Sobre la enseñanza de la trigonometría

Set-Theoretic Basic Relations

Recall that the natural logic formalism as described in MacCartney (2009) defines seven basic entailment relations, which capture all of the possible ways that two sets defined in the same universe might relate to one another (see Section 2.2.3). Using these basic entailment relations to partition the space of possible relationships betweenJHKandJM HKresults in a

more fine-grained characterization ofM H compositions than does the traditional taxonomy of modifiers just described. Table 6 depicts the relationship between these two set-theoretic classifications of modified noun phrases.

Subsective _JM H_K=_JH_K Equiv. (≡) JM HK⊂JHK JM HK6=JHK Forward (@) Privative JM HK∪JHK=U Negation (ˆ) Non-Sub. JM HK∩JHK=∅ JM HK∪JHK6=U Alt. (|) JM HK6⊂JHK Plain JHK⊂JM HK Reverse (A) JM HK∩JHK=6 ∅ JHK6⊂JM HK JM HK∪JHK=U Cover (^) JM HK∪JHK6=U Indep. (#)

Table 6: Relationship between natural logic relations and formal semantics adjective classes. Table reads as a decision tree from left to right.

Basic Relations Defined on String Operations

We discussed in Section 2.2.3 that while the basic relations are defined between sets, it is often preferable, especially for NLP applications, to focus on the inference rules implied by the underlying set relations, rather than on the set relations themselves. In natural logic

(MacCartney (2009)), this is accomplished by focusing on the relationsgenerated by atomic edits applied to natural language strings. For example, while formal semantics focuses on specifying the relationship between _J“brown dog”_K and _J“dog”_K in the abstract (across all possible worlds), natural logic focuses on determining the entailment relationship between a sentence s containing the word “dog” and a sentence e(s) into which the word “brown” has been inserted in front of “dog”.

Table 7 shows examples of sentences and edits in which the composition of a modifier M

with a noun H can generate each of the basic entailment relations previously described. The relation (β(e)) generated by the atomic edit is determined by the inferences that hold between s and e(s). See Table 5 from Section 2.2.3 for a summary of the inference rules associated with each relation.

M H s e β(e)

entire world It is her favorite book in the world. IN S(“entire”) ≡

brown dog Fido is a dog. IN S(“brown”) A

potential successor She is the president’s successor. IN S(“potential”) @

former senator She is a senator. IN S(“former”) |

alleged hacker She is a hacker. IN S(“alleged”) #

Table 7: Basic entailment relations generated by modifier-noun composition–i.e. inserting modifiers in front of nouns in context.

Note that we do not offer examples for whichM H composition generates the Cover (^) or the Negation (ˆ) relation, as these two relations have the requirement that_JH_K∪_JM H_K=U, a difficult condition to meet in practice. I discuss our treatment of this “exhaustivity” constraint as it pertains to our work in Section 2.4.

Comparison to Formal Semantics Approach

Regarding modifier-noun composition, the atomic edit approach taken in natural logic makes no attempt to assign meaning to the modifier itself. This is unlike the function application approach taken in formal semantics. In formal semantics, a subsective modifier like“brown” carries some intrinsic meaning, which is used, for example, to discriminate the set of“brown

dogs” from the set of “dogs” more generally. MacCartney (2009)’s formulation does not require we assign any intrinsic meaning to words nor that we ground words to entities and relations either concretely (in terms of the real world) or abstractly (in terms of possible worlds).

This decrease in representational power, however, makes natural logic quite flexible for tasks like RTE, as it is able to avoid the need to address difficult theoretical problems, like the issue of domains and function types for non-subsective modifiers (Section 2.3.1). The edit-based formalization makes it possible for the natural logic framework to support the handling of issues like word sense, context, and pragmatics when reasoning about entailment, without needing a complete logical formalization of these complex phenomena. The examples in Table 7 illustrate how, by focusing on the relation generated by an edit in specific context, the natural logic framework sidesteps any formal treatment of issues such as definite and indefinite reference, temporal reasoning, and hyperbole. That is not to say that systems based on natural logic would not need to deal with these issues, but rather that the general natural logic framework outsources4 _{these issues to whatever subroutine determines β(e)} for a given edit e and context s. Thus, the goal is simply to recognize that the context generates a particular relation, not to model why the context warrants that relation. It is an open question as to whether automatic systems can master the former without the latter. We return to this distinction again in our discussion of modifier-noun composition and semantic containment in Chapter 4.