This paper looks closely at previously enunciated axioms that specifically include phenomenology as the sense of a self in a perceptual world. This, we suggest, is an appropriate way of doing science on a first-person phenomenon. The axioms break consciousness down into five key components: presence, imagination, attention, volition and emotions. The paper examines anew the mechanism of each and how they interact to give a single sensation. An abstract architecture, the Kernel Architecture, is introduced as a starting point for (...) building computational models. The thrust of the paper is to relate the axioms to the kernel architecture and indicate that this opens a way of discussing some first-person issues: tests for consciousness, animal consciousness and Higher Order Thought. (shrink)

In the epicenter of his attempt to justify the “objective validity” of morality and freedom in the Critique of Practical Reason, Kant introduces a so-called fact of reason, which is rendered as the fact that human beings are consciou s of the moral ought’s categorical authority. However, few parts of Kant’s thinking have bemused commentators as much as this. In this article, the author explores a set of intersecting problems related to the fact of reason: (1) the problem of its (...) general argumentative role in Kant’s practical philosophy, (2) the problem of the fact as a brute fact. He argues that both problems can be understood and resolved only if we regard Kant’s introduction of the fact of reason as an implicit attempt to articulate intrinsic problems with reductive explanations of morality. His main claim, which has been surprisingly absent in most interpretations of Kant’s fact of reason, will be that the fact of reason functions as an undeniable yet improvable fact (similar to mathematical principles) from which other practical truths (such as the objective reality of freedom) can be derived. As Kant says unambiguously in the Jäsche-Logic : the “reality” of the moral law “is an axiom.”. (shrink)

Economic models describe individuals in terms of underlying characteristics, such as taste for some good, sympathy level for another player, time discount rate, risk attitude, and so on. In real life, such characteristics change through experiences: taste for Mozart changes through listening to it, sympathy for another player through observing his moves, and so on. Models typically ignore change, not just for simplicity but also because it is unclear how to incorporate change. I introduce a general axiomatic framework for (...) defining, analysing and comparing rival models of change. I show that seemingly basic postulates on modelling change together have strong implications, like irrelevance of the order in which someone has his experiences and ‘linearity’ of change. This is a step towards placing the modelling of change on solid axiomatic grounds and enabling non-arbitrary incorporation of change into economic models. (shrink)

In this paper we discuss two approaches to the axiomatization of scientific theories in the context of the so called semantic approach, according to which (roughly) a theory can be seen as a class of models. The two approaches are associated respectively to Suppes’ and to da Costa and Chuaqui’s works. We argue that theories can be developed both in a way more akin to the usual mathematical practice (Suppes), in an informal set theoretical environment, writing the set theoretical predicate (...) in the language of set theory itself or, more rigorously (da Costa and Chuaqui), by employing formal languages that help us in writing the postulates to define a class of structures. Both approaches are called internal , for we work within a mathematical framework, here taken to be first-order ZFC. We contrast these approaches with an external one, here discussed briefly. We argue that each one has its strong and weak points, whose discussion is relevant for the philosophical foundations of science. (shrink)

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained (...) by Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space. (shrink)

We present axiom systems, and provide soundness and strong completeness theorems, for classes of Kripke models with restricted extension rules among the node structures of the model. As examples we present an axiom system for the class of cofinal extension Kripke models, and an axiom system for the class of end-extension Kripke models. We also show that Heyting arithmetic is strongly complete for its class of end-extension models. Cofinal extension models of HA are models of Peano arithmetic.

We shift attention from the development of model theory for demarcated languages to the development of this theory for fragments of a language. Although it is often assumed that model theory for demarcated languages is not compatible with a universalist conception of logic, no one has denied that model theory for fragments of a language can be compatible with that conception. It thus seems unwarranted to ignore the universalist tradition in the search for the origins and development (...) of model theory. This point is illustrated by Carnap’s early semantics and model theory, which he developed within a type theoretical framework and which stand out both for their universalistic treatment and for certain idiosyncratic technicalities by which the construction is supported. One special property is that individuals are context relative in Carnap’s system. This leads to a model theory in which the model domains are more flexible than has been suggested in the literature. (shrink)

This book provides a systematic and comprehensive description of Non-Axiomatic Logic, which is the result of the author's research for about three decades.Non-Axiomatic Logic is designed to provide a uniform logical foundation for Artificial Intelligence, as well as an abstract description of the “laws of thought” followed by the human mind. Different from “mathematical” logic, where the focus is the regularity required when demonstrating mathematical conclusions, Non-Axiomatic Logic is an attempt to return to the original aim of (...) logic, that is, to formulate the regularity in actual human thinking. To achieve this goal, the logic is designed under the assumption that the system has insufficient knowledge and resources with respect to the problems to be solved, so that the “logical conclusions” are only valid with respect to the available knowledge and resources. Reasoning processes according to this logic covers cognitive functions like learning, planning, decision making, problem solving, etc.This book is written for researchers and students in Artificial Intelligence and Cognitive Science, and can be used as a textbook for courses at graduate level, or upper-level undergraduate, on Non-Axiomatic Logic. (shrink)

