What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has been endorsed by many authors (cf. van Benthem 1991). We will investigate these fragments, and find that, illuminating and interesting though they are, they lack the required nice behaviour in our sense. (Several new negative results support this claim.) (...) As a counterproposal, then, we define a large fragment of predicate logic characterized by its use of only bounded quantification. This so-called guarded fragment enjoys the above nice properties, including decidability, through an effectively bounded finite model property. (These are new results, obtained by generalizing notions and techniques from modal logic.) Moreover, its own internal finite variable hierarchy turns out to work well. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantifiers, but quantifiers are also like modal operators. This observation inspires a generalized semantics for first-order predicate logic with accessibility constraints on available assignments (cf. N´emeti 1986, 1992) which moves the earlier quantifier restrictions into the semantics. This provides a fresh look at the landscape of possible predicate logics, including candidates sharing various desirable features with basic modal logic – in particular, its decidability. (shrink)

Glymour and Quine propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we "derive" an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist.

The aim of this paper is to present a new logic-based understanding of the connection between classical kinematics and relativistic kinematics. We show that the axioms of special relativity can be interpreted in the language of classical kinematics. This means that there is a logical translation function from the language of special relativity to the language of classical kinematics which translates the axioms of special relativity into consequences of classical kinematics. We will also show that if we distinguish a class (...) of observers in special relativity and exclude the non-slower-than light observers from classical kinematics by an extra axiom, then the two theories become definitionally equivalent. Furthermore, we show that classical kinematics is definitionally equivalent to classical kinematics with only slower-than-light inertial observers, and hence by transitivity of definitional equivalence that special relativity theory extended with “Ether” is definitionally equivalent to classical kinematics. So within an axiomatic framework of mathematical logic, wee xplicitly show that the transition from classical kinematics to relativistic kinematics is the knowledge acquisition that there is no “Ether”, accompanied by a redefinition of the concepts of time and space. (shrink)

We give two theories, Th1 and Th2, which are explicitly definable over each other , but are not definitionally equivalent. The languages of the two theories are disjoint.

For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to non-disjoint languages and they show that their generalization is not equivalent to intertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce another formalization of definitional equivalence due to Andréka and Németi which is equivalent to the Barrett–Halvorson (...) generalization in the case of disjoint languages. We show that the Andréka–Németi generalization is the smallest equivalence relation containing the Barrett–Halvorson generalization and it is equivalent to intertranslatability, which is another definition for definitional equivalence, even for non-disjoint languages. Finally, we investigate which definitions for definitional equivalences remain equivalent when we generalize them for theories in non-disjoint languages. (shrink)