Distancia entre ternas

can be true of the situation described.

Our solution (first proposed in [47]) will use the event calculus, but before we go into this, we discuss Michaelis’ attempt in [78] to explain the problem away; this will then show why an elaborate machinery is necessary. Explicitly denying that the progressive creates an intensional context, Michaelis writes

Under the present proposal, the Progressive sentence She is drawing a circledenotes a state which is a subpart not of the accomplishment type She- draw a circle but of the activity type which is entailed by the causal representation of the ac- complishment type. Since this activity can be identified with the preparatory activity that circle drawing entails, circle draw- ing can in principle be distinguished from square drawing etc. within the narrow window afforded by the Progressive con- strual [and] does not require access to culmination points either in this world or a possible world . . . [78, p. 38].

We find this rather doubtful. Without access to a person’s intention it may be very hard to tell initially whether she is drawing a circle or a square, building a barn or building a house. But that person’s intention in performing an activity is characterised precisely by the associated consequent state, even though the latter cannot yet be inferred from the available data.

Here the event calculus comes to our rescue, because the notion of goal or intention is built in from the start. In the event calculus, an activity comes with a scenario which describes a plan for reaching the goal. However, un- like approaches such as Parsons’ [87], where one quantifies existentially over events, the scenario is a universal theory and does not posit the occur- rence of the intended consequences, i.e. the attainment of the goal. Even if the plan is appropriate for the goal, attaining the goal is guaranteed only in minimal models of the scenario combined with the axioms for the event calculus, in which no unforeseen obstacles occur. Thus, the meaning of an accomplishment (as embodied in the scenario) involves a culminating event type(which therefore must exist); but there are no existential claims about the corresponding event token3. Type and token are handled by different mechanisms.

We will now make the above considerations slightly more formal. Con- sider the sentence

(17) Carlos is building a house.

In this example, lexical material (‘build a house’) is combined with the present progressive to create the sentence (17). Semantically, the lexical material is represented by a scenario, in fact the scenario introduced in Sec- tion 2.1 of Chapter 7, which is reproduced here for convenience:

158 10. GRAMMATICAL ASPECT (1) Initially(house(a))

(2) Initiates(start,build,t) (3) Initiates(finish,house(c),t) (4) Terminates(finish,build,t)

(5) HoldsAt(build,t)∧HoldsAt(house(c),t)→Happens(finish,t) (6) Releases(start,house(x),t)

(7) HoldsAt(house(x),t)→ Trajectory(build,t,house(x+g(d)),d) The present progressive is applicable because the scenario features a dy- namics. The temporal contribution of the present progressive is the integrity constraint

?HoldsAt(build,now)succeeds,

where ‘build’ is the activity that drives the dynamics, in accordance with the crucial presupposition for the use of the progressive4 . The following the- orem then provides information on the default character of the progressive. A computational proof is given in Section 3.

THEOREM 6. Let P be the logic program consisting of EC and the

scenario given in 2.1 of chapter 7. Suppose P is extended to P! _{so that}

the query ?HoldsAt(build, now) succeeds in P!_{. Suppose lim}

d→∞g(d) ≥

c. Then comp(P!_{) has a unique model, and in this model there is a time}

t _≥ now for which HoldsAt(house(c), t). By virtue of the stipulation that House(house(c)), there will be a house at timet.

A by now familiar argument shows that the integrity constraint has the effect of introducing a time at which the start event happens. The effect of taking the completion is that in a model, only those events occur which are forced to happen due to the scenario; similarly, only those influences of events are considered which are explicitly mentioned in the scenario. In this particular case, since the scenario does not mention an event ‘Accident’ with its attendant consequences, the completion excludes this possibility; and similarly for other possible impediments to completing the construc- tion. It is then a consequence of the general result corollary 1 of theorem 4 in chapter 5 that the completion actually has a unique model, thus proving theorem 6. One may reformulate the preceding theorem as a result on en- tailment between integrity constraints, following the ‘dynamic’ meaning of

4_{Darrin Hindsill drew our attention to the Papua language Kalam, where to express} the mono-clausal accomplishmentI am building a house for youthree separate clauses are required :

(i) kotp gy, np ñnp gspyn

house having-built-SS you intending-to-give-SS I-am-doing I am building a house for you

Her ‘SS’ means ‘same subject’. Note that there are separate clauses for the finished house (in the future) and the building–in–progress. Languages such as Kalam, in which event structure is coded much more explicitly than in English, provide some evidence for the appropriateness of the above representation.

