Event Polarity

This section introduces a notion of event polarity under the assumption that we have ordinary models against which the truth of a sentence can be evaluated. Since it will become obvious that this leads to technical complications, a revised version in the spirit of section2.2.3and section2.2.4will then be given in section4.1.2.

The philosophical and logical discussions surrounding negation from Aristotle to Rus-sell and contemporary linguists could fill whole libraries. Even a cursory account of

the history of negation fills such stately an erudite volumes as Horn (2001), of which I am going to discuss at least a little bit soon.

However, in logics as applied to natural language semantics, negation is virtually al-ways implemented by means of the truth-functional negation operator¬. For senten-tial (boolean) negation, this is trivially adequate, and for property negation, a simple pointwise definition based on boolean negation does the trick. A definition of property negation is demonstrated in (1), taken from Carpenter (1997:91).

(1) ¬P^{de f}= λx.¬P(x)

In standard Event Semantics, we can assume a negated existential quantifier over the event variable, such as ¬∃e.φ (usually with at least one occurrence of e in φ^{). This,} if true, excludes all events for which φ does not hold from models compatible with this sentence. It makes the sentence false in a concrete model if there is at least one event for whichφ holds. Similar pointwise definitions could take care of non-sentential negation, down to negation of nominal predicates (such as the English nominal prefix non-).

As far as I can see, there is one semantic framework under which negation is modelled differently, namely Situation Semantics. My proposals owes a lot to Situation Seman-tics as an inspirational source, although maybe less on the technical side.

In Infon/Situation Semantics (Barwise and Perry 1983 etc. as cited above), infons are taken (informally speaking) to be the minimal building blocks of information. They are essentially tuples of relations (relation-objects) and objects which fill argument places of the relation. Infons describe situations, and in turn situations support (or do not sup-port) infons. Parameters are open places within a given infon, which must be linked to concrete objects. Parameters are sorted corresponding to the types in the ontology of the framework (types are RELⁿfor n-player relations, IND for individuals, T IM for times, etc.).

Most importantly, every infon is straightforwardly specified for a polarity of 0 or 1.

As also discussed by Devlin (1991), there are certain similarities between situations and Kimian events (Kim 1976) as property exemplifications.¹ Also in standard Event Semantics with full event individuation by parameters (as presented here, especially in

1 I think the parallel could also be made between infons and Kimian events, but lack the space to discuss the implications. Since I am not presently concerned with the cognitive and computational questions which are much under focus in Situation Semantics, and since the strength of Situation Semantics lies in the modelling of context-dependence (which is also not required for this study), I will not elaborate further on similarities and dissimilarities with Situation Semantics past this para-graph. However, I think that a more thorough investigation of the parallels could eventually be of some profit, mostly to enrich the present theory with solutions to problems of context-dependency.

chapter2), where events have no easily graspable ontological status outside the seman-tics of natural language, events are much better understood as abstract bits of infor-mation about configurations of objects or the relations between them. A single event compresses all information relevant to a specified collection of objects, in specified places, at a specified time, etc., with a certain primary action, process, etc. going on be-tween them, etc. The main difference to infons as I see it is that the compressed object (the event) itself can be assigned properties directly which otherwise would have to be encoded as more complex relations between objects, times, etc. Parameters instantiated by adverbials like manner or intensity are prime examples. This is possible because the abstract events we are dealing with are explicitly not infons, but they still have physi-cal properties like temporal stretch, spatial coordinates, manners of execution, etc. An event as constructed here might be the closest thing to an infon for people who prefer working with (albeit abstract) ontological objects and a more or less traditional model theory. Put differently, although events as presented here are very similar to infons, infons are not model-theoretic entities in the same manner as events.

Where does all this lead us with respect to negation? I propose that it backs up the postulation of events with a polarity parameter. Such a parameter encodes whether an event is factual or not (in a secondary, more “realistic” sense). A classical model with event polarities would therefore have to contain for every possible permutation of event parameters (except polarity) either the positive or the negative event. Despite the involvement of the word possible here, the notion of possible event is a completely extensional notion, and possible events can easily be calculated by the following line of reasoning. Language conceptualizes the individuals (objects and events) of the world by providing linguistic expressions which allow the language-user to discern between these individuals. Looking only at event individuals now, this means that the events we can possibly talk about are those discernible by linguistic expressions.

