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"Predicate (Logic)"
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AN ALGEBRAIC PROOF OF COMPLETENESS FOR MONADIC FUZZY PREDICATE LOGIC \\( MMTL \\)
Monoidal t-norm based logic \\( MTL\\) is the weakest t-norm based residuated fuzzy logic, which is a \\([0,1]\\)-valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic \\( mMTL \\) that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic \\( MTL \\), which is indeed the predicate version of monoidal t-norm based logic \\( MTL\\). The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic \\( mMTL \\) and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic \\( mMTL \\) and S5-like fuzzy modal logic \\( S5(MTL)\\), we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic \\( mMTL \\), giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic \\( mMTL \\), and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic \\( mIMTL \\) via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.
Journal Article
CRAIG INTERPOLATION THEOREM FAILS IN BI-INTUITIONISTIC PREDICATE LOGIC
In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication
$\\phi \\rightarrow \\psi $
with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article is about the property more correctly named ‘deductive interpolation’ (see Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent.
Journal Article
Formal modelling of a sheet metal smart manufacturing system by using Petri nets and first-order predicate logic
This study introduces a developed method to a smart computer-aided design/manufacturing (CAD/CAM) system, where layout design, process planning, and comprehensive computerized numerical control (CNC) code generation can be implemented to satisfy laser cutting holes, tapping, irregular and complicated profile processing, engraving, and burr back-scraping. The smart CAD/CAM(SCAM) system is developed as a commercial software product or application and firstly applied to flexible sheet metal machining center (BGL 130R). In this study, a formal modeling method involving Petri nets and first-order predicate logic is proposed to develop the smart manufacturing system. High-level Petri nets are employed to achieve the formal application architecture design of data flow for various functions, and the first-order logic used to represent the process plan is defined and deduced according to the machining methods. The developed system possesses the following characteristics: (1) a sound and complete deductive system to implement various types of trajectory planning, automatic generation, and validation of the CNC code; (2) a convenient design input environment and readiness for re-design and modification by adding specific design functions and using standard design procedures on a widely used CAD/CAM package; (3) helpful for designers in sheet metal layout designing, layout interference detection, process planning validation, preprocess manufacturing operation of CNC code generation, and autodefinition of storable file names; and (4) formal and simple in human–computer interaction, automatic and intelligent in process operations, and satisfactory in terms of the requirements of the flexible sheet metal machining center (BGL 130R).
Journal Article
Logics of Organization Theory
Building theories of organizations is challenging: theories are partial and \"folk\" categories are fuzzy. The commonly used tools--first-order logic and its foundational set theory--are ill-suited for handling these complications. Here, three leading authorities rethink organization theory. Logics of Organization Theory sets forth and applies a new language for theory building based on a nonmonotonic logic and fuzzy set theory. In doing so, not only does it mark a major advance in organizational theory, but it also draws lessons for theory building elsewhere in the social sciences. Organizational research typically analyzes organizations in categories such as \"bank,\" \"hospital,\" or \"university.\" These categories have been treated as crisp analytical constructs designed by researchers. But sociologists increasingly view categories as constructed by audiences. This book builds on cognitive psychology and anthropology to develop an audience-based theory of organizational categories. It applies this framework and the new language of theory building to organizational ecology. It reconstructs and integrates four central theory fragments, and in so doing reveals unexpected connections and new insights.
eBook
A rendezvous approach for correcting accumulative errors of multiple robots
Wireless communication with no range and bandwidth limitations is desired for coordination and information sharing among multiple robots. However, the perfect communication is not available for a few of reasons. This paper proposed a simple yet effective scheme for correcting odometer errors existed in each robot of a multi-robot system. A contribution is that additional communication bandwidth is needed only if a rendezvous for two robots happened. Implementation of the error correction scheme is addressed in detail. Moreover, rendezvous is formulated by a set of predicate logic reasoning implications for each robot at upper level of soft architecture. The proposed approach was validated by computer simulations.
Journal Article
A semantic study of the first-order predicate logic with uncertainty involved
In this paper, we provide a semantic study of the first-order predicate logic for situations involving uncertainty. We introduce the concepts of uncertain predicate proposition, uncertain predicate formula, uncertain interpretation and degree of truth in the framework of uncertainty theory. Compared with classical predicate formula taking true value in
{
0
,
1
}
, the degree of truth of uncertain predicate formula may take any value in the unit interval
[
0
,
1
]
. We also show that the uncertain first-order predicate logic is consistent with the classical first-order predicate logic on some laws of the degree of truth.
Journal Article
The validity degree vectors of formulae in two-valued predicate logic
By means of infinite product of uniformly distributed probability spaces of cardinal
n
, the concept of
n
-validity degrees and validity degree vectors of formulae in two-valued predicate logic are introduced. It is proved that the validity degree vectors of formulae can preserve the logical relation between formulae. Moreover, a consistency theorem is obtained which says that the
n
-validity degree
τn(A
) of the quantifier-free first-order formula
A
without any repeated predicate symbols or terms is independent of the natural number
n
, and is a constant equal to the validity degree
τ
(
A
0
) of the corresponding proposition
A
0
in classical propositional logic.
Journal Article
A CONSTRUCTIVE INTERPRETATION OF THE LOGICAL CONSTANTS
Heyting’s intuitionistic predicate logic describes very general regularities observed in constructive mathematics. The intended meaning of the logical constants is clarified through Heyting’s proof interpretation. A re-evaluation of proof interpretation and predicate logic leads to the new constructive Basic logic properly contained in intuitionistic logic. We develop logic and interpretation simultaneously by an axiomatic approach. Basic logic appears to be complete. A brief historical overview shows that our insights are not all new.
Journal Article
Continuous first order logic and local stability
We develop continuous first order logic, a variant of the logic described by Chang and Keisler (1966). We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson’s logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.
Journal Article