Two Early Arabic Applications of Model-Theoretic Consequence

We trace two logical ideas further back than they have previously been traced. One is the idea of using diagrams to prove that certain logical premises do—or don’t—have certain logical consequences. This idea is usually credited to Venn, and before him Euler, and before him Leibniz. We find the idea correctly and vigorously used by Abū al-Barakāt in 12th century Baghdad. The second is the idea that in formal logic, P logically entails Q if and only if every model of P is a model of Q. This idea is usually credited to Tarski, and before him Bolzano. But again we find Abū al-Barakāt  already exploiting the idea for logical calculations. Abū al-Barakāt’s work follows on from related but inchoate research of Ibn Sīnā in eleventh century Persia. We briefly trace the notion of model-theoretical consequence back through Paul the Persian (sixth century) and in some form back to Aristotle himself.     

Herbrand's theorem and term induction

We study the formal first order system TIND in the standard language of Gentzen's LK . TIND extends LK by the purely logical rule of term-induction, that is a restricted induction principle, deriving numerals instead of arbitrary terms. This rule may be conceived as the logical image of full induction.     

A case study in logical deconstruction: Formalizing J.D. Thompson's Organizations in Action in a multi-agent action logic

Logic is a popular word in the social sciences, but it is rarely used as a formal tool. In the past, the logical formalisms were cumbersome and difficult to apply to domains of purposeful action. Recent years, however, have seen the advance of new logics specially designed for representing actions. We present such a logic and apply it to a classical organization theory, J.D. Thompson's Organizations in Action. The working hypothesis is that formal logic draws attention to some finer points in the logical structure of a theory, points that are easily neglected in the discursive reasoning typical for the social sciences. Examining Organizations in Action we find various problems in its logical structure that should, and, as we argue, could be addressed.     

Logic, language games and ludics

Wittgenstein’s language games can be put into a wider service by virtue of elements they share with some contemporary opinions concerning logic and the semantics of computation. I will give two examples: manifestations of language games and their possible variations in logical studies, and their role in some of the recent developments in computer science. It turns out that the current paradigm of computation that Girard termed Ludics bears a striking resemblance to members of language games. Moreover, the kind of interrelations that are emerging could be scrutinised from the viewpoint of logic that virtually necessitates game-theoretic conceptualisations, demonstrating the fact that the meaning of utterances may, in many situations, be understood as Wittgenstein’s language games of ‘showing or telling what one sees’. This provides motivation for the use of games in relation to logic and formal semantics that some commentators have called for. Many of the ideas can be traced to C.S. Peirce, for whom signs were vehicles of strategic communication. The conclusion about Wittgenstein is that the notions of saying and showing converge in his late philosophy.     

Theories and Theories of Truth

Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory.     

On the Meaning of Connectives (Apropos of a Non-Necessitarianist Challenge)

According to logical non-necessitarianism, every inference may fail in some situation. In his defense of logical monism, Graham Priest has put forward an argument against non-necessitarianism based on the meaning of connectives. According to him, as long as the meanings of connectives are fixed, some inferences have to hold in all situations. Hence, in order to accept the non-necessitarianist thesis one would have to dispose arbitrarily of those meanings. I want to show here that non-necessitarianism can stand, without disposing arbitrarily of the meanings of connectives, based on a minimalist view on the meanings of connectives.     

Gevrey character of formal solutions of differential—difference equations with polynomials coefficients

In this paper we prove that formal factorial series as well as formal power series in 1/x solutions of a differential—difference equation with polynomial coefficients are Gevrey of some order which can be determined from a suitable Newton polygon     

Leveraging Structure: Logical Necessity in the Context of Integer Arithmetic

Looking for, recognizing, and using underlying mathematical structure is an important aspect of mathematical reasoning. We explore the use of mathematical structure in children’s integer strategies by developing and exemplifying the construct of logical necessity. Students in our study used logical necessity to approach and use numbers in a formal, algebraic way, leveraging key mathematical ideas about inverses, the structure of our number system, and fundamental properties. We identified the use of carefully chosen comparisons as a key feature of logical necessity and documented three types of comparisons students made when solving integer tasks. We believe that logical necessity can be applied in various mathematical domains to support students to successfully engage with mathematical structure across the K–12 curriculum.     

A constructive investigation of satisfiability

We present a constructive analysis of the logical notions of satisfiability and consistency for first-order intuitionistic formulae. In particular, we use formal topology theory to provide a positive semantics for satisfiability. Then we propose a “co-inductive” logical calculus, which captures the positive content of consistency.     