Two conflicting interpretations of modern axiomatics will be considered. The logico-analytical interpretation goes back to Pasch, while the model-theoretical approach stems from Hilbert. This perspective takes up the distinction between logic as calculus ratiocinator versus lingua characterica that Heijenoort and Hintikka placed emphasis on. It is argued that the Heijenoort-Hintikka distinction can be carried over from logic to mathematical axiomatics. In particular, the model-theoretical viewpoint is deeply connected to a philosophy of mathematics that is not committed to a (...) foundational perspective, but oriented more at applications and at mathematical practice. (shrink)

Two conflicting interpretations of modern axiomatics will be considered. The logico-analytical interpretation goes back to Pasch, while the model-theoretical approach stems from Hilbert. This perspective takes up the distinction between logic as calculus ratiocinator versus lingua characterica that Heijenoort and Hintikka placed emphasis on. It is argued that the Heijenoort-Hintikka distinction can be carried over from logic to mathematical axiomatics. In particular, the model-theoretical viewpoint is deeply connected to a philosophy of mathematics that is not committed to a (...) foundational perspective, but oriented more at applications and at mathematical practice. (shrink)

Maxmin Expected Utility was first axiomatized by Gilboa and Schmeidler in an Anscombe–Aumann setup Anscombe and Aumann which includes exogenous probabilities. The model was later axiomatized in a purely subjective setup, where no exogenous probabilities are assumed. The purpose of this note is to show that in all these axiomatizations, the only assumptions that are needed are the basic ones that are used to extract a cardinal utility function, together with the two typical Maxmin assumptions, Uncertainty Aversion and Certainty (...) Independence, applied only to 0.5:0.5 mixtures. For the purely subjective characterizations, this means that assumptions involving an unbounded number of variables can be replaced with assumptions that involve only a finite number thereof. (shrink)

As is well known, some paradoxes arise through inadequate analysis of the meanings of terms in a language, an adequate analysis showing that the paradoxes arise through a lack of separation of an object theory and a metatheory. Under such an adequate analysis in which parts of the metatheory are modelled in the object theory, the paradoxes give way to remarkable theorems establishing limitations of the object theory.Such a modelling is often accomplished by a Gödel numbering. Here we shall use (...) a somewhat different technique in axiomatic set theory, from which we shall reap a few results having the effect of comparing the strength of various axiom schema of comprehension for sets and classes. Similar results were obtained by A. Mostowski [7] using Gödel numbering. (shrink)

Ernst Mayr argued that the emergence of biology as a special science in the early nineteenth century was possible due to the demise of the mathematical model of science and its insistence on demonstrative knowledge. More recently, John Zammito has claimed that the rise of biology as a special science was due to a distinctive experimental, anti-metaphysical, anti-mathematical, and anti-rationalist strand of thought coming from outside of Germany. In this paper we argue that this narrative neglects the important role (...) played by the mathematical and axiomatic model of science in the emergence of biology as a special science. We show that several major actors involved in the emergence of biology as a science in Germany were working with an axiomatic conception of science that goes back at least to Aristotle and was popular in mid-eighteenth-century German academic circles due to its endorsement by Christian Wolff. More specifically, we show that at least two major contributors to the emergence of biology in Germany—Caspar Friedrich Wolff and Gottfried Reinhold Treviranus—sought to provide a conception of the new science of life that satisfies the criteria of a traditional axiomatic ideal of science. Both C.F. Wolff and Treviranus took over strong commitments to the axiomatic model of science from major philosophers of their time, Christian Wolff and Friedrich Wilhelm Joseph Schelling, respectively. The ideal of biology as an axiomatic science with specific biological fundamental concepts and principles thus played a role in the emergence of biology as a special science. (shrink)

