Our topic is logical inference in natural language, as it is done by people and computers.
The first main topic will be monotonicity inference, arguably the best of the simple ideas
in the area. Monotonicity can be incorporated in running systems whereby one can take
parsed real-life sentences and see simple inferences in action. I will present some of the
theory, related to higher-order monotonicity and the syntax-semantics interface offered by
In a different direction, these days monotonicity inference can be done by machines as well
as humans. The talk also discusses this development along with some ongoing work on the
borderline of natural logic and machine learning.
The second direction in the talk will be an overview of the large number of logical systems for
various linguistic phenomena. This work begins as an updating of traditional syllogistic logic,
but with much greater expressive power.
Overall, the goal of the talk is to persuade you that the research program of “natural logic”
leads to a lively research area with connections to many areas both inside and outside of more
mainstream areas of logic.
(joint work with Norbert Gratzl, MCMP, Munich)
Free logics is a family of first-order logics which came about as a result of examining the existence assumptions of classical logic. What those assumptions are varies, but the central ones are that (i) the domain of interpretation is not empty, (ii) every name denotes exactly one object in the domain and (iii) the quantifiers have existential import. Free logics usually reject the claim that names need to denote in (ii), and of the systems considered in this paper, the positive free logic concedes that some atomic formulas containing non-denoting names (including self-identity) are true, while negative free logic rejects even the latter claim.
These logics have complex and varied axiomatizations and semantics, and the goal of the present work is to offer an orderly examination of the various systems and their mutual relations. This is done by first offering a formalization, using sequent calculi which possess all the desired structural properties of a good proof system, including admissibility of contraction and cut, while streamlining free logics in a way no other approach has. We then present a simple and unified system of generalized semantics, which allows for a straightforward demonstration of the meta-theoretical properties, while also offering insights into the relationship between different logics (free and classical). Finally, we extend the system with modalities by using a labeled sequent calculus, and here we are again able to map out the different approaches and their mutual relations using the same framework.
This paper adapts Lewis’s “Ptolemaic Astronomy” from Counterfactuals for use in thinking about social hierarchy and subordination.