Additional Axioms Research Articles

Interpreting First-order Theories into a Logic of Records

Features are unary operators used to build record-like expressions. The resulting term algebras are encountered in linguistic computation and knowledge representation. We present a general description of feature logic and of a slightly restricted version, called record logic. It is shown that every first-order theory can be faithfully interpreted in a record logic with various additional axioms. This fact is used elsewhere [15] to extend a result of Tarski and Givant [14] on expressing first order theories in relation algebra.

A CO-MARGINALISTIC CONTRIBUTION VALUE FOR SET GAMES

In this paper, a new value for set games, named co-marginalistic contribution value, is characterised by the standard axioms of global efficiency and equal treatment property, and another axiom of co-coalitional marginalistic monotonicity, which is a modification of the axiom of monotonicity. The co-marginalistic contribution value for set games is analogous to the solidarity value for n-person TU-games. Since the co-marginalistic contribution value does not satisfy the axiom of additivity, the proof of the characterisation is much different from that for TU-games.

The semireactive bargaining set of a cooperative game

The semireactive bargaining set, a solution for cooperative games, is introduced. This solution is in general a subsolution of the bargaining set and a supersolution of the reactive bargaining set. However, on various classes of transferable utility games the semireactive and the reactive bargaining set coincide. The semireactive prebargaining set on TU games can be axiomatized by one-person rationality, the reduced game property, a weak version of the converse reduced game property with respect to subgrand coalitions, and subgrand stability. Furthermore, it is shown that there is a suitable weakening of subgrand stability, which allows to characterize the prebargaining set. Replacing the reduced game by the imputation saving reduced game and employing individual rationality as an additional axiom yields characterizations of both, the bargaining set and the semireactive bargaining set.

Realism, nonstandard set theory, and large cardinals

Mathematicians justify axioms of set theory “intrinsically”, by reference to the universe of sets of their intuition, and “extrinsically”, for example, by considerations of simplicity or usefullness for mathematical practice. Here we apply the same kind of justifications to Nonstandard Analysis and argue for acceptance of BNST + (Basic Nonstandard Set Theory plus additional Idealization axioms). BNST + has nontrivial consequences for standard set theory; for example, it implies existence of inner models with measurable cardinals. We also consider how to practice Nonstandard Analysis in BNST +, and compare it with other existing nonstandard set theories.

The algebra of stream processing functions

Data flow networks are a model of concurrent computation. They consist of a collection of concurrent asynchronous processes which communicate by sending data over FIFO channels. In this paper we study the algebraic structure of the data flow networks and base their semantics on stream processing functions. Our algebraic theory is based on the calculus of flownomials. With an additive (or cantorian) interpretation the calculus gives a unified presentation of the classical algebraic models for control structures, that is, regular algebra and iteration theories. The kernel of the calculus is an equational axiomatization called basic network algebra (BNA) for flow graphs modulo graph isomorphism. We show that the algebra of stream processing functions called SPF (used for deterministic networks) and the algebra of sets of stream processing functions called P SPF (used for nondeterministic networks) are BNA algebras. Actually they give a multiplicative (or cartesian) interpretation of the calculus of flownomials. As a byproduct this shows that both semantic models are compositional. This means the semantics of a network may be described in terms of the semantics of its components. (As it is well known this is not true for the input–output relational semantics of nondeterministic networks.) We also identify additional axioms satisfied by the branching constants in these two algebraic theories. For the deterministic case we study in addition the coarser equivalence relation on networks given by the input–output behavior and provide a correct and complete axiomatization.

The Normal Distribution Derived from Qualitative Conditions

The normal distribution is characterized in a measurement theoretic framework. The qualitative conditions guarantee that representations can be regarded as random variables. Additional axioms, also qualitative in the measurement sense, yield the normal. One characterization draws on a limit theorem. The main result derives the normal distribution from conjoint measurement axioms. This approach consists of formulating properties of a linear model as a component structure with error as one component. The normal distribution of errors is shown to be a consequence of the measurement theoretic assumptions. The possible impact of these results on statistical models is discussed.