While much of semantic theorizing is based on intuitions about logical phenomena associated with linguistic constructions—phenomena such as consistency and entailment—it is rare to see axiomatic treatments of linguistic fragments. Given a fragment interpreted in some class of formally specified models, it is often possible to ask for a characterization of the reasoning patterns validated by the class of models. Axiomatizations provide such a characterization, often in a perspicuous and efficient manner. In this paper, we highlight some of the (...) benefits of providing axiomatizations for the purpose of semantic theorizing. We illustrate some of these benefits using three examples from the study of modality. (shrink)

sBuilding models as a practical aspect of ecological theory has as a principal purpose the determination of relations in formal language. In this paper, the authors provide a formalization of ecological models based on impure systems theory. Impure systems contain objects and subjects: subjects are human beings. We can distinguish a person as an observer that by definition is the subject himself and part of the system. In this case he acquires the category of object. Objects are significances, which are (...) the consequence of perceptual beliefs on the part of the subject about material or energetic objects with certain characteristics. The impure system approach is as follows: objects are perceptual significances of material or energetic objects. The set of these objects will form an impure set of the first order. The existing relations between these relative objects will be of two classes: transactions of matter and/or energy and inferential relations. Transactions can have alethic modality: necessity, possibility, impossibility and contingency. In this work we define measures which let us choose the more suitable variables to relate both the model with the ecosystem and with different models. In this way we define different comparison indexes. (shrink)

We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in (...) KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system RA< ωω ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system RA<ε₀ of ramified analysis up to ε₀. (shrink)

Distributive justice deals with allocations of goods and bads within a group. Different principles and results of distributions are seen as possible ideals. Often those normative approaches are solely framed verbally, which complicates the application to different concrete distribution situations that are supposed to be evaluated in regard to justice. One possibility in order to frame this precisely and to allow for a fine-grained evaluation of justice lies in formal modelling of these ideals by metrics. Choosing a metric that is (...) supposed to map a certain ideal has to be justified. Such justification might be given by demanding specific substantiated axioms, which have to be met by a metric. This paper introduces such axioms for metrics of distributive justice shown by the example of needs-based justice. Furthermore, some exemplary metrics of needs-based justice and a three dimensional method for visualisation of non-comparative justice axioms or evaluations are presented. Therewith, a base worth discussing for the evaluation and modelling of metrics of distributive justice is given. (shrink)

Axiomatic rationality is defined in terms of conformity to abstract axioms. Savage limited axiomatic rationality to small worlds, that is, situations in which the exhaustive and mutually exclusive set of future states S and their consequences C are known. Others have interpreted axiomatic rationality as a categorical norm for how human beings should reason, arguing in addition that violations would lead to real costs such as money pumps. Yet a review of the literature shows little evidence that (...) violations are actually associated with any measurable costs. Limiting axiomatic rationality to small worlds, I propose a naturalized version of rationality for situations of intractability and uncertainty, all of which are not in. In these situations, humans can achieve their goals by relying on heuristics that may violate axiomatic rationality. The study of ecological rationality requires formal models of heuristics and an analysis of the structures of environments these can exploit. It lays the foundation of a moderate naturalism in epistemology, providing statements about heuristics we should use in a given situation. Unlike axiomatic rationality, ecological rationality can explain less-is-more effects, formalize when one should move from ‘is’ to ‘ought,’ and be evaluated by goals beyond coherence, such as predictive accuracy, frugality, and efficiency. Ecological rationality can be seen as a formalization of means–end instrumentalist rationality, based on Herbert Simon’s insight that rational behavior is a function of the mind and its environment. (shrink)

In the literature, different axiomatizations of Public Announcement Logic (PAL) have been proposed. Most of these axiomatizations share a “core set” of the so-called “reduction axioms”. In this paper, by designing non-standard Kripke semantics for the language of PAL, we show that the proof system based on this core set of axioms does not completely axiomatize PAL without additional axioms and rules. In fact, many of the intuitive axioms and rules we took for granted could not be derived from the (...) core set. Moreover, we also propose and advocate an alternative yet meaningful axiomatization of PAL without the reduction axioms. The completeness is proved directly by a detour method using the canonical model where announcements are treated as merely labels for modalities as in normal modal logics. This new axiomatization and its completeness proof may sharpen our understanding of PAL and can be adapted to other dynamic epistemic logics. (shrink)

Definitional and axiomatic theories of truth -- Objects of truth -- Tarski -- Truth and set theory -- Technical preliminaries -- Comparing axiomatic theories of truth -- Disquotation -- Classical compositional truth -- Hierarchies -- Typed and type-free theories of truth -- Reasons against typing -- Axioms and rules -- Axioms for type-free truth -- Classical symmetric truth -- Kripke-Feferman -- Axiomatizing Kripke's theory in partial logic -- Grounded truth -- Alternative evaluation schemata -- Disquotation -- Classical logic (...) -- Deflationism -- Reflection -- Ontological reduction -- Applying theories of truth. (shrink)

We give a sound and complete axiomatization for the full computation tree logic, CTL*, of R-generable models. This solves a long standing open problem in branching time temporal logic.

