Bochvar algebras consist of the quasivariety $\mathsf {BCA}$ playing the role of equivalent algebraic semantics for Bochvar (external) logic, a logical formalism introduced by Bochvar [4] in the realm of (weak) Kleene logics. In this paper, we provide an algebraic investigation of the structure of Bochvar algebras. In particular, we prove a representation theorem based on Płonka sums and investigate the lattice of subquasivarieties, showing that Bochvar (external) logic has only one (...) proper extension (apart from classical logic), algebraized by the subquasivariety $\mathsf {NBCA}$ of $\mathsf {BCA}$. Furthermore, we address the problem of (passive) structural completeness ((P)SC) for each of them, showing that $\mathsf {NBCA}$ is SC, while $\mathsf {BCA}$ is not even PSC. Finally, we prove that both $\mathsf {BCA}$ and $\mathsf {NBCA}$ enjoy the amalgamation property (AP). (shrink)

Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic ‘of nonsense’ introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic $$\textbf{K}_{\textbf{3}}^{\textbf{w}}$$ by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions (...) are attached; and (iii) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules. (shrink)

A three-valued propositional logic is presented, within which the three values are read as ?true?, ?false? and ?nonsense?. A three-valued extended functional calculus, unrestricted by the theory of types, is then developed. Within the latter system, Bochvar analyzes the Russell paradox and the Grelling-Weyl paradox, formally demonstrating the meaninglessness of both.

Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic `of nonsense' introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic K3W by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions (...) are attached; and (iii) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules. (shrink)

[This résumé was published in English in Matematicheskii Sbornik along with the article.]The present paper contains an investigation of a three-valued logical calculus (the system) previously described by the author [Recueil Mathématique 4 (46), 2 (1938)].

We propose a combination of Bochvar's strict three-valued logic, McCarthy's sequential three-valued logic, and process algebra via the conditional guard construct. This combination entails the introduction of a new constant meaningless in process algebra. We present an operational semantics in SOS-style, and a completeness result for ACP with conditional guard construct and the proposed logic.

In an essential way I make use of Frege’s and Vasilev’s ideas. N. A. Vasilev distinguished two levels in a logic. The abstract logic depends on gnoeologic assumptions while the empirical part of logic depends on ontological ones. Vasilev did not change the external logic but he did change the internal one. His system can be viewed as a non-standard syllogistics based on classical propositional logic . Vasilev’s ideas become plain if we discern cleary (...) acts of predication and acts of assertion, the relation of mental contents with the way things are. The second sourse of the idea seems to Frege’s differentation of mental process and assertion, statement . As a symbol of assertion for Frege serves a vertical stroke |. A horizontal stroke following it is correlated with holding a proposition, it makes the sign determined everywhere. In the string of signs the one following the vertical stroke does not express a proposition, it describes a possible state of affairs. (shrink)

This article is an investigation into Hegel's claim that teleology is the truth of mechanism, which Hegel puts forward in the objectivity section in the Science of Logic. Contrary to most accounts of this section of the Logic, I make a case for a reading of Hegel's conception of external purposiveness according to which the latter makes a positive contribution to the structural development of the concepts of the Logic. I argue that external purposiveness plays (...) a major role in understanding the Hegelian claim of teleology as the truth of mechanism. More specifically, I argue that structures of external purposiveness provide the conditions for the individuation of mechanical objects. (shrink)

Intra-theoretical logical pluralism is a form of meaning-invariant pluralism about logic, articulated recently by Hjortland :355–373, 2013). This version of pluralism relies on it being possible to define several distinct notions of provability relative to the same logical calculus. The present paper picks up and explores this theme: How can a single logical calculus express several different consequence relations? The main hypothesis articulated here is that the divide between the internal and external consequence relations in Gentzen systems generates (...) a form of intra-theoretical logical pluralism. (shrink)

