Modus Ponens Rule Research Articles

Properties of implication in effect algebras

Abstract Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain “unsharp” residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

Set Theory Inc# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.Ii) Hyper Inductive Definitions [Enter Paper Title

Set Theory Inc# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.Ii) Hyper Inductive Definitions [Enter Paper Title

Resolution as intersection subtyping via Modus Ponens

Resolution and subtyping are two common mechanisms in programming languages. Resolution is used by features such as type classes or Scala-style implicits to synthesize values automatically from contextual type information. Subtyping is commonly used to automatically convert the type of a value into another compatible type. So far the two mechanisms have been considered independently of each other. This paper shows that, with a small extension, subtyping with intersection types can subsume resolution. This has three main consequences. Firstly, resolution does not need to be implemented as a separate mechanism. Secondly, the interaction between resolution and subtyping becomes apparent. Finally, the integration of resolution into subtyping enables first-class (implicit) environments. The extension that recovers the power of resolution via subtyping is the modus ponens rule of propositional logic. While it is easily added to declarative subtyping, significant care needs to be taken to retain desirable properties, such as transitivity and decidability of algorithmic subtyping, and coherence. To materialize these ideas we develop λiMP, a calculus that extends a iprevious calculus with disjoint intersection types, and develop its metatheory in the Coq theorem prover.

Equality propositional logic and its extensions

We introduce a new formal logic, called equality propositional logic. It has two basic connectives, $boldsymbol{wedge}$ (conjunction) and $equiv$ (equivalence). Moreover, the $Rightarrow$ (implication) connective can be derived as $ARightarrow B:=(Aboldsymbol{wedge}B)equiv A$. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such as Modus Ponens(MP) rule, Hypothetical Syllogism(HS) rule and completeness, etc. Especially, we provide two ways to prove the completeness of this logic system. We also introduce two extensions of equality propositional logic. The first one is involutive equality propositional logic, which is equality propositional logic with double negation. The second one adds prelinearity which is rich enough to enjoy the strong completeness property. Finally, we introduce additional connective $Delta$(delta) in equality propositional logic and demonstrate that the resulting logic holds soundness and completeness.

Конечная аксиоматизируемость квазинормальных модальных логик

Квазиномальными модальными логиками называют логики в модальном языке, которые содержат логику ${\bf K}$, замкнуты по правилу modus ponens и для которых не постулирована
 замкнутость относительно правила Гёделя. До последнего времени этим логикам уделялось мало внимания, несмотря на то, что среди первых систем модальных логик, сформулированных К. И. Льюисом, содержались и квазинормальные логики. Здесь мы рассмотрим вопрос о конечной аксиоматизируемости квазинормальных модальных логик.
 
 Как известно, квазинормальный напарник логики ${\bf K}$ не имеет конечной аксиоматизации. Кроме того, существуют и другие модальные нормальные конечно-аксиоматизируемые логики, квазинормальные напарники которых не имеют конечной аксиоматизации, например логика ${\bf D}$. Поэтому вопрос о конечной аксиоматизируемости той или иной модальной квазинормальной логики нетривиален.
 
 Отметим, что известные частные критерии конечной аксиоматизируемости квазинормальных логик сформулированы только для квазинормальных напарников нормальных модальных логик.
 
 В данной работе получено обобщение этих частных критериев на случай произвольных квазинормальных модальных логик, попутно указана возможная аксиоматизация этих логик. Таким образом, получен частный критерий конечной аксиоматизируемости, общий как для квазинормальных напарников нормальных логик, так и для квазинормальных логик, которые таковыми не являются.
 
 Также в работе приведен алгоритм, который по относительной аксиоматизации квазинормальной логики $L$ над квазинормальным вариантом логики ${\bf K}$ дает абсолютную аксиоматизацию логики $L$.
 
 Отдельно рассмотрены аксиоматизации расширений логики ${\bf K4}$. Сформулирован частный критерий конечной аксиоматизируемости расширений этой логики. Приведен алгоритм, который по относительной аксиоматизации квазинормальной логики $L$ над квазинормальным вариантом логики ${\bf K4}$ дает абсолютную аксиоматизацию логики $L$.

Logics with Impossibility as the Negation and Regular Extensions of the Deontic Logic D2

In [1] J.-Y. Bèziau formulated a logic called Z. Bèziau’s idea was generalized independently in [6] and [7]. A family of logics to which Z belongs is denoted in [7] by K. In particular; it has been shown in [6] and [7] that there is a correspondence between normal modal logics and logics from the class K. Similar; but only partial results has been obtained also for regular logics (see [8] and [9]). In (Došen; [2]) a logic N has been investigated in the language with negation; implication; conjunction and disjunction by axioms of positive intuitionistic logic; the right-to-left part of the second de Morgan law; and the rules of modus ponens and contraposition. From the semantical point of view the negation used by Došen is the modal operator of impossibility. It is known this operator is a characteristic of the modal interpretation of intuitionistic negation (see [3; p. 300]). In the present paper we consider an extension of N denoted by N+. We will prove that every extension of N+ that is closed under the same rules as N+; corresponds to a regular logic being an extension of the regular deontic logic D21 (see [4] and [13]). The proved correspondence allows to obtain from soundnesscompleteness result for any given regular logic containing D2, similar adequacy theorem for the respective extension of the logic N+.

