The first paper in this pair (Bickhard in Axiomathes, 2015) developed a model of the nature of representation and cognition, and argued for a model of the micro-functioning of the brain on the basis of that model. In this sequel paper, starting with part III, this model is extended to address macro-functioning in the CNS. In part IV, I offer a discussion of an approach to brain functioning that has some similarities with, as well as differences from, the model (...) presented here: sometimes called the Predictive Brain approach. (shrink)

This is the continuation of the paper :417–446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, we consider extensions (...) of the language with lattice connectives and generalized disjunctions, study their interplay with implication and obtain axiomatizations and further descriptions of semilinear logics in terms of disjunctions and the proof by cases property. (shrink)

A typical interpreted formal language has (first‐order) variables that range over a collection of objects, sometimes called a domain‐of‐discourse. The domain is what the formal language is about. A language may also contain second‐order variables that range over properties, sets, or relations on the items in the domain‐of‐discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second‐order variable ranging over qualities. Similarly, (...) the sentence ‘there is a property that holds of all and only the prime numbers’ has a variable ranging over properties of natural numbers. Third‐order variables range over properties of properties, sets of sets, functions from properties to sets, etc. For example, according to some logicist accounts, the number 4 is the property shared by all properties that apply to exactly four objects in the domain. Accordingly, the number 4 is a third‐order item. Fourth‐order variables, and beyond, are characterized similarly. The phrase ‘higher‐order variable’ refers to the variables beyond first‐order. (shrink)

[p.225] Introduction (i) Although the following essay attempts to deal in a connected way with a number of connected conceptual tangles, it is by no means monolithic in design. It divides roughly in two, with the first half (Parts I and II) devoted to certain puzzles which have their source in a misunderstanding of the more specific structure of the language in which we describe and explain natural phenomena; while the second half (Parts III and IV) attempts to resolve the (...) more sweeping controversy over the nature of the connection between 'cause' and 'effect,' or, in modem dress, the logical status of 'lawlike statements.' (ii) The essay begins with a case analysis of a puzzle, taken from recent philosophical literature, relating to the analysis of counterfactual conditionals, statements of the form "If that lump of salt had been put in water, it would have dissolved." The diagnosis of this puzzle, which occupies the whole of Part I, shows it to rest on a misunderstanding of the conceptual framework in terms of which we speak of what things do when acted upon in certain ways in certain kinds of circumstance. Although the puzzle is initially posed in terms of examples taken from everyday life, the logical features of these examples which, misunderstood, generate the puzzle, are to be found in even the more theoretical levels of the language of science, and the puzzle is as much at home in the one place as in the other. For the framework in which things of various kinds (e.g. matches, white rats) behave ('respond') in various ways (catch fire, leap at a door) when acted upon ('submitted to such and such stimuli') under given conditions (presence of oxygen, 24 hours of food deprivation) is far more basic than the distinctions between metrical and non-metrical concepts, molar and micro-things, [p.226] observable and unobservable.. (shrink)

These notes contain the material covered in the second level logic course which has been offered at the Institut für Maschinelle Sprachverarbeitung of the University of Stuttgart on an annual basis since 1992. The course is aimed at students who are familiar with the notation and use of the first order predicate calculus but have had little or no previous exposure to metamathematics.

