Synthetic Logic

Synthetic logic Alex J. Djalali1 The role of inference as it relates to natural language (NL) semantics has often been neglected. Recently, there has been a move away by some NL semanticists from the heavy machinery of, say, Montagovian- style semantics to a more proof-based approach. Although researchers tend to study each type of system independently, MacCartney (2009) and MacCartney and Manning (2009) (henceforth M&M) recently developed an algorithmic approach to natural logic that attempts to combine insights from both monotonicity calculi and various syllogistic fragments to derive compositionally the relation be- tween two NL sentences from the relations of their parts. At the heart of their system, M&M begin with seven intuitive lexical- semantic relations that NL expressions can stand in, e.g., synonymy and antonymy, and then ask the question: if ' stands in some lexical- semantic relation to ; and stands in (a possibly di↵erent) lexical- semantic relation to ✓; what lexical-semantic relation (if any) can be concluded about the relation between ' and ✓? This type of reasoning has the familiar shape of a logical inference rule. However, the logical properties of their join table have not been ex- plored in any real detail. The purpose of this paper is to give M&M’s table a proper logical treatment. As I will show, the table has the underlying form of a syllogistic fragment and relies on a sort of gen- eralized transitive reasoning. 1 Introduction The role of inference as it relates to natural language (NL) semantics has oft been neglected. Recently, there has been a move away by some NL semanticists from the heavy machinery of, say, Montagovian-style semantics to a more proof-based approach. This represents a belief that the notion of derivability plays as central a role in NL semantics 1Stanford University 151 LiLT Volume 9 Perspectives on Semantic Representations for Textual Inference. Copyright c ⃝ 2014, CSLI Publications. 152 / Alex J. Djalali as that of entailment. Beginning with van Benthem (1986), and contin- uing on with Valencia (1991), Dowty (1994), Gilad and Francez (2005), Moss (2008), Moss (2010), van Benthem (2008), Moss (2009) and Moss (2012) among others, the study of various natural logics has become commonplace. Natural logicians place an emphasis on the development and study of proof theories that capture the sort of inferences speakers of a par- ticular NL like English make in practice. It should be said, though, that ‘natural logic’ is a catchall term that refers to either the study of various monotonicity calculi `a la van Benthem or Aristotelean-style syl- logistic fragments `a la Moss. Although researchers tend to study each type of system independently, MacCartney (2009) and MacCartney and Manning (2009) (henceforth M&M) recently developed an algorithmic approach to natural logic that attempts to combine insights from both monotonicity calculi and various syllogistic fragments to derive compo- sitionally the relation between two NL sentences from the relations of their parts. At the heart of their system, M&M begin with seven intuitive lexical- semantic relations that NL expressions …

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