3. **A COMPUTATIONAL PROOF 159 integrity constraints as explained in Section 1 of Chapter 8. Recall Dowty’s intuitive explanation of the progressive as:

‘Carlos is building a house’ entails that ‘Carlos will have built a house’ in all inertia worlds.

Now read ‘minimal model’ for ‘inertia world’: note that whereas ‘inertia world’ is of necessity an informal concept, ‘minimal model’ is defined pre- cisely and moreover computable. We get, using definition 40

COROLLARY4. Under the same assumptions as theorem 6: the query

?HoldsAt(house(c), R), R >now

succeeds.

We leave the proof of the corollary as an exercise to the reader (cf. 13 below). A glance back at Section 1.3 shows that the query in corollary 4 is precisely the one occurring in the integrity constraint for the future perfect. Notice, however, that the existence of a time at which the house is fin- ished is only guaranteed in a minimal model. Thus, this inference is non- monotonic: if we obtain more information, the conclusion may fail. For example, Carlos’ story as given in (15-b) expands the scenario with

(8) Terminates(Accident,build,t) (9) the integrity constraint

?Happens(Accident, t), t <now,_¬HoldsAt(house(c), t)succeeds

Together, these imply that building will be clipped before the house is com- pleted. However, there is no longer an imperfective paradox. The meaning of ‘build a house’ is the same in both (17) and (15-b), since essentially the same scenario is involved in both cases. It is true thatTerminates(Accident, build, t) was not included among (1–7), but it would have made no differ- ence had we done so, sinceAccidentdid not occur as argument ofHappens there. The crucial point is that in both cases the successful completion of the building process is present as agoal, and not as an eventtoken.

3. **A computational proof

In this Section we provide a logic programming proof showing the truth of theorem 6. This is included as an illustration of the computational content of the theory – it can be skipped without loss. The proof has the following structure. On the basis of the scenario and the axioms for the event calculus, we have to show that the query

?HoldsAt(house(c), t1), t1 ≥now)

is satisfiable, given the integrity constraint that

?HoldsAt(build,now)succeeds.

We start with a derivation having?HoldsAt(house(c), t1), t1 ≥now as the

top query (figures 2 and 3) and we apply the integrity constraint when that derivation cannot be developed any further.

160 10. GRAMMATICAL ASPECT ?HoldsAt(house(c),t1), now _≤t1 (1) (((((( (((((( (2) 2 2 2 2 2 2 2 2 2 2 2 ?HoldsAt(house(c),t1), now <t1 ?HoldsAt(house(c),t1), now = t1 Axiom 4 ... .... ?Happens(start,t0), t0<now< t1, t1 = t0+d, Initiates(start,build,t0), Trajectory(build,t0,house(c),d), ¬Clipped(t0,build,t1) Formula 2 of scenario 3333 3333 3333 3333 3 ?Happens(start,t0), t0<now< t1, t1 = t0+d, Trajectory(build,t0,house(c),d), ¬Clipped(t0,build,t1) Formula 7 in scenario 3333 3333 3333 333 ?Happens(start,t0), t0<now< t1, t1 = t0+d, HoldsAt(house(a),t0), c = a+g(d), ¬Clipped(t0,build,t 1) Axioms 1 & 2 3333 3333 3333 3333 3 ?Happens(start,t0), t0 <now <t1, t1 = t0+d, c = a+g(d), ¬Clipped(t0,build,t1) ?Clipped fails %%%% %%%% %%%% %%%% ?Happens(start,t0), t0 <now <t1, t1 = t0+d, c = a+g(d)

FIGURE 2. Proof of theorem 6: left branch

In both cases we must ensure that the query?Happens(start, t0), t0 <

now succeeds. We know that ?HoldsAt(build,now)succeeds, and by ap- plying axiom 3 and formula 2 of the scenario, the latter query reduces to the former. We are thus left with two constraints: ?t0 < now < t1, t1 =

t0+d, c=a+g(d)for the left branch and?t0 <now =t0+d, c=a+g(d)

4. COMMENTS ON THE LITERATURE 161