Related papers: Implicit Commitment in a General Setting
In this paper we investigate the complexity of abduction, a fundamental and important form of non-monotonic reasoning. Given a knowledge base explaining the world's behavior it aims at finding an explanation for some observed manifestation.…
The handling of probabilities in the form of uncertainty or partial information is an essential task for LLMs in many settings and applications. A common approach to evaluate an LLM's probabilistic reasoning capabilities is to assess its…
An implicit operation of a class of similar algebras $\mathsf{K}$ is a collection of first order definable partial functions on the members of $\mathsf{K}$ that is globally preserved by homomorphisms. For instance, "taking inverses" can be…
In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion…
We give a survey of current research on G\"{o}del's incompleteness theorems from the following three aspects: classifications of different proofs of G\"{o}del's incompleteness theorems, the limit of the applicability of G\"{o}del's first…
The compactness theorem for a logic states, roughly, that the satisfiability of a set of well-formed formulas can be determined from the satisfiability of its finite subsets, and vice versa. Usually, proofs of this theorem depend on the…
We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original…
Uniform interpolation is the property that, for any formula and set of atoms, there exists the strongest consequence omitting those atoms. It plays a central role in knowledge representation and reasoning tasks such as knowledge update and…
Classical complexity theory measures the cost of computing a function, but many computational tasks require committing to one valid output among several. We introduce determination depth -- the minimum number of sequential layers of…
System F, the polymorphic lambda calculus, features the principle of impredicativity: polymorphic types may be (explicitly) instantiated at other types, enabling many powerful idioms such as Church encoding and data abstraction.…
We give a reframing of Godel's first and second incompleteness theorems that applies even to some undefinable theories of arithmetic. The usual Hilbert-Bernays provability conditions and the diagonal lemma are replaced by a more direct…
All scientific interpretations of statistical outputs depend on background (auxiliary) assumptions that are rarely delineated or explicitly interrogated. These include not only the usual modeling assumptions, but also deeper assumptions…
In this essay we'll prove G\"odel's incompleteness theorems twice. First, we'll prove them the good old-fashioned way. Then we'll repeat the feat in the setting of computation. In the process we'll discover that G\"odel's work, rightly…
It is well-known that relatively pseudocomplemented lattices can serve as an algebraic semantics of intuitionistic logic. To extend the concept of relative pseudocomplementation to non-distributive lattices, the first author introduced…
The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…
The prevalent interpretation of G\"odel's Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of…
In this paper we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain…
To what extent can a neural network systematically reason over symbolic facts? Evidence suggests that large pre-trained language models (LMs) acquire some reasoning capacity, but this ability is difficult to control. Recently, it has been…
An inductive logic can be formulated in which the elements are not propositions or probability distributions, but information systems. The logic is complete for information systems with binary hypotheses, i.e., it applies to all such…
Propositional term modal logic is interpreted over Kripke structures with unboundedly many accessibility relations and hence the syntax admits variables indexing modalities and quantification over them. This logic is undecidable, and we…