We discuss in this paper the scope of abduction in Economics. The literature on this type of inference shows that it can be interpreted in different ways, according to the role and nature of its outcome. We present a formal model that allows to capture these various meanings in different economic contexts.

Dung’s argumentation frameworks have been applied for over twenty years to the analysis of argument justification. This representation focuses on arguments and the attacks among them, abstracting away from other features like the internal structure of arguments, the nature of utterers, the specifics of the attack relation, etc. The model is highly attractive because it reduces most of the complexities involved in argumentation processes. It can be applied to different settings, like the argument evaluation of an individual agent or the (...) case of dialectic disputes between two agents (pro and con), or even in multi-agent collective argumentation. The latter case involves agents with possibly different arguments and/or opinions on how to evaluate them, leading to the possibility of considering multiple sets of arguments and attack relations. Two basic questions can be asked here, namely ‘what to aggregate’ and ‘how to aggregate’. The former concerns what kinds of entities do the agents intend to choose (arguments, attacks, assessments, etc.), while the second one focuses on which aggregation mechanisms yield rational choices (voting on arguments, merging procedures to obtain a common argumentation framework, deliberation processes, etc.). In particular, the question about the rationality of a collective argument choice relates this topic to Social Choice and Judgment Aggregation theories, while its associated strategic issues relate it to Game Theory. The research efforts on the disparate problems elicited by collective argumentation have generated a considerable corpus of literature that deserves an orderly evaluation. This survey is intended as a contribution to that end. (shrink)

The problems that arise from the presence of self-attacking ar- guments and odd-length cycles of attack within argumentation frameworks are widely recognized in the literature on defeasible argumentation. This paper introduces two simple semantics to capture different intuitions about what kinds of arguments should become justified in such scenarios. These semantics are modeled upon two extensions of argumentation frameworks, which we call sustainable and tolerant. Each one is constructed on the common ground of the powerful concept of admissibility introduced by (...) Dung in [10]. The novelty of this approach consists in viewing the admissibility of a subset of arguments as relative to potentially challenging subsets of arguments. Both sustainable and tolerant semantics are more credulous than preferred semantics (i.e. they justify at least the same arguments, and possibly more). Given certain sufficient conditions they coincide among them as well as with other semantics introduced by Dung. (shrink)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic (Peirce’s Existential Graphs α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}). The straightforward construction of a presheaf category EGα∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_{\alpha ^{*}}$$\end{document} of cuts-only Existential Graphs (equivalent to the well-studied category of finite forests) provides a basis for the further construction of the category EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {E}}}{{\mathcal {G}}}_\alpha $$\end{document} which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the formal investigation of relations between syntax and semantics in such diagrammatic systems. (shrink)

We present a formal analysis of Douglas Hofstadter’s concept of superrationality. We start by defining superrationally justifiable actions, and study them in symmetric games. We then model the beliefs of the players, in a way that leads them to different choices than the usual assumption of rationality by restricting the range of conceivable choices. These beliefs are captured in the formal notion of type drawn from epistemic game theory. The theory of coalgebras is used to frame type spaces and to (...) account for the existence of some of them. We find conditions that guarantee superrational outcomes. (shrink)

We explore the relationship between evidence and knowledge when knowledge is described by a standard partition over a finite state space, and evidence is represented by a state-dependent collection of finite sets of messages. When the collection is measurable with respect to the partition, there is evidence for an event only if the event is self-evident—i.e., known at every one of its states. Thus, an event is commonly known in and only if there is _mutual_ evidence that implies that the (...) event has occurred, and all knowledge can be proved only when the agent is informed about the state or knows nothing. The existence of partial provability outside these two extremes hinges on the non monotonic nature of provability. (shrink)

In this paper we present an embedding of abstract argumentation systems into the framework of Barwise and Seligmans logic of information flow. We show that, taking P.M. Dungs characterization of argument systems, a local logic over states of a deliberation may be constructed. In this structure, the key feature of non-monotonicity of commonsense reasoning obtains as the transition from one local logic to another, due to a change in certain background conditions. Each of Dungs extensions of argument systems leads to (...) a corresponding ordering of background conditions. The relations among extensions becomes a relation among partial orderings of background conditions. This introduces a conceptual innovation in Barwise and Seligmans representation of commonsense reasoning. (shrink)

Peirce’s diagrammatic system of Existential Graphs (EGα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha })$$\end{document} is a logical proof system corresponding to the Propositional Calculus (PL). Most known proofs of soundness and completeness for EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document} depend upon a translation of Peirce’s diagrammatic syntax into that of a suitable Frege-style system. In this paper, drawing upon standard results but using the native diagrammatic notational framework of the graphs, we present (...) a purely syntactic proof of soundness, and hence consistency, for EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document}, along with two separate completeness proofs that are constructive in the sense that we provide an algorithm in each case to construct an EGα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$EG_{\alpha }$$\end{document} formal proof starting from the empty Sheet of Assertion, given any expression that is in fact a tautology according to the standard semantics of the system. (shrink)

The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the (...) formal investigation of relations between syntax and semantics in such diagrammatic systems. (shrink)

