A certain type of inference rules in modal logics, generalizing Gabbay's Irreflexivity rule, is introduced and some general completeness results about modal logics axiomatized with such rules are proved.

We study the general problem of axiomatizing structures in the framework of modal logic and present a uniform method for complete axiomatization of the modal logics determined by a large family of classes of structures of any signature.

We study the general problem of axiomatizing structures in the framework of modal logic and present a uniform method for complete axiomatization of the modal logics determined by a large family of classes of structures of any signature.

(...) However, whether we chose a weak or strong approximation, the set would not make any sense at all, if (once more) this choice would not be justified in either temporal or spatial sense or given the context of possible applicability of the set in different circumstances. This would obviously represent a dualism in itself as we would (for instance) posit and apply a full identity-equality-equivalence of x and y when applying Newtonian physics to certain observations we make (it would (...) be the case of neural correlates), and we would posit and apply a non - identity-equality-equivalence of x and y when applying Quantum mechanics to other observations. Following this dualism in and of theories, the same sate would need to be slightly modified: U2:(x~y)∪(x≈y); U3:(x~y)∩(x≈y); U4a:(x~y)⊆(x≈y) vs. U4b:(x~y)⊇(x≈y). (shrink)

This is part one of a two-part paper, in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. This allows us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths of arithmetic and (...) show that the theory is a proof-theoretically conservative extension of the theory PT of positive truth. We construct models for the theory and draw some conclusions for the semantics of conceptualist ground. (shrink)

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory (...) is a proof-theoretically conservative extension of the ramified theory of positive truth up to. (shrink)

Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a (...) classical interpretation of the connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators. (shrink)

In the paper we examine the method of axiomatic rejection used to describe the set of nonvalid formulae of Aristotle's syllogistic. First we show that the condition which the system of syllogistic has to fulfil to be ompletely axiomatised, is identical to the condition for any first order theory to be used as a logic program. Than we study the connection between models used or refutation in a first order theory and rejected axioms for that theory. We show (...) that any formula of syllogistic enriched with classical connectives is decidable using models in the domain with three members. (shrink)

Edward Nieznański developed two logical systems in order to deal with a version of the problem of evil associated with two formulations of religious determinism. The aim of this research was to revisit these systems, providing them with a more appropriate formalization. The new resulting systems, namely, N1 and N2, were reformulated in first-order modal logic; they retain much of their original basic structures, but some additional results were obtained. Furthermore, our research found that an underlying minimal set of (...) axioms is enough to settle the questions proposed. Thus, we developed a minimal system, called N3, that solves the same issues tackled by N1 and N2, but with less assumptions than these systems. All of the systems developed here are proposed as solutions to the logical problem of evil through the refutation of two versions of religious determinism, showing that the attributes of God in Classical Theism, namely, those of omniscience, omnipotence, infallibility, and omnibenevolence, when formalized, are consistent with the existence of evil, providing one more response to this traditional issue. (shrink)

We investigate an enrichment of the propositional modal language L with a "universal" modality ■ having semantics x ⊧ ■φ iff ∀y(y ⊧ φ), and a countable set of "names" - a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒ $_{c}$ proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ(⍯) of ℒ, where ⍯ is an additional modality with the semantics x ⊧ ⍯φ (...) iff Vy(y ≠ x → y ⊧ φ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in ℒ $_{c}$ . Strong completeness of the normal ℒ $_{c}$ logics is proved with respect to models in which all worlds are named. Every ℒ $_{c}$ -logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from ℒ to ℒ $_{c}$ are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched. (shrink)

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in (...) particular, Löwenheim and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

