Craig interpolation theorem for intuitionistic logic and extensions Part III

1977 ◽  
Vol 42 (2) ◽  
pp. 269-271 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Possible Worlds ◽  
Predicate Logic ◽  
Interpolation Property ◽  
Interpolation Theorem ◽  
Kripke Structure ◽  
Unary Predicate ◽  
The World ◽  
Image Position ◽  
Constant Domains ◽  
Positive Integers

This is a continuation of two previous papers by the same title [2] and examines mainly the interpolation property for the logic CD with constant domains, i.e., the extension of the intuitionistic predicate logic with the schemaIt is known [3], [4] that this logic is complete for the class of all Kripke structures with constant domains.Theorem 47. The strong Robinson consistency theorem is not true for CD.Proof. Consider the following Kripke structure with constant domains. The set S of possible worlds is ω0, the set of positive integers. R is the natural ordering ≤. Let ω0 0 = , Bn, is a sequence of pairwise disjoint infinite sets. Let L0 be a language with the unary predicates P, P1 and consider the following extensions for P,P1 at the world m.(a) P is true on ⋃i≤2nBi, and P1 is true on ⋃i≤2n+1Bi for m = 2n.(b) P is true on ⋃i≤2nBi, and P1 for ⋃i≤2n+1Bi for m = 2n.Let (Δ,Θ) be the complete theory of this structure. Consider another unary predicate Q. Let L be the language with P, Q and let M be the language with P1, Q.

A new version of Beth semantics for intuitionistic logic

1977 ◽  
Vol 42 (2) ◽  
pp. 306-308 ◽  
Author(s):  
Dov M. Gabbay
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Tree Structure ◽  
Truth Conditions ◽  
Truth Definition ◽  
Image Position ◽  
Constant Domains ◽  
Condition B

We use the notation of Kripke's paper [1]. Let M = (G, K, R) be a tree structure and D a domain and η a Beth model on M. The truth conditions of the Beth semantics for ∨ and ∃ are (see [1]):(a) η (A ∨ B, H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B, either η(A, H′) = T or η(B, H′) = T.(b) η(∃xA(x), H) = T iff for some B ⊆ K, B bars H and for each H′ ∈ B there exists an a ∈ D such that η(A (a), H′) = T.Suppose we change the truth definition η to η* by replacing condition (b) by the condition (b*) (well known from the Kripke interpretation):Call this type of interpretation the new version of Beth semantics. We proveTheorem 1. Intuitionistic predicate logic is complete for the new version of the Beth semantics.Since Beth structures are of constant domains, and in the new version of Beth semantics the truth conditions for ∧, →, ∃, ∀, ¬ are the same as for the Kripke interpretation, we get:Corollary 2. The fragment without disjunction of the logic CD of constant domains (i.e. with the additional schema ∀x(A ∨ B(x))→ A ∨ ∀xB(x), x not free in A) equals the fragment without disjunction of intuitionistice logic.

scholarly journals Failure of Interpolation in Constant Domain Intuitionistic Logic

2013 ◽  
Vol 78 (3) ◽  
pp. 937-950 ◽  
Author(s):  
Grigori Mints ◽  
Grigory Olkhovikov ◽  
Alasdair Urquhart
Keyword(s):  
Intuitionistic Logic ◽  
Interpolation Property ◽  
Interpolation Theorem ◽  
Constant Domain ◽  
Constant Domains

AbstractThis paper shows that the interpolation theorem fails in the intuitionistic logic of constant domains. This result refutes two previously published claims that the interpolation property holds.

A second paper “On the interpolation theorem for the logic of constant domains”

1983 ◽  
Vol 48 (3) ◽  
pp. 595-599 ◽  
Author(s):  
E.G.K. López-Escobar
Keyword(s):  
Interpolation Theorem ◽  
Cut Elimination ◽  
Property A ◽  
Formula One ◽  
Correct Rule ◽  
Tableau System ◽  
Image Position ◽  
The Right ◽  
Constant Domains ◽  
Subformula Property

It was brought to our attention by M. Fitting that Beth's semantic tableau system using the intuitionistic propositional rules and the classical quantifier rules produces a correct but incomplete axiomatization of the logic CD of constant domains. The formulawhere T is a truth constant, being an instance of a formula which is valid in all Kripke models with constant domains but which is not provable without cut.From the Fitting formula one can immediately obtain that the sequentalthough provable in the system GD outlined in [3], does not have a cut-free proof (in the system GD).If the only problem with GD were the sequent S0, then we could extend GD to the system GD+ by adding the following (correct) rule:Since the new rule still satisfies the subformula property a cut elimination theorem for GD+ would be a step in the right direction for a syntactical proof for the interpolation theorem for the logic of constant domains (cf. Gabbay [2]; see also §4). Unfortunately, one can show that the sequentwhere P is a propositional parameter (or formula without x free) has a derivation in GD+, but does not have a cut-free derivation (in GD+). Of course, we could extend GD+ to GD++ by adding the following correct (and with the subformula property) rule:But then we can find a sequent S2 which, although provable with cut in GD++, does not have a cut-free derivation in GD++.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

scholarly journals Monochromatic solutions to equations with unit fractions

1991 ◽  
Vol 43 (3) ◽  
pp. 387-392 ◽  
Author(s):  
Tom C. Brown ◽  
Voijtech Rödl
Keyword(s):  
Solutions To Equations ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

