New New Modification of the Subformula Property for a Modal Logic

A modified subformula property for the modal logic KD with the additional axiom $\Box\Diamond(A\vee B)\supset\Box\Diamond A\vee\Box\Diamond B$ is shown. A new modification of the notion of subformula is proposed for this purpose. This modification forms a natural extension of our former one on which modified subformula property for the modal logics K5, K5D and S4.2 has been shown (Bull Sect Logic 30:115--122, 2001 and 48:19--28, 2019). The finite model property as well as decidability for the logic follows from this.

Equations of quantum theory in the space of randomly joint quantum events

The dynamics of the system in the space of random joint events is considered. The symmetric difference of events is introduced in space based on the Kolmogorov axioms. To describe quantum effects in the dynamics of the system, an additional axiom is introduced for random joint events: “the symmetric sum of random events.” In the generated space of random joint events, an equation is constructed for the probability of a system transition between two events. It is shown that for pairwise joint events it is equivalent to the equation of quantum mechanics.

Boole, George (1815–64)

George Boole, a British mathematician, is credited with making a fundamental contribution to modern logic. If Leibniz’s manuscript essays on logic, effectively unknown until the end of the nineteenth century, are excluded, then Boole’s algebra of logic (1847, 1854) was the first successful mathematical treatment of one part of logic. The treatment was mathematical in the broad sense of using a formal language expressed in symbols with definite rules. It was also mathematical in a narrow sense of being closely modelled after numerical algebra, from which it differed by an additional axiom, x2=x. Letter symbols of this algebra were conceived as representing classes, 1 standing for a ‘universe’ of objects and 0 for the empty class. By identifying logical terms with their extensions, that is, with classes, inferences of a much more general character than those of the traditional syllogistic could be carried out. Boole also showed how this algebra could be used in propositional logic, presenting its earliest systematic general formulation.

Strong Syntax Splitting for Iterated Belief Revision

AGM theory is the most influential formal account of belief revision. Nevertheless, there are some issues with the original proposal. In particular, Parikh has pointed out that completely irrelevant information may be affected in AGM revision. To remedy this, he proposed an additional axiom (P) aiming to capture (ir)relevance by a notion of syntax splitting. In this paper we generalize syntax splitting from logical sentences to epistemic states, a step which is necessary to cover iterated revision. The generalization is based on the notion of marginalization of epistemic states. Furthermore, we study epistemic syntax splitting in the context of ordinal conditional functions. Our approach substantially generalizes the semantical treatment of (P) in terms of faithful preorders recently presented by Peppas and colleagues.

CATEGORICITY IN QUASIMINIMAL PREGEOMETRY CLASSES

Quasiminimal pregeometry classes were introduced by [6] to isolate the model theoretical core of several interesting examples. He proves that a quasiminimal pregeometry class satisfying an additional axiom, called excellence, is categorical in all uncountable cardinalities. Recently, [2] showed that the excellence axiom follows from the rest of the axioms. In this paper we present a direct proof of the categoricity result without using excellence.

Generalized geometric theories and set-generated classes

We introduce infinitary propositional theories over a set and their models which are subsets of the set, and define a generalized geometric theory as an infinitary propositional theory of a special form. The main result is thatthe class of models of a generalized geometric theory is set-generated. Here, a class$\mathcal{X}$of subsets of a set is set-generated if there exists a subsetGof$\mathcal{X}$such that for each α ∈$\mathcal{X}$, and finitely enumerable subset τ of α there exists a subset β ∈Gsuch that τ ⊆ β ⊆ α. We show the main result in the constructive Zermelo–Fraenkel set theory (CZF) with an additional axiom, called the set generation axiom which is derivable inCZF, both from the relativized dependent choice scheme and from a regular extension axiom. We give some applications of the main result to algebra, topology and formal topology.

NONEMPTY CORE-TYPE SOLUTIONS OVER BALANCED COALITIONS IN TU-GAMES

In this paper we introduce two related core-type solutions for games with transferable utility (TU-games) the [Formula: see text]-core and the [Formula: see text]-core. The elements of the solutions are pairs [Formula: see text] where x, as usual, is a vector representing a distribution of utility and [Formula: see text] is a balanced family of coalitions, in the case of the [Formula: see text]-core, and a minimal balanced one, in the case of the [Formula: see text]-core, describing a plausible organization of the players to achieve the vector x. Both solutions extend the notion of classical core but, unlike it, they are always nonempty for any TU-game. For the [Formula: see text]-core, which also exhibits a certain kind of "minimality" property, we provide a nice axiomatic characterization in terms of the four axioms nonemptiness (NE), individual rationality (IR), superadditivity (SUPA) and a weak reduced game property (WRGP) (with appropriate modifications to adapt them to the new framework) used to characterize the classical core. However, an additional axiom, the axiom of equal opportunity is required. It roughly states that if [Formula: see text] belongs to the [Formula: see text]-core then, any other admissible element of the form [Formula: see text] should belong to the solution too.

THE CLASS OF EFFICIENT LINEAR SYMMETRIC VALUES FOR GAMES IN PARTITION FUNCTION FORM

In this paper we study linear symmetric solutions for the space of games in partition function form with n players. In particular, we provide an expression for all linear, symmetric and efficient solutions. Furthermore, adding an additional axiom, we identify a unique value satisfying these properties.

The Ganea and Whitehead Variants of the Lusternik–Schnirelmann Category

The Lusternik–Schnirelmann category has been described in different ways. Two major ones, the first by Ganea, the second by Whitehead, are presented here with a number of variants. The equivalence of these variants relies on the axioms of Quillen's model category, but also sometimes on an additional axiom, the so-called “cube axiom”.

Can quantum cryptography imply quantum mechanics?

It has been suggested that the ability of quantum mechanics to allow secure distribution of secret key together with its inability to allow bit commitment or communicate superluminally might be sufficient to imply the rest of quantum mechanics. I argue using a toy theory as a counterexample that this is not the case. I further discuss whether an additional axiom (key storage) brings back the quantum nature of the theory.