The philosophy of science of Patrick Suppes is centered on two important notions that are part of the title of his recent book (Suppes 2002): Representation and Invariance. Representation is important because when we embrace a theory we implicitly choose a way to represent the phenomenon we are studying. Invariance is important because, since invariants are the only things that are constant in a theory, in a way they give the “objective” meaning of that theory. Every scientific theory gives a (...) representation of a class of structures and studies the invariant properties holding in that class of structures. In Suppes’ view, the best way to define this class of structures is via axiomatization. This is because a class of structures is given by a definition, and this same definition establishes which are the properties that a single structure must possess in order to belong to the class. These properties correspond to the axioms of a logical theory. In Suppes’ view, the best way to characterize a scientific structure is by giving a representation theorem for its models and singling out the invariants in the structure. Thus, we can say that the philosophy of science of Patrick Suppes consists in the application of the axiomatic method to scientific disciplines. What I want to argue in this paper is that this application of the axiomatic method is also at the basis of a new approach that is being increasingly applied to the study of computer science and information systems, namely the approach of formal ontologies. The main task of an ontology is that of making explicit the conceptual structure underlying a certain domain. By “making explicit the conceptual structure” we mean singling out the most basic entities populating the domain and writing axioms expressing the main properties of these primitives and the relations holding among them. So, in both cases, the axiomatization is the main tool used to characterize the object of inquiry, being this object scientific theories (in Suppes’ approach), or information systems (for formal ontologies). In the following section I will present the view of Patrick Suppes on the philosophy of science and the axiomatic method, in section 3 I will survey the theoretical issues underlying the work that is being done in formal ontologies and in section 4 I will draw a comparison of these two approaches and explore similarities and differences between them. (shrink)

The debate about the foundations of mathematical sciences traces back to Greek antiquity, with Euclid and the foundations of geometry. Through the flux of history, the debate has appeared in several shapes, places, and cultural contexts. Remarkably, it is a locus where logic, philosophy, and mathematics meet. In mathematical astronomy, Nicolaus Copernicus’s axiomatic approach toward a heliocentric theory of the universe has prompted questions about foundations among historians who have studied Copernican axioms in their terminological and logical aspects (...) but never examined them as a question of mathematical practice. Copernicus provides seven unproved assumptions in the introduction of the brief treatise entitled Nicolaus Copernicus’s draft on the models of celestial motions established by himself, better known as Commentariolus (ca. 1515), published circa 30 years before the final composition of his heliocentric theory (On the revolutions of the heavenly spheres, 1543). The assumptions deal with the renowned Copernican hypothesis of considering the Earth in motion and the Sun, not affected by motion, near the center of the universe. Although Copernicus decides to omit the proofs for the sake of brevity, the deductions in the Commentariolus are supposed to be drawn from the initial seven assumptions. Questions on the nature (are they postulates or axioms?) and the logic (is there an internal rigor?) of those assumptions have yet to be fully explored. By examining Copernicus’s seven assumptions as a question of mathematical practice, it is possible to hold historical, philosophical, and logical aspects of Copernican axiomatics together and understand them as part of Copernicus’s intuition and creativity. (shrink)

Study represents an application of the neutrosophic method, for solving the contradiction between communication and information. In addition, it recourse to an appropriate method of approaching the contradictions: Extensics, as the method and the science of solving the contradictions. The research core is the reality that the scientific research of communication-information relationship has reached a dead end. The bivalent relationship communicationinformation, information-communication has come to be contradictory, and the two concepts to block each other. After the critical examination of conflicting (...) positions expressed by many experts in the field, the extensic and inclusive hypothesis is issued that information is a form of communication. The object of communication is the sending of a message. The message may consist of thoughts, ideas, opinions, feelings, beliefs, facts, information, intelligence or other significational elements. When the message content is primarily informational, communication will become information or intelligence. The arguments of supporting the hypothesis are: a) linguistic (the most important being that there is "communication of information" but not "information of communication"; also, it is clarified and reinforced the over situated referent, that of the communication as a process), b) systemic-procedural (in the communication system is developing an information system; the informing actant is a type of communicator, the information process is a communication process), c) practical (the delimitation eliminates the efforts of disparate and inconsistent understanding of the two concepts), d) epistemological arguments (the possibility of intersubjective thinking of reality is created), linguistic arguments, e) logical and realistic arguments (it is noted the situation that allows to think coherently in a system of concepts - derivative series or integrative groups) f) and arguments from historical experience (the concept of communication has temporal priority, it appears 13 times in Julius Caesar’s writings ). In an axiomatic conclusion, the main arguments are summarized in four axioms: three are based on the pertinent observations of specialists, and the fourth is a relevant application of Florentin Smarandache’s neutrosophic theory. (shrink)