Cognitive maps and fuzzy implications

A cognitive map is a collection of nodes linked by some arcs. Up to this point, there is unanimity in the literature about the previous definition. But if we look closer at the meaning of the nodes and links, we can see that there are crucial differences between the various authors. And these differences are not always explicit. In spite of this, it seems that many authors perform on the maps the same kind of analysis (strongly inspired by the book of Axelrod), even if these analyses are not consistent with their conception of a cognitive map. That is why it is important to clearly and formally define the kind of map used. In this paper, we propose a formal definition of a cognitive map relying on the concept of fuzzy implication. Thus in our framework, a node is a logical proposition and a link is an implication. Starting from our definition, we show some properties of this kind of maps and some analysis techniques.     

The Structuring Causes of Behavior: Has Dretske Saved Mental Causation?

Fred Dretske’s account of mental causation, developed in Explaining Behavior and defended in numerous articles, is generally regarded as one of the most interesting and most ambitious approaches in the field. According to Dretske, meaning facts, construed historically as facts about the indicator functions of internal states, are the structuring causes of behavior. In this article, we argue that Dretske’s view is untenable: On closer examination, the real structuring causes of behavior turn out to be markedly different from Dretske’s meaning facts. Our argument proceeds in three steps. First, we set forth the problem of meaning individuation: We argue that the proposal that meaning facts are structuring causes of behavior commits Dretske to a very fine-grained individuation of meanings that is deeply counterintuitive. In a second step, we show that even these finely individuated meaning facts cannot do the job that they are supposed to do, since information facts—which are constitutive of, but distinct from Dretske’s meaning facts—are better candidates for the role of structuring causes. Finally, we argue that it is not even information facts, but facts of co-instantiation which are the real structuring causes of behavior. In concluding, we briefly consider the options that are left for Dretske if our arguments succeed.     

New Logics for Quantum Non-individuals?

According to a very widespread interpretation of the metaphysical nature of quantum entities—the so-called Received View on quantum non-individuality—, quantum entities are non-individuals. Still according to this understanding, non-individuals are entities for which identity is restricted or else does not apply at all. As a consequence, it is said, such approach to quantum mechanics would require that classical logic be revised, given that it is somehow committed with the unrestricted validity of identity. In this paper we examine the arguments to the inadequacy of classical logic to deal with non-individuals, as previously defined, and argue that they fail to make a good case for logical revision. In fact, classical logic may accommodate non-individuals in that specific sense too. What is more pressing for the Received View, it seems, is not a revision of logic, but rather a more adequate metaphysical characterization of non-individuals.     

Carnap,Goguen, and the Hyperontologies: Logical Pluralism and Heterogeneous Structuring in Ontology Design

This paper addresses questions of universality related to ontological engineering, namely aims at substantiating (negative) answers to the following three basic questions: (i) Is there a ‘universal ontology’?, (ii) Is there a ‘universal formal ontology language’?, and (iii) Is there a universally applicable ‘mode of reasoning’ for formal ontologies? To support our answers in a principled way, we present a general framework for the design of formal ontologies resting on two main principles: firstly, we endorse Rudolf Carnap’s principle of logical tolerance by giving central stage to the concept of logical heterogeneity, i.e. the use of a plurality of logical languages within one ontology design. Secondly, to structure and combine heterogeneous ontologies in a semantically well-founded way, we base our work on abstract model theory in the form of institutional semantics, as forcefully put forward by Joseph Goguen and Rod Burstall. In particular, we employ the structuring mechanisms of the heterogeneous algebraic specification language HetCasl for defining a general concept of heterogeneous, distributed, highly modular and structured ontologies, called hyperontologies. Moreover, we distinguish, on a structural and semantic level, several different kinds of combining and aligning heterogeneous ontologies, namely integration, connection, and refinement. We show how the notion of heterogeneous refinement can be used to provide both a general notion of sub-ontology as well as a notion of heterogeneous equivalence of ontologies, and finally sketch how different modes of reasoning over ontologies are related to these different structuring aspects.     

Logic of agreement: Foundations,semantic system and proof theory

In this paper a multi-valued propositional logic — logic of agreement — in terms of its model theory and inference system is presented. This formal system is the natural consequence of a new way to approach concepts as commonsense knowledge, uncertainty and approximate reasoning — the point of view of agreement. Particularly, it is discussed a possible extension of the Classical Theory of Sets based on the idea that, instead of trying to conceptualize sets as “fuzzy” or “vague” entities, it is more adequate to define membership as the result of a partial agreement among a group of individual agents. Furthermore, it is shown that the concept of agreement provides a framework for the development of a formal and sound explanation for concepts (e.g. fuzzy sets) which lack formal semantics. According to the definition of agreement, an individual agent agrees or not with the fact that an object possesses a certain property. A clear distinction is then established, between an individual agent — to whom deciding whether an element belongs to a set is just a yes or no matter — and a commonsensical agent — the one who interprets the knowledge shared by a certain group of people. Finally, the logic of agreement is presented and discussed. As it is assumed the existence of several individual agents, the semantic system is based on the perspective that each individual agent defines her/his own conceptualization of reality. So the semantics of the logic of agreement can be seen as being similar to a semantics of possible worlds, one for each individual agent. The proof theory is an extension of a natural deduction system, using supported formulas and incorporating only inference rules. Moreover, the soundness and completeness of the logic of agreement are also presented.     

