Formalizing the Institution for Event-B in the Coq Proof Assistant

We formalize a fragment of the theory of institutions sufficient to establish basic facts about the institution "Image missing" for Event-B, and its relationship with the institution "Image missing" for first-order predicate logic. We prove the satisfaction condition for "Image missing" and encode the institution comorphism "Image missing" embedding "Image missing" in "Image missing" .

A Logical Reconstruction of Leibniz’s Argument for His Complete Concept Conception of the Nature of Substance in Discours §8

This paper develops a valid reconstruction in first-order predicate logic of Leibniz’s argument for his complete concept definition of substance in §8 of the Discours de Métaphysique. Following G. Rodriguez-Pereyra, it construes the argument as resting on two substantial premises, the “merely verbal” Aristotelian definition and Leibniz’s concept containment theory of truth, and it understands the resulting “real” definition as saying not that an entity is a substance iff its complete concept contains every predicate of that entity, but iff its complete concept contains every predicate of any subject to which that concept is truly attributable. An account is suggested of why Leibniz criticises the Aristotelian definition as merely nominal and how he takes his own definition to overcome this shortcoming: while on the Aristotelian basis the predication relation could generate endless chains, so that substances as endpoints of predication would be impossible, Leibniz’s definition reveals lowest species as such endpoints, which he therefore identifies with individual substances. Since duplicate lowest species make no sense, the Identity of Indiscernibles for substances follows. The reading suggests a Platonist interpretation according to which substances do not so much have but are individual essences, natures or forms.

Description Logics That Count, and What They Can and Cannot Count

Simple counting quantifiers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a fixed natural number have been employed in Description Logics (DLs) for more than two decades under the respective names of number restrictions and cardinality restriction on concepts. Recently, we have considerably extended the expressivity of such quantifiers by allowing to impose set and cardinality constraints formulated in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) on sets of role successors and concepts, respectively. We were able to prove that this extension does not increase the complexity of reasoning.In the present paper, we investigate the expressive power of the DLs obtained this way, using appropriate bisimulation characterizations and 0--1 laws as tools for distinguishing the expressiveness of different logics. In particular, we show that, in contrast to most classical DLs, these logics are no longer expressible in first-order predicate logic (FOL), and we characterize their first-order fragments. In most of our previous work on DLs with QFBAPA-based set and cardinality constraints we have employed finiteness restrictions on interpretations to ensure that the obtained sets are finite. Here we dispense with these restrictions to make the comparison with classical DLs, where one usually considers arbitrary models rather than finite ones, easier. It turns out that doing so does not change the complexity of reasoning.

Deontic Logics as Axiomatic Extensions of First-Order Predicate Logic: An Approach Inspired by Wolniewicz’s Formal Ontology of Situations

The aim of this article is to present a method of creating deontic logics as axiomatic theories built on first-order predicate logic with identity. In the article, these theories are constructed as theories of legal events or as theories of acts. Legal events are understood as sequences (strings) of elementary situations in Wolniewicz′s sense. On the other hand, acts are understood as two-element legal events: the first element of a sequence is a choice situation (a situation that will be changed by an act), and the second element of this sequence is a chosen situation (a situation that arises as a result of that act). In this approach, legal rules (i.e., orders, bans, permits) are treated as sets of legal events. The article presents four deontic systems for legal events: AEP, AEPF, AEPOF, AEPOFI. In the first system, all legal events are permitted; in the second, they are permitted or forbidden; in the third, they are permitted, ordered or forbidden; and in the fourth, they are permitted, ordered, forbidden or irrelevant. Then, we present a deontic logic for acts (AAPOF), in which every act is permitted, ordered or forbidden. The theorems of this logic reflect deontic relations between acts as well as between acts and their parts. The direct inspiration to develop the approach presented in the article was the book Ontology of Situations by Boguslaw Wolniewicz, and indirectly, Wittgenstein’s Tractatus Logico-Philosophicus.

Finite countermodels as invariants. A case study in verification of parameterized mutual exclusion protocol

We present a case study of the verification of parameterized mutual exclusion protocol using finite model finder Mace4. Thhe verification follows an approach based on modeling of reachability between states of the protocol as deducibility between appropriate encodings of states by first-order predicate logic formulae. The result of successful verification is a finite countermodel, a witness of non-deducibility, which represents a system invariant.