In this paper, we introduce a new approach to independent quantifiers, as originally introduced in Informational independence as a semantic phenomenon by Hintikka and Sandu [9] under the header of independence-friendly languages. Unlike other approaches, which rely heavily on compositional methods, we shall analyze independent quantifiers via equilibriums in strategic games. In this approach, coined equilibrium semantics, the value of an IF sentence on a particular structure is determined by the expected utility of the existential player in any (...) of the game’s equilibriums. This approach was suggested in Henkin quantifiers and complete problems by Blass and Gurevich [2] but has not been taken up before. We prove that each rational number can be realized by an IF sentence. We also give a lower and upper bound on the expressive power of IF logic under equilibrium semantics. (shrink)

In this paper it is argued that functional role semantics can be saved from criticisms, such as those raised by Putnam and Fodor and Lepore, by indicating which beliefs and inferences are more constitutive in determining mental content. The Scylla is not to use vague expressions; the Charybdis is not to endorse the analytic/synthetic distinction. The core idea is to use reflective equilibrium as a strategy to pinpoint which are the beliefs and the inferences that constitute the content (...) of a mental state. The beliefs and the inferences that are constitutive are those that are in reflective equilibrium in the process of attributing mental states to others. (shrink)

This paper contains a survey of the main definitions and results obtained to date related to Temporal Equilibrium Logic, a nonmonotonic hybrid approach that combines Equilibrium Logic (the best-known logical characterisation for the stable models semantics of logic programs) with Linear-Time Temporal Logic.

We present a compositional semantics for first-order logic with imperfect information that is equivalent to Sevenster and Sandu’s equilibrium semantics (under which the truth value of a sentence in a finite model is equal to the minimax value of its semantic game). Our semantics is a generalization of an earlier semantics developed by the first author that was based on behavioral strategies, rather than mixed strategies.

The paper focuses on conceptual images of the market built on the core idea of equilibrium. It argues that logical difficulties and dead ends were encountered in building equilibrium models of the market. A heavy toll was paid in loss of realism, failure to understand history, and ineptitude in addressing relevant problems. Equilibrium economics is marked by the exclusion of a wide range of economic phenomena, with a codified lack of attention to change, transition and learning. The (...) loss of relevance involved was a consequence of the primitive idea of conceiving of the market within the semantic field of equilibrium. The most evident failure of the equilibrium imagery was in understanding economic development as a process of social and structural change in history. The paper argues that is time to free economic theory from the obsessive centrality of the equilibrium idea. Attention should shift to the market economy as a complex network of rules, organizations and institutions, which make viable economic change. This approach invites scholars in economics to a fruitful dialogue with other fields of knowledge, to understand both crises and successes in processes of economic change.L’article traite des images conceptuelles du marché construites à partir de l’idée d’équilibre. Il est suggéré que la construction des modèles d’équilibre du marché a souffert de difficultés logiques et a aboutit à des impasses. Il en a résulté une perte de réalisme, et une incapacité à comprendre l’histoire et à aborder les questions pertinentes. L’économie de l’équilibre est caractérisée par l’exclusion d’un grand nombre de phénomènes économiques mais aussi par un manque d’attention au changement, à la transition, et à l’apprentissage. Le manque de pertinence qui en découle est le résultat de l’idée primitive de concevoir le marché dans le champ sémantique de l’équilibre. L’échec le plus évident de l’image d’équilibre réside dans la compréhension du développement économique comme un processus du changement social et structurel au cours de l’histoire. L’article suggère qu’il est temps de libérer la théorie économique de cette idée obsessionnelle d’équilibre. Il faut plutôt concevoir le marché comme un réseau complexe de règles, d’organisations et d’institutions qui rendent le changement économique viable. Cette approche invite l’économie à un dialogue fructueux avec les autres domaines de la connaissance pour comprendre à la fois les crises et les réussites des processus de changement économique. (shrink)