Locatedness and overt sublocales

Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected.Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact set is Bishop compact iff it is located. We translate this result to formal topology. ‘Bishop compact’ is translated as compact and overt. We propose a definition of locatedness on subspaces of a formal topology, and prove that a closed subspace of a compact regular formal space is located iff it is overt. Moreover, a Bishop-closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt.Finally, we show by elementary methods that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales.We work constructively, predicatively and avoid the use of the axiom of countable choice.     

Entropy,spatial interaction models and discrete choice analysis: Static and dynamic analogies

This paper aims at analysing the existence of a formal correspondence between spatial interaction models emanating from entropy theory and micro-economic discrete choice theory (in particular, multinomial logit models.). After a concise review of the literature on this issue, the emphasis is placed on an interpretation of formal analogies between both classes of models in a dynamic context. A simple dynamic spatial interaction model—based on optimal control theory—is proposed, and it is shown that the results confirm also the existence of a formal analogy between (macro) dynamic interaction models and (micro) choice models. Similar results are also derived for Alonso's general theory of movement in a spatial system.     

A multidimensional knapsack model for asset-backed securitization

Securitization is a financial operation which allows a financial institution to transform financial assets, for instance mortgage assets or lease contracts, into marketable securities. We focus the analysis on a real case of a bank for the leasing. Once the securitization characteristics, such as size and times of the operation, have been defined, the profit for the financial institution—Italease Bank for the Leasing in our case—depends on how the financial assets to use in the securitization are selected. We show that the selection problem can be modelled as a multidimensional knapsack problem (MDKP). Some formal arguments suggest that there may exist a prevailing constraint in the MDKP. Such an idea is used in the design of some simple heuristics which turn out to be very effective.     

Development of an Expert System: Translating Higher-Level Decisions into Lower-Level Aircraft Flights

This note describes two phases in the development of an expert system that translates European-wide directives into individual flight orders. This expert system was built for a military computer simulation, but parts of the logic can apply — and many of the illustrations in the text have been tailored — to creating and organizing commercial flights. The first phase of the development — a proof of principle expert system — emulated the logical deductions of an air traffic control supervisor or a flight planner. The second phase of development led to a working prototype decision support system that employed more mathematical comparisons of aircraft attributes than logical deductions.     

An introduction to the perplex number system

The perplex number system is a generalization of the abstract logical relationships among electrical particles. The inferential logic of the new number system is homologous to the inferential logic of the progression of the atomic numbers. An electrical progression is defined categorically as a sequence of objects with teridentities. Each identity infers corresponding values of an integer, units and a correspondence relation between each unit and its integer. Thus, in this logical system, each perplex numeral contains an exact internal representational structure; it carries an internal message. This structure is a labeled bipartite graph that is homologous to the internal electrical structure of a chemical atom. The formal logical operations are conjunctions and disjunctions. Combinations of conjunctions and disjunctions compose the spatiality of objects. Conjunctions may include the middle term of pairs of propositions with a common term, thereby creating new information. The perplex numerals are used as a universal source of diagrams.The perplex number system, as an abstract generalization of concrete objects and processes, constitutes a new exact notation for chemistry without invoking alchemical symbols. Practical applications of the number system to concrete objects (chemical elements, simple ions and molecules, and the perplex isomers, ethanol and dimethyl ether) are given. In conjunction with the real number system, the relationships between the perplex number system and scientific theories of concrete systems (thermodynamics, intra-molecular dynamics, molecular biology and individual medicine) are described.     

Expressivist Perspective on Logicality

Various attempts at demarcating logic were undertaken, many of them based on specific understanding of how logical knowledge is formal and not material. MacFarlane has persuasively shown that general idea of formality of logic can be understood in various ways. I take two of the accounts of formality, namely the requirement of conservativity and the requirement of schematicity of logical vocabulary, into consideration as promising candidates to make the all too unclear notion of formality more precise and study to what degree they could be considered as either necessary or sufficient conditions for logicality of some piece of vocabulary. Finding both notion unsatisfactory, as they stand, I propose combining them and envisage a hierarchy of logicality of expressions of a given language. Such a hierarchy is complicated and not linear, yet still offers a valuable explication of both the range and pragmatic significance of logic, if we combine it with logical expressivism.