Sequent-systems for modal logic

1985 ◽  
Vol 50 (1) ◽  
pp. 149-168 ◽  
Author(s):  
Kosta Došen
Keyword(s):  
Order Logic ◽  
Philosophical Logic ◽  
Logical Constants ◽  
First Order ◽  
Sequent Systems ◽  
Left And Right ◽  
Level 1 ◽  
The Right ◽  
Level 2

AbstractThe purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 2 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else).This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1.These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical.

Second-order logic, philosophical issues in

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Practice ◽  
Expressive Power ◽  
Deductive System ◽  
Second Order ◽  
Order Logic ◽  
Mathematical Structures ◽  
First Order ◽  
Correct Reasoning ◽  
Second Order Logic

Typically, a formal language has variables that range over a collection of objects, or domain of discourse. A language is ‘second-order’ if it has, in addition, variables that range over sets, functions, properties or relations on the domain of discourse. A language is third-order if it has variables ranging over sets of sets, or functions on relations, and so on. A language is higher-order if it is at least second-order. Second-order languages enjoy a greater expressive power than first-order languages. For example, a set S of sentences is said to be categorical if any two models satisfying S are isomorphic, that is, have the same structure. There are second-order, categorical characterizations of important mathematical structures, including the natural numbers, the real numbers and Euclidean space. It is a consequence of the Löwenheim–Skolem theorems that there is no first-order categorical characterization of any infinite structure. There are also a number of central mathematical notions, such as finitude, countability, minimal closure and well-foundedness, which can be characterized with formulas of second-order languages, but cannot be characterized in first-order languages. Some philosophers argue that second-order logic is not logic. Properties and relations are too obscure for rigorous foundational study, while sets and functions are in the purview of mathematics, not logic; logic should not have an ontology of its own. Other writers disqualify second-order logic because its consequence relation is not effective – there is no recursively enumerable, sound and complete deductive system for second-order logic. The deeper issues underlying the dispute concern the goals and purposes of logical theory. If a logic is to be a calculus, an effective canon of inference, then second-order logic is beyond the pale. If, on the other hand, one aims to codify a standard to which correct reasoning must adhere, and to characterize the descriptive and communicative abilities of informal mathematical practice, then perhaps there is room for second-order logic.

A phase IB trial of 5-azacitidine (5AC) and suberoylanilide hydroxamic acid (SAHA) in patients with metastatic or locally recurrent nasopharyngeal carcinoma (NPC) and NK-T cell lymphoma.

2013 ◽  
Vol 31 (15_suppl) ◽  
pp. e17017-e17017
Author(s):  
Wen Son Hsieh ◽  
Eng Huat Tan ◽  
Wan-Teck Lim ◽  
Ross A. Soo ◽  
Anthony T. C. Chan ◽  
...  
Keyword(s):  
Gene Expression ◽  
Viral Load ◽  
Dose Level ◽  
Clinical Benefit ◽  
Tumor Tissue ◽  
Level 3 ◽  
Level 1 ◽  
Grade 3 ◽  
Level 2

e17017 Background: Epigenetic up-regulation of EBV and cellular genes via demethylation and histone deacetylase inhibition can induce EBV lytic replication enhancing immune mediated tumor killing and up-regulation of tumor suppressor genes resulting in tumor apoptosis. Methods: Patients (Pt) with relapsed or refractory NPC and NK-T cell lymphomas were enrolled to determine safety, tolerability, pharmacokinetics (PK), pharmacodynamics and preliminary anti-tumor activity using a dose escalation design. 5AC was administered on days 1 to 10 sub-cutaneously while SAHA was administered on days 1 to 14 orally. PK for SAHA, EBV viral load, characterization of circulating EBV, Immunohistochemistry (IHC) and EBV promoter methylation analysis in tumor tissue were performed. Results: 11 pt have been treated (M:F 8:3, median age 48, R: 35-71) at 3 dose levels – 5AC 50 mg/m2 and SAHA 200 mg b.i.d. (dose level 1), 5AC 37.5 mg/m2 and SAHA 200 mg q. am and 100 mg q. pm (dose level 2), and 5AC 25 mg/m2 and SAHA 100 mg b.i.d (dose level 3). Median number of previous treatment regimens was 3 (R:1-6). Dose limiting toxicities (DLT) were seen in 2/2 pts at dose level 1: grade 4 thrombocytopenia (1 pt), grade 3 nausea, vomiting and fatigue (2 pts), and grade 5 hepatic failure (1 pt). Two of six patients at dose level 2 experienced DLT: grade 3 fatigue (1 pt) and worsening of pre-existing Sweet’s Syndrome (1 pt). Common AEs (G1/2) included fatigue (73%), cough (64%), anorexia (55%), and injection site reaction (45%). One minor response was seen and 5 pt had prolonged stable disease (>16 weeks), including one patient for 88 weeks. Analysis of post-treatment tumor biopsies showed demethylation of EBV lytic cycle gene promoters after treatment. SAHA PK, IHC results for EBV gene expression in tumor tissue, EBV viral load and characterization of circulating EBV will be presented. Conclusions: 5AC/SAHA appears to be tolerable at dose level 3 with suggestion of clinical benefit. Analysis of post-treatment tumor and blood samples suggests that modulation of EBV gene expression may play a role in the mechanism underlying clinical benefit. Continued accrual at dose level 3 is ongoing. Clinical trial information: NCT00336063.

