arXiv subjects
Publications and source records attributed to Seyed Mohammad Amin Khatami.
In this article, a model-theoretic approach is proposed to prove that the first-order Gödel logic, $\mathbf{G}$, as well as its extension $\mathbf{G}^Δ$ associated with first-order relational languages enjoy the Craig interpolation property. These results partially provide an affirmative answer to a question posed in [Aguilera, Baaz, 2017, Ten problems in Gödel logic].
This paper is devoted to introduce a topology on BL-algebras, makes them semitopological algebras. For any BL-algebra $\mathcal{L}=(L, \wedge, \vee, *, \to , 0, 1)$, the introduced topology is defined by a distance-like function between elements of $L$ which is defined by $a \leftrightarrow b=(a\to b)*(b\to a)$. We will show that when the continuous scale $[0,1]$ is endowed to be a BL-algebra, then this topology admits some of the most important properties of the metric topology. Finally, we will show that this topology can be examined by a similar topology on dual of BL-algebras as well.
A generalization of the triangle inequality is introduced by a mapping similar to a t-conorm mapping. This generalization leads us to a notion for which we use the $\star$-metric terminology. We are interested in the topological space induced by a $\star$-metric. Considering some examples of non-trivial $\star$-metrizable topological spaces, we also study the product topology for a finite family of $\star$-metrizable topological spaces.
One of the nice properties of the first-order logic is the compactness of satisfiability. It state that a finitely satisfiable theory is satisfiable. However, different degrees of satisfiability in many-valued logics, poses various kind of the compactness in these logics. One of this issues is the compactness of $K$-satisfiability. Here, after an overview on the results around the compactness of satisfiability and compactness of $K$-satisfiability in many-valued logic based on continuous t-norms (basic logic), we extend the results around this topic. To this end, we consider a reverse semantical meaning for basic logic. Then we introduce a topology on $[0,1]$ and $[0,1]^2$ that the interpretation of all logical connectives are continuous with respect to these topologies. Finally using this fact we extend the results around the compactness of satisfiability in basic ogic.
We study an extension of \g propositional logic whose corresponding algebra is an ordered Abelian group. Then we expand the ideas to first-order case of this logic.
We study three kinds of compactness in some variants of Gödel logic: compactness, entailment compactness, and approximate entailment compactness. For countable first-order underlying language we use the Henkin construction to prove the compactness property of extensions of first-order Gödel logic enriched by nullary connective or the Baaz's projection connective. In the case of uncountable first-order language we use the ultraproduct method to derive the compactness theorem
This paper is devoted to systematic studies of some extensions of first-order Gödel logic. The first extension is the first-order rational Gödel logic which is an extension of first-order Gödel logic, enriched by countably many nullary logical connectives. By introducing some suitable semantics and proof theory, it is shown that the first-order rational Gödel logic has the completeness property, that is any (strongly) consistent theory is satisfiable. Furthermore, two notions of entailment and strong entailment are defined and their relations with the corresponding notion of proof is studied. In particular, an approximate entailment-compactness is shown. Next, by adding a binary predicate symbol $d$ to the first-order rational Gödel logic, the ultrametric logic is introduced. This serves as a suitable framework for analyzing structures which carry an ultrametric function $d$ together with some functions and predicates which are uniformly continuous with respect to the ultrametric $d$. Some model theory is developed and to justify the relevance of this model theory, the Robinson joint consistency theorem is proven.