Constructing strict left (right)-disjunctive left (right) semi-uninorms and coimplications satisfying the order property

In this paper, we further study the constructions of left (right) semi-uninorms and coimplications on a complete lattice. We firstly give out the formulas for calculating the upper and lower approximation strict left (right)-disjunctive left (right) semi-uninorms of a binary operation. Then, we lay out the formulas for calculating the upper and lower approximation coimplications, which satisfy the order property, of a binary operation. Finally, we investigate the relationships between the lower approximation strict left (right)-disjunctive left (right) arbitrary ∧-distributive left (right) semi-uninorms and upper approximation right arbitrary ∨-distributive coimplications which satisfy the order property, and give some conditions such that the upper approximation strict left (right)-disjunctive left (right) semi-uninorms of a binary operation and lower approximation coimplication, which satisfies the order property, of the left (right) deresiduum of the binary operation satisfy the generalized dual modus ponens rule.

Undecidable problems for propositional calculi with implication

In this article, we deal with propositional calculi over a signature containing the classical implication $\to$ with the rules of modus ponens and substitution. For these calculi we consider few recognizing problems such as recognizing derivations, extensions, completeness, and axiomatizations. The main result of this paper is to prove that the problem of recognizing extensions is undecidable for every propositional calculus, and the problems of recognizing axiomatizations and completeness are undecidable for propositional calculi containing the formula $x \to ( y \to x )$. As a corollary, the problem of derivability of a fixed formula $A$ is also undecidable for all $A$. Moreover, we give a historical survey of related results.

Undecidability of the problem of recognizing axiomatizations for propositional calculi with implication

In this paper we consider propositional calculi, which are finitely axiomatizable extensions of intuitionistic implicational propositional calculus together with the rules of modus ponens and substitution. We give a proof of undecidability of the following problem for these calculi: whether a given finite set of propositional formulas constitutes an adequate axiom system for a fixed propositional calculus. Moreover, we prove the same for the following restriction of this problem: whether a given finite set of theorems of a fixed propositional calculus derives all theorems of this calculus. The proof of these results is based on a reduction of the undecidable halting problem for the tag systems introduced by Post.

Undecidability of the Problem of Recognizing Axiomatizations of Superintuitionistic Propositional Calculi

We give a new proof of the following result (originally due to Linial and Post): it is undecidable whether a given calculus, that is a finite set of propositional formulas together with the rules of modus ponens and substitution, axiomatizes the classical logic. Moreover, we prove the same for every superintuitionistic calculus. As a corollary, it is undecidable whether a given calculus is consistent, whether it is superintuitionistic, whether two given calculi have the same theorems, whether a given formula is derivable in a given calculus. The proof is by reduction from the undecidable halting problem for the so-called tag systems introduced by Post. We also give a historical survey of related results.

The algebra of Fourier-dual operations: Logic with exception

In the development of an approach to the implementation of fuzzy logics on neural networks with interconnections via the scheme of Fourier holography, a model of logic with exception is proposed, which is associated with the basic Generalized Modus Ponens rule. The exception is recalled from the associative memory by an inference formed by the basic rule and it modifies the original inference. The results of numerical simulation are based on the experimental data.

The simplest protoalgebraic logic

The logic is the sentential logic defined in the language with just implication → by the axiom of reflexivity or identity “” and the rule of Modus Ponens “from φ and to infer ψ”. The theorems of this logic are exactly all formulas of the form . We argue that this is the simplest protoalgebraic logic, and that in it every set of assumptions encodes in itself not only all its consequences but also their proofs. In this paper we study this logic from the point of view of abstract algebraic logic, and in particular we use it as a relatively natural counterexample to settle some open problems in this theory. It appears that this logic has almost no properties: it is neither equivalential nor weakly algebraizable; it does not have an algebraic semantics; it does not satisfy any form of the Deduction Theorem, other than the most general parameterized and local one that all protoalgebraic logics satisfy; it is not filter-distributive; and so on. It satisfies some forms of the interpolation property but in a rather trivial way. Very few things are known about its algebraic counterpart, save that its intrinsic variety is the class of all algebras of the similarity type.

On axiomatization of Łukasiewicz's four-valued modal logic

Formal aspects of various ways of description of Jan Łukasiewicz’s four-valued modal logic £ are discussed. The original Łukasiewicz’s description by means of the accepted and rejected theorems, together with the four-valued matrix, is presented. Then the improved E.J. Lemmon’s description based upon three specific axioms, together with the relational semantics, is presented as well. It is proved that Lemmon’s axiomatic is not independent: one axiom is derivable on the base of the remanent two. Several axiomatizations, based on three, two or one single axiom are provided and discussed, including S. Kripke’s axiomatics. It is claimed that (a) all substitutions of classical theorems, (b) the rule of modus ponens, (c) the definition of “◊” and (d) the single specific axiom schema: ⬜A ∧ B → A ∧ ⬜B, called the jumping necessity axiom, constitute an elegant axiomatics of the system £.