A note on Spector's quantifier-free rule of extensionality

In this note we show that the so-called weakly extensional arithmetic in all finite types, which is based on a quantifier-free rule of extensionality due to C. Spector and which is of significance in the context of Godel"s functional interpretation, does not satisfy the deduction theorem for additional axioms. This holds already for Π0 1-axioms. Previously, only the failure of the stronger deduction theorem for deductions from (possibly open) assumptions (with parameters kept fixed) was known.

On the Convergence of ω1 Sequences of Real Functions

We study sets of points at which ω1 sequences of real functions from a given class F converge. As F we consider continuous functions, first class of Baire, Borel measurable functions, functions with Baire property and Lebesgue measurable functions. Connections of those problem with additional set-theoretic axioms are discussed.

Inferring the statistical interpretation of quantum mechanics from the classical limit

It is widely believed that the statistical interpretation of quantum mechanics cannot be inferred from the Schrodinger equation itself, and must be stated as an additional independent axiom. Here I propose that the situation is not so stark. For systems that have both continuous and discrete degrees of freedom (such as coordinates and spin respectively), the statistical interpretation for the discrete variables is implied by requiring that the system's gross motion can be classically described under circumstances specified by the Schrodinger equation. However, this is not a full-fledged derivation of the statistical interpretation because it does not apply to the continuous variables of classical mechanics.

Axiomatic etal maps and a theory of spectrum

We consider and develop the axioms introduced by A. Joyal that define an abstract notion of an etal class A of arrows in a Grothendieck topos E . The axioms are intended to be sufficient in the sense that the category of objects etal over the terminal object (and etal maps between them) should be a topos, the ‘Etal’ Topos Et . It can be shown that Et ⊂ E is a full subcategory, closed under all colimits and finite limits, and so it is almost a topos. However, the problem of the existence of generators for Et and the existence of a right adjoint for the inclusion making it the inverse image of a geometric morphism E → Et is still open. We introduce an additional axiom that we call the etal topology condition or ETC, and for a topos E equipped with such a class, we develop a general construction of germs which yields a new point associated to any given point F → E . A particular case of this furnishes a right adjoint for the inclusion Et ⊂ E , and it follows that in this case Et is a subtopos of E , the center of a local geometric morphism of topoi. E → Et . Also, we introduce a new general theory of Spectrum where etal classes take care of the role assigned to the admissible morphisms, and prove a general theorem of existence based in the construction of germs. This theorem includes all the known results in the theory of Cole's spectrum. This theory is more general since it is associated to any geometric morphism rather than only to the inclusion of subtopoi, and conceptually it is independent from the notion of geometric theory.

The additivity and dummy axioms in the discrete cost sharing model

The paper considers the discrete cost sharing model first studied in Moulin’s paper [Moulin, H., 1995. On additive methods to share joint costs. Japanese Economic Review 46, 303–332]. It shows that the set of additive rules satisfying the dummy axiom is the set of all convex combinations of the path generated rules.

Irreflexive Modality in the Intuitionistic Propositional Logic and Novikov Completeness

A. Kuznetsov considered a logic which extended intuitionistic propositional logic by adding a notion of 'irreflexive modality'. We describe an extension of Kuznetsov's logic having the following properties: (a) it is the unique maximal conservative (over intuitionistic propositional logic) extension of Kuznetsov's logic; (b) it determines a new unary logical connective w.r.t. Novikov's approach, i.e., there is no explicit expression within the system for the additional connective; (c) it is axiomatizable by means of one simple additional axiom scheme.

A Note on Spector’s Quantifier-Free Rule of Extensionality

In this note we show that the so-called weakly extensional arithmetic<br />in all finite types, which is based on a quantifier-free rule of<br />extensionality due to C. Spector and which is of significance in the<br />context of G¨odel's functional interpretation, does not satisfy the deduction<br />theorem for additional axioms. This holds already for PI^0_1-<br />axioms. Previously, only the failure of the stronger deduction theorem<br />for deductions from (possibly open) assumptions (with parameters<br />kept fixed) was known.

