We study asymptotic probabilities of formal languages specified by logical formulas. We focus on first-order sentences. We prove several results related to the asymptotic tractability of the two-variable fragment. Those results amount to show that asymptotic reasoning with two variables is feasible. We study some applications that are related to temporal reasoning.
It is proposed that sentential complements such as that S or whether S denote sentential quantifiers where a sentential quantifier is a set of sentential predicates (like is a rumour, is strongly believed) and a sentential predicate is a set of (declarative) sentences. Some properties of sentential predicates are studied and an analogy with nominal quantifiers is pointed out. An application to the analysis of questions (interrogative sentences) is made. Questions are rogative sentential predicates (which are semantically defined). No reference to possible worlds is made in the proposed analysis.
It is usually considered that A ↣ (B ↣ C) (i) implies and (ii) is implied by (A ∧ B) ↣ C . (i) is called Import (IM) and (ii) Export (EX). IM seems to be valid for natural language conditionals, but here we present counterexamples to EX, such as: If he uses cocaine every weekend and hides it from his family, he is already addicted, but not: If he uses cocaine every weekend, then if he hides it from his family, he is already addicted. Gibbard’s collapse theorem depends on IM-EX and Simplification (Conjunction Elimination). Here we use the implicative conditional for a logical analysis of IM-EX. We show that possibilistic versions of IM and EX are respectively valid and invalid for the implicative conditional, in accordance with what is observed in natural language conditionals. We conclude that this supports the adequacy of the implicative conditional for analysing the implicative use of natural language conditionals. We also conclude that the implicative conditional is immune to Gibbard’s collapse theorem, since it invalidates both IM-EX and Simplification.
Tabular intermediate logics are intermediate logics characterized by finite posets treated as Kripke frames. For a poset ℙ , let L(ℙ) denote the corresponding tabular intermediate logic. We investigate the complexity of the following decision problem LogContain: given two finite posets ℙ and ℚ , decide whether L(ℙ) ⊆ L(ℚ) . By Jankov’s and de Jongh’s theorem, the problem LogContain is related to the problem SPMorph: given two finite posets ℙ and ℚ , decide whether there exists a surjective p-morphism from ℙ onto ℚ . Both problems belong to the complexity class NP. We present two contributions. First, we describe a construction which, starting with a graph 𝔾 , gives a poset (𝔾) such that there is a surjective locally surjective homomorphism (the graph-theoretic analog of a p-morphism) from 𝔾 onto ℍ if and only if there is a surjective p-morphism from (𝔾) onto (ℍ) . This allows us to translate some hardness results from graph theory and obtain that several restricted versions of the problems LogContain and SPMorph are NP-complete. Among other results, we present a 18-element poset ℚ such that the problem to decide, for a given poset ℙ , whether L(ℙ)⊆ L(ℚ) is NP-complete. Second, we describe a polynomial-time algorithm that decides LogContain and SPMorph for posets 𝕋 and ℚ , when 𝕋 is a tree.
Utilizing abstractions of large conceptual models may enhance their clarity and comprehensibility. This work assesses an existing algorithm for generating abstractions of ontology-driven conceptual models. Although the algorithm has been empirically evaluated using the FAIR catalog of such models, it still lacks formal semantics. This paper addresses this gap by formalizing the basic transformations underlying the abstraction process in 𝒮ℛ𝒪ℐ𝒬 —the expressive and decidable description logic that underpins the Web Ontology Language (OWL 2). Specifically, it demonstrates that, under certain natural assumptions, these transformations are obtained by a formal procedure known as axiom weakening.