Our main result is that if G(x1, …, xn) = 0 is a system of homogeneous equations such that for every partition of the positive integers into finitely many classes there are distinct y1,…, yn in one class such that G(y1, …, yn) = 0, then, for every partition of the positive integers into finitely many classes there are distinct Z1, …, Zn in one class such thatIn particular, we show that if the positive integers are split into r classes, then for every n ≥ 2 there are distinct positive integers x1, x1, …, xn in one class such thatWe also show that if [1, n6 − (n2 − n)2] is partitioned into two classes, then some class contains x0, x1, …, xn such that(Here, x0, x2, …, xn are not necessarily distinct.)

Ibn al-Ṭiqṭaqā and the Ta'rīkh-i-Jakān-Gushāy of Juvaynī

1952 ◽  
Vol 14 (1) ◽  
pp. 175-177
Author(s):  
J. A. Boyle
Keyword(s):  
Direct Access ◽  
The World ◽  
Image Position

Had Ihn al-Ṭiqṭaqā direct access to the Ta'rīkh-i-Jahān-Gushāy ? He twice quotes it by name and there are two other passages for which it must have been his source; but the question is whether he made use of Juvaynī's work at first hand. That he knew Persian is evident from his twice quoting Persian verses. Of these the following are also in Juvaynī:“O king, what will come of (drinking) strong wine ?And what will come of continual drunkenness ?The king drunk, the world in rums and enemies before and behind— It is plain to see what will come of this! ”These verses were, according to Ibn al-Ṭiqṭaqā, addressed to Jalāl al-Dīn Khwārazmshāh by his court poet, whereas all that Juvaynī says is that they were composed by some unspecified person à propos of Jalāl al-Dīn's conduct. But the discrepancy is not an important one, and it seems not unreasonable to suppose that Ibn al-Ṭiqṭaqā copied these verses from Juvaynī and that, in consequence, he examined the latter's work in the original.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

Ternary Quadratic Forms and Eta Quotients

2015 ◽  
Vol 58 (4) ◽  
pp. 858-868 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Dedekind Eta Function ◽  
Ternary Quadratic Forms ◽  
Sum Formula ◽  
Image Position ◽  
Positive Integers

AbstractLet denote the Dedekind eta function. We use a recent productto- sum formula in conjunction with conditions for the non-representability of integers by certain ternary quadratic forms to give explicitly ten eta quotientssuch that the Fourier coefficients c(n) vanish for all positive integers n in each of infinitely many non-overlapping arithmetic progressions. For example, we show that if we have c(n) = 0 for all n in each of the arithmetic progressions

scholarly journals ¿El Brasil se encuentra aquí? La construcción de mundos posibles a través de la marca Brasil por jóvenes catalanes / Is Brazil Found Here? The construction of possible worlds through of the Brazil brand by young Catalans

2019 ◽  
Vol 6 (1) ◽  
pp. 15-22
Author(s):  
Adriano De Oliveira Sampaio ◽  
Inés Martins
Keyword(s):  
Discourse Analysis ◽  
Focus Group ◽  
Focus Groups ◽  
Possible Worlds ◽  
Brand Positioning ◽  
Advertising Campaign ◽  
The World

ABSTRACTThis article analyzes the sense production that a group of young Catalans came up with after seeing some advertising pieces about Brazil’s self-promotion campaign (2011-2014) abroad and after some posters’ presentations about the advertising campaign “Brazil. The world is here”. Discourse Analysis and focus groups were used as a technique to analyze and to collect the interviews. The results show us that the binomial similarity / difference - essential for a brand positioning construction - is practically non-existent in those campaigns, which causes a homogenization of them in several countries.RESUMENEl artículo se centra en el análisis sobre los significados que un grupo de jóvenes catalanes elaboran de las campañas turísticas de autopromoción de Brasil (2011-2014) en el exterior antes y después de la presentación de los carteles de la campaña publicitaria “Brasil. O mundo se encontra aquí”. Fueron utilizados el análisis del discurso y el focus-group como técnica de recolección de las entrevistas. Los resultados obtenidos nos han permitido comprobar que el binomio semejanza/diferencia –esencial para la construcción do posicionamiento de marcas- es prácticamente inexistente en las campañas analizadas, lo que provoca la homogenización de las campañas turísticas de diferentes países.

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