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The basic entailment relations defined by MacCartney (2009)’s natural logic (summarized in Section 2.2.3) provide a clear and simple vocabulary for talking about entailment in natural language. In Chapters 3 and 4, we would like to frame our analyses largely in terms of these relations. In order to do so, we make a few modifications to MacCartney (2009)’s definitions, as described below.

2.4.1. Relaxing Requirements of Exhaustivity

The definitions of two of MacCartney (2009)’s basic entailment relations, Negation (ˆ) and Cover (^), require that the union of the sets standing in the relation be “exhaustive”. That is, in order for x and y to stand in either of these two relations, everything in the universe U must be either xory (or possibly both). While this is a meaningful theoretical distinction to make, its relevance to natural language, in practice, is arguably very limited. We therefore disregard these two relations, as described below.

Negation Relation. Recall the definition of Negation (xˆy) is that (X∩Y =∅)∧(X∪Y =

U). That is, everything in the universe U is either x or it is y, and it cannot be both simultaneously. This Negation relation allows one to make the strong inference that not only doesx⇒ ¬ybut also¬x⇒y. This relation is primarily used to deal with explicit negation, e.g. it is often the basic entailment relation generated by a DEL(“not”) or IN S(“not”) edit. However, in this thesis, we focus on lexical substitutions (Chapter 3) and on modifier- noun compositions (Chapter 4). It is difficult to come up with cases of in either of these

two settings for which ˆ is relevant. For example, in the context of lexical substitution, even indisputably antonymous words do not generate the negation relation: “not good” 6⇒“bad” and “not bad” 6⇒ “good”. Even for words which are explicitly negated through a prefix (“intelligent”/“unintelligent”), people may perceive the dimensions under discussion as having a “middle ground”: just because“she isn’t intelligent”, we don’t necessarily assume “she is unintelligent”. There is undoubtedly room to explore the pragmatic circumstances under which a lexical substitution may or may not yield a ˆ relation, but that is well beyond the scope of this thesis.

Therefore, we only use one relation to represent mutual exclusion, which we will refer to asExclusion and represent with the asymbol. Our a relation signifies semantic exclusion in which x ⇒ ¬y and y ⇒ ¬x but it is not necessarily the case that ¬x ⇒ y or that

¬y⇒x. Note that our exclusion relation is definitionally equivalent to MacCartney (2009)’s Alternation (|); the new name and symbol are for clarity only, so that we can still refer to MacCartney (2009)’s symbols when needed without confusion.

We acknowledge that removing the distinction between | and ˆ weakens the strength of the inferences we are able to make. For example, using the natural logic framework, and given the premise/hypothesis pair “The claim is not true”/“The claim is false”, a sys- tem which models only non-exhaustive exclusion (“true”a“false”) can only concludenon- entailment. In contrast, a system which models exhaustive exclusion (“true”ˆ“false”) can draw the stronger conclusion of contradiction. In practice, however, these types of cases are rare, and we therefore see little disadvantage to simplifying the exclusion relations to remove the focus on exhaustivity.

Cover Relation. Recall the definition of Cover (x ^ y) is that (X∩Y 6=∅)∧(X∪Y =U). Like for the ˆ relation, the^ relation requires that everything in the universeU is eitherx

ory, but in^ it is possible for something in the universe to be bothxandy. MacCartney (2009) describes a canonical case of this relation as a word x paired with the negation of a hyponym of x, e.g. “animal”^“nonhuman”. MacCartney (2009) acknowledges that the

application of this relation is “not immediately obvious.” The relation has since been shown to be applicable to the analysis of insertions and deletions when such edits involve one-way implicative verbs (Karttunen (2016)). However, for simple lexical substitutions or modifier insertions, which is the focus of this thesis, the Cover relation is unlikely to arise. Thus, we disregard this relation completely going forward.

2.4.2. Definitions

We define five basic entailment relations which we use to describe the relationship between two natural language strings: Equivalence (≡), Forward Entailment (@), Reverse Entail- ment (A), Exclusion (a), and Independent (#).

Within our definition of Exclusion (a), there are some pairs which are intuitively interpreted as falling along a single dimension and thus are naturally interpreted as “opposites” (e.g. “good/bad”) and others which are clearly mutually exclusive but more categorical than bipolar (e.g. “dog/cat”). When it is necessary to distinguish, we useaoppto refer to natural

opposites anda_alt to refer to mutually-exclusive alternatives under a common category. Within our definition of Independent (#), there are some paris which are semantically related but not through entailment, for example meronyms (“eye”/“face”) or derivationally related terms (“academy”/“academia”). When relevant, we will refer to these pairs as “Otherwise Related” and denote this type of relatedness with the ∼symbol. We will refer to the remaining independent relations as “Unrelated”, denoted using the 6∼symbol. Throughout this thesis, unless otherwise specified, the below names, symbols, and associated inference rules will refer to the definitions given here.

• Equivalence (x≡y): x⇔y. E.g. “couch”/“sofa”

• Forward Entailment (x@y): x⇒y E.g. “couch”/“furniture”

• Exclusion (xay): y⇒ ¬x∧y ⇒ ¬x

– Opposites (x aoppy): xay, and humans typically view x and y as “opposites”,

or the two ends of a single bipolar dimension. E.g. “good”/“bad”

– Alternatives (x aalt y): x a y, and humans typically view x and y as mutually

exclusive types under a common category (and there are typically more than two alternatives under that category). E.g. “couch”/“table”

• Independent (x#y) : x6⇒y∧y6⇒x

– Otherwise Related (x∼y) : x#ybut there is some natural relationship between

x and y that can’t be captured by entailment. E.g. “couch”/“cushion”