John Venn and logical theory

This paper proposes grounding for the business process modeling (BPM) based on general time theory providing axiomatic system. First order logic is used to give a clear definition of abstract business process and corresponding temporal relations including derived relations using a single “Meets” relation. Temporal logic used here treats time interval and time points on equal footing. We use model theoretic approach, in which abstract business process is represented as a formal system and mapped to an instance/concrete realization. Also, we used resolution theorem to provide its soundness and completeness properties. A Process temporal graph as a directed graph is introduced with graphical notation defined to represent the temporal knowledge. A real world realization of the corresponding graph is considered an instance of an abstract business process. Sound and completeness properties of the process temporal graph using reachability analysis. However, Arcs representing time elements, vertex representing the ‘Meets’ relation and also allows expression of both logical AND and OR.

The information in data depends on the subjective value system that the receiver of the data uses to interpret them. This paper looks at the information in a theory of first order logic (a knowledge base) from the perspective of a decision maker for whom the validation of formulae (facts and rules) have varying importance. The decision maker's preferences and prior knowledge are both incorporated into the information measure. The value of information is determined by what it conveys about the formulae of importance to the decision maker. The information measure is applied as a heuristic in commonsense reasoning; in relevance assessment ; and as a preference function in belief revision.

The paper is a study of the logic of existence, negation, and order in the Neoplatonic tradition. The central idea is that Neoplatonists assume a logic in which the existence predicate is a comparative adjective and in which monadic predicates function as scalar adjectives that nest the background order. Various scalar predicate negations are then identifiable with various Neoplatonic negations, including a privative negation appropriate for the lower orders of reality and a hyper-negation appropriate for the higher. Reversion to the One can then be explained as the logical inference of hyper-negations from mundane knowledge. Part I develops the relevant linguistic and logical theory, and Part II defends Wolfson and the scalar interpretation against the more traditional Aristotelian understanding of Whittaker and others of reversion as intensional abstraction

A new first-order logic, functional logic, was proposed recently by Staples, Robinson and Hazel. The logic provides a formal means of describing and reasoning about dependence on an implicit parameter, a prime motivation being the unification of the Hoare logic used to reason about procedural programs with the powerful and well established techniques of classical logic. Viewed more abstractly, independently of possible applications, functional logic may be described as a logic with primitives, axioms and inference rules appropriate for reasoning about the properties of mathematical functions. In this paper, the completeness of functional logic is proved; that is, it is shown that any term of a theory in the logic which is true in all models is a theorem.

1 Introduction Program schemata (also called schemes, or abstract programs) are widely known as devices for treating properties of computer programs which are invariant across a range of interpretatio ns. Since the intent of schemata is to formalize a specific aspect of effective computation, the properties studied are usually those which have been prescribed in other models of computation: e.g., totality, equivalence, decision power. It is perhaps for this reason that the methodology for dealing with schemata has evolved around constructive mathematics and direct demonstration rather than continuous mathematics, transfinite mathematics and indirect proof. Recently, however, it has been pointed out (Kfoury [3], DeMillo [2]) that useful properties of schemata can be established using arguments which rest on nonconstructi ve foundations; in this case the model theory of first order logic. The outcome of this activity has been to recast some deep questions regarding, for instance, what can and cannot be proved about computer programs into a form which is similar to the corresponding logical questions, where some of the answers are known. In this note,* we present three such applications of model theory. We allow a free wheeling notion of schemata (allowing, for example, first order oracles of any sort desired) and show that direct limitations exist in the ability to draw inferences about schemata by examining finite entailment and axiomatizability. The third application shows that even the expansive concept of effectiveness allowed here does not admit more relative decision power than the standard models of computation. 2 Notation Logical notation follows such standard sources as Bell and Slomson [1]. In particular, such notions as first order language, similarity type, relational structure, and interpretation are assumed in this presentation. A given first order language always talks about

