We introduce now for the first time the neutrosophic modal logic. The Neutrosophic Modal Logic includes the neutrosophic operators that express the modalities. It is an extension of neutrosophic predicate logic and of neutrosophic propositional logic.

This paper is devoted to the formulation and investigation of a dynamic semantic interpretation of the language of first-order predicate logic. The resulting system, which will be referred to as ‘dynamic predicate logic’, is intended as a first step towards a compositional, non-representational theory of discourse semantics. In the last decade, various theories of discourse semantics have emerged within the paradigm of model-theoretic semantics. A common feature of these theories is a tendency to do away with (...) the principle of compositionality, a principle which, implicitly or explicitly, has dominated semantics since the days of Frege. Therefore the question naturally arises whether non-compositionality is in any way a necessary feature of discourse semantics. Since we subscribe to the interpretation of compositionality as constituting primarily a methodological principle, we consider this to be a methodological rather than an empirical question. As a consequence, the emphasis in the present paper lies on developing an alternative compositional semantics of discourse, which is empirically equivalent to its non-compositional brethren, but which differs from them in a principled methodological way. Hence, no attempts are made to improve on existing theories empirically. Nevertheless, as we indicate in section 5, the development of a compositional alternative may in the end have empirical consequences, too. First of all, it can be argued that the dynamic view on interpretation developed in this paper suggests natural and relatively easy to formulate extensions which enable one to deal with a wider range of phenomena than can be dealt with in existing theories. Moreover, the various approaches to the model-theoretic semantics of discourse that have been developed during the last decade, have constituted a ‘fresh start’ in the sense that much of what had been accomplished before was ignored, at least for a start. Of course, this is a justified strategy if one feels one is trying to develop a radically different approach to recalcitrant problems. However, there comes a time when such new approaches have to be compared with the older one, and when an assessment of the pros and cons of each has to be made. One of the main problems in semantics today, we feel, is that a semantic theory such as Montague grammar, and an approach like Kamp’s discourse representation theory, are hard to compare, let alone that it is possible to unify their insights and results.. (shrink)

We investigate a rst-order predicate logic based on Wittgenstein's suggestion to express identity of object by identity of sign, and difference of objects by difference of signs. Hintikka has shown that predicate logic can indeed be set up in such a way; we show that it can be done nicely. More specically, we provide a perspicuous cut-free sequent calculus, as well as a Hilbert-type calculus, for Wittgensteinian predicate logic and prove soundness and completeness theorems.

We develop a predicate logical extension of a subintuitionistic propositional logic. Therefore a Hilbert type calculus and a Kripke type model are given. The propositional logic is formulated to axiomatize the idea of strategic weakening of Kripke''s semantic for intuitionistic logic: dropping the semantical condition of heredity or persistence leads to a nonmonotonic model. On the syntactic side this leads to a certain restriction imposed on the deduction theorem. By means of a Henkin argument strong completeness (...) is proved making use of predicate logical principles, which are only classically acceptable. (shrink)

A logic-enriched type theory is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named and , which we claim correspond closely to the classical predicative systems of second order arithmetic and . We justify this claim by translating each second order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different (...) approaches to the foundations of mathematics.The two LTTs we construct are subsystems of the logic-enriched type theory , which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system has also been claimed to correspond to Weyl’s foundation. By casting and as LTTs, we are able to compare them with . It is a consequence of the work in this paper that is strictly stronger than .The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work. (shrink)

In our previous paper [5], we have studied Kripke-type semantics for propositional logics without the contraction rule. In this paper, we will extend our argument to predicate logics without the structure rules. Similarly to the propositional case, we can not carry out Henkin's construction in the predicate case. Besides, there exists a difficulty that the rules of inference () and () are not always valid in our semantics. So, we have to introduce a notion of normal models.

Predicates are supposed to slice reality neatly in two halves, one for which the predicate holds, the other for which it fails. Yet far from being razors, predicates tend to be dull knives that mangle reality. If reality is a tomato and predicates are knives, then when these knives divide the tomato, plenty of mush remains unaccounted for. Of course some knives are sharper than others, just as some predicates are less vague than others. “x is water” is certainly (...) sharper than “x is beautiful.” But perfect sharpness, perfect boundaries, and perfect separation seem only to obtain in mathematics. The vagueness inherent in many predicates became particularly evident in the twentieth century. Quantum mechanics, the revival of certain ancient paradoxes, and the philosophy of science all contributed to a growing awareness that vagueness was ineliminable from many predicates. Quantum mechanical superposition seems to allow mutually exclusive simultaneous states. In searching for plausible interpretations of quantum mechanics, some researchers attempted to do away with classical bivalent logic. In its place they substituted multivalent quantum logics. Any logic with more than two values forces its predicates to slice reality into more than two parts. Ancient paradoxes involving heaps and baldness also pointed up the 1 Random Predicate Logic 2.. (shrink)