scholarly journals Morpho-phonological levels and grammaticalization in Karimojong: A review of the evidence

2012 ◽  
Vol 41 (1) ◽  
pp. 99-169
Author(s):  
Diane Lesley- Lesley-Neuman
Keyword(s):  
Phonological Processes ◽  
Linguistic Structure ◽  
Surface Irregularities ◽  
Level 3 ◽  
Level 1 ◽  
The Right ◽  
Level 2 ◽  
Different Levels ◽  
Phonological Rules ◽  
Verbal Complex

An analysis of the affixation processes and the phonological rules governing [ATR] harmony in Karimojong verbs permits the organization of derivational processes into three levels. On each level specific groups of morphemes are affixed and the resultant derived words undergo defined sets of phonological processes. On Level 1 a feature filling [±ATR] harmony rule applies, in which the [ATR] feature spreads bi-directionally from the principal root vowel across the root and all Level 1 affixes. However, there are also localized disharmonic domains created by phonologized co-articulation effects of consonants and dissimilation rules for vowels. On Level 2, suffixation of Tense-Mood-Aspect (TMA) markers at the right edge of the verbal complex triggers [+ATR] feature spreading leftward across the derived word. On Level 3 affixation does not trigger [ATR] harmony processes, and both affix vowels and the derivational complexes to which they are added retain their [ATR] features. Some Karimojong affixes exhibit behaviors characteristic of two different levels, depending on context; these affixes are proposed to be in transition between levels. The three levels are proposed to result from diachronic evolution, and their relative chronological development can be established by (1) correspondences to landmarks within the cross-linguistically attested agreement system grammaticalization cline, which groups affixes according to ordering universals, and (2) the existence of successive evolutionary cycles of frequentive morphology. A model of the morphology-phonology interface is proposed in which linguistic structure internal to the morpheme channels the spread of [ATR] features. The model provides an explanation for surface irregularities that originated at the time of areal vowel mergers in Nilotic languages (Dimmendaal, 2002).

scholarly journals K-MEANS CLUSTERING UNTUK PEMETAAN DAERAH RAWAN DEMAM BERDARAH

2019 ◽  
Vol 7 (1) ◽  
Author(s):  
Suprihatin Suprihatin ◽  
Yustina Retno Wahyu Utami ◽  
Didik Nugroho
Keyword(s):  
Hemorrhagic Fever ◽  
Flow Diagram ◽  
Literature Study ◽  
System Use ◽  
Level 3 ◽  
Data Flow Diagram ◽  
Level 1 ◽  
The Right ◽  
Level 2 ◽  
Collection Methods

District Nogosari is one of the dengue-prone areas in Boyolali District. During the period of 2012 to 2014, there was a significant increase in dengue cases at Boyolali district. For the reasons above, the study is focused on how to cluster Areas DHF-Prone using K-Means method. Clustering is based on the parameter of the number of dengue cases in the sporadic and endemic zones. There are several types of data collection methods that include: observation, interview, and literature study. Design of this proposed system use modeling language the context diagram and data flow diagram. The system is implemented using PHP Programming Language and MYSQL database. This system cluster 3 level zones of Endemic and 3 level zones of Sporadic based on geographic information systems. The result of system testing using the silhouette coefficient on the sporadic zone is the average coefficient for level 1 is 0.837, level 2 is 0.858, and level 3 is 0.773 that means the object has been in the right group. The proposed system is expected to be a consideration in preventing, controlling and eradicating dengue hemorrhagic fever.Keywords: Endemic, Sporadic, K-Means clustering, Dengue hemorrhagic fever.

SKEPTICAL ABDUCTION

1993 ◽  
Vol 02 (04) ◽  
pp. 511-540 ◽  
Author(s):  
P. MARQUIS
Keyword(s):  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Real Problem ◽  
Logical Framework ◽  
First Order ◽  
Selective Generation ◽  
The Given ◽  
Reasonable Prospect

Abduction is the process of generating the best explanation as to why a fact is observed given what is already known. A real problem in this area is the selective generation of hypotheses that have some reasonable prospect of being valid. In this paper, we propose the notion of skeptical abduction as a model to face this problem. Intuitively, the hypotheses pointed out by skeptical abduction are all the explanations that are consistent with the given knowledge and that are minimal, i.e. not unnecessarily general. Our contribution is twofold. First, we present a formal characterization of skeptical abduction in a logical framework. On this ground, we address the problem of mechanizing skeptical abduction. A new method to compute minimal and consistent hypotheses in propositional logic is put forward. The extent to which skeptical abduction can be mechanized in first—order logic is also investigated. In particular, two classes of first-order formulas in which skeptical abduction is effective are provided. As an illustration, we finally sketch how our notion of skeptical abduction applies as a theoretical tool to some artificial intelligence problems (e.g. diagnosis, machine learning).