ARE CONDITIONAL AND DISJUNCTION REALLY COMPARABLE?

Liu and Chou theorize that the rules of ModusPonens and ModusTollenshave equivalent disjunctive forms. 5 ey present an experiment using these forms and conclude that, if applied to disjunctive inferences, the rule of ModusTollens is not more di: cult than the rule of ModusPonens. 5 eir explanation is based on a successive<conditionalization framework, but, in this paper, it is argued that their results can be interpreted according to, at least, another two theories, the mental models theory and the formal rules theory, and that, therefore, their experiment does not necessarily prove their theses.

Logics preserving degrees of truth from varieties of residuated lattices

Let K be a variety of (commutative, integral) residuated lattices. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.e., 1 is the only truth value preserved by the inferences of the logic. In this paper we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences. We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus, and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth preserving logic is the purely inferential extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.

Intuitionistic propositional logic with Galois connections

In this work, an intuitionistic propositional logic with a Galois connection (IntGC) is introduced. In addition to the intuitionistic logic axioms and inference rule of modus ponens, the logic contains only two rules of inference mimicking the performance of Galois connections. Both Kripke-style and algebraic semantics are presented for IntGC, and IntGC is proved to be complete with respect to both of these semantics. We show that IntGC has the finite model property and is decidable, but Glivenko’s Theorem does not hold. Duality between algebraic and Kripke semantics is presented, and a representation theorem for Heyting algebras with Galois connections is proved. In addition, an application to rough L-valued sets is presented.

Logics from Galois connections

In this paper, Information Logic of Galois Connections (ILGC) suited for approximate reasoning about knowledge is introduced. In addition to the three classical propositional logic axioms and the inference rule of modus ponens, ILGC contains only two auxiliary rules of inference mimicking the performance of Galois connections of lattice theory, and this makes ILGC comfortable to use due to the flip-flop property of the modal connectives. Kripke-style semantics based on information relations is defined for ILGC. It is also shown that ILGC is equivalent to the minimal tense logic K t , and decidability and completeness of ILGC follow from this observation. Additionally, relationship of ILGC to the so-called classical modal logics is studied. Namely, a certain composition of Galois connection mappings forms a lattice-theoretical interior operator, and this motivates us to axiomatize a logic of these compositions. It turns out that this logic satisfies the axioms of the non-normal logic EMT4. Hence, EMT4 can be viewed to be embedded in ILGC. EMT4 is complete with respect to the neighbourhood semantics. Here, we introduce an alternative semantics for EMT4. This is done by defining the so-called interior models, and completeness of EMT4 is proved with respect to the interior semantics.

A complete fuzzy logical system to deal with trust management systems

In this paper we approach trust management systems in a fuzzy logical setting. The idea is to provide a generalization of the classical framework, where trust is understood via the dichotomy “true–false”. In order to overcome the classical approach proposed by Weeks, following the ideas used by Hájek, Esteva, Godo and others to deal with probability, possibility, and necessity in a many-valued logical setting, we introduce the modal logic FTn(ŁΠ12) built up over the many-valued logic ŁΠ12. In particular, we enlarge the ŁΠ12 language by means of a binary modality says acting on pairs (pi,ϕ) of principals and assertions, where a principal is a propositional variable and an assertion is a propositional formula of a suited many-valued logic. The idea is to regard the evaluation of the modal formula says(pi,ϕ) as the degree of confidence the principalpiputs in the assertionϕ. For FTn(ŁΠ12) we introduce a syntax, a semantic and we show completeness. Then we discuss the validity of generalized modus ponens rule in our setting. Finally we deal with a Pavelka-style extension of our logic, and we also extend FTn(ŁΠ12) to allow principals to be hierarchically organized.

On the infinite-valued Łukasiewicz logic that preserves degrees of truth

Łukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Łukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Łukasiewicz algebra by using a “truth-preserving” scheme. This deductive system is algebraizable, non-selfextensional and does not satisfy the deduction theorem. In addition, there exists no Gentzen calculus fully adequate for it. Another presentation of the same deductive system can be obtained from a substructural Gentzen calculus. In this paper we use the framework of abstract algebraic logic to study a different deductive system which uses the aforementioned algebra under a scheme of “preservation of degrees of truth”. We characterize the resulting deductive system in a natural way by using the lattice filters of Wajsberg algebras, and also by using a structural Gentzen calculus, which is shown to be fully adequate for it. This logic is an interesting example for the general theory: it is selfextensional, non-protoalgebraic, and satisfies a “graded” deduction theorem. Moreover, the Gentzen system is algebraizable. The first deductive system mentioned turns out to be the extension of the second by the rule of Modus Ponens.

SCHIFFER ON VAGUENESS

Abstract: I go through, and criticize, Stephen Schiffer's account of vagueness and the sorites paradox. I discuss his notion of a happy‐face solution to a paradox, his appeal to vagueness‐related partial belief, his claim that indeterminacy is a psychological notion, and his view that the sorites premise and the inference rule of modus ponens are indeterminate.