Domain theory is the mathematical theory underlying denotational semantics. This thesis presents a formalization of domain theory in the Higher Order Logic (HOL) theorem proving system along with a mechanization of proof functions and other tools to support reasoning about the denotations of functional programs. By providing a fixed point operator for functions on certain domains which have a special undefined (bottom) element, this extension of HOL supports the definition of recursive functions which are not also primitive recursive. Thus, it provides an approach to the long-standing and important problem of defining non-primitive recursive functions in the HOL system.<br /> <br />Our philosophy is that there must be a direct correspondence between elements of complete partial orders (domains) and elements of HOL types, in order to allow the reuse of higher order logic and proof infrastructure already available in the HOL system. Hence, we are able to mix domain theoretic reasoning with reasoning in the set theoretic HOL world to advantage, exploiting HOL types and tools directly. Moreover, by mixing domain and set theoretic reasoning, we are able to eliminate almost all reasoning about the bottom element of complete partial orders that makes the LCF theorem prover, which supports a first order logic of domain theory, difficult and tedious to use. A thorough comparison with LCF is provided.<br /> <br />The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm. A major part of the proof is conducted in the set theoretic setting of higher order logic, and only at a late stage of the proof domain theory is introduced to give a recursive definition of the algorithm, which is not primitive recursive. Furthermore, a total well-founded recursive unification function can be defined easily in pure HOL by proving that the unification algorithm (defined in domain theory) always terminates; this proof is conducted by a non-trivial well-founded induction. In such applications, where non-primitive recursive HOL functions are defined via domain theory and a proof of termination, domain theory constructs only appear temporarily.

In logical function analysis, the analysis deals with clause complexes form which is the meaning relation of each clause in the clause complex. This thesis is about the logical function analysis used in the thread about vaccines in social media twitter. The objective of this research is to find out the types of logical function analysis in the threads about vaccines which is in clause complexes form. This research is conducted by qualitative descriptive approach which means this research makes the description of the situation. The data of this research is from the thread 1 about vaccine by Prof. Christine Pagel on her twitter account. The thread consists of 10 tweets that are 35 clause complexes. And the other data is from the thread 2 by Nida Qadir, MD on her twitter account. The thread consists of 5 tweets that are 21 clause complexes. The finding of this research is indicated that there are 2 main types of logical function; expansion and projection. In analyzing thread 1, there were 19 elaboration, 3 extension, 12 enhancement, and 1 types of idea. There is no locution type in thread 1. In analyzing the thread 2, there are 9 elaboration, 3 extension, 7, enhancement, 1 locution, and 1 idea.

Montague grammarians claim that the logical analysis of language sketched in standard textbooks of first order logic is too coarse to be appropriate to such natural languages as English. In the following we shall defend the opposite view - at least as far as a very narrow fragment of English is concerned, traditional logical analysis is perfectly adequate provided we are ready to pay the price of abandoning classical logic in favour of logics containing a sufficiently rich stock of logical constants. Within the framework of Casari's comparative logic, we shall outline a model of a restricted fragment of English. We shall address such issues as the construction of complex noun phrases, the distinction between gradable and vague adjectives, the structure of comparative sentences and of antonym adjectival pairs

Event-B method, as an evolution of B-method, is a formal method for system-level modelling and analysis based on extended first order logic and set theory, which provides flexible approaches of refinement and decomposition to construct large systems gradually with some successful applications in their formal verifications. However, it is hard to manipulate and grasp this method for many researchers because of its highly abstraction. In order to reduce the burden of developers’ work and the complexity of Event-B model, a graphic modeling approach, called Event-B graph, is introduced in this paper, which is used as an alternative way to clearly describe the state flow of the model and provide a graphic way of system-level modelling to bridge the real problem and the Event-B model construction. After introducing some new concepts and structure of Event-B graph, the transformation algorithm from Event-B graph to Event-B model is provided and the equivalence of the transformation is then proved. An example is finally provided to illustrate the procedure of Event-B graph construction.