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem''s modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap''s display logic by introduction rules for the existential and the universal quantifier. These rules for x and x are analogous to the display introduction rules for the modal operators and and do not themselves allow the Barcan (...) formula or its converse to be derived. En route from the minimal modal predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules. (shrink)

• 1st try: Free variables in PL (Predicate Logic) (1) Jim1 came in. He1 sat down. (antecedent Jim1 … anaphoric he1) |=M, g cm ιx(x = z1  z1 = jim)  sit z1 iff g(z1) ∈ cm & g(z1) = jim & g(z1) ∈ sit.

In this paper, we show that the predicate logics of consistent extensions of Heyting's Arithmetic plus Church's Thesis with uniqueness condition are complete $\Pi _{2}^{0}$. Similarly, we show that the predicate logic of HA*, i.e. Heyting's Arithmetic plus the Completeness Principle (for HA*) is complete $\Pi _{2}^{0}$. These results extend the known results due to Valery Plisko. To prove the results we adapt Plisko's method to use Tennenbaum's Theorem to prove 'categoricity of interpretations' under certain assumptions.

For each ordinal $\alpha > 0, L(\alpha)$ is the intermediate predicate logic characterized by the class of all Kripke frames with the poset α and with constant domain. This paper will be devoted to a study of logics of the form L(α). It will be shown that for each uncountable ordinal of the form α + η with a finite or a countable $\eta (> 0)$ , there exists a countable ordinal of the form β + η such (...) that L(α + η) = L(β + η). On the other hand, such a reduction of ordinals to countable ones is impossible for a logic L(α) if α is an uncountable regular ordinal. Moreover, it will be proved that the mapping L is injective if it is restricted to ordinals less than ω ω , i.e. α ≠ β implies L(α) ≠ L(β) for each ordinal $\alpha,\beta. (shrink)

A new formalism for predicate logic is introduced, with a non-standard method of binding variables, which allows a compositional formalization of certain anaphoric constructions, including donkey sentences and cross-sentential anaphora. A proof system in natural deduction format is provided, and the formalism is compared with other accounts of this type of anaphora, in particular Dynamic Predicate Logic.

The paper presents predicate logical extensions of some subintuitionistic logics. Subintuitionistic logics result if conditions of the accessibility relation in Kripke models for intuitionistic logic are dropped. The accessibility relation which interprets implication in models for the propositional base subintuitionistic logic considered here is neither persistent on atoms, nor reflexive, nor transitive. Strongly complete predicate logical extensions are modeled with a second accessibility relation, which is a partial order, for the interpretation of the universal quantifier.

This clearly written book makes logic interesting and easier to learn without sacrificing content or rigor. It covers symbolization, proofs, counterexamples, and truth trees. These topics are presented in graded steps, beginning with the symbolization of categorical propositions and concluding with the properties of relations. Logic is applied to materials with which readers will be familiar; both examples and exercises are drawn from newspapers, television, and other popular sources. For individuals intrigued by the formal study of logic.

After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell's Principia Mathematica with the axiom of infinity but without the axiom of reducibility.

In this paper we introduce a notion of context for Groenendijk & Stokhof's Dynamic Predicate Logic DPL. We use these contexts to give a characterization of the relations on assignments that can be generated by composition from tests and random resettings in the case that we are working over an infinite domain. These relations are precisely the ones expressible in DPL if we allow ourselves arbitrary tests as a starting point. We discuss some possible extensions of DPL and (...) the way these extensions interact with our notion of context. (shrink)

D2.1 (PL models and assignments) i. A PL model is a pair M = 〈DM, ·M〉 such that (a) DM is a non-empty set, and (b) ·M maps each A ∈ Con to AM ∈ DM, and each B ∈ Prdn to BM  (DM)n. ii. GM = {g| g: Var  DM} is the set of M-assignments. For any g ∈ GM, u ∈ Var, d ∈ DM, g[u/d] := (g\{u, g(u)})  {u, d} is the u-to-d alternative to (...) g. (shrink)

Monoidal t-norm based logic $\mathbf {MTL}$ 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 $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal (...) t-norm based logic $\mathbf {MTL}$. The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ 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 $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, 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 $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras. (shrink)