We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the (...) formal investigation of relations between syntax and semantics in such diagrammatic systems. (shrink)

A formal model of abductive inference is provided in which abduction is conceived as expansive and contractive movements through a topological space of theoretical and practical commitments. A pair of presheaves over the space of commitments corresponds to communities sharing commitments on the one hand and possible obstructions to commitments on the other. In this framework, abductive inference is modeled by the dynamics of redistributed communities of commitment made in response to obstructive encounters. This semantic-pragmatic model shows how elementary category (...) theory tools can be used to formalize abductive inference while hewing close to ordinary intuitions about collective agency and reasoning. (shrink)

Kaushik Basu presenta el dilema del viajero como un desafío a la teoría de juegos. El desafío ha sido investigado experimentalmente. Al enfrentarse a la versión de Basu del DV o similares, los participantes se comportan como sugiere Basu. Sin embargo, un pequeño cambio en el juego tiene como consecuencia revertir las tendencias de elección. La cuestión es, entonces, si es posible brindar una explicación de los principales hallazgos empíricos como consecuencias de decisiones racionales. Hay varias propuestas en la literatura (...) pero ninguna provee una explicación satisfactoria de por qué expertos en teoría de juegos jugando entre sí usualmente rechazan la única estrategia no dominada del DV. El objetivo de este artículo es sugerir una propuesta alternativa que solucione este problema. (shrink)

Many authors in the discipline as well as outsiders have claimed that the main results from Mathematical Economics are far removed from real world phenomena. A more precise version of this position is that one of the main reasons for this unrealistic stance is the use of the wrong formal tools. So, for example, it has been pointed out that the computability of choice functions as well as the existence of economic equilibria and of states of the world may not (...) be ensured in general if the assumed set theory is ZFC. We will show that there exists a very natural set theory that overcomes some formal limitations of contemporary economic theory. A switch to an alternative set theory helps to obtain in a more natural way results widely accepted by mathematical economists. Moreover, alternative set-theoretical frameworks convey different intuitions about how agents behave when solving problems. We claim that AFA−+AD+DC is the adequate alternative set-theoretical universe for economic theory. (shrink)

Iterated admissibility embodies a minimal criterion of rationality in interactions. The epistemic characterization of this solution has been actively investigated in recent times: it has been shown that strategies surviving \ rounds of iterated admissibility may be identified as those that are obtained under a condition called rationality and m assumption of rationality in complete lexicographic type structures. On the other hand, it has been shown that its limit condition, with an infinity assumption of rationality ), might not be satisfied (...) by any state in the epistemic structure, if the class of types is complete and the types are continuous. In this paper we analyze the problem in a different framework. We redefine the notion of type as well as the epistemic notion of assumption. These new definitions are sufficient for the characterization of iterated admissibility as the class of strategies that indeed satisfy \. One of the key methodological innovations in our approach involves defining a new notion of generic types and employing these in conjunction with Cohen’s technique of forcing. (shrink)

The usual procedure in the theory of social choice consists in postulating some desirable properties which an aggregation procedure should verify and derive from them the features of a corresponding social choice function and the outcomes that arise at each possible profile of preferences. In this paper we invert this line of reasoning and try to infer, up from what we call social situations the criteria verified in the implicit aggregation procedure. This inference process, which extracts intensional from extensional information (...) can be seen as an exercise in “qualitative statistics”. (shrink)

In economically meaningful interactions negotiations are particularly important because they allow agents to improve their information about the environment and even to change accordingly their own characteristics. In each step of a negotiation an agent has to emit a message. This message conveys information about her preferences and endowments. Given that the information she uses to decide which message to emit comes from beliefs generated in previous stages of the negotiation, she has to cope with the uncertainty associated with them. (...) The assessment of the states of the world also evolves during the negotiation. In this paper we analyze the intertwined dynamics of beliefs and decision, in order to determine conditions on the agents that allow them to reach agreements. The framework for decision making we consider here is based on defeasible evaluation of possibilities: an argument for a choice defeats another one if it is based on a computation that better uses all the available information. (shrink)

In this paper we present a concept of similarity in games, on which to ground alternative solution concepts, some of which differ from the classical notions in the field. In order to do this we impose a constraint on players’ beliefs that amounts to a variant of the well-known symmetry principle in classical bargaining theory. We show how this similarity relation helps to identify different Nash equilibria in games, and how these “similar Nash equilibria” can be extended to non-symmetric games. (...) While the notion is normative, it is nonetheless inspired by phenomena in which similarities between players lead to outcomes detected in behavioral studies. We study the strategic properties of the concept of similarity and discuss its relationships with Hofstadter’ notion of superrationality. (shrink)

In this paper we show that any reasoning process in which conclusions can be both fallible and corrigible can be formalized in terms of two approaches: (i) syntactically, with the use of defeasible reasoning, according to which reasoning consists in the construction and assessment of arguments for and against a given claim, and (ii) semantically, with the use of partial structures, which allow for the representation of less than conclusive information. We are particularly interested in the formalization of scientific reasoning, (...) along the lines traced by Lakatos’ methodology of scientific research programs. We show how current debates in cosmology could be put into this framework, shedding light on a very controversial topic. (shrink)