Turing machines and the spectra of first-order formulas

1974 ◽  
Vol 39 (1) ◽  
pp. 139-150 ◽  
Author(s):  
Neil D. Jones ◽  
Alan L. Selman
Keyword(s):  
Order Logic ◽  
Automata Theory ◽  
Turing Machines ◽  
First Order ◽  
Context Sensitive ◽  
Open Questions ◽  
Special Cases ◽  
Computational Aspects ◽  
The Difference

H. Scholz [11] defined the spectrum of a formula φ of first-order logic with equality to be the set of all natural numbers n for which φ has a model of cardinality n. He then asked for a characterization of spectra. Only partial progress has been made. Computational aspects of this problem have been worked on by Gunter Asser [1], A. Mostowski [9], and J. H. Bennett [2]. It is known that spectra include the Grzegorczyk class and are properly included in . However, no progress has been made toward establishing whether spectra properly include , or whether spectra are closed under complementation.A possible connection with automata theory arises from the fact that contains just those sets which are accepted by deterministic linear-bounded Turing machines (Ritchie [10]). Another resemblance lies in the fact that the same two problems (closure under complement, and proper inclusion of ) have remained open for the class of context sensitive languages for several years.In this paper we show that these similarities are not accidental—that spectra and context sensitive languages are closely related, and that their open questions are merely special cases of a family of open questions which relate to the difference (if any) between deterministic and nondeterministic time or space bounded Turing machines.In particular we show that spectra are just those sets which are acceptable by nondeterministic Turing machines in time 2cx, where c is constant and x is the length of the input. Combining this result with results of Bennett [2], Ritchie [10], Kuroda [7], and Cook [3], we obtain the “hierarchy” of classes of sets shown in Figure 1. It is of interest to note that in all of these cases the amount of unrestricted read/write memory appears to be too small to allow diagonalization within the larger classes.

Towards a characterization of order-invariant queries over tame graphs

2009 ◽  
Vol 74 (1) ◽  
pp. 168-186 ◽  
Author(s):  
Michael Benedikt ◽  
Luc Segoufin
Keyword(s):  
Linear Order ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Finite Graphs ◽  
Bounded Treewidth ◽  
First Order ◽  
A New Technique ◽  
Order Invariant

AbstractThis work deals with the expressive power of logics on finite graphs with access to an additional “arbitrary” linear order. The queries that can be expressed this way are the order-invariant queries for the logic. For the standard logics used in computer science, such as first-order logic, it is known that access to an arbitrary linear order increases the expressiveness of the logic. However, when we look at the separating examples, we find that they have satisfying models whose Gaifman Graph is complex – unbounded in valence and in treewidth. We thus explore the expressiveness of order-invariant queries over well-behaved graphs. We prove that first-order order-invariant queries over strings and trees have no additional expressiveness over first-order logic in the original signature. We also prove new upper bounds on order-invariant queries over bounded treewidth and bounded valence graphs. Our results make use of a new technique of independent interest: the application of algebraic characterizations of definability to show collapse results.

Finite Axiomatizability using additional predicates

1958 ◽  
Vol 23 (3) ◽  
pp. 289-308 ◽  
Author(s):  
W. Craig ◽  
R. L. Vaught
Keyword(s):  
Order Logic ◽  
The Other ◽  
First Order Logic ◽  
Logical Constant ◽  
Finite Axiomatizability ◽  
Logical Constants ◽  
First Order

By a theory we shall always mean one of first order, having finitely many non-logical constants. Then for theories with identity (as a logical constant, the theory being closed under deduction in first-order logic with identity), and also likewise for theories without identity, one may distinguish the following three notions of axiomatizability. First, a theory may be recursively axiomatizable, or, as we shall say, simply, axiomatizable. Second, a theory may be finitely axiomatizable using additional predicates (f. a.+), in the syntactical sense introduced by Kleene [9]. Finally, the italicized phrase may also be interpreted semantically. The resulting notion will be called s. f. a.+. It is closely related to the modeltheoretic notion PC introduced by Tarski [16], or rather, more strictly speaking, to PC∩ACδ.For arbitrary theories with or without identity, it is easily seen that s. f. a.+ implies f. a.+ and it is known that f. a.+ implies axiomatizability. Thus it is natural to ask under what conditions the converse implications hold, since then the notions concerned coincide and one can pass from one to the other.Kleene [9] has shown: (1) For arbitrary theories without identity, axiomatizability implies f. a.+. It also follows from his work that : (2) For theories with identity which have only infinite models, axiomatizability implies f. a.+.

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