In this work we extend results from [4], [3] and [2] about propositional calculi with graded modalities to the predicative level. Our semantic is based on Kripke models with a single domain of interpretation for all the worlds. Therefore the axiomatic system will need a suitable generalization of the Barcan formula. We haven't considered semantics with world-relative domains because they don't present any new difficulties with respect to classical case. Our language will have, as in [1], constant and function symbols, (...) but they will have a rigid interpretation. In this instance the terms will be considered rigid designators, that is, their interpretations will be the same in all possible worlds (cfr. [5]). Nevertheless we will not consider languages with identity: in this way we avoid the problems of identity in modal contexts. On the other hand, we think that in that context would arise essentially the same problems which occur in classical predicative modal logic. Finally we will consider only denumerable languages because the extension to non denumerable ones isn't conceptually problematic. (shrink)

An intermediate predicate logicS + n (n>0) is introduced and investigated. First, a sequent calculusGS n is introduced, which is shown to be equivalent toS + n and for which the cut elimination theorem holds. In § 2, it will be shown thatS + n is characterized by the class of all linear Kripke frames of the heightn.

This paper establishes model-theoretic properties of \, a variation of monadic first-order logic that features the generalised quantifier \. We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality and \, respectively). For each logic \ we will show the following. We provide syntactically defined fragments of \ characterising four different semantic properties of \-sentences: being monotone and continuous in a given set of monadic predicates; having (...) truth preserved under taking submodels or being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence \ to a sentence \ belonging to the corresponding syntactic fragment, with the property that \ is equivalent to \ precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for \-sentences. (shrink)

Abstract.Given a class C of t-norm BL-algebras, one may wonder which is the complexity of the set Taut(C∀) of predicate formulas which are valid in any algebra in C. We first characterize the classes C for which Taut(C∀) is recursively axiomatizable, and we show that this is the case iff C only consists of the Gödel algebra on [0,1]. We then prove that in all cases except from a finite number Taut(C∀) is not even arithmetical. Finally we consider (...) class='Hi'>predicate monadic logics TautM(C∀) of classes C of t-norm BL-algebras, and we prove that (possibly with finitely many exceptions) they are undecidable. (shrink)

The article deals with the interpretation of propositional attitudes in the framework of modal predicate logic. The first part discusses the classical puzzles arising from the interplay between propositional attitudes, quantifiers and the notion of identity. After comparing different reactions to these puzzles it argues in favor of an analysis in which evaluations of de re attitudes may vary relative to the ways of identifying objects used in the context of use. The second part of the article gives (...) this analysis a precise formalization from a model-and proof-theoretic perspective. (shrink)

An intermediate predicate logic L is called finite iff it is characterized by a finite partially ordered set M, i.e., iff L is the logic of the class of all predicate Kripke frames based on M. In this paper we study axiomatizability of logics of this kind. Namely, we consider logics characterized by finite trees M of a certain type (levelwise uniform trees) and establish the finite axiomatizability criterion for this case.

A notion of strictly primitive recursive realizability is introduced by Damnjanovic in 1994. It is a kind of constructive semantics of the arithmetical sentences using primitive recursive functions. It is of interest to study the corresponding predicate logic. It was argued by Park in 2003 that the predicate logic of strictly primitive recursive realizability is not arithmetical. Park’s argument is essentially based on a claim of Damnjanovic that intuitionistic logic is sound with respect to strictly (...) primitive recursive realizability, but that claim was disproved by the author of this article in 2006. The aim of this paper is to present a correct proof of the result of Park. (shrink)

In [Ono 1987] H. Ono put the question about axiomatizing the intermediate predicate logicLFin characterized by the class of all finite Kripke frames. It was established in [ Skvortsov 1988] thatLFin is not recursively axiomatizable. One can easily show that for any finite posetM, the predicate logic characterized byM is recursively axiomatizable, and its axiomatization can be constructed effectively fromM. Namely, the set of formulas belonging to this logic is recursively enumerable, since it is embeddable in (...) the two-sorted classical predicate calculusCPC 2. Thus the logicLFin is II 2 0 -arithmetical.Here we give a more explicit II 2 0 -description ofLFin: it is presented as the intersection of a denumerable sequence of finitely axiomatizable Kripke-complete logics. Namely, we give an axiomatization of the logicLB n P m + characterized by the class of all posets of the finite height m and the finite branching n. A finite axiomatization of the predicate logicLP m + characterized by the class of all posets of the height m is known from [Yokota 1989]. We prove thatLB n P m + =,B n being the propositional axiom of the branching n. (shrink)