A well known conception of axiomatization has it that an axiomatized theory must be interpreted, or otherwise coordinated with reality, in order to acquire empirical content. An early version of this account is often ascribed to key figures in the logical empiricist movement, and to central figures in the early “formalist” tradition in mathematics as well. In this context, Reichenbach’s “coordinative definitions” are regarded as investing abstract propositions with empirical significance. We argue that over-emphasis on the abstract elements of this (...) approach fails to appreciate a rich tradition of empirical axiomatization in the late nineteenth and early twentieth centuries, evident in particular in the work of Moritz Pasch, Heinrich Hertz, David Hilbert, and Reichenbach himself. We claim that such over-emphasis leads to a misunderstanding of the role of empirical facts in Reichenbach’s approach to the axiomatization of a physical theory, and of the role of Reichenbach’s coordinative definitions in particular. (shrink)

_Framing effects_ concern the having of different attitudes towards logically or necessarily equivalent contents. Framing is of crucial importance for cognitive science, behavioral economics, decision theory, and the social sciences at large. We model a typical kind of framing, grounded in (i) the structural distinction between beliefs activated in working memory and beliefs left inactive in long term memory, and (ii) the topic- or subject matter-sensitivity of belief: a feature of propositional attitudes which is attracting growing research attention. We introduce (...) a class of models featuring (i) and (ii) to represent, and reason about, agents whose belief states can be subject to framing effects. We axiomatize a logic which we prove to be sound and complete with respect to the class. (shrink)

The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplification of set theories (...) due to Scott, Montague, Derrick, and Potter. (shrink)

ABSTRACT A complete axiomatic system CTLrp is introduced for a temporal logic for finitely branching ω+ -trees in a language extended with so called reference pointers. Syntactic and semantic interpretations are constructed for the branching time computation tree logic CTL* into CTLrp. In particular, that yields a complete axiomatization for the translations of all valid CTL*-formulae. Thus, the temporal logic with reference pointers is brought forward as a simpler (with no path quantifiers), but in a way (...) more expressive medium for reasoning about branching time. (shrink)

A complete axiomatic system CTL$_{rp}$ is introduced for a temporal logic for finitely branching $\omega^+$-trees in a temporal language extended with so called reference pointers. Syntactic and semantic interpretations are constructed for the branching time computation tree logic CTL$^{*}$ into CTL$_{rp}$. In particular, that yields a complete axiomatization for the translations of all valid CTL$^{*}$-formulae. Thus, the temporal logic with reference pointers is brought forward as a simpler (with no path quantifiers), but in a way more (...) expressive medium for reasoning about branching time. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...) for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics. (shrink)

Weak enough relevant logics are often closed under depth substitutions. To determine the breadth of logics with this feature, we show there is a largest sublogic of R closed under depth substitutions and that this logic can be recursively axiomatized.

In the substitutional framework, validity is truth under all substitutions of the nonlogical vocabulary. I develop a theory where □ is interpreted as substitutional validity. I show how to prove soundness and completeness for common modal calculi using this definition.

The paper contains an overview of the most important results presented in the monograph of the author "Teorie Językow Syntaktycznie-Kategorialnych" ("Theories of Syntactically-Categorial Languages" (in Polish), PWN, Warszawa-Wrocław 1985. In the monograph four axiomatic systems of syntactically-categorial languages are presented. The first two refer to languages of expression-tokens. The others also takes into consideration languages of expression-types. Generally, syntactically-categorial languages are languages built in accordance with principles of the theory of syntactic categories introduced by S. Leśniewski [1929,1930]; they are (...) connected with- the Ajdukiewicz’s work [1935] which was a continuation of Leśniewski’s idea and further developed and popularized in the research on categorial grammars, by Y. Bar-Hillel [1950,1953,1964]. To assign a suitable syntactic category to each word of the vocabulary is the main idea of syntactically-categorial approach to language. Compound expressions are built from the words of the vocabulary and then a suitable syntactic-category is assigned to each of them. A language built in this way should be decidable, which means that there should exist an algorithm for deciding about each expression of it, whether it is well-formed or is syntactically connected . The traditional, originating from Husserl, understanding of the syntactic category confronts some difficulties. This notion is defined by abstraction using the concept of affiliation of two expressions to the same syntactic category. (shrink)

