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1
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On the completeness of decomposable properties of graphs

100%
Hałuszczak M., Vateha P.
Discussiones Mathematicae Graph Theory
|
1999
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tom 19
|
nr 2
229-236
EN
Let 𝓟₁,𝓟₂ be additive hereditary properties of graphs. A (𝓟₁,𝓟₂)-decomposition of a graph G is a partition of E(G) into sets E₁, E₂ such that induced subgraph $G[E_i]$ has the property $𝓟_i$, i = 1,2. Let us define a property 𝓟₁⊕𝓟₂ by {G: G has a (𝓟₁,𝓟₂)-decomposition}. A property D is said to be decomposable if there exists nontrivial additive hereditary properties 𝓟₁, 𝓟₂ such that D = 𝓟₁⊕𝓟₂. In this paper we determine the completeness of some decomposable properties and we characterize the decomposable properties of completeness 2.
2
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On some properties of Musielak-Orlicz sequence spaces

100%
Shragin I. V.
Commentationes Mathematicae
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2008
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tom 48
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nr 2
EN
We consider a nontrivial vector space \(X\) and a semimodular \(M\colon X\tp [0, \infty]\) with property: \((\forall\ x \in X) (\exists\ \alpha \gt 0)\ M (\alphax) \lt \infty\) (in other words, \(M\) is normal (i.e. \((\forall\ x\in X \setminus \{0\}) (\exists \alpha \gt 0)\ M (\alphax) \gt 0)\) pregenfunction). The function \(M\) generates in \(X\) a metric \(d\) with \[ d(x, y) := inf \{a \gt 0: M (a^{-1} (x-y)) \leq a\}. \] At the same time \(M\) generates a metric \(\rho\) in Musielak-Orlicz sequence space \(l_M\), namely \[ \rho(\varphi, \psi) := inf \{a \gt 0 : I(a^{-1} (\varphi - \psi)) \leq a\} \] with \(I(\varphi) = \sum_{n \geq 1} M (\varphiφ(n))\). It is proved that the space \((l_M,\rho)\) is complete if and only if the space \((X, d)\) is complete. We consider also the closed subspace \(G_M \subset l_M\) of sequences \(\varphi = \{\varphi(n)\}\) such that \((\forall \alpha \gt 0) (\exists m \in N) \sum_{n\geq m} M(\alpha\varphi(n)) \lt \infty\) and prove that \((G_M ,\rho)\) is separable if and only if \((X, d)\) is the same. Several examples are considered.
3
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The Axiomatization of Propositional Logic

100%
Giero M.
Formalized Mathematics
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2016
|
tom 24
|
nr 4
281-290
EN
This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes α ⇒ (β ⇒ α), (α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)), (¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ β). Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved. The proof of the completeness theorem is carried out by a counter-model existence method. In order to prove the completeness theorem, Lindenbaum’s Lemma is proved. Some most widely used tautologies are presented.
4
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Propositional Linear Temporal Logic with Initial Validity Semantics1

100%
Giero M.
Formalized Mathematics
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2015
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tom 23
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nr 4
379-386
EN
In the article [10] a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of “until” operator in a very strict version. The very strict “until” operator enables to express all other temporal operators. In this article we construct a formal system for LTLB with the initial semantics [12]. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [10] and the one introduced in this article. Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in [10] and in this article can be used to carry out such verifications in Mizar [4].
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Completeness, Categoricity and Imaginary Numbers: The Debate on Husserl

88%
Aranda V.
Bulletin of the Section of Logic
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2020
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tom 49
|
nr 2
EN
Husserl's two notions of "definiteness" enabled him to clarify the problem of imaginary numbers. The exact meaning of these notions is a topic of much controversy. A "definite" axiom system has been interpreted as a syntactically complete theory, and also as a categorical one. I discuss whether and how far these readings manage to capture Husserl's goal of elucidating the problem of imaginary numbers, raising objections to both positions. Then, I suggest an interpretation of "absolute definiteness" as semantic completeness and argue that this notion does not suffice to explain Husserl's solution to the problem of imaginary numbers.
6
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Identity, Equality, Nameability and Completeness

63%
Manzano M., Moreno M. C.
Bulletin of the Section of Logic
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2017
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tom 46
|
nr 3/4
EN
This article is an extended promenade strolling along the winding roads of identity, equality, nameability and completeness, looking for places where they converge. We have distinguished between identity and equality; the first is a binary relation between objects while the second is a symbolic relation between terms. Owing to the central role the notion of identity plays in logic, you can be interested either in how to define it using other logical concepts or in the opposite scheme. In the first case, one investigates what kind of logic is required. In the second case, one is interested in the definition of the other logical concepts (connectives and quantifiers) in terms of the identity relation, using also abstraction. The present paper investigates whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic a reliable definition of identity is possible. However, the definition needs the standard semantics and we know that with this semantics completeness is lost. We have also studied the relationship of equality with comprehension and extensionality and pointed out the relevant role played by these two axioms in Henkin’s completeness method. We finish our paper with a section devoted to general semantics, where the role played by the nameable hierarchy of types is the key in Henkin’s completeness method.
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Identity, equality, nameability and completeness. Part II

63%
Manzano M., Moreno M. C.
Bulletin of the Section of Logic
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2018
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tom 47
|
nr 3
EN
This article is a continuation of our promenade along the winding roads of identity, equality, nameability and completeness. We continue looking for a place where all these concepts converge. We assume that identity is a binary relation between objects while equality is a symbolic relation between terms. Identity plays a central role in logic and we have looked at it from two different points of view. In one case, identity is a notion which has to be defined and, in the other case, identity is a notion used to define other logical concepts. In our previous paper, [16], we investigated whether identity can be introduced by definition arriving to the conclusion that only in full higher-order logic with standard semantics a reliable definition of identity is possible. In the present study we have moved to modal logic and realized that here we can distinguish in the formal language between two different equality symbols, the first one shall be interpreted as extensional genuine identity and only applies for objects, the second one applies for non rigid terms and has the characteristic of synonymy. We have also analyzed the hybrid modal logic where we can introduce rigid terms by definition and can express that two worlds are identical by using the nominals and the @ operator. We finish our paper in the kingdom of identity where the only primitives are lambda and equality. Here we show how other logical concepts can be defined in terms of the identity relation. We have found at the end of our walk a possible point of convergence in the logic Equational Hybrid Propositional Type Theory (EHPTT), [14] and [15].
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