CONTENTS0. Preliminaries....................................................................... 71. Adding propositional connectives to $L_{\omega }{1}_{\omega }$............... 82. The propositional part of $L_{\omega }{1}_{\omega }$ (S)............................. 103. The operation S and the Boolean algebra $B_S$............... 114. General model-theoretic properties of $L_{\omega }{1}_{\omega }$(S)...... 175. Hanf number computations...................................................... 226. Negative results for $L_{\omega }{1}_{\omega }$(S)...........................................

In this paper we present a topological duality for a certain subclass of the $F_{\omega }$-structures defined by M. M. Fidel, which conform to a non-standard semantics for the paraconsistent N. C. A. da Costa logic $C_{\omega }$. Actually, the duality introduced here is focused on $F_{\omega }$-structures whose supports are chains. For our purposes, we characterize every $F_{\omega }$-chain by means of a new structure that we will call (DCC) here. This characterization will allow us to prove the dual equivalence between the category...

T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the ${ℒ}_n^m$-propositional calculus, denoted by ${\ell }_n^m$, is introduced in terms of the binary connectives $\to$ (implication), $\twoheadrightarrow$ (standard implication), $\wedge$ (conjunction), $\vee$ (disjunction) and the unary ones $f$ (negation) and $D_i$, $1\le i\le n-1$ (generalized Moisil operators). It is proved that ${\ell }_n^m$ belongs to the class of standard systems...

CONTENTSIntroduction..............................................................................................................................................5I. Naive set theory.....................................................................................................................................61. The formal system................................................................................................................................62. Inversion and reduction...

Contents Introduction.................................................................................................................................................. 3 § 1. System $𝒮$ of a propositional calculus...................................................................... 4 § 2. System $𝒮*$..................................................................................................................... 5 § 3. $𝒮*$-algebras.....................................................................................................................

It is well known that, in forward inference in fuzzy logic, the generalized modus ponens is guaranteed by a functional inequality called the law of $T$-conditionality. In this paper, the $T$-conditionality for $T$-power based implications is deeply studied and the concise necessary and sufficient conditions for a power based implication $I^T$ being $T$-conditional are obtained. Moreover, the sufficient conditions under which a power based implication $I^T$ is $T^{*}$-conditional are discussed, this discussions...

CONTENTSIntroduction........................................................................................................................................................................................................................50. Terminology and preliminaries......................................................................................................................................................................................121. The extent of cut elimination by...

Ogni logica $L$ genera canonicamente la $L$-equivalenza ${\equiv }_L$e la $L$-immersione $\underset{𝐿}{\to }$ proprio come la logica del primo ordine genera l’equivalenza elementare $\equiv$ e l’immersione elementare $\lesssim$. Astraendo da $L$, è interessante studiare in sè relazioni d’equivalenza e di immersione generali tra strutture. Mostriamo che esiste una corrispondenza biunivoca tra relazioni d’equivalenza con la proprietà di Robinson e relazioni d’immersione con la proprietà di Amalgamazione Forte ($A{P}^{+}$). Caratterizziamo algebricamente...

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of $L_{\lambda ⁺\omega }$, with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with $⟨D^{ℳ},{\prec }^{ℳ}⟩$ a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where $⟨D^{{ℳ}^{\text{'}}},{\prec }^{{ℳ}^{\text{'}}}⟩$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{\lambda ⁺\omega }$ is exactly ${\beth }_{\delta \left(\lambda \right)}$. It was proved in [BK71] that if ℵ₀ < λ < κ$areregularcardinalnumbers,thenthereisaforcingextension,preservingcofinalities,suchthatintheextension$2λ = κ$and\delta \left(\lambda \right)<\lambda ⁺⁺.Weimprovethisresultbyprovingthefollowing:Suppose\aleph ₀<\lambda <\theta \le \kappa arecardinalnumberssuchthat$∙ ${\lambda }^{<\lambda }=\lambda$; ∙ cf(θ) ≥ λ⁺ and ${\mu }^{\lambda }<\theta$ whenever μ < θ; ∙ ${\kappa }^{\lambda }=\kappa$. Then there...

CONTENTS0. Introduction.......................................................................... 51. Preliminaries............................................................................... 72. Basic facts to be used in the sequel....................................... 113. Predicates OD(.,.) and CL(.,.).................................................... 174. Predicate Sels............................................................................. 185. Strong ${\sum }_n^1$-collection...........................................................

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence ${\varphi }_{ℳ}$ if ${\varphi }_{ℳ}$ has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if ${\aleph }_{\beta }$ is characterized by a Scott sentence, then $2^{{\aleph }_{\beta +\beta ₁}}$ is (homogeneously) characterized by a Scott sentence, for all 0 <...

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.