There is a curious bifurcation in the literature on ground and its logic. On the one hand, there has been a great deal of work that presumes that logical complexity invariably yields grounding. So, for instance, it is widely presumed that any fact stated by a true conjunction is grounded in those stated by its conjuncts, that any fact stated by a true disjunction is grounded in that stated by any of its true disjuncts, and that any fact stated (...) by a true double negation is grounded in that stated by the doubly-negated formula. This commitment is encapsulated in the system GG axiomatized and semantically characterized by [deRosset and Fine, 2023] (following [Fine, 2012]). On the other hand, there has been a great deal of important formal work on “flatter” theories of ground, yielding logics very different from GG [Correia, 2010] [Fine, 2016, 2017b]. For instance, these theories identify the fact stated by a self-conjunction $$(\phi \wedge \phi )$$ with that stated by its conjunct $$\phi $$. Since, in these systems, no fact grounds itself, the “flatter” theories are inconsistent with the principles of GG. This bifurcation raises the question of whether there is a single notion of ground suited to fulfill the philosophical ambitions of grounding enthusiasts. There is, at present, no unified semantic framework employing a single conception of ground for simultaneously characterizing both GG and the “flatter” approaches. This paper fills this gap by specifying such a framework and demonstrating its adequacy. (shrink)

Logics of joint strategic ability have recently received attention, with arguably the most influential being those in a family that includes Coalition Logic (CL) and Alternating-time Temporal Logic (ATL). Notably, both CL and ATL bypass the epistemic issues that underpin Schelling-type coordination problems, by apparently relying on the meta-level assumption of (perfectly reliable) communication between cooperating rational agents. Yet such epistemic issues arise naturally in settings relevant to ATL and CL: these logics are standardly interpreted on structures where (...) agents move simultaneously, opening the possibility that an agent cannot foresee the concurrent choices of other agents. In this paper we introduce a variant of CL we call Two-Player Strategic Coordination Logic (SCL2). The key novelty of this framework is an operator for capturing coalitional ability when the cooperating agents cannot share strategic information. We identify significant differences in the expressive power and validities of SCL2 and CL2, and present a sound and complete axiomatization for SCL2. We briefly address conceptual challenges when shifting attention to games with more than two players and stronger notions of rationality. (shrink)

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic from the (...) perspective of abstract algebraic logic . We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices. (shrink)

The idea of science being the best – or the only – way to reach the truth about our cosmos has been a major belief of modern civilization. Yet, science has grown tall on fragile legs of clay. Every scientific theory uses axioms and assumptions that by definition cannot be proved. This poses a serious limitation to the use of science as a tool to find the truth. The only way to search for the latter is to redefine the former (...) to its original glory. In the days well before Galileo and Newton, science and religion were not separated. They worked together to discover the truth and while the latter had God as its final destination, the former had God as its starting point. Science is based on the irrational (unproven) belief that the world is intelligible along many other assumptions. This poses a serious limitation to science that can only be overcome if we accept the irrationality of the cosmos. The motto “Credo quia absurdum” holds more truth than one can ever realize at first glance. There is nothing logical in logic, whereas there is deep wisdom in the irrational. For while the former tries to build castles on moving sand, the latter digs deep inside the depths of existence itself in order to build on the most concrete foundations that there can be: the cosmos itself. The only way forward is backwards. Backwards to a time when religion led the quest for knowledge by accepting what we cannot know, rather than trying to comprehend what we do not. Science was anyway based on that in the first place. (shrink)

Every domain-specific ontology must use as a framework some upper-level ontology which describes the most general domain-independent categories of reality. In the present paper we sketch a new type of upper-level ontology, and we outline an associated knowledge modelling language called GOL – for: General Ontological Language. It turns out that the upper-level ontology underlying well-known standard modelling languages such as KIF, F-Logic and CycL is restricted to the ontology of sets. In a set theory which allows Urelements, however, (...) there will be ontological relations between these Urelements which the set-theoretic machinery cannot capture. In contrast to standard modelling and representation formalisms, GOL provides a machinery for representing and analysing such ontologically basic relations. GOL is thus a genuine extension of KIF and of similar languages. In GOL entities are divided into sets and Urelements, the latter being divided in their turn into individuals and universals. Foremost among the individuals are things or substances, tropes or moments, and situoids: entities containing facts as components. (shrink)