The author attempts to link the social and biological sciences in a new way. It appears that there are Social Logics which are cross-cultural and seem to have a biological substrate. This book shows how the biological baseline can be used for research into social, cultural and other forms of organization. It includes three case studies which use Logics Analysis to illustrate how this method can be successfully applied to group structure and process. Parker discusses evidence for Social Logics through (...) principles of language, art, myth, literature and music. Contents: The Social Logics Paradigm; Issues about Interpretation and Method; Social Logics: Analysis of "Real Situations"; Conclusion. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Croire et savoir: Les principes de la connaissance selon Nicolas d’AutrécourtDallas G. Denery IIChristophe Grellard. Croire et savoir: Les principes de la connaissance selon Nicolas d’Autrécourt. Paris: J. Vrin, 2005. Pp. 313. Paper, €38,00.Nicholas of Autrecourt has often seemed to be one of those philosophers doomed to be best known for everything but their own ideas. Famously, if inaccurately, dubbed "the Medieval Hume" by one of his first (...) modern critics, the threat or fear of skepticism has in one way or another shaped most subsequent studies of Nicholas's work. While understandable, this really is unfortunate. Nicholas himself clearly states that his philosophical goal was to avoid the skeptical consequences that he believed were latent in the writings of his scholastic peers. In this excellent new work, the first monograph-length study devoted to Nicholas's thought in nearly 60 years, Christophe Grellard goes a long way towards recovering Nicholas's very real contributions to fourteenth-century intellectual life.Any attempt to interpret Nicholas's thought runs up against one significant problem—few of his works survive. All that remains are a fascinating and lengthy (not to mention, confusing and unfinished) work of natural philosophy, Exigit ordo, and several letters touching on issues in epistemology, cognition, and logic. As a result, even situating Nicholas's thought within the proper intellectual currents of the first decades of the fourteenth century has proven difficult. Whose work influenced him? Who were the objects of his criticism? What were his motives? Until recently, most scholars have attempted to trace the influence of William Ockham on Nicholas's thought. Drawing from recent work on the University of Paris during the fourteenth century, Grellard not only downplays Ockham's influence on Nicholas, but also Nicholas's membership in the faculty of theology. Rather, he stresses Nicholas's ongoing involvement in the arts faculty, where he continued to teach even as he studied for his degree in theology. This has two significant consequences. First, it allows Grellard to treat all of Nicholas's writings as contributions to the practice of natural philosophy. Indeed, in his compelling first chapter, Grellard contends that the initial sections of the Exigit, in which Nicholas famously condemns scholastic Aristotelianism as nothing more than an institutionalized pedagogy of error and untruth, must be read as a plea for a distinction between philosophy and theology, between speculation and faith. Even if Nicholas was not alone at the time in pushing for these distinctions (Ockham and John Buridan had argued along similar lines), Grellard demonstrates that Nicholas's approach was certainly the more radical and extreme. Second, the shift in focus from theology to arts brings with it a shift in Nicholas's most important interlocutors. To begin, Aristotle and Averroes become constant background forces in Nicholas's thought. Moreover, as the final section of Grellard's work makes clear, many of Nicholas's ideas are most usefully understood in light of writings of two other arts masters, John Buridan and Nicole Oresme, rather than in the context of debates among his contemporaries in the theology faculty. [End Page 119]These initial interpretive decisions shape the overall structure of Grellard's book, which moves, in three parts, from a deft analysis of Nicholas's theory of evidence and probability to his account of the objects of cognition and, finally, to a comparison of Nicholas's and Buridan's epistemological theories. Unlike so many prior studies that begin with Nicholas's famous critique of a fellow theologian's cognitive theories, thereby framing Nicholas's project in terms of theological debates concerning the intuitive cognition of non-existents and God's ability to cause false visions, Grellard begins with Nicholas's positive theory of knowledge. This effectively sidesteps the entire question of Nicholas's skepticism, a question Nicholas himself explicitly tried to avoid. Nicholas comes across as a philosopher keenly aware of the limits of human cognition, yet eager to develop a viable and wholly rational epistemology within those limits. Those limits, Grellard makes clear, have everything to do with appearances, with what we can rationally and legitimately infer about the world based upon what we see... (shrink)

This paper gives an account of Anderson and Belnap’s selection criteria for an adequate theory of entailment. The criteria are grouped into three categories: criteria pertaining to modality, those pertaining to relevance, and those related to expressive strength. The leitmotif of both this paper and its prequel is the relevant legitimacy of disjunctive syllogism. Relevant logics are commonly held to be paraconsistent logics. It is shown in this paper, however, that both E and R can be extended to explosive logics (...) which satisfy all of Anderson and Belnap’s selection criteria, provided the truth-constant known as the Ackermann constant is available. One of the selection criteria related to expressive strength is having an “enthymematic” conditional for which a deduction theorem holds. I argue that this allows for a new interpretation of Anderson and Belnap’s take on logical consequence, namely as committing them to pluralism about logical consequence. (shrink)

Suppose there are several experts, with some dominating others (expert A dominates expert B if B says something is true whenever A says it is). Suppose, further, that each of the experts has his or her own view of what is possible — in other words each of the experts has their own Kripke model in mind (subject, of course, to the dominance relation that may hold between experts). How will they assign truth values to sentences in a common modal (...) language, and on what sentences will they agree? This problem can be reformulated as one about many-valued Kripke models, allowing many-valued accessibility relations. This is a natural generalization of conventional Kripke models that has only recently been looked at. The equivalence between the many-valued version and the multiple expert one will be formally established. Finally we will axiomatize many-valued modal logics, and sketch a proof of completeness. (shrink)