In Language and Equilibrium, Prashant Parikh offers a new account of meaning for natural language. He argues that equilibrium, or balance among multiple interacting forces, is a key attribute of language and meaning and shows how to derive the meaning of an utterance from first principles by modeling it as a system of interdependent games.His account results in a novel view of semantics and pragmatics and describes how both may be integrated with syntax. It considers many aspects (...) of meaning--including literal meaning and implicature--and advances a detailed theory of definite descriptions as an application of the framework.Language and Equilibrium is intended for a wide readership in the cognitive sciences, including philosophers, linguists, and artificial intelligence researchers as well as neuroscientists, psychologists, and economists interested in language and communication. (shrink)

There has frequently been taken to be a tension, if not an incompatibility, between "externalist" theories of content (which allow the make-up of one's physical environment and the linguistic usage of one's community to contribute to the contents of one's thoughts and utterances) and the "methodologically individualist" intuition that whatever contributes to the content of one's thoughts and utterances must ultimately be grounded in facts about one's own attitudes and behavior. In this dissertation I argue that one can underwrite such (...) externalist theories within a methodologically individualistic framework by understanding semantic norms in terms of the need to reach, for each of one's terms, a type of "equilibrium." Each speaker's commitment to making her _own_ beliefs and applications consistent allows one to incorporate these 'external' factors into the contents of their thoughts and utterances in a way that remains methodologically individualistic. Methodologically individualistic accounts are typically taken to be unable to incorporate 'external' factors such as the world's physical make-up or communal usage because of arguments suggesting that the individual's own beliefs and usage underdetermine or even misidentify what, according to externalist accounts, they mean by their terms. These arguments, however, only seem plausible if one presupposes a comparatively impoverished conception of the individual's beliefs. The beliefs a speaker associates with a given term extend far beyond the handful of sentences they would produce if asked to list such beliefs. In particular, speakers have an implicit, but rich, understanding of their language, their world, and the relation between them. Speakers typically understand languages as shared temporally extended practices about which they can be, both individually and collectively, mistaken. Once this conception of language is taken into account, the ascriptions which purportedly forced 'non-individualistic' conceptions of content upon us (particularly ascriptions which seemed to tie what we meant to social use rather than our own beliefs) turn out to be ultimately grounded in the individual's own beliefs. Indeed, our self-conception does much more than merely underwrite 'non-individualistic' ascriptions.. (shrink)

We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. ‘‘Truth values’’ are interpreted as deviations from a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statements, for which reason there is only one distinguished truth value, namely the one that represents equilibrium. The main results are that the system Bal is algebraizable in the sense of (...) [5] and its equivalent algebraic semantics BAL is definitionally equivalent to the variety of abelian lattice ordered groups, that is, the categories of the algebras in BAL and of ℓ–groups are isomorphic (see [10], Ch.4, 4). We also prove the deduction theorem for Bal and we study different kinds of semantic consequence associated to Bal. Finally, we prove the co-NP-completeness of the tautology problem of Bal. (shrink)

In the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are developed.We begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations (...) in the newly introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Łukasiewicz Infinitely-valued Logic.Then, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Łukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically detected.Lastly, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional Logic and Łukasiewicz Infinitely-valued Logic depending on whether only pure equilibria or also mixed equilibria are allowed. Such reductions rely upon the properties of semantics modulo satisfiability. We provide complexity and algorithmic discussion for the coherence problem and, also, for the problem of computing maximal and minimal probabilistic constraints on actions that preserves coherence.prepared by Sandro Márcio da Silva Preto.E-mail: [email protected]:https://doi.org/10.11606/T.45.2021.tde-17062021-163257. (shrink)