This paper presents a formalization of the classical proof of completeness in Henkin-style developed by Troelstra and van Dalen for intuitionistic logic with respect to Kripke models. The completeness proof incorporates their insights in a fresh and elegant manner that is better suited for mechanization. We discuss details of our implementation in the Lean theorem prover with emphasis on the prime extension lemma and construction of the canonical model. Our implementation is restricted to a system of intuitionistic propositional (...) class='Hi'>logic with implication, conjunction, disjunction, and falsity given in terms of a Hilbert-style axiomatization. As far as we know, our implementation is the first verified Henkin-style proof of completeness for intuitionistic logic following Troelstra and van Dalen's method in the literature. (shrink)

The criticism of Hilbert's axiomatic system of geometry by Mate Meršić (Merchich, 1850-1928), presented in his work "Organistik der Geometrie" (1914, also in "Modernes und Modriges", 1914), is analyzed and discussed. According to Meršić, geometry cannot be based on its own axioms, as a logical analysis of spatial intuition, but must be derived as a "spatial concretion" using "higher" axioms of arithmetic, logic, and "rational algorithmics." Geometry can only be one, because space is also only one. It cannot (...) be reduced to manipulation with symbols but has to deal with concepts and provide real knowledge. According to Meršić, aiming at some particular proof of the consistency of the whole axiomatic system is meaningless because the consistency follows from the truth of individual axioms based on spatial intuition. It is impossible, according to Meršić, for the geometric axioms to be mutually independent because they must be interconnected as essential parts of a whole. Although Meršić did not recognize all the advantages of Hilbert's formalization of geometry, he clearly pointed out some crucial limitations of the axiomatic method. (shrink)

In previous work, I introduced a complete axiomatization of classical non-tautologies based essentially on Łukasiewicz’s rejection method. The present paper provides a new, Hilbert-type axiomatization (along with related systems to axiomatize classical contradictions, non-contradictions, contingencies and non-contingencies respectively). This new system is mathematically less elegant, but the format of the inferential rules and the structure of the completeness proof possess some intrinsic interest and suggests instructive comparisons with the logic of tautologies.

Gravitation is interpreted to be an “ontomathematical” force or interaction rather than an only physical one. That approach restores Newton’s original design of universal gravitation in the framework of “The Mathematical Principles of Natural Philosophy”, which allows for Einstein’s special and general relativity to be also reinterpreted ontomathematically. The entanglement theory of quantum gravitation is inherently involved also ontomathematically by virtue of the consideration of the qubit Hilbert space after entanglement as the Fourier counterpart of pseudo-Riemannian space. Gravitation can be (...) also interpreted as purely mathematical or logical “force” or “interaction” as a corollary from its ontomathematical (rather than physical) realization. The ontomathematical approach to gravitation is implicit in general relativity equating it to operators in pseudo-Riemannian space obeying the Einstein field equation and also well-known by the “geometrization of physics”. Quantum mechanics shares the same by the separable complex Hilbert space and defining “physical quantity” by the Hermitian operators on it. One can interpret special Minkowski space involved by special relativity and the qubit Hilbert space of quantum information as Fourier counterparts immediately noticing that general relativity means gravitation as the Fourier counterpart of non-Hermitian operators implying non-unitarity and the violation of energy conservation and thus destroying Pauli’s particle paradigm. Since the Standard model obeys it, this explains the impossibility of “quantum gravitation” in any framework conservatively generalizing the Standard model so that it would include gravitation along with electromagnetic, weak, and strong interactions. Einstein’s geometrization of gravitation can be continued into a purely mathematical theory of it following Euclid’s realization for geometry to be exhaustively built in a deductive and axiomatic way as well as Riemann’s parametrization of all the class of Euclidean and non-Euclidean geometries by “space curvature”, then being generalized to Minkowski space as the operators on pseudo-Riemannian space as the Einstein field equation means gravitation. The transition from mathematical gravitation to logical one can rely on the historical lesson of the pair of Lobachevski’s and Riemann’s approaches now “reversely”, i.e., from the latter to the former. Logical gravitation is linkable to Hegel’s dialectical logic and ontological dialectics abandoning their interpretations as a new zero logic substituting classical propionyl logic. The approach of ontomathematics generalizing that of ontology, traceable even to Aristotle’s reformation of Plato’s doctrine, needs Hegel’s doctrine to be formalized as a first-order logic naturally containing Boolean algebra, isomorphic to both classical propositional logic and set theory being the class of all first-order logics, as a sub-logic along with Peano arithmetic as another sub-logic. The first-order logic at issue is called Hilbert arithmetic and elaborated in detail in other papers. It allows for both self-foundation of mathematics to be internally proved as complete and furthermore, quantum mechanics reinterpreted as quantum information to be included by the qubit Hilbert space interpretable in turn as a dual and physical counterpart of Hilbert arithmetic in a narrow sense, that is, both counterparts constitute Hilbert arithmetic in a wide sense, being mathematical and physical simultaneously and thus overcoming the Cartesian dualism of “body” gapped from “mind” by an abyss. Then, the proper philosophical interpretation of gravitation to be the fundamental ontomathematical force or interaction overcomes the ridiculous belief of the Big Bang wrongly alleged to be a scientific theory. Ontomathematical gravitation suggests an omnipresent and omnitemporal medium of “God’s” creation “ex nihilo” following only the natural necessity of quantum-information conservation particularly and locally manifested as energy conservation. (shrink)

Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bar-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; specifically, with (...) Level Theory, as developed in Part 1. (shrink)

A sound and complete axiomatization of two tabloid blogs is presented, Leiter Logic (KB) and Deontic Leiter Logic (KDB), the latter of which can be extended to Shame Game Logic for multiple agents. The (B) schema describes the mechanism behind this class of tabloids, and illustrates the perils of interpreting a provability operator as an epistemic modal. To mark this difference, and to avoid sullying Brouwer's good name, the (B) schema for epistemic modals should be called the (...) Blog Schema. (shrink)

This paper discusses the theoretical assumptions behind the conception of the logic of faith and deed and outlines its formal-axiomatic frame and its method of construction, which enable us to understand it as a kind of deductive science. The paper is divided into several sections, starting with the logical analysis of the ambiguous terms of ‚faith’ and ‚action’, and focusing in particular on the concepts of religious faith and deed as a type of conscious activity relating to a (...) matter or matters of social importance. After outlining the main ideas and basic assumptions of the theoretical conception of the LF&D as an axiomatic theory, the author introduces some axiom systems for: 1) the logics of faith LF, 2) the logic of deed LD, and 3) certain logics of norms DL connected with „duties” and concerning actions/deeds. Lastly, the paper outlines the scientific LF&D based on the three types of logic 1)–3). (shrink)

We study the modal logic M L r of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of M L r and show that it is not finitely axiomatizable. Then we describe the finite frames of that logic and show (...) that it has the finite frame property and its satisfiability problem is in EXPTIME. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic. (shrink)

Since the early days of physics, space has called for means to represent, experiment, and reason about it. Apart from physicists, the concept of space has intrigued also philosophers, mathematicians and, more recently, computer scientists. This longstanding interest has left us with a plethora of mathematical tools developed to represent and work with space. Here we take a special look at this evolution by considering the perspective of Logic. From the initial axiomatic efforts of Euclid, we revisit the (...) major milestones in the logical representation of space and investigate current trends. In doing so, we do not only consider classical logic, but we indulge ourselves with modal logics. These present themselves naturally by providing simple axiomatizations of different geometries, topologies, space-time causality, and vector spaces. (shrink)

The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge- whether individual or common- is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in Sections (...) 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought- only monotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the "infinitary" nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they are decidable in the logician's sense. (shrink)