We examine some assumptions about the nature of 'levels of reality' in the light of examples drawn from physics. Three central assumptions of the standard view of such levels (for instance, Oppenheim and Putnam 1958) are (i) that levels are populated by entities of varying complexity, (ii) that there is a unique hierarchy of levels, ranging from the very small to the very large, and (iii) that the inhabitants of adjacent levels are related by the parthood relation. Using examples from (...) physics, we argue that it is more natural to view the inhabitants of levels as the behaviors of entities, rather than entities themselves. This suggests an account of reduction between levels, according to which one behavior reduces to another if the two are related by an appropriate limit relation. By considering cases where such inter-level reduction fails, we show that the hierarchy of behaviors differs in several respects from the standard hierarchy of entities. In particular, while on the standard view, lower-level entities are 'micro' parts of higher-level entities, on our view, a system's macro-level behavior can be seen as a ('non-spatial') part of its micro-level behavior. We argue that this second hierarchy is not really in conflict with the standard view and that it better suits examples of explanation in science. (shrink)

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of (...) meet-prime logics only. (shrink)

This paper surveys the various forms of Deduction Theorem for a broad range of relevant logics. The logics range from the basic system B of Routley-Meyer through to the system R of relevant implication, and the forms of Deduction Theorem are characterized by the various formula representations of rules that are either unrestricted or restricted in certain ways. The formula representations cover the iterated form,A 1 .A 2 . ... .A n B, the conjunctive form,A 1&A 2 & ...A n (...) B, the combined conjunctive and iterated form, enthymematic version of these three forms, and the classical implicational form,A 1&A 2& ...A n B. The concept of general enthymeme is introduced and the Deduction Theorem is shown to apply for rules essentially derived using Modus Ponens and Adjunction only, with logics containing either (A B)&(B C) .A C orA B .B C .A C. (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)

This is the second paper in a two-part series introducing logics for reasoning about the dynamics of knowledge and beliefs. Part I introduced different logical systems that can be used to reason about the knowledge and beliefs of a group of agents. In this second paper, I show how to adapt these logical systems to reason about the knowledge and beliefs of a group of agents during the course of a social interaction or rational inquiry. Inference, communication and observation are (...) typical examples of informative events, which have been subjected to a logical analysis. The main goal of this article is to introduce the key conceptual and technical issues that drive much of the research in this area. (shrink)

This is a supplement to the paper “Finitary Algebraic Logic” [1]. It includes corrections for several errors and some additional results. MSC: 03G15, 03G25.

In this paper, possible objections to the propensity microrealistic version of quantum mechanics proposed in Part I are answered. This version of quantum mechanics is compared with the statistical, particle microrealistic viewpoint, and a crucial experiment is proposed designed to distinguish between these to microrealistic versions of quantum mechanics.

The categoricity problem for a system of logic reveals an asymmetry between the model-theoretic and the proof-theoretic resources of that logic. In particular, it reveals prima facie that the proof-theoretic instruments are insufficient for matching the envisaged model-theory, when the latter is already available. Among the proposed solutions for solving this problem, some make use of new proof-theoretic instruments, some others introduce new model-theoretic constrains on the proof-systems, while others try to use instruments from both sides. On the (...) proof-theoretical side, two main approaches for solving the categoricity problem for propositional classical logic consist in the enforcement of the formal systems of this logic by introducing rules of inference of a new kind. A multiple-conclusions logic allows rules of inference with more than one conclusion, while a bilateralist system contains rules of inference formulated in terms of assertion and rejection. Both these approaches reveal interesting formal features of logical reasoning. This paper analyses some of the advantages and limitations of these two approaches as solutions for a full formalization of logic, i.e., for a categorical formalization. (shrink)

The paper studies closure properties of classes of fuzzy structures defined by fuzzy implicational theories, i.e. theories whose formulas are implications between fuzzy identities. We present generalizations of results from the bivalent case. Namely, we characterize model classes of general implicational theories, finitary implicational theories, and Horn theories by means of closedness under suitable algebraic constructions.