Imagine this scenario: If we have a language which only expresses two main param-eters of events, walk and talk, two expressions referring to individuals (kermit and scooter), two manner expressions (quickly and slowly), two expressions about spatial locations in the universe (onstage and backstage), and no more than two points in time encoded by generalized operators i₀ and i₁. Whether the real world is actually as min-imalist as this language suggests does not matter. As long as language-users reserve language to talk about Kermit or Scooter, on stage or backstage, talking or walking quickly or slowly on two famous historic occasions, our models are accordingly mini-malistic. The reduced L_GOA language of these language-users can generate a reduced array of maximally specific expressions, cf. (2) (inL_GOA notation).

(2) a. kermit^Ext(quickly(onstage(i0(walk)))) b. scooter^Ext(quickly(onstage(i₀(walk)))) c. kermit^Ext(slowly(onstage(i0(walk)))) d. kermit^Ext(quickly(backstage(i0(walk)))) e. kermit^Ext(quickly(onstage(i₁(walk))))

f. kermit^Ext(quickly(onstage(i0(talk)))) g. . . .

Since this language has no means of introducing recursion, the number of sentences will be finite, all in all we get a bounded number of possible maximally specific expres-sions, each denoting (under^GOA) a singleton set of singleton sets of events. But even if there was an infinite number of compositionally derivable expressions (in a language with recursion), the maximum number of informative statements about the existence of events will always depend on the number of parameters expressible by the finite num-ber of lexical entries.

Quantification introduces sets of events into the world of denotations, but the events in those sets must characteristically be expressible by simple statements without quan-tification (referring to single events), much as quanquan-tificational formulae in traditional semantics based on predicate logic can be expressed by conjunctions and disjunctions of non-quantificational formulae at least in finite models.

There is thus a possibly infinite but bounded set of events users of this language can talk about. These are the possible events maximally present in the models of that language.

It is obvious that in a real world each of these possible events must be either factual or not, which is encoded straightforwardly by requiring that it exists with either positive polarity (i.e., it exists in the classical sense) or negative polarity (i.e., it does not exist under the classical model-theoretic view) in fully specified realistic mental models. For reasons to become clear soon, we must also posit that any operator (besides negation and maybe operators which can be decomposed involving negation) only and exclu-sively maps sets of events with polarityα to sets of events with polarityα^.

Now, in how far are events with a polarity parameter conceptually justified and tech-nically feasible? I think they are both if either (i) one is willing to give up a strictly correspondence-theoretic view of models for the interpretation of language, or (ii) if one is willing to accept that in the real world the absence of some event is actually rather the existence of a negative event.

I tend to opt for the first variant as can probably be guessed from what was said in previous chapters where I argued for a procedure which interprets sentences against an

abstract informational ontology, then constructing a mental model of the world.² As I mentioned in section2.2.3, an uninformed language-user who has not received any information is in a state where he must consider all possible events. That is, knowing that there are Miss Piggy and Kermit, that there are such things as hitting events, and that there was a point in time (say, Monday, August 7, 1978, 1:30 p.m.), he must at least entertain the possibility that there was a hitting event at that point in time with Miss Piggy as the agent and Kermit as the theme, or that there wasn’t. Also, regardless of his beliefs, any language-user can talk both about the positive event and about the otherwise identical negative event, although if he does so affirmatively and in a single discourse, he will probably contradict himself. The fact remains that we can talk about non-factual events, and that we therefore absolutely need to have them in the ontology if we interpret sentences as (sets of) events.

This concludes the first introduction of the concept of positive and negative polarity for events, and I will now turn to questions of the technical implementation, starting with a more precise definition of negative events in the discourse-level update proce-dure introduced in section2.2.4, including a recapitulation of the mutual exclusiveness of positive and negative events in section4.1.2.