ABSTRACT: This 1974 paper builds on our 1969 paper (Corcoran-Weaver [2]). Here we present three (modal, sentential) logics which may be thought of as partial systematizations of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of these three logics coincide with one another and with those of standard formalizations of Lewis's S5. These logics, when regarded as logistic systems (cf. Corcoran [1], p. 154), are seen to be equivalent; (...) but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be as closely linked as previously thought. -/- This 1974 paper uses the linear notation for natural deduction presented in [2]: each two-dimensional deduction is represented by a unique one-dimensional string of characters. Thus obviating need for two-dimensional trees, tableaux, lists, and the like—thereby facilitating electronic communication of natural deductions. The 1969 paper presents a (modal, sentential) logic which may be thought of as a partial systematization of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of this logic coincides those of standard formalizations of Lewis’s S4. Among the paper's innovations is its treatment of modal logic in the setting of natural deduction systems--as opposed to axiomatic systems. The author’s apologize for the now obsolete terminology. For example, these papers speak of “a proof of a sentence from a set of premises” where today “a deduction of a sentence from a set of premises” would be preferable. 1. Corcoran, John. 1969. Three Logical Theories, Philosophy of Science 36, 153–77. J P R -/- 2. Corcoran, John and George Weaver. 1969. Logical Consequence in Modal Logic: Natural Deduction in S5 Notre Dame Journal of Formal Logic 10, 370–84. MR0249278 (40 #2524). 3. Weaver, George and John Corcoran. 1974. Logical Consequence in Modal Logic: Some Semantic Systems for S4, Notre Dame Journal of Formal Logic 15, 370–78. MR0351765 (50 #4253). (shrink)

In this paper a propositional logic of viewpoints is presented. The language of this logic consists of the usual modal operatorsL (of necessity) andM (of possibility) as well as of two new operatorsA andR. The intuitive interpretations ofA andR are from all viewpoints and from some viewpoint, respectively. Semantically the language is interpreted by using Kripke models augmented with sets of viewpoints and with a new alternativeness relation for the operatorA. Truth values of formulas are evaluated with respect (...) to a world and a viewpoint. Various axiomatizations of the logic of viewpoints are presented and proved complete. Finally, some applications are given. (shrink)

The theories of belief change developed within the AGM-tradition are not logics in the proper sense, but rather informal axiomatic theories of belief change. Instead of characterizing the models of belief and belief change in a formalized object language, the AGM-approach uses a natural language — ordinary mathematical English — to characterize the mathematical structures that are under study. Recently, however, various authors such as Johan van Benthem and Maarten de Rijke have suggested representing doxastic change within a formal (...) logical language: a dynamic modal logic. Inspired by these suggestions Krister Segerberg has developed a very general logical framework for reasoning about doxastic change: dynamic doxastic logic (DDL). This framework may be seen as an extension of standard Hintikka-style doxastic logic with dynamic operators representing various kinds of transformations of the agent's doxastic state. Basic DDL describes an agent that has opinions about the external world and an ability to change these opinions in the light of new information. Such an agent is non-introspective in the sense that he lacks opinions about his own belief states. Here we are going to discuss various possibilities for developing a dynamic doxastic logic for introspective agents: full DDL or DDL unlimited. The project of constructing such a logic is faced with difficulties due to the fact that the agent’s own doxastic state now becomes a part of the reality that he is trying to explore: when an introspective agent learns more about the world, then the reality he holds beliefs about undergoes a change. But then his introspective (higher-order) beliefs have to be adjusted accordingly. In the paper we shall consider various ways of solving this problem. (shrink)

In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of (...) the system is deducible by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)

According to Jago (2014a), logical omniscience is really part of a deeper paradox. Jago develops an epistemic logic with principles of indeterminate closure to solve this paradox, but his official semantics is difficult to navigate, it is motivated in part by substantive metaphysics, and the logic is not axiomatized. In this paper, I simplify this epistemic logic by adapting the hyperintensional semantic framework of Sedlár (2021). My first goal is metaphysical neutrality. The solution to the epistemic paradox (...) should not require appeal to a metaphysics of truth-makers, situations, or impossible worlds, by contrast with Jago’s official semantics. My second goal is to elaborate on the proof theory. I show how to axiomatize a family of logics with principles of indeterminate epistemic closure. (shrink)

According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)