Aldo Antonelli offers a novel view on abstraction principles in order to solve a traditional tension between different requirements: that the claims of science be taken at face value, even when involving putative reference to mathematical entities; and that referents of mathematical terms are identified and their possible relations to other objects specified. In his view, abstraction principles provide representatives for equivalence classes of second-order entities that are available provided the first- and second-order domains are in the equilibrium dictated (...) by the abstraction principles, and whose choice is otherwise unconstrained.entities are the referents of abstraction terms: such referents are to an extent indeterminate, but we can still quantify over them, predicate identity or non-identity, etc. Our knowledge of them is limited, but still substantial: we know whatever has to be true no matter how the representatives are chosen, i.e., what is true in all models of the corresponding abstraction principles. This view is backed up by an “austere” conception of universals, according to which these are first-order objects, i.e., ways of collecting first-order objects. Antonelli thus claims that second-order logic does not import any novel ontological commitment beyond ontology of first-order, naturalistically acceptable, objects. Moreover, in the case of arithmetic, even if Hume’s Principle is construed as described above, Frege’s Theorem goes through unaffected, since there is nothing in its proof that depends on an account of the “true nature” of numbers. A viable construal of logicism is then given by the combination of semantic nominalism and a naturalistic conception of abstraction. [Editors note]. (shrink)

Metalogic is an open-ended cognitive, formal methodology pertaining to semantics and information processing. The language that mathematizes metalogic is known as metalanguage and deals with metafunctions purely by extension on patterns. A metalogical process involves an effective enrichment in knowledge as logical statements, and, since human cognition is an inherently logic–based representation of knowledge, a metalogical process will always be aimed at developing the scope of cognition by exploring possible cognitive implications reflected on successive levels of abstraction. Indeed, it (...) is basically impracticable to maintain logic–and–metalogic without paying heed to cognitive theorizing. For it is cognitively irrelevant whether possible conclusions are deduced from some premises before the premises are determined to be true or whether the premises themselves are determined to be true first and, then, the conclusions are deduced from them. In this paper we consider the term metalogic as inherently embodied under the framework referred to as cognitive science and mathematics. We propose a metalogical interpretation of Arrow’s impossibility theorem and, to that end, choice theory is understood as a mental course of action dealing with logic and metalogic issues, in which a possible mathematical approach to model a mental course of action is adopted as a systematic operating method. Nevertheless, if we look closely at the core of Arrow’s impossibility theorem in terms of metalogic, a second fundamental contribution to this framework is represented by the Nash equilibrium. As a result of the foregoing, therefore, we prove the metalogical equivalence between Arrow’s impossibility theorem and the existence of the Nash equilibrium. More specifically, Arrow’s requirements correspond to the Nash equilibrium for finite mixed strategies with no symmetry conditions. To demonstrate this proof, we first verify that Arrow’s set and Nash’s set are isomorphic to each other, both sets being under stated conditions of non–symmetry. Then, the proof is completed by virtue of category theory. Indeed, the two sets are categories that correspond biuniquely to one another and, thus, it is possible to define a covariant functor that preserves their mutual structures. According to this, we show the proof–dedicated metalanguage as a precursor to a special equivalence theorem. (shrink)

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics, which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a game, than (...) the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics. (shrink)

The aim of this paper is twofold: (1) to assess whether the construct of neural representations plays an explanatory role under the variational free-energy principle and its corollary process theory, active inference; and (2) if so, to assess which philosophical stance - in relation to the ontological and epistemological status of representations - is most appropriate. We focus on non-realist (deflationary and fictionalist-instrumentalist) approaches. We consider a deflationary account of mental representation, according to which the explanatorily relevant contents of neural (...) representations are mathematical, rather than cognitive, and a fictionalist or instrumentalist account, according to which representations are scientifically useful fictions that serve explanatory (and other) aims. After reviewing the free-energy principle and active inference, we argue that the model of adaptive phenotypes under the free-energy principle can be used to furnish a formal semantics, enabling us to assign semantic content to specific phenotypic states (the internal states of a Markovian system that exists far from equilibrium). We propose a modified fictionalist account: an organism-centered fictionalism or instrumentalism. We argue that, under the free-energy principle, pursuing even a deflationary account of the content of neural representations licenses the appeal to the kind of semantic content involved in the aboutness or intentionality of cognitive systems; our position is thus coherent with, but rests on distinct assumptions from, the realist position. We argue that the free-energy principle thereby explains the aboutness or intentionality in living systems and hence their capacity to parse their sensory stream using an ontology or set of semantic factors. (shrink)

1. A. Heyting (1930) introduced an intermediate logic whose semantics is based on a pair of states (‘here’ and ‘there’). This logic was axiomatized by Hosoi (1966), using the sequence of intermedia...