This is part one of a two-part paper, in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. This allows us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths of arithmetic and (...) show that the theory is a proof-theoretically conservative extension of the theory PT of positive truth. We construct models for the theory and draw some conclusions for the semantics of conceptualist ground. (shrink)

We investigate intuitionistic propositional modal logics in which a modal operator is equivalent to intuitionistic double negation. Whereas ¬¬ is divisible into two negations, is a single indivisible operator. We shall first consider an axiomatization of the Heyting propositional calculus H, with the connectives →,∧,∨ and ¬, extended with . This system will be called Hdn . Next, we shall consider an axiomatization of the fragment of H without ¬ extended with . This system will be called Hdn + . (...) We shall show that these systems are sound and complete with respect to specific classes of Kripke-style models with two accessibility relations, one intuitionistic and the other modal. This type of models is investigated in [2] and [3], and here we try to apply the techniques of these papers to an intuitionistic modal operator with a natural interpretation. The full results of our investigation will be published in [4] and [1]. (shrink)

Boolean connexive logic is an extension of Boolean logic that is closed under Modus Ponens and contains Aristotle’s and Boethius’ theses. According to these theses a sentence cannot imply its negation and the negation of a sentence cannot imply the sentence; and if the antecedent implies the consequent, then the antecedent cannot imply the negation of the consequent and if the antecedent implies the negation of the consequent, then the antecedent cannot imply the consequent. Such a logic was first introduced (...) by Jarmużek and Malinowski, by means of so-called relating semantics and tableau systems. Subsequently its modal extension was determined by means of the combination of possible-worlds semantics and relating semantics. In the following article we present axiomatic systems of some basic and modal Boolean connexive logics. Proofs of completeness will be carried out using canonical models defined with respect to maximal consistent sets. (shrink)

For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets. We discuss here the corresponding situation for fuzzy set theory.Our emphasis will be on various approaches toward (more or less naively formed)universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.

More attention has been paid in recent years to the relationship between Aristotle’s science and his ethics, but little effort has been directed toward constructing a concrete model of a science of Aristotle’s ethics. I offer a proposal about how we might go about constructing a science of Aristotle’s ethics. I argue that constructing an axiomatic model for a portion of Aristotle’s ethics is not only possible, but helpful in making explicit relationships among concepts at the core (...) of Aristotle’s theory. The model of an axiomatic approach to Aristotle’s ethics, which I propose in this paper, is only a small first step in constructing a full-blown science of Aristotle’s ethics, but taking this first step goes a long way toward showing that this project is promising. (shrink)

More attention has been paid in recent years to the relationship between Aristotle’s science and his ethics, but little effort has been directed toward constructing a concrete model of a science of Aristotle’s ethics. I offer a proposal about how we might go about constructing a science of Aristotle’s ethics. I argue that constructing an axiomatic model for a portion of Aristotle’s ethics is not only possible, but helpful in making explicit relationships among concepts at the core (...) of Aristotle’s theory. The model of an axiomatic approach to Aristotle’s ethics, which I propose in this paper, is only a small first step in constructing a full-blown science of Aristotle’s ethics, but taking this first step goes a long way toward showing that this project is promising. (shrink)

In a recent paper Berto introduces a semantic system for a logic of imagination, intended as positive conceivability, and aboutness of imaginative acts. This system crucially adopts elements of both the semantics of conditionals and the semantics of analytical implications in order to account for the central logical traits of the notion of truth in an act of imagination based on an explicit input. The main problem left unsolved is to put forward a complete set of axioms for the proposed (...) system. In the present paper I offer a solution to this problem by providing a complete axiomatization of a generalization of the original semantics. The difficulty in proving completeness lies in the fact that the modalities that capture the notion of truth in an act of imagination are neither standard nor minimal, so that the construction of the canonical model and the proof of the truth lemma are to be substantially modified. (shrink)

The aim of the present paper is to provide a logical reconstruction of the Marxian Theory of Value. This reconstruction is based upon the concept of abstract as opposed to that of homogeneous labor.labour is a social relation that holds in a market economy, whenever the profit rate is uniform; it consists in a comparison of the different labors through the exchange relationship. The author proved in a different place that there exists a mathematical representation of abstract labor which is (...) unique up to similarity transformations. The existence of this function -- which is none other than the value function -- is the basis of the present axiomatization, which includes detailed reconstructions of two of the most important models of the theory. (shrink)

We formulate a Hilbert-style axiomatic system and a tableau calculus for the STIT-based logic of imagination recently proposed in Wansing. Completeness of the axiom system is shown by the method of canonical models; completeness of the tableau system is also shown by using standard methods.

Abstract.We use a model theoretic approach to investigate properties of local-global principles for positive primitive formulas in spaces of orderings, such as the existence of bounds and the axiomatizability of local-global principles. As a consequence we obtain various classes of special groups satisfying local-global principles for all positive primitive formulas, and we show that local-global principles are preserved by some natural constructions in special groups.