W tym artykule zaproponuję konstrukcję trójwartościowej logiki przekonań, którą nazywam: LSB3_1. Podam również i udowodnię kompletność LSB3_1 w odniesieniu do danej semantyki. LSB3_1 opiera się na preformalnych założeniach i intuicjach, które przedstawiono w sekcji 1. Sekcja 2 zawiera składnię i podział instrukcji LSB3_1 na wewnętrzne i zewnętrzne. Rozdział 3 przedstawia semantykę LSB3_1, a także szereg tautologii i formuł nietautologicznych w LSB3_1 wraz z ich intuicyjną interpretacją. System aksjomatyczny dla LSB3_1 i jego porównanie z silną logiką Kleene'a są przedstawione w sekcji (...) 4. Twierdzenie o kompletności dla LSB3_1 jest przedstawione w rozdziale 5. Zdefiniuję termin koniunkcyjna forma normalna i przedstawię lematy, które prowadzą do udowodnienia redukcji języka LSB3_1 wzory przed udowodnieniem twierdzenia o zupełności. (shrink)

The paper presents the semantic model of logical content processing in the environment of tabooed sentences. Such sentences occur both in academic and colloquial discourses. They can be interpreted as infected formulas with the value of an insult. The proposed model assumes that the inferential mechanism of argumentative activities realised in discourses in the environment of such formulas is based on Bochvar's logical matrices and Kripke’s possible worlds semantic structures. This mechanism is determined by semantic consequence operators of three (...) types. The presented theory can be viewed as a formal tool for reconstructing the logical mental mechanism which stops deductive actions aiming at the development of narrative worlds in discourse practices during epistemic wars. It postulates that these practices, which occur in processes of developing science, are governed by special logical mechanisms of inference. (shrink)

The paper presents the semantic model of logical content processing in the environment of tabooed sentences. Such sentences occur both in academic and colloquial discourses. They can be interpreted as infected formulas with the value of an insult. The proposed model assumes that the inferential mechanism of argumentative activities realised in discourses in the environment of such formulas is based on Bochvar's logical matrices and Kripke’s possible worlds semantic structures. This mechanism is determined by semantic consequence operators of three (...) types. The presented theory can be viewed as a formal tool for reconstructing the logical mental mechanism which stops deductive actions aiming at the development of narrative worlds in discourse practices during epistemic wars. It postulates that these practices, which occur in processes of developing science, are governed by special logical mechanisms of inference. (shrink)

In this paper, I present the discussive adaptive logic DLI r . As is the case for other discussive logics, the intended application context of DLI r is the interpretation of discussions. What is new about the system is that it does not lead to explosion when some of the premises are self-contradictory. It is argued that this is important in view of the fact that human reasoners are not logically omniscient, and hence, that it may not be evident (...) to discover the inconsistencies in one's beliefs. In addition to this, DLI r can handle cases in which different participants contradict each other. It is shown that, in both kinds of cases, DLI r leads to an interpretation of the discussion that is as rich as possible (even though no discussive connectives are introduced). (shrink)

This article develops a logical (or semantic) response to scepticism about the existence of an external world. Specifically, it is argued that any doubt about the existence of an external world can be proved to be false, but whatever appears to be doubt about the existence of an external world that _cannot_ be proved to be false is nonsense, insofar as it must rely on the assertion of something that is logically impossible. The article further suggests that (...) both G. E. Moore and Ludwig Wittgenstein worked towards the same solution but left their work unfinished. (shrink)

The years 1909-1913 were among the most productive, philosophically speaking, of Bertrand Russell's entire career. In addition to the papers reprinted in this volume, he brought Principia Mathematica to its finished form and wrote The Problems of Philosophy, Theory of Knowledge and Our Knowledge of the External World . In October 1910, Russell began teaching at Cambridge, having accepted an appointment as lecturer in logic and the principles of mathematics at Trinity College for a term of five years. (...) The following year, Ludwig Wittgenstein began to attend his lectures. Within a few months, Wittgenstein had exerted a major influence on Russsell's philosophical thinking, perhaps even more than Russell had influenced his thought. (shrink)

