Bul Game: Playing With Knights and Knaves

Introduction

This chapter describes the features and possible use of Bul Game, a free online software designed to support didactic activities about logic at all school levels. The name Bul comes from the Italian pronunciation of Boole, the well-known English logician who is among the founders of modern mathematical logic. The characters of the game are inspired by the inhabitants of Smullyan Island (Smullyan, 1987), and indeed the aim of the game is to make correct choices based on statements made by knights, who always tell the truth, and by knaves, who always lie. As we will see, the game allows for increasing difficulties, which make it suitable for all grade levels. The author experimented with the game in two primary schools, specifically in two second-grade classes of an Italian school with many foreign-language students and a fourth-grade class of a French school in Rome, within a wider program to introduce logic at primary school.

The purpose of this program is to introduce students to the concepts of negation, true and false statements, predicates, and connectives, always underlining the connections with everyday language. These concepts are not explicitly mentioned by Italian National Indications, it is worth noting that these Indications are not compulsory for the teacher to be followed. The characters of the knight and the knaves were used in all phases of the program. A more detailed description of this activity can be found at (OILER, 2021a) and (OILER, 2021c) and will be discussed in (Bernardi, in press). In this chapter, we only refer to some of the main features of the program in order to describe how Bul Game was used in this context.

Bul Game shares the same purposes of the aforementioned program; indeed, the activity with Bul Game aims to consolidate and revise the concepts discussed in class. The author believes that the program helps students’ awareness of the structure of everyday language, without giving native speakers an overly privileged position.

Background

The relationship between logic, language, and mathematical reasoning is a crucial one. With reference to it, Durand-Guerrier (2021) claims that “the difficulties related to questions of logic and language in the mathematics class are most often insufficiently taken into account by secondary and university teachers” (p. 2). This leads to the paradox that mathematical formalism—which should serve to clarify concepts—becomes instead an obstacle to students’ learning. In relation to the same issue, Ferrari (2021) discusses similarities and differences between the communicative mechanisms inherent in natural language and the interpretation of symbolic notations in mathematical logic—with particular emphasis on connectives. Both authors advocate not only for an explicit introduction of logic in the mathematical school curriculum from the earliest years of primary school—thus refuting the idea that an understanding of basic logical concepts can be acquired automatically through standard mathematical teaching—but also for an appropriate use of symbolism.

In a multilingual context, the links between logic and language are connected to the fact that the structure of language can affect the thought process. “Logical connectives are one of the resources that a language uses to link and sequence ideas” (Edmonds-Whaten et al., 2021, p. 33) and their knowledge in the language of instruction seems to be the most important variable in deductive reasoning for bilinguals from different countries (Dawe, 1983). According to Durand-Guerrier et al. (2021) in a multilingual context “predicate logic can be used to unpack the logic of a given statement by identifying the logical categories, connectors, quantifiers, and their respective scopes” (p. 88). These concepts often remain implicit or are hidden through linguistic means depending on the language, and this can lead to ambiguities.

Natural languages are distinguished by the alphabet, by the words composed with the alphabet, and by the structure—that is, the rules that indicate how to arrange words to construct sentences. The fundamental characterization of a language may seem, at first glance, to be related to words only. However, structure also plays a fundamental role in a language because it clarifies how we express ourselves. Indeed, the importance of language rules is recognized. Moreover, as most of us have experienced, the translation of single words is not sufficient to interpret a sentence in a language unknown to us.