At present, problems related to data mining are aris-ing in various poorly formalizable areas of research(genetics, archaeology, medicine, etc.). Such problemsare characterized by insufficient knowledge about theobjects under study, which hampers the development ofmathematical models for them; by a large number ofmultitype (quantitative and qualitative) factors and arelatively small amount of data; and by the requirementthat analysis results be represented in a form clear toexperts in applied areas.A promising approach to solving such problems isbased on the class of logical decision functions. Theyare frequently represented in the form of a decisiontree. A detailed description of this class and referencescan be found in [1].The construction of decision functions with a mini-mal risk of erroneous predictions is an important task inmethods based on logical decision functions and inother data analysis techniques. It is well known that thecomplexity of the class of decision functions (thecapacity characteristic of the class [2]) is a importantfactor affecting the quality of decisions. To achieve thebest quality, we need a certain tradeoff between thecomplexity of the class and the accuracy of decisionsobtained on the training sample. Thus, the problemarises of choosing an optimal complexity of the class.In numerous applications, empirical data (trainingsample) are supplemented with expert knowledge,which is not tightly related to a distribution model(expert knowledge can be specified as upper bounds forthe risk; they can express constraints on the class of dis-tributions or the class of decision functions, specifypreference rules used in decision making, etc.). Whenan optimal complexity of the class is chosen, availableempirical data and expert knowledge have to be jointlytaken into account.This problem can be solved within the framework ofthe Bayesian learning theory. The idea behind thisapproach is to use a priori knowledge about the prob-lem in question so that each possible distribution (strat-egy, the state of nature) is assigned a certain weight.This weight reflects the expert’s intuitive confidence inwhether the unknown true distribution coincides withthat under consideration.In this paper, we propose a Bayesian recognitionmodel based on a finite set of events. The model is usedto develop and analyze methods for constructing logi-cal decision functions. While choosing an optimal com-plexity of the class, the model takes into account empir-ical data and expert knowledge.In pattern recognition, the task is to predict the pat-tern’s index for an arbitrary object of the populationdescribed by a set of variables. The prediction is madeby analyzing a training sample consisting of the valuesof these variables, together with the index of the corre-sponding pattern for each object. The variables can beof different types; i.e., some of them can be quantita-tive, while the others can be qualitative. As a rule, theproblem is solved in a class of decision functions inwhich an optimal function is sought according to agiven criterion. The class of logical decision functionsis defined on a set of partitions of the feature space intoa finite number of subdomains described by conjunc-tions of simple predicates. The number of subdomainsdetermines the degree of complexity of a logical func-tion.A Bayesian pattern recognition model on a finite setof events is defined by formulating certain statementsthat avoid the local metric properties of the featurespace (the transition from points of the space to events,where an event is understood as that of the original vari-ables taking values from a certain subdomain of thepartition). Moreover, the pattern recognition problem isconsidered on the values of a discrete unordered vari-able, the concept of a learning method (a mapping from

The book on the Eastern Han Dynasty's ”Jinguiyaolue”- ”Golden Chamber” concise text difficult, zhushu were nearly a thousand, and their insight distinct. When we on the ”Golden Chamber” a careful examination of the language, you will find ”Golden Chamber” in the sentence is called a number of ”observation sentences”, ”Records of sentence” or ”question and answer period,” the statement, We rarely find someone from the ”name debate” or ”logical” perspective study ”Golden Chamber” of the reasoning and validity. In the main part, which focused on ”Golden Chamber” and ”Treatise on Febrile Diseases” The logic of the method, it would be to ”Logic” of vision research ”Golden Chamber” square cards on the rule system, research is no longer restricted to a logical theory.

In this paper, a comprehensive review is presented which is classified as analysis, design and optimization of current mode logic. Literature review reveals that amongst available logic design techniques used for mixed signal applications; current mode logic (CML) is most efficient logic design technique. Current-Mode Logic (CML) is most promising in reducing power consumption, switching/transient noise and also provides analog friendly environment. Thus, CML is a natural candidate for researchers to explore and implementing digital blocks in mixed-signal circuits. Further, operation of inverter/buffer in MCML and PFSCL style, basic terminologies, static and delay parameters related to MCML and PFSCL circuits are also presented in this paper. Realization approaches to implement various logic functions is described for both MCML and PFSCL styles. Static and dynamic variant of MCML and PFSCL circuits are simulated for performance comparison and their suitability in high performance logic designs.