Curry’s paradox is well known. The original version employed a conditional connective, and is not forthcoming if the conditional does not satisfy contraction. A newer version uses a validity predicate, instead of a conditional, and is not forthcoming if validity does not satisfy structural contraction. But there is a variation of the paradox which uses “external validity”. And since external validity contracts, one might expect the appropriate version of the Curry paradox to be inescapable. In this paper we (...) show that this is not the case. We consider two ways of formalising the notion of external validity, and show that in both of these the paradox is not forthcoming without the appropriate forms of contraction. (shrink)

In this paper I will draw attention to an important route to external world skepticism, which I will call confidence skepticism. I will argue that we can defang confidence skepticism (though not a meeker ‘argument from might’ which has got some attention in the 20th century literature on external world skepticism) by adopting a partially psychologistic answer to the problem of priors. And I will argue that certain recent work in the epistemology of mathematics and logic provides (...) independent support for such psychologism. (shrink)

We discuss the externally definable Ramsey property, a weakening of the Ramsey property for relational structures, where the only colourings considered are those that are externally definable: that is, definable with parameters in an elementary extension. We show a number of basic results analogous to the classical Ramsey theory, and show that, for an ultrahomogeneous structure M with countable age, the externally definable Ramsey property is equivalent to the dynamical statement that, for all $$n \in \mathbb {N} $$, every subflow (...) of the space $$S_n(M)$$ of n-types has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures. (shrink)

This paper adds evidence structure to standard models of belief, in the form of families of sets of worlds. We show how these more fine-grained models support natural actions of “evidence management”, ranging from update with external new information to internal rearrangement. We show how this perspective leads to new richer languages for existing neighborhood semantics for modal logic. Our main results are relative completeness theorems for the resulting dynamic logic of evidence.

We examine the relationship between the logics of nonsense of Bochvar and Halldén and the containment logics in the neighborhood of William Parry’s A I. We detail two strategies for manufacturing containment logics from nonsense logics—taking either connexive and paraconsistent fragments of such systems—and show how systems determined by these techniques have appeared as Frederick Johnson’s R C and Carlos Oller’s A L. In particular, we prove that Johnson’s system is precisely the intersection of Bochvar’s B 3 and (...) Graham Priest’s non-symmetrized connexive logic and that Oller’s system lies just beneath the intersection of B 3 and Priest’s paraconsistent L P. We conclude by examining Oller’s system in more depth, giving it a characterization in terms of L P and showing that it plays the same role to Harry Deutsch’s paraconsistent containment logic S that Aleksandr Zinov'ev’s S 1 plays with respect to A I. (shrink)

Hegel's philosophy of biology is one of the strongest chapters of Hegel'sPhilosophy of Nature. It can be argued that Hegel's understanding of organicity underscores the explanatory power of ‘dialectical thinking’, as Hegel himself claims. Hegel's interpretation of organicity is based upon the logical development of categories in his chapter on Objectivity of his Logic. If we compare Hegel's treatment of teleology in the Logic with his interpretation of organicity in his Philosophy of Nature, a mismatch can be found. (...) In the Logic, Hegel introduces the notion of life (Leben) after the chapter on Objectivity, i.e. after elaborating on external teleology. In his Philosophy of Nature, the notion of the inner purposiveness of organicity belongs to the externality of nature. This article compares Hegel's treatment of teleology in his Logic and his Philosophy of Nature, reconstructs possible explanations for mismatches and offers a possible solution to the question whether the notion of life should be regarded as the last part of the chapter on teleology or the first part of the chapter on the Idea. (shrink)

In the original publication of the article, in Definition 4, the sixth line which reads as.