We consider a syntax and semantics of modal logics based on parametrized modal connectives with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \forall $$\end{document}-satisfaction definitions, we axiomatically introduce different parametrized modal logics, we prove their completeness with respect to appropriate classes of parametrized relational structures and we show the decidability of some related satisfiability problems.

In part I, we presented an algebraic-style of semantics, which we called “content semantics,” for quantified relevant logics based on the weak systemBBQ. We showed soundness and completeness with respect to theunreduced semantics ofBBQ. In part II, we proceed to show soundness and completeness for extensions ofBBQ with respect to this type of semantics. We introducereduced semantics which requires additional postulates for primeness and saturation. We then conclude by showing soundness and completeness forBB d Q and its extentions with respect (...) to this reduced semantics. (shrink)

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew. The main result of Part I of this series [41] shows that the equivalent variety semantics of N and the equivalent variety semantics of NFL ew are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting (...) of deductive systems to establish the definitional equivalence of the logics N and NFL ew. It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logic. (shrink)

A view of evolution is presented in this paper (a two paper series), intended as a methodological infrastructure for modeling spatio-cultural systems (the design outline of such a model is presented in paper II). A motivation for the re-articulation of evolution as information dynamics is the phenomenologically discovered prerequisite of embedding a meaning-attributing apparatus in any and all models of spatio-cultural systems. An evolution is construed as the dynamics of a complex system comprised of memory devices, connected in an ordered (...) fashion (not randomly) by information-exchanges. An information-exchange transpires when the recipient system adopts a strategy (a continuum of events) that eventually changes its structure; namely, after the exchange, it contains and conveys different information. These memory devices—sub-systems—are also similarly constructed complex system. Only a part of the information is retained by a system in its physical-memory storage, which eventually loses this function too, when the ability to retrieve a common enough structure is lost. The entire amount of information is a system’s structure of connections (information exchanges); it is contained (apparently stored) dynamically when a system is observed in a temporarily stable state. This temporary permanence—robustness and resilience—is attained dynamically; namely, enabled by changes taking place in its sub-systems and in each of their sub-systems. Therefore, for modeling such a system a multi-layer/multi-scale approach is preferable. It enables the addition and subtraction of an interim scale and the consideration of such a scale as a micro or a macro (thus initiating a maneuver up or down scale) according to an ad-hoc requirement of the model, which imitates an envisaged sub system (called an ‘Inner World’), to which a certain range of decisions is relegated. This dynamics is driven (in time) by dynamically originated information growth, as defined by Shannon; i.e. by the fact that each of certain state transitions have occurred sometime in the past of the system just so and not otherwise, and by the fact that they have occurred in a certain order. Therefore, each ‘history’ and memory retrieval availability of information is unique, and thus can be used to differentiate meanings. Hence, there cannot be a comprehensive solution to the meaning attribution model-design challenge. However, the observation that at the core of each envisaged complex system, moving in time according to a rounded logic, there is an information manipulating device, operating necessarily according to a Boolean logic, can be copied into the design of a model. This observation enables, therefore, the embedding of a specific, locally fitted, meaning attributing device, which is an information manipulating mechanism (it splices/attaches one segment of information to another—its meaning). However, this is just a framework; the actual solution has still to be found locally—for each subject system. Such a solution is demonstrated for the change in location or layout in the Israeli city in paper II. (shrink)

The?-admissibility is one of the most important problems in the realm of relevant logics. To prove the 7-admissibility, either the method of normal models or the method using metavaluations may be employed. The?-admissibility of a wide class of relevant modal logics has been discussed in Part I based on a former method, but the?-admissibility based on metavaluations has not hitherto been fully considered. Sahlqvist axioms are well known as a means of expressing generalized forms of formulas with modal operators. This (...) paper shows that? is admissible for relevant modal logics with restricted Sahlqvist axioms in terms of the method using metavaluations. (shrink)

The main goal of this paper is to explain the link between the algebraic models and the Kripke-style models for certain classes of propositional non-classical logics. We consider logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and present a way of constructing topological and non-topological Kripke-style models for these types of logics. Moreover, we show that, under certain additional assumptions on the variety of (...) the algerabic models of the given logics, soundness and completeness with respect to these classes of Kripke-style models follows by using entirely algebraical arguments from the soundness and completeness of the logic with respect to its algebraic models. (shrink)

First the expansion of the Łukasiewicz logic by the unary connectives of dividing by any natural number is studied; it is shown that in the predicate case the expansion is conservative w.r.t. witnessed standard 1-tautologies. This result is used to prove that the set of witnessed standard 1-tautologies of the predicate product logic is Π2-hard.