This thesis consists of four independent papers. In the first paper, joint with Kechris, we study the global aspects of structurability in the theory of countable Borel equivalence relations. For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We show that K-structurability interacts well with various preorders commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K; in particular, we characterize the K such that every K-structurable equivalence relation is smooth. In the second paper, we consider the classes of Kn-structurable equivalence relations, where Kn is the class of n-dimensional contractible simplicial complexes. We show that every Kn-structurable equivalence relation Borel embeds into one structurable by complexes in Kn with the further property that each vertex belongs to at most Mn := 2n-1(n2+3n+2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n=1. In the third paper, we consider the amalgamation property from model theory in an abstract categorical context. A category is said to have the amalgamation property if every pushout diagram has a cocone. We characterize the finitely generated categories I such that in every category with the amalgamation property, every I-shaped diagram has a cocone. In the fourth paper, we prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic Lω1ω: every countable Lω1ω-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories (L,T) and (L',T'), every Borel functor Mod(L',T') → Mod(L,T) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L'ω1ω-interpretation of T in T', which generalizes a recent result of Harrison-Trainor, Miller, and Montalban in the case where T, T' are ℵ0-categorical.

The purpose of this note is to show that a known and natural four-valued logic co-exists with classical two-valued logic in the familiar context of truth tables. The tool required is the power construction. The context of this paper is that of 2-valued truth tables in n propositional variables, say p1, p2, . . . , pn. The two classical truth values are 0 and 1. The truth value assignments (or valuations) are the 2 {0, 1}-vectors of length n. Propositional formulae are freely generated from the propositional variables by application of the usual classical connectives ∼ (not), & (and) and ∨ (or), each of which is defined by a given truth table. Accordingly, each propositional formula has a truth table. Formulae with the same truth table are logically equivalent, and we do not distinguish between them. The valuations form a Boolean algebra Val = (2,∧,∨, ′ ,0,1), where the Boolean operations ∧, ∨ and ′ are defined componentwise from their counterparts in the 2-element Boolean algebra 2 = {0, 1}, and 0 and 1 are respectively the vectors (0, 0, . . . , 0) and (1, 1, . . . , 1) of length n. The elements of 2 are denoted by x,y,z, . . . Up to equivalence, formulae may be regarded as truth functions from 2 to 2, and the connectives are characterised by operations in the image algebra 2. That is, if bool(A,−) : 2 → 2 is the truth function corresponding to a propositional formula A, then bool (∼A,x) = bool (A,x)′ , bool (A & B,x) = bool (A,x) ∧ bool (B,x) , (1) bool (A ∨ B,x) = bool (A,x) ∨ bool (B,x) . 1991 Mathematics Subject Classification: 03B46, 03B05. The paper is in final form and no version of it will be published elsewhere.

Semantic Analyses for Dyadic Deontic Logic.- Some Remarks Concerning Many-Valued Propositional Logics.- Conditional Obligation.- Remarks on Interpersonal Utility Theory.- On the Proper Treatment of Quantifiers in Montague Semantics.- Extracting Information from Logical Proofs.- A New Approach to the Logical Theory of Actions and Causality.- Some Basic Concepts of Action.- Some Remarks Concerning Logical and Ontological Theories.- Combined Evidence.- Solution to a Problem Raised by Stig Kanger and a Set Theoretical Statement Equivalent to the Axiom of Choice.- On Characterizing Elementary Logic.- Rules and Derived Rules.- A Program for Pragmatics.- Models.- Remarks on Logic and Probability.- Analytic and Synthetic Arithmetical Statements.- Index of Names.- Tabula Gratulatoria.