It is proved in this paper that the predicate logic of each complete constructive arithmetic theory T having the existential property is Π1T-complete. In this connection, the techniques of a uniform partial truth definition for intuitionistic arithmetic theories is used. The main theorem is applied to the characterization of the predicate logic corresponding to certain variant of the notion of realizable predicate formula. Namely, it is shown that the set of irrefutable predicate formulas is (...) recursively isomorphic to the complement of the set ︀. The notion of Σn-realizability is defined on the basis of the notion of Σn-function. It is proved that the predicate logic of Σn-realizability is Πω+1-hard. (shrink)

In this paper, a semantics for predicate logics without the contraction rule will be investigated and the completeness theorem will be proved. Moreover, it will be found out that our semantics has a close connection with Beth-type semantics.

In this paper a semantics for dynamic predicate logic is developed that uses sequence valued assignments. This semantics is compared with the usual relational semantics for dynamic predicate logic: it is shown that the most important intuitions of the usual semantics are preserved. Then it is shown that the refined semantics reflects out intuitions about information growth. Some other issues in dynamic semantics are formulated and discussed in terms of the new sequence semantics.

Shehtman and Skvortsov introduced Kripke bundles as semantics of non-classical first-order predicate logics. We show the structural equivalence between Kripke bundles for intermediate predicate lógics and Kripke-type frames for intuitionistic modal propositional logics. This equivalence enables us to develop the semantical study of relations between intermediate predicate logics and intuitionistic modal propositional logics. New examples of modal counterparts of intermediate predicate logics are given.

The complexity of any logical modeling reflects both the intrinsic structure of a topic described and the weight of the formal tools. Some of this weight seems inherent in even the most basic logical systems. Notably, standard predicate logic is undecidable. In this paper, we investigate ‘lighter’ versions of this general purpose tool, by modally ‘deconstructing’ the usual semantics, and locating implicit choice points in its set up. The first part sets out the interest of this program and (...) the modal techniques employed, while the second part provides technical elaborations demonstrating its viability. (shrink)

In this paper we study the notion of forcing for Łukasiewicz predicate logic (Ł∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for Ł∀, while for the latter, we study the generic and existentially complete standard models of Ł∀.

In spite of the existence of many examples of incomplete logics, it is an important problem to find intermediate predicate logics complete with respect to Kripke frame (or Kripke sheaf) semantics because they are closed under substitution. But, most of known completeness proofs of finitely axiomatizable logics are difficult to apply to other logics since they are highly dependent on the specific properties of given logics. So, it is preferable to find a general methods of completeness proof. We give (...) some results on this problem using canonical formulas of "propositional" logics 8,2ESS. (shrink)

Basic Predicate Logic, BQC, is a proper subsystem of Intuitionistic Predicate Logic, IQC. For every formula in the language {, , , , , , }, we associate two sequences of formulas 0,1,... and 0,1,... in the same language. We prove that for every sequent , there are natural numbers m, n, such that IQC , iff BQC n m. Some applications of this translation are mentioned.

The core language can be extended by defining additional logical constants. E.g., we can add ‘→’ (implication), ‘∨’ (disjunction), and ‘∀x’ (universal quantifiers). The choice of logical primitives is not as optional in DPL as it is in standard 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. (shrink)

An example of finite tree Mo is presented such that its predicate logic (i.e. the intermediate predicate logic characterized by the class of all predicate Kripke frames based on Mo) is not finitely axiomatizable. Hence it is shown that the predicate analogue of de Jongh - McKay - Hosoi's theorem on the finite axiomatizability of every finite intermediate propositional logic is not true.

It is a little understood fact that the system of formal logic presented in Wittgenstein?s Tractatusprovides the basis for an alternative general semantics for a predicate calculus that is consistent and coherent, essentially independent of the metaphysics of logical atomism, and philosophically illuminating in its own right. The purpose of this paper is threefold: to describe the general characteristics of a Tractarian-style semantics, to defend the Tractatus system against the charge of expressive incompleteness as levelled by Robert Fogelin, (...) and to give a semantics for a formal language that is the Tractarian equivalent of a first-order predicate calculus. Of note in regard to the latter is the fact that a Tractatusstyle truth-definition makes no appeal to the technical trick of defining truth in terms of the satisfaction of predicates by infinite sequences of objects, yet is materially equivalent to the usual Tarski-style truth-definitions. (shrink)