This paper adds evidence structure to standard models of belief, in the form of families of sets of worlds. We show how these more fine-grained models support natural actions of “evidence management”, ranging from update with external new information to internal rearrangement. We show how this perspective leads to new richer languages for existing neighborhood semantics for modal logic. Our main results are relative completeness theorems for the resulting dynamic logic of evidence.

In the first part of the paper the authors describe Hrušovský’s model of the development of scientific knowledge, which, due to the influence of logical empiricism, he saw as cumulative, hypothetical-confirmationist and internalistic, i.e. taking into account only scientific factors. In the second part it is showed, that Hrušovský acknowledged the influence of the external factors, emphasizing at the same time the fundamental independence of scientific knowledge. He dismissed the vulgar eco nomism as well as the extreme internalism.

The theory of visual perception that Irvin Rock develops and supports in this book with numerous original experiments, views perception as the outcome of a process of unconscious inference, problem solving, and the building of structural descriptions of the external world.

In considering the very possibility of deviant logic, we face the following question: what makes us see an operator of one logical system as a deviant version of an operator of another system? Why not see it simply as a different operator? Why do we see, say, intuitionist implication as an operator 'competing' with classical implication? Is it only because both happen to be called implications?1 It is clear that if we want to make cross-systemic comparisons, we need an (...) 'Archimedean point' external to the systems compared. Some logicians and philosophers, including Quine (1986), come close to saying that no such Archimedean point is available, and hence that there can be no deviant logics, for if two operators are governed by different axioms, then they are simply two different operators (or, if you prefer, operators with different meanings). From this viewpoint, intuitionist implication is no less different from classical implication than, say, intuitionist or classical conjunction. This conclusion is indeed plausible if we consider logical calculi simply as algebraic structures2; however, things are different if we see them as a means of accounting for something that is already 'there' before we establish the structures and is to be explicated by them. From such a perspective, two operators of different systems may be variants of the same operator in virtue of the fact that they are both means of capturing the same pretheoretical item. What might these items be? Sometimes it seems that logicians tacitly assume that there are some mythical archetypes of implication, conjunction etc., located somewhere in some Platonic heaven or somehow underlying the a priori structures of our mind, which logic tries to capture (for better or worse). However, when it comes to the comparison of the concrete outcomes of our logical efforts, say the classical and the intuitionist implications with the archetypal Implication, the latter can never be materialized so distinctly as to be of any help. A more constructive proposal is that the operators are related to elements or constructions of our language.. (shrink)

Fourteen of the American philosopher's most influential essays appear here, offering profound reflections on many different aspects of knowledge, reality, and epistemology. These papers on experimental logic are rooted in the implication that possession of knowledge implies a judgment, resulting from an inquiry or investigation. The presence of this "inquiry stage" suggests an intermediate and mediating phase between the external world and knowledge, an area conditioned by other factors. Expanding upon this basis, these essays consider the relationship of (...) thought and its subject matter; the antecedents and stimuli of thought, data, and meanings; the objects of thought; control of ideas by facts; and similar topics. Unabridged republication of the classic 1916 edition. (shrink)

Does pure consciousness exist without being hooked to a physical mechanism? Can such claim be proven logically? The magnitude of asking this sort of question is similar to asking: Is it logical that matter exists out of the total void? The answer to both questions is yes. The aim of this paper is to show that, the existence of pure consciousness is a logical state, it is not energy, and it exists timelessly and can be experienced beyond the physical body. (...) Furthermore, we show that pure consciousness is a virtual junction on the metaphysical virtual net where each junction incorporates a set of cords of consciousness. The set can be limited, unlimited or uncountable. Cords of consciousness is treated as brain neural patterns and as such the brain can easily be manipulated by an external device which may transmit neural patterns in order to generate artificial scenes. A computation example was provided to elaborate the neural patterns and cords of consciousness. Finally, we discuss the opportunity rather than the possibility to visit the metaphysical virtual space. In that sense, if it is possible, should we grab the opportunity and expend our science beyond physics, into the metaphysics? (shrink)

Multiple-conclusion Hilbert-style systems allow us to finitely axiomatize every logic defined by a finite matrix. Having obtained such axiomatizations for Paraconsistent Weak Kleene and Bochvar–Kleene logics, we modify them by replacing the multiple-conclusion rules with carefully selected single-conclusion ones. In this way we manage to introduce the first finite Hilbert-style single-conclusion axiomatizations for these logics.