In a series of joint papers, Teppo Felin and Nicolai J. Foss recently launched a microfoundations project in the field of strategic management. Felin and Foss observe that extant explanations in strategic management are predominantly collectivist or macro. Routines and organizational capabilities, which are supposed to be properties of firms, loom large in the field of strategic management. Routines figure as explanantia in explanations of firm behavior and firm performance, for example. Felin and Foss plead for a replacement of such (...) macro-explanations by micro-explanations. Such a replacement is needed, Felin and Foss argue, because macro-explanations are necessarily incomplete: they miss out on crucial links in the causal chain that connect macro phenomena with each other. I argue that this argument is flawed. It is based on a doubtful if not outright incorrect understanding and use of Coleman’s diagram. In a sense to be explained below, only if Coleman’s diagram is squared it can accurately account for the relations between individual action and interaction, routines and firm behavior and firm performance. Once Coleman’s diagram is squared, one can see why and how macro-explanations need not miss out on any link in the causal chains that connect macro phenomena. Micro-analyses are still needed, not to highlight and specify causal links that macro-explanations miss out on, but to check whether the many properties that are ascribed to routines in macro-explanations of firm behavior are warranted. (shrink)

This experimental study provides further support for a theory of meaning first put forward by Bar-Hillel and Carnap in 1953 and foreshadowed by Asimov in 1951. The theory is the Popperian notion that the meaningfulness of a proposition is its a priori falsity. We tested this theory in the first part of this paper by translating to logical form a long, tightly written, published text and computed the meaningfulness of each proposition using the a priori falsity measure. We then selected (...) the top propositions—by a priori falsity—and strung them together to form ad hoc abstracts and compared these abstracts with the published summary. The results were startling: translation to logical form, followed by application of the Asimovian idea and Bar-Hillel/Carnap mathematics as elaborated into an AI/NLP proposal in Fulda (1986, 1988), produced excellent abstracts, thereby providing a proof-of-concept that merely by knowing the logical form of long text passages, one can produce reasonable abstracts of them—without actually understanding the text. We here report on a second experiment analyzing, in the exact same manner, the correspondence that followed the published text of the first experiment. While the results of this confirming experiment are less startling, they nevertheless provide additional confidence in the promise of the technique. In other words, were the results of these two experiments to generalize, that would show that logical form captures much more semantics than has heretofore been considered likely. Far from (as is commonly supposed) being merely the syntactical rewrite of text into formal notation, translation to logical form, even when undertaken with almost no knowledge about the particular predicates, individual constants, or other objects referred to in that form, might capture the core of the meaning in some important sense. -/- Note: There is a key difference in the /style/ of the texts analyzed in I and II in this long paper, divided into two only for practical reasons—space; the first text analyzed is written in a tight mathematical style, while the second is written in the discursive style of philosophy. -/- Finally, the unusual (for me) length of this two-part paper is justified by its being data-driven. (shrink)

It is shown how to model propositional constants within the simplified Routley-Meyer semantics. Various axioms and rules allowingthe definition of modal operators, implicative negations, enthymematical conditionals, and propositions expressing various infinite conjunctions anddisjunctions are set forth and shown to correspond to specific frame conditions. Two propositional constants which are both often designated as “the Ackermann constant” are shown to capture two such “infinite” propositions: The conjunction of every logical law and the conjunction of every truth – what Anderson and Belnap (...) called the “world” constant. (shrink)

We introduce a new operator – belief fusion– which aggregates the beliefs of two agents, each informed by a subset of sources ranked by reliability. In the process we definepedigreed belief states, which enrich standard belief states with the source of each piece of information. We note that the fusion operator satisfies the invariants of idempotence, associativity, and commutativity. As a result, it can be iterated without difficulty. We also define belief diffusion; whereas fusion generally produces a belief state with (...) more information than is possessed by either of its two arguments, diffusion produces a state with less information. Fusion and diffusion are symmetric operators, and together define a distributive lattice. Finally, we show that AGM revision can be viewed as fusion between a novice and an expert. (shrink)