Related papers: Gradualist descriptionalist set theory
We construct a formal theory, which we call reflectica, whose language possesses the following properties of natural language: it is a self-reflecting language and an intensional language. By a self-reflecting language we understand an…
The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to…
We first partly develop a mathematical notion of stable consistency intended to reflect the actual consistency property of human beings. Then we give a generalization of the first and second G\"odel incompleteness theorem to stably…
Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true…
G\"odel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit…
We give a new proof of the well-known fact that all functions $(\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$ which are definable in G\"odel's System T are continuous via a syntactic approach. Differing from the usual syntactic method, we…
Existential rules, a.k.a. dependencies in databases, and Datalog+/- in knowledge representation and reasoning recently, are a family of important logical languages widely used in computer science and artificial intelligence. Towards a deep…
We describe a mathematical structure that can give extensional denotational semantics to higher-order probabilistic programs. It is not limited to discrete probabilities, and it is compatible with integration in a way the models that have…
The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of…
As mathematical induction is applied to prove statements on natural numbers, {\it continuous induction} (or, {\it real induction}) is a tool to prove some statements in real analysis.(Although, this comparison is somehow an overstatement.)…
We present a family of paraconsistent counterparts of the constructive modal logic CK. These logics aim to formalise reasoning about contradictory but non-trivial propositional attitudes like beliefs or obligations. We define their…
Gradually typed programming languages, which allow for soundly mixing static and dynamically typed programming styles, present a strong challenge for metatheorists. Even the simplest sound gradually typed languages feature at least…
A process algebra is proposed, whose semantics maps a term to a nondeterministic finite automaton (NFA, for short). We prove a representability theorem: for each NFA $N$, there exists a process algebraic term $p$ such that its semantics is…
We see how nested sequents, a natural generalisation of hypersequents, allow us to develop a systematic proof theory for modal logics. As opposed to other prominent formalisms, such as the display calculus and labelled sequents, nested…
In this paper we extend our previous results on sets of graded attribute implications with witnessed non-redundancy. We assume finite residuated lattices as structures of truth degrees and use arbitrary idempotent truth-stressing linguistic…
Gradually typed languages are designed to support both dynamically typed and statically typed programming styles while preserving the benefits of each. While existing gradual type soundness theorems for these languages aim to show that…
We define a class of languages of infinite words over infinite alphabets, and the corresponding automata. The automata used for recognition are a generalisation of deterministic Muller automata to the setting of nominal sets. Remarkably,…
We show that first-order logic can be translated into a very simple and weak logic, and thus set theory can be formalized in this weak logic. This weak logical system is equivalent to the equational theory of Boolean algebras with three…
Knowledge distillation is a key technique for compressing large language models (LLMs), but most existing methods align representations at fixed layers or token-level outputs, ignoring how representations evolve across depth. As a result,…
This paper offers an approach to extensible knowledge representation and reasoning for a family of formalisms known as Description Logics. The approach is based on the notion of adding new concept constructors, and includes a heuristic…