
+ In Truth, Interpretation and Information, Foris, Dordrecht
+ (1984): 43-63, or
+ Veltman, Frank. "Defaults in update semantics." Journal of
+ philosophical logic 25.3 (1996): 221-261.
That is, Groenendijk and Stokhof have a well-known theory of dynamic
semantics, and Veltman has a well-known theory of epistemic modality,
and this fragment brings both of those strands together into a single
system.
-We will be interested in this paper both from a theoretical point of
-view and from a practical engineering point of view. On the
-theoretical level, these scholars are proposing a strategy for
-managing the connection between variables and the objects they
-designate in way that is flexible enough to be useful for describing
-natural language.
-
## Basics of GSV's fragment
The fragment in this paper is unusually elegant. We'll present it on
-its own terms, with the exception that we will not use pegs. See the
-digression below concerning pegs for an explanation. After presenting
-the paper, we'll re-engineering the fragment using explicit monads.
-
-In this fragment, points of evaluation are not just worlds, but a pair
-of a world and an assginment function. This is familiar from Heim's
-1983 File Change Semantics. We'll follow GSV and call a
-world-assignment pair a "possibility". Then a context is a set (an
-"information state") is a set of possiblities. Infostates
-simultaneously track both information about the world (which possible
-worlds are live possibilities?) as well as information about the
-discourse (which objects to the variables refer to?).
-
-Worlds in general settle all matters of fact in the world. In
-particular, they determine the extensions of predicates and relations.
+its own terms, with the exception that we will not use GSV's "pegs".
+See the discussion below below concerning pegs for an explanation.
+After presenting the paper, we'll re-engineering the fragment using
+explicit monads.
+
+In this fragment, points of evaluation are not just worlds, but pairs
+consisting of a world and an assginment function. This conception of
+an evaluation point is familiar from Heim's 1983 File Change
+Semantics. Following GSV, we'll call a world-assignment pair a
+"possibility", and so a context (an "information state") will be set
+of possiblities. As GSV emphasize, infostates simultaneously track
+information about the world (which possible worlds are live
+possibilities?) as well as information about the discourse (which
+objects to the variables refer to?).
The formal language the fragment interprets is Predicate Calculus with
-equality, existential and universal quantification, along with one
-unary modality (box and diamond, corresponding to epistemic necessity
-and epistemic possibility).
+equality, existential and universal quantification, and on unary
+modality (box and diamond, corresponding to epistemic necessity and
+epistemic possibility).
An implementation in OCaml is available [[here|code/gsv.ml]]; consult
that code for details of syntax, types, and values. [[An implementation
else variables. So in general, the referent of a term can depend on a
possibility:
- ref(i, t) = t if t is an individual, and
+ ref (i,t) = t if t is an individual, and
g(t) if t is a variable, where i = (w,g)
-Here are the main clauses for update (their definition 3.1).
-
-Following GSV, we'll write `update(s, φ)` (the update of information
-state `s` with the information in φ) as `s[φ]`.
+Immediately following are the recipes for context update (GSV's
+definition 3.1). Following GSV, we'll write `update(s, φ)` (the
+update of information state `s` with the information in φ) as `s[φ]`.
- s[P(t)] = {i in s | w(P)(ref(i,t))}
+ s[P(t)] = {(w,g) in s | w P (ref((w,g),t))}
-So `man(x)` is the set of live possibilities `i = (w,g)` in s such that
+So `man(x)` is the set of live possibilities `(w,g)` in s such that
the set of men in `w` given by `w(man)` maps the object referred to by
`x`, namely, `g("x")`, to `true`. That is, update with "man(x)"
discards all possibilities in which "x" fails to refer to a man.