Monotonic Inference for Underspecified Episodic Logic (2021)

Abstract: We present a method of making natural logic inferences from Unscoped Logical Form of Episodic Logic. We establish a correspondence between inference rules of scope resolved Episodic Logic and the natural logic treatment by Sánchez Valencia (1991a), and hence demonstrate the ability to handle foundational natural logic inferences from prior literature as well as more general nested monotonicity inferences.

## introduction natural logic is an approach to generating inferences from language directly over the grammatical structure through knowledge of entailment monotonicity in the lexicon. monotonicity is a characteristic of functions within an ordering, i.e. f is upward monotone if @xmath0. for example, not is downward monotone in entailment since it flips the entailment ordering of "fido is a dog" entails "fido is an animal" to "fido is not an animal" entails "fido is not a dog". natural logic can be seen as an extension of aristotelian syllogistic reasoning @xcite and was first formally related to higher-order logic entailments by sánchez @xcite . @xcite and @xcite later construed natural logic as a formal system of its own, independent of a separate logical formalism. unscoped logical form (ulf) of episodic logic was developed with the aim to integrate machine learning into automatic natural language inference by simplifying the semantic parsing task that presupposes symbolic inference @xcite . ulfs retain certain ambiguities in the sentence while strictly defining the core semantic type structure that is necessary to specify the compositional structure. this results in a parsing task that is similar in form and complexity to constituency parsing, for which the community has built effective parsers @xcite . promising preliminary results show parsability of ulf from a small dataset and minimal representation-specific knowledge @xcite . automatic inference generation from ulf has been demonstrated for dialogue-focused structural inferences which correspond to simple presuppositions and implicatures over questions, requests, counterfactual constructions, and clausetaking verbs @xcite . here we present a proof-based natural logic inference formalism for ulf. we show that this method covers inferences presented by sánchez @xcite and can support rule instantiation from episodic logic (el) inference for nested polarity inference. the contributions of this paper are two-fold: (1) this marks the first formalized inference procedure for ulf and (2) we present an alternative to parsing full logical formulas in symbolic natural logic inference through the use of an underspecified representation. an implementation of the inference procedure that we describe is beyond the scope of this paper, but is an important next step for empirically evaluating the efficacy of this approach against existing natural logic inference systems. due to space limitations, we leave fully formalized definitions and proofs to the appendix and use the main document for condensed explanations and demonstrative examples. ## natural logic we will limit our discussion of natural logic to that presented by sánchez valencia (1991a) and follow his notation and terminology. 1 the semantics of sánchez-valencia's natural logic is rooted in an undirected, typed lambda calculus constructed 1 much of the recent work in natural logic uses the terminology and notation introduced by @xcite . marking x a, y x a * , y a, x y a * , x y negation x, n eg(a) y x a, y ## episodic logic: unscoped logical form episodic logic (el) is an extended fol that closely matches the form and expressivity of nat-(|ali| (do.aux-s not @xcite x )))) ) ))) ural language, using type-shifters and a liberal ontology of individuals (e.g. basic individuals, situations, propositions, kinds, etc.) to keep the logic first-order while allowing for intensionality, general quantifiers, etc. @xcite . el supports deductive and uncertain inference, including forward and goal-chaining inference that uses polarity-based substitution in a natural logic-like manner @xcite @xcite . the forward inference rules are basically the same as nested inference rules proposed by @xcite . ulf fully specifies the semantic type structure of el by specifying the types of the atoms and all of the predicate-argument relationships while leaving operator scope, anaphora, and word sense unresolved @xcite . the name, unscoped logical form, is a label for its stage in the interpretation process of el and does not mean that scoping is the only unresolved aspect of the logical form. @xcite describe the role of ulf in the interpretation process. the types of ulf atoms that correspond to surface words and are not logical or macro symbols are marked with suffixed tags resembling the partof-speech (e.g. .v, .n, .pro, .d for verbs, nouns, pronouns, and determiners). case-sensitive symbols such as names and titles are marked with pipes (e.g. |john|). pipe-marked symbols may be left without a type tag in which case they default to having an entity type. a closed set of logical and macro symbols have unique types so the type marking is omitted. each suffix indicates a set of possible semantic denotations, e.g. .pro always denotes an entity and .v denotes an n-ary predicate where n can vary. type shifters in ulf maintain coherence of the semantic type structure. for example, the type shifter adv-a maps a predicate into a predicate modifier as in the prepositional phrase "with a dog" in figure 2 , as opposed to its predicative use "i am with my dog". the syntactic structure is closely reflected in ulf even under syntactic movement through the use of simple rewriting macros which explicitly mark these occurrences and upon expansion make available the exact semantic argument structure. the ordering of operator-argument relations in ulf can have the operator in the first or second position, disambiguated by the types of the participating expressions. the el type system only allows function application for combining types, a, b , @xmath1, much like montagovian semantics @xcite without type-raising. scoped logical