Mojtaba Mojtahedi

Ghent University | UGhent · Mathematics: Analysis, Logic and Discrete Mathematics

In this paper we introduce a modal theory $H_{\sigma}$, which is sound and complete for arithmetical $\Sigma$_1 substitutions in ${\bf HA}$, in other words, we will show that $H_{\sigma}$ is the $\Sigma$_1-provability logic of ${\bf HA}$. Moreover we will show that $H_{\sigma}$ is decidable. As a by-product of these results, we show that ${\bf HA}...

We show that the provability logic of PA, GL and the truth provability logic, i.e. the provability logic of PA relative to the standard model ℕ, GLS are reducible to their Σ1-provability logics, GLV and GLSV, respectively, by only propositional substitutions.

Let denote a first-order logic in a language that contains infinitely many constant symbols and also containing intuitionistic logic . By , we mean the associated logic axiomatized by the double negation of the universal closure of the axioms of plus . We shall show that if is strongly complete for a class of Kripke models , then is strongly comple...

We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A(p 1,..., p n ) built up of atoms p 1,..., p n , BPC ⊢ A(p 1,..., p n ) if and only if for all arithmetical sentences B 1,..., B n , BA ⊢ A(B 1,..., B n ). The technique used in our proof can easily be applied to some known extensions of BA.

The \(\Sigma _1\)-provability logic of Peano Arithmetic \(\textsf{PA}{} \), is characterized by (Visser, 1982) as \(\mathsf{GLC_a}\), the Gödel-Löb logic \(\textsf{GL}\) plus the completeness principle for atomic variables. Also the \(\Sigma _1\)-provability logic of the Heyting Arithmetic \(\textsf{HA}{} \), is characterized by (Ardeshir & Mojtahe...

We examine the interplay between projectivity (in the sense that was introduced by S. Ghilardi) and uniform post-interpolant for the classical and intuitionistic propositional logic. More precisely, we explore whether a projective substitution of a formula is equivalent to its uniform post-interpolant, assuming the substitution leaves the variables...

We axiomatize the provability logic of HA and show that it is decidable. Moreover we axiomatize the preservativity and relative admissibility for several modal logics extending iK4. As a main tool, we also provide some sort of semantics, called provability semantics, for modal logics extending iGL, which is a mixture of usual Kripke semantics and p...

We prove that $\textbf {K}5$ and some of its extensions that do not contain $\textbf {K}4$ are of unification type $1$.

In this paper, we show that the implication fragment of classical propositional logic is finitary for unification with parameters.

Let \(\mathsf{PL}(T,T')\) and \(\mathsf{PL}_{_{\Sigma _1}}(T,T')\) respectively indicate the provability logic and \(\Sigma _1\)-provability logic of \(T\) relative in \(T'\). In this paper we characterise the following relative provability logics: \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{HA},\mathbb {N})\), \(\mathsf{PL}_{_{\Sigma _1}}(\mathsf{HA},\ma...

This volume is a collection of essays in honour of Professor Mohammad Ardeshir. It examines topics which, in one way or another, are connected to the various aspects of his multidisciplinary research interests. Based on this criterion, the book is divided into three general categories. The first category includes papers on non-classical logics, inc...

Let $\mathcal{PL}({\sf T},{\sf T}')$ and $\mathcal{PL}_{\Sigma_1}({\sf T},{\sf T}')$ respectively indicates the provability logic and $\Sigma_1$-provability logic of ${\sf T}$ relative in ${\sf T}'$. In this paper we characterize the following relative provability logics: $\mathcal{PL}_{\Sigma_1}({\sf HA},\mathbb{N})$, $\mathcal{PL}_{\Sigma_1}({\sf...

For the Heyting Arithmetic HA, $HA^{\text{*}} $ is defined [14, 15] as the theory $\left\{ {A|HA \vdash A^\square } \right\}$ , where $A^\square $ is called the box translation of A (Definition 2.4). We characterize the ${\text{\Sigma }}_1 $ -provability logic of $HA^{\text{*}} $ as a modal theory $iH_\sigma ^{\text{*}} $ (Definition 3.17).

In this paper we introduce a modal theory iHσ which is sound and complete for arithmetical Σ1-interpretations in HA, in other words, we will show that iHσ is the Σ1-provability logic of HA. Moreover we will show that iHσ is decidable. As a by-product of these results, we show that HA+□⊥ has de Jongh property.

For the Heyting Arithmetic HA, HA* is defined as the theory $\{A\mid {\sf HA}\vdash A^{\Box}\}$, where $A^{\Box}$ is called the box translation of $A$. We characterize the $\Sigma_1$-provability logic of HA* as a modal theory ${\sf iH}_\sigma^*$.

We can look at a first-order (or propositional) intuitionistic Kripke model as an ordered set of classical models. In this paper, we show that for a finite-depth Kripke model in an arbitrary first-order language or propositional language, local (classical) truth of a formula is equivalent to non-classical truth (truth in the Kripke semantics) of a...

We can look at a first-order (or propositional) intuitionistic Kripke model as an ordered set of classical models. In this paper, we show that for a finite-depth Kripke model in an arbitrary first-order language or propositional language, local (classical) truth of a formula is equivalent to non-classical truth (truth in the Kripke semantics) of a...