We analyze the behaviour of declarations of independence between existential quantifiers in quantifier prefixes of Independence-Friendly (IF) sentences; we give a syntactical criterion to decide whether a sentence beginning with such prefix exists, such that its truth values may be affected by removal of the declaration of independence. We extend the result also to equilibrium semantics values for undetermined IF sentences.

In recent work, semantic games of independence-friendly logic were studied in strategic form in terms of Nash equilibria. The class of strategic games of independence-friendly logic is contained in the class of win-loss, zero-sum two-player games. In this note we draw on the theory of linear programming to develop tools to analyze the value of such games. We give two applications of these tools to independence-friendly logic under the so-called equilibrium semantics.

The article addresses two closely related questions: What are the criteria of adequacy of logical formalization of natural language arguments, and what gives logic the authority to decide which arguments are good and which are bad? Our point of departure is the criticism of the conception of logical formalization put forth, in a recent paper, by M. Baumgartner and T. Lampert. We argue that their account of formalization as a kind of semantic analysis brings about more problems than it solves. (...) We also argue that the criteria of adequate formalization need not be based on truth conditions associated with logical formulas; in our view, they are better based on structural (inferential) grounds. We then put forward our own version of the criteria. The upshot of the discussion that follows is that the quest for an adequate formalization in a suitable logical language is best conceived of as the search for a Goodmanian reflective equilibrium. (shrink)

Having its seeds in the 2nd International Lauener Symposium held in honour of Dagfinn Follesdal, the present collection contains a rich, kaleidoscopic ensemble of previously unpublished contributions by leading authors, representing diverse approaches to a variety of philosophical themes on which Follesdal has had a longstanding, formative impact. Follesdal himself contributes an orientating essay continuing to develop his pioneering theory of reference as well as in-depth commentaries on each of the other authors elaborated papers plus candid answers in the added (...) interview. The volume assembles a wealth of original articles containing in part direct discussions of Follesdal s work and covering a broad range of topics like subjectivity, intersubjectivity, objectivity, rationality, logics and mathematics, choice theory, values, modalities, intentionality, individuation, perception, communication, meaning, reference, the slingshot, one- and two-sorted semantics, evidence, neuropsychology, space and time, science and society, methodology, fallibilism, the relative a priori, justification, holism, the life-world, reflective equilibrium, empathy, and ethics. Moreover the book includes an incisive memoir of Follesdal the philosopher as well as a spanning interview with him, which are both critically directing toward Follesdal s subtly differentiated understanding of the dynamic philosophical horizon he shares in. With contributions from Dagfinn Follesdal, Charles Parsons, Patrick Suppes, Jon Elster, John Perry, Michael Friedman, Dag Prawitz, Wilhelm K. Essler, David Woodruff Smith, Olav Gjelsvik, Graciela De Pierris, Nils Roll-Hansen, Christian Beyer, Oystein Linnebo, Michael Frauchiger. Excerpt Open publication". (shrink)

Is conceptual engineering feasible? Answering that question requires a theory of semantic change, which is sometimes thought elusive. Fortunately, much is known about semantic change as it occurs in the wild. While usage is chaotic and complex, changes in a word’s use can produce changes in its meaning. There are several under-appreciated empirical constraints on how meanings change that stem from the following observation: word use finely reflects equilibrium between various communicative pressures. Much of the relevant work in linguistics (...) has employed the methods of empirical pragmatics and diachronic semantics. In this way, the study of meaning change can be brought to bear on the conceptual engineer’s normative project. The picture that emerges tells against the sorts of engineering projects most likely to appeal to philosophers. Some may stand to succeed, but they have significantly different contours than the typical ones. (shrink)