form (slf) slf is ulf with explicit scoping. since polarity propagates through scoped operator relations, scopes must be fully specified before adding polarities. while inferences will interface with ulfs, auxiliary slfs are necessary to model the polarities and bookkeep scope-related assumptions in the inferences. scoped operator orderings are represented using parentheses, and are lifted around the sentence that it scopes around. scoped determiners are represented as (δ ν: φ ψ) where δ is a determiner, ν is a variable, φ is the restrictor wff, and ψ is the scope wff. figure 2 shows examples of the scoping process. for a ulf, ψ, that contains a quantified expression ϙ of form the (δ π), where δ is a determiner and π is a predicate, the corresponding formula with (δ π) at the top-level scope is (δ x: (x π) ψ[ϙ/x]). top-level scope marking process given the slf that defines the scope ordering, the constituent of the form (δ π) in ψ at the position of x in ψ[ϙ/x] is marked with # as the top scope of ψ. 3 below is 2 in accordance with sánchez-valencia's treatment, we do not address the possible scoping complexities of including sentential modifiers, tenses, and aspect as scoped operators. scoping operators (s1) not (-) , never.adv (-) verbs know.v (+,•) determiners (s2) a.d (+,+) , every.d (-,+) , some.d (+,+) , many.d (+,+) , most.d (+,•) an example to help illustrate the mapping. polarity marking we perform polarity marking in a two stage process that mirrors the process used by sánchez valencia (1991a). first we classify lexical entries according to their monotonicity properties-in what entailment contexts they place their arguments-and mark them in the slf with parenthesized subscripts. the possible entailment options are + for upward,for downward, and • for none. figure 3 provides a few examples. 4 using the lexical annotations, we mark the local entailment direction of the constituents in the slf using subscripts without parentheses. finally, the global polarity is derived from these local entailment directions and marked with superscripts. the global polarity is computed by traversing the slf from the root and counting the number of occurrences of negative and flat entailments, with the following rules. 1. node a has no polarity if any node in the path from the root to a is marked with •. 2. else, node a has negative polarity if there are an odd number of nodes between the root and a (inclusive) marked with -. 3. otherwise, node a has positive polarity. figure 4 shows all of these markings in a tree format. we then mark the global polarity in the ulf according to the corresponding slf to get. inference rules figure 5 lists the ulf versions of the monotonicity and conversion rules from figure 1 , but in a standard rule of inference format. sánchez-valencia's marking and negation rules are specific to the tableau system and not relevant ## integration with machine learning while a working inference system is beyond the scope of this paper, in this section we discuss some ways in which machine learning can be leveraged in conjunction with the inference formalism that we describe in this paper. an obvious and important role for machine learning in building a ulf-based inference system is to train a semantic parser to provide ulfs for english sentences. our preliminary work in this direction @xcite using an annotated dataset has shown promising results. our current method is to train an lstm to parse action sequences for a cache transition parser @xcite . including contextual embeddings such as bert @xcite and roberta @xcite as inputs to such as model will allow the parser to use the representational power of these embeddings to select the most appropriate parse. similarly, we can expect polarity labeling algorithms to improve with the introduction of contextual embeddings, though we are unaware of any work that has tried to do this. this labeling could also have a collaborative effect with a symbolic polarity labeling. with a partially complete lexicon of negative polarity inducing operators, a ulf could verify parts of the sequentially labeled polarities or correct them if inconsistencies are found in the graph where lexical knowledge is available. for tasks like sick @xcite which rely largely on lexical specializations, we envision using a lexical resource like wordnet @xcite . there still remains the issue of word sense, which is not resolved in ulf. again distributional word representations could be used here to select the most appropriate word sense or set of word senses. tasks that provide all the necessary relationships such as fracas @xcite do not require any additional axioms beyond inference rules for basic logical operators and for introducing and eliminating macro operators. for example, modeling relative clauses, which appear frequently in fracas, simply requires properly handling the relativizer and post-nominal modification macros to get a monotonicity ordering between predicates "a" and "a that b" that is fully modeled by logical conjunction "a" and "(λ x: ((x a) ∧ (x b)))". the near-syntactic nature of ulf allows accurate generation of english sentences corresponding to formulas @xcite ). an interesting question is whether generative language models could be used to enhance inference generation. such models are trained to learn the patterns of language use, and as such do not necessarily reflect valid entailments. but anchoring the use of language models to a symbolic representation like ulf would potentially enable constraining inferences to interpretable ones. ## conclusion we have presented a proof-based formalism for making natural logic inferences from ulfs with inproof scoping assumption declarations using less ambiguous, scoped lfs. this inferential capacity of ulf, in conjunction with its ease of parsing, positions ulf as a promising representational basis for automatically generating natural logic inferences. machine learning tools can then be deployed for semantic parsing (ulfs) and sequence labeling (polarities), both well-researched paradigms, rather than building a model of natural logic directly on top of statistical tools.