Diego Pinheiro Fernandes

In this note I present a Lindström theorem characterizing the hybrid logic $$\mathcal {H}(\exists )$$ as the most expressive logic having compactness, the Tarski union property, and invariance under _quasi_-generated substructures. The logic $$\mathcal {H}(\exists )$$ is rather interesting, as it mixes the expressive power brought by the availability of world variables with an “almost local” quantification, which gives it a counting ability. However, $$\mathcal {H}(\exists )$$ did not receive the same attention as the other logics in the hybrid family, and only quite recently bisimulation-related invariance results were obtained for it. The Lindström theorem presented here helps clarify further its characteristics and situate better its place among extensions. The result is based on a characterization of first-order logic obtained by Lindström and employs a strategy different from the one used in recent Lindström theorems for modal and intuitionistic logics, which require a characterization of the logic at issue with respect to some invariance notion derived from bisimulation. For $$\mathcal {H}(\exists )$$ the strategy used in this note arguably provides a better Lindström theorem, as the invariance notion used gives a clearer perspective on the distinguishing capacities of the competing logics.

In this note I present a Lindström theorem characterizing the hybrid logic H(∃) as the most expressive logic having compactness, the Tarski union property, and invariance under quasi-generated substructures. The logic H(∃) is rather interesting, as it mixes the expressive power brought by the availability of world variables with an “almost local” quantification, which gives it a counting ability. However, H(∃) did not receive the same attention as the other logics in the hybrid family, and only quite recently bisimulation-related invariance results were obtained for it by Badia et al. (2023). The Lindström theorem presented here helps clarify further its characteristics and situate better its place among extensions. The result is based on (Lindström, 1973) and employs a strategy different from the one used in recent Lindström theorems for modal and intuitionistic logics, which require a characterization of the logic at issue with respect to some invariance notion derived from bisimulation. For H(∃) the strategy from (Lindström, 1973) arguably provides a better Lindström theorem, as the invariance notion used gives a clearer perspective on the distinguishing capacities of the competing logics.

It is possible to understand the expressive power of a logic as issuing from its capacity to express properties of its models. There are some ways to formally capture whether a property of models is expressible, among them is one based on the notion of definability, and one based on the notion of discrimination. If the logics to be compared are defined within the same class of models, one can employ the notions of definability and discrimination directly to obtain formal conditions for relative expressiveness. This paper studies generalizations of these formal conditions to cases where the compared logics are defined within different classes of models. There have been proposed in the literature formal conditions of two main kinds: with forward and with backward model-mappings. It is shown that none of them is adequate, despite their initial reasonableness. Moreover, we argue that general and reasonable formal conditions for relative expressiveness involving forward mappings are not likely to be found, given that they turn out to be highly dependent on specific features of the compared logics. On the other hand, it will be argued that there is a reasonable formal condition involving backward model-mappings.

This paper investigates the question “when is a logic more expressive than another?” In order to approach it, “logic” is understood in the model-theoretic sense and, contrary to other proposals in the literature, it is argued that relative expressiveness between logics is best framed with respect to the notion of expressing properties of models, a notion that can be captured precisely in various ways. It is shown that each precise rendering can give rise to a formal condition for relative expressiveness that has appeared in the literature. Five such conditions are exposed, tested for some properties and compared to each other. As the formal conditions for relative expressiveness have various levels of stringency, some results on lifting some conditions to stricter ones are explored. Finally, a discussion on the properties of these formal conditions is presented. Special attention is given to notion of meaning equivalence, and how one may consider that it holds or not, depending on the weight attributed to logical and non-logical constants in expressiveness comparisons.

In this paper a proposal by Henkin of a nominalistic interpretation for second and higher-order logic is developed in detail and analysed. It was proposed as a response to Quine’s claim that second and higher-order logic not only are committed to the existence of sets, but also are committed to the existence of more sets than can ever be referred to in the language. Henkin’s interpretation is rarely cited in the debate on semantics and ontological commitments for these logics, though it has many interesting ideas that are worth exploring. The detailed development will show that it employs an early strategy of using substitutional quantification in order to reduce ontological commitments. It will be argued that the perspective adopted for the predicate variables renders it a natural extension of Quine’s nominalistic interpretation for first-order logic. However, we will argue that, with respect to Quine’s nominalistic program and his notion of ontological commitment, still holds and thus Henkin’s interpretation is not nominalistic. Nevertheless, it will be seen that is addressed successfully and this provides further insights on the so-called “Skolem Paradox”. Moreover, the interpretation is ontologically parsimonious and, in this respect, it arguably fares better than a recent proposal by Bob Hale.

The aim of this text is to present the evolution of the relation between the concept of number and magnitude in ancient Greek mathematics. We will briefly revise the Pythagorean program and its crisis with the discovery of incommensurable magnitudes. Next, we move to the work of Eudoxus and present its advances. He improved the Pythagorean theory of proportions, so that it could also treat incommensurable magnitudes. We will see that, as the time passed by, the existence of incommensurable magnitudes was no longer something strange. Already in the period of Plato and Aristotle, their existence was common place: up to the point of being considered absurd that all magnitudes were commensurable. Aristotle criticized the Pythagorean program and defended that, though belonging to the same category, number and magnitude are of distinct species: number is discrete and magnitude is continuous. We finish by presenting briefly how the concept of number was amplified throughout the centuries until it came also to include the notion of continuity.