In [4] the lattice of all subvarieties of the variety G n Ex defined by so called externally compatible identities of Abelian groups together with the identity x n ≈ y n , for any n ∈ N and n ≥ 1 was described. In that paper classes of models of the type (2,1) where considered. It appears that diagrams of lattices of subvariaties defined by externally compatible identities satisfied in a given equational theory depend on the language of the (...) considered class of algebras. A question was asked to what extent the diagram of the lattice of subvarieties of the variety defined by externally compatible identities of a given variety will depend on changing the type of algebras. In general case, the answer to this question seems to be very complicated. In this paper we describe the variety of Abelian groups of exponent p·q, where p,q are different primes of type (2,1,0). (shrink)

This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete (...) sequent calculi for S*fde, dS*fde, crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define containment logics, we explore the single-premise/single-conclusion fragment of S*fde, dS*fde, crossS*fdeand the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues. (shrink)

The most natural and straightforward step beyond two‐valued logic is to introduce more logical values, thereby rejecting the principle of bivalence. Another, indirect, way consists in challenging the classical laws concerning the sentence connectives and introducing other non‐two‐valued connectives into the language. Either way, prepositional logic seems fundamental to many‐valuedness, rather than its first‐order extension. Hence, although there has been interesting research into first‐order many‐valued logics, we shall confine our discussion here to the 0‐order case.

This paper presents a semantical analysis of the Weak Kleene Logics Kw3 and PWK from the tradition of Bochvar and Halldén. These are three-valued logics in which a formula takes the third value if at least one of its components does. The paper establishes two main results: a characterisation result for the relation of logical con- sequence in PWK – that is, we individuate necessary and sufficient conditions for a set.

In his book Rethinking the Good: Moral Ideals and the Nature of PracticalReasoning Larry Temkin contrasts two views of ideals for evaluating outcomes:the Internal Aspects View and the Essentially Comparative View. He claimsthat the latter view can make the relation of being better/worse than all thingsconsidered nontransitive, while the former can’t. This paper argues that theInternal Aspects View can also be a source of nontransitivity. The gist of theargument is that perfect similarity as regards supervenient properties, likevalue, is compatible with (...) differences as regards their subvenient propertiesand that it’s logically possible that such sets of insufficient differences add upto differences that are sufficient for supervenient differences. Thus, perfectsimilarity or identity is nontransitive as regards the supervenient property ofvalue, and this implies that the relation of being better/worse than all thingsconsidered is also nontransitive. (shrink)

A long series of conversations interweaving mathematical, historical and philosophical aspects of categoricity in model theory took place between the author and Saharon Shelah in 2016 and 2017. In this excerpt of that long conversation, we explore the relationship between explicit and implicit aspects of categoricity. We also discuss the connection with definability issues.

A good metaphysical account of logical form, must make clear why logical form is logical. However, this task has proved to be very elusive. Here, I analyze different attempts to meet this challenge and defend an inferential externalism where logical form is grounded on external logical relations as our most promising option.