Since the time of Aristotle's students, interpreters have considered Prior Analytics to be a treatise about deductive reasoning, more generally, about methods of determining the validity and invalidity of premise-conclusion arguments. People studied Prior Analytics in order to learn more about deductive reasoning and to improve their own reasoning skills. These interpreters understood Aristotle to be focusing on two epistemic processes: first, the process of establishing knowledge that a conclusion follows necessarily from a set of premises (that is, on the (...) epistemic process of extracting information implicit in explicitly given information) and, second, the process of establishing knowledge that a conclusion does not follow. Despite the overwhelming tendency to interpret the syllogistic as formal epistemology, it was not until the early 1970s that it occurred to anyone to think that Aristotle may have developed a theory of deductive reasoning with a well worked-out system of deductions comparable in rigor and precision with systems such as propositional logic or equational logic familiar from mathematical logic. When modern logicians in the 1920s and 1930s first turned their attention to the problem of understanding Aristotle's contribution to logic in modern terms, they were guided both by the Frege-Russell conception of logic as formal ontology and at the same time by a desire to protect Aristotle from possible charges of psychologism. They thought they saw Aristotle applying the informal axiomatic method to formal ontology, not as making the first steps into formal epistemology. They did not notice Aristotle's description of deductive reasoning. Ironically, the formal axiomatic method (in which one explicitly presents not merely the substantive axioms but also the deductive processes used to derive theorems from the axioms) is incipient in Aristotle's presentation. Partly in opposition to the axiomatic, ontically-oriented approach to Aristotle's logic and partly as a result of attempting to increase the degree of fit between interpretation and text, logicians in the 1970s working independently came to remarkably similar conclusions to the effect that Aristotle indeed had produced the first system of formal deductions. They concluded that Aristotle had analyzed the process of deduction and that his achievement included a semantically complete system of natural deductions including both direct and indirect deductions. Where the interpretations of the 1920s and 1930s attribute to Aristotle a system of propositions organized deductively, the interpretations of the 1970s attribute to Aristotle a system of deductions, or extended deductive discourses, organized epistemically. The logicians of the 1920s and 1930s take Aristotle to be deducing laws of logic from axiomatic origins; the logicians of the 1970s take Aristotle to be describing the process of deduction and in particular to be describing deductions themselves, both those deductions that are proofs based on axiomatic premises and those deductions that, though deductively cogent, do not establish the truth of the conclusion but only that the conclusion is implied by the premise-set. Thus, two very different and opposed interpretations had emerged, interestingly both products of modern logicians equipped with the theoretical apparatus of mathematical logic. The issue at stake between these two interpretations is the historical question of Aristotle's place in the history of logic and of his orientation in philosophy of logic. This paper affirms Aristotle's place as the founder of logic taken as formal epistemology, including the study of deductive reasoning. A by-product of this study of Aristotle's accomplishments in logic is a clarification of a distinction implicit in discourses among logicians--that between logic as formal ontology and logic as formal epistemology. (shrink)

It is shown that a complete axiomatization of classical non-tautologies can be obtained by taking F (falsehood) as the sole axiom along with the two inference rules: (i) if A is a substitution instance of B, then A |– B; and (ii) if A is obtained from B by replacement of equivalent sentences, then A |– B (counting as equivalent the pairs {T, ~F}, {F, F&F}, {F, F&T}, {F, T&F}, {T, T&T}). Since the set of tautologies is also specifiable by (...) purely syntactic means, the resulting picture gives an improved syntactic account of classical sentential logic. The picture can then be completed by considering related systems that prove adequate to specify the set of contradictions, the set of non-contradictions, and the set of contingencies respectively. (shrink)

We introduce and study hierarchies of extensions of the propositional modal and temporal languages with pairs of new syntactic devices: point of reference-reference pointer which enable semantic references to be made within a formula. We propose three different but equivalent semantics for the extended languages, discuss and compare their expressiveness. The languages with reference pointers are shown to have great expressive power (especially when their frugal syntax is taken into account), perspicuous semantics, and simple deductive systems. For instance, Kamp's and (...) Stavi's temporal operators, as well as nominals (names, clock variables), are definable in them. Universal validity in these languages is proved undecidable. The basic modal and temporal logics with reference pointers are uniformly axiomatized and a strong completeness theorem is proved for them and extended to some classes of their extensions. (shrink)

Do other earthly forms of life evolved to the level of intelligent life? Cancer and resistance to antibiotics obliged to ask this question. Signs of intelligence are found at its simplest levels. We try to see if logic is used at these levels. Peter of Spain’s suppositio materialis is applied to the chemical signals of cells. Dynamic Logic is used to understand these chemical communications. <System of communications> is used, instead of “language”. The development of life appears as (...) the development of an axiomatic system. The rights of unborn humans and God’s rights over His own creation appear as the most powerful arguments for conservationism. (shrink)