Related papers: Logic Lectures: G\"odel's Basic Logic Course at No…
This paper proposes an alternative to standard first-order logic that seeks greater naturalness, generality, and semantic self-containment. The system removes the first-order restriction, avoids type hierarchies, and dispenses with external…
Consider the following story: A teacher announces to her students a test for the following week, such that the test will be ``surprising''. The students use this as the basis for a ``logical derivation'' and reach a contradiction, which…
We present the first class of mathematically rigorous, general, fully self-referential, self-improving, optimally efficient problem solvers. Inspired by Kurt Goedel's celebrated self-referential formulas (1931), such a problem solver…
In 1931, G\"odel presented in K\"onigsberg his famous Incompleteness Theorem, stating that some true mathematical statements are unprovable. Yet, this result gives us no idea about those independent (that is, true and unprovable)…
These lecture notes are based on a set of six lectures that I gave in Edinburgh in 2008/2009 and they cover some topics in the interface between Geometry and Physics. They involve some unsolved problems and conjectures and I hope they may…
Most non-classical logics are subclassical, that is, every inference/theorem they validate is also valid classically. A notable exception is the three-valued propositional Logic of Ordinary Discourse (OL) proposed and extensively motivated…
Recent published work has addressed the Shalqvist correspondence problem for non-distributive logics. The natural question that arises is to identify the fragment of first-order logic that corresponds to logics without distribution, lifting…
These Course Notes provide an introduction to mathematical proofs for undergraduate students transitioning from computational calculus to abstract mathematics. Topics include propositional logic, proof techniques, mathematical induction,…
There is a sharp disconnect between the programming and mathematical portions of the standard undergraduate computer science curriculum, leading to student misunderstanding about how the two are related. We propose connecting the subjects…
The use of lab notebooks for scientific documentation is a ubiquitous part of physics research. However, it is common for undergraduate physics laboratory courses not to emphasize the development of documentation skills, despite the fact…
These notes are concerned with the existence and the basic properties of the set-theoretic universes for nonstandard analysis, compiled by a beginner in the subject. It assumes a basic background in first-order logic, though the necessary…
Until recently, punctuation has received very little attention in the linguistics and computational linguistics literature. Since the publication of Nunberg's (1990) monograph on the topic, however, punctuation has seen its stock begin to…
Different from the view that information is objective reality, this paper adopts the idea that all information needs to be compiled by the interpreter before it can be observed. From the traditional complexity definition, this paper defines…
We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies…
G\"odel's Dialectica interpretation is a fundamental tool for the extraction of computational content from proofs, and plays a central role in today's proof mining program. In the past decades, it has also been studied from the perspective…
We introduce and study single-conclusioned nested sequent calculi for a broad class of intuitionistic multi-modal logics known as "intuitionistic grammar logics (IGLs)." These logics serve as the intuitionistic counterparts of classical…
The study of propositional logic -- fundamental to the theory of computing -- is a cornerstone of the undergraduate computer science curriculum. Learning to solve logical proofs requires repeated guided practice, but undergraduate students…
We explore the problem of explaining observations in contexts involving statements with truth degrees such as `the lift is loaded', `the symptoms are severe', etc. To formalise these contexts, we consider infinitely-valued {\L}ukasiewicz…
The formal construction of the second-order logic or predicate calculus essentially adds quantifiers to propositional logic. Why second-order logic cannot be reduced to that of the first order? How to demonstrate that certain predicates are…
Several different proof translations exist between classical and intuitionistic logic (negative translations), and intuitionistic and linear logic (Girard translations). Our aims in this paper are (1) to consider extensions of…