Intensional logic is the technical study of such intensional phenomena in human reasoning as modality, knowledge, or flow of time. These all require a richer semantic picture than standard truth values in one static environment. Such a picture is provided by so-called possible worlds semantics, a paradigm which is surveyed in this book, both as to its external sources of motivation and as to the internal dynamics of the resulting program. In particular, Manual of Intensional Logic presents (...) the major classical topics, including modal logic, tense logic, and conditional logic, all of which illustrate motivations coming from philosophy and linguistics. The Book also discusses recent computational applications in computer science and AI. Finally, Manual of Intensional Logic takes up recent developments in the study of language and information making themselves felt in the area. The Book examines the role of partial information--with illustrations drawn from different branches of Intensional Logic--and various influences stemming from current theories of the semantics of natural language, involving generalized quantifiers and theories of types. (shrink)

Even in mathematical logic the proposition as an arbitrary technical device can be eliminated and replaced by the constative. The items of experience of the external world—objects, properties, relationships, and the like— are processed in consciousness on the basis of a situational demand that defines the situational context which in turn elicits a particular constatival attitude. The output of this mental processing is the appropriate pragmatic constatival response that generally is publicly communicated.

Morality is the effort to guide one’s conduct by reasons, that is, to do what there are the best reasons for doing. From a cognitive perspective, there are many types of moral hypotheses that provide good reasons in practical and moral deliberation and action. They can take the form of principles, rules, prototypes, previous analogical cases, examples, images, feelings, metaphors, narratives, and so on. I will address the central problems of the logical structure of reasons and of inferring reasons taking (...) advantage of the fundamental distinction, suggested by Searle, between internal and external reasons. What, then, is the role of internal reasons? The answer is that they serve, in a certain way, to mediate between external and internal obligations by connecting them both logically and causally in practical and moral cognition and illustrating the relative importance of abductive cognition. Internal reasons might be based on a desire or an intention, for instance, while external reasons may be grounded in external obligations and duties that we may recognize as such. (shrink)

In this paper, I take a closer look at Hans Reichenbach’s relation to metaphysics and work out some interesting parallels between his account and that of the proponents of inductive metaphysics, a tradition that emerged in the mid- and late 19th century and the early 20th century in Germany. It is in particular Hans Reichenbach’s conception of the relation between the natural sciences and metaphysics, as displayed in his treatment of the question of the existence of the external world, (...) that shows some very interesting similarities with inductive metaphysics. By a comparison with the position of the inductive metaphysician Erich Becher and his handling of the problem of realism, I work out the parallels between Reichenbach’s program and inductive metaphysics. I come to the conclusion that while there are certainly some respects in which Reichenbach’s logical empiricism is closer to the positions of the representatives of the Vienna Circle, it turns out that with regard to his views on metaphysics there is a greater affinity with the program of inductive metaphysics. (shrink)

The paper presents a logical treatment of actions based on dynamic logic. This approach makes it possible to reflect clearly the differences between static and dynamic elements of the world, a distinction which seems crucial to us for a representation of actions.Starting from propositional dynamic logic a formal system (DLA) is developed, the programs of which are used to model action types. Some special features of this system are: Basic aspects of time are incorporated in DLA as far (...) as they are needed for our purpose. Names for states and for instants are simulated by formulas. It is possible to express formally that a formula is satisfiable or valid. A special program is introduced to reflect developments which are not caused by an official agent but by external influences. (shrink)

This chapter contains sections titled: Mill on Our Knowledge of the External World Mill on Our Knowledge of “Necessary” Truths Mill's “Reduction” of Deductive Reasoning to Inductive Reasoning Mill on the Ground of Inductive Reasoning Mill's Methods Further Reading.

In continuation of our study of HST, Hrbaek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st--language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe.We show that, given a standard cardinal , any set R * generates an "internal" class S(R) of all sets standard relatively to elements of R, and an " (...) class='Hi'>external" class L[S(R)] of all sets constructible (in a sense close to the Gödel constructibility) from sets in S(R). We prove that under some mild saturation-like requirements for R the class L[S(R)] models a certain -version of HST including the principle of +-saturation; moreover, in this case L[S(R)] is an elementary extension of L[S(R)] in the st--language whenever sets R R satisfy the requirements. (shrink)