Algebraic Structures Arising in Axiomatic Unsharp Quantum Physics

This article presents and compares various algebraic structures that arise in axiomatic unsharp quantum physics. We begin by stating some basic principles that such an algebraic structure should encompass. Following G. Mackey and G. Ludwig, we first consider a minimal state-effect-probability (minimal SEFP) structure. In order to include partial operations of sum and difference, an additional axiom is postulated and a SEFP structure is obtained. It is then shown that a SEFP structure is equivalent to an effect algebra with an order determining set of states. We also consider σ-SEFP structures and show that these structures distinguish Hilbert space from incomplete inner product spaces. Various types of sharpness are discussed and under what conditions a Brouwer complementation can be defined to obtain a BZ-poset is investigated. In this case it is shown that every effect has a best lower and upper sharp approximation and that the set of all Brouwer sharp effects form an orthoalgebra.

Finite Functions and the Necessary Use of Large Cardinals

Finite Functions and the Necessary Use of Large Cardinals

Basic fuzzy logic and BL-algebras

The many-valued propositional logic BL (basic fuzzy logic) is investigated. It is known to be complete for tautologies over BL-algebras (particular residuated lattices). Each continuous t-norm on [0,1] determines a BL-algebra; such algebras are called t-algebras. Two additional axioms B1, B2 are found such that BL+(B1,B2) is complete for tautologies over t-algebras. It remains open whether B1, B2 are provable in BL.

The Supporting Line Property and the Additive Choice Function for Two‐Dimensional Choice Problems

The paper provides an axiomatic characterization of the additive choice function using the additivity property. It is seen that, along with Pareto optimality, symmetry, and a supporting line property, the additivity axiom uniquely characterizes the additive choice function. The additive choice function as defined in this paper does not satisfy Nash's assumption of the independence of irrelevant alternatives. This raises the question of when a choice function is representable, an issue addressed in an appendix.

Some Consequences of Cryptographical Conjectures forS12and EF

We show that there is a pair of disjoint N P -sets, whose disjointness is provable in S 1 2 and which cannot be separated by a set in P /poly, if the cryptosystem RSA is secure. Further we show that factoring and the discrete logarithm are implicitly definable in any extension of S 1 2 admitting an N P -definition of primes about which it can prove that no number satisfying the definition is composite. As a corollary we obtain that the Extended Frege (EF) proof system does not admit a feasible interpolation theorem unless the RSA cryptosystem is not secure and that an extension of EF by tautologies τ p ( p primes), formalizing that p is not composite, as additional axioms do not admit a feasible interpolation theorem unless factoring and the discrete logarithm are in P /poly.

Imprecise sampling and direct decision making

In decision theory, uncertainty is generally modeled through probabilities or probabilityintervals. Data however, when collected by sampling, do not provide probabilities (resp.,lower/upper probabilities) but frequencies (resp., lower/upper frequencies). Discrepanciesbetween the former and the latter are taken into account by the model presented: axiomaticrequirements are shown to imply that the ordering of the decisions must only depend on aquadruple (GEU, u, U, N), where GEU is the generalized expected utility evaluation of thedecision that would result from the assimilation of frequencies to lower/upper probabilities;u and U are the utility levels of, respectively, the worst and the best reachable outcomes;and N is the size of the sample. Additional axioms are given that ensure the existence of anadditive utility representing the ordering.

Informational independence: Models and normal forms

The concept of informational independence plays a key role in most knowledge-based systems. J. Pearl and his co-researchers analysed the basic properties of the concept and formulated an axiomatic system for informational independence. This axiomatic system focuses on independences among mutually disjoint sets of variables. We show that in the context of probabilistic independence a focus on disjoint sets of variables can hide various interesting properties. To capture these properties, we enhance Pearl's axiomatic system with two additional axioms. We investigate the set of models of the thus enhanced system and show that it provides a better characterization of the concept of probabilistic independence than Pearl's system does. In addition, we observe that both Pearl's axiomatic system and our enhanced system offer inference rules for deriving new independences from an initial set of independence statements and as such allow for a normal form for representing independence. We address the normal forms ensuing from the two axiomatic systems for informational independence.