This course aims at (1) developing and training the ability to recognize and evaluate arguments and a variety of forms of reasoning, and to tell apart good arguments from bad arguments, according to the definitions provided through the course; (2) developing the capability of solving reasoning problems that refer to the many different forms of reasoning that we discuss in the course; (3) securing a suitable understanding of basics aspect of propositional logic and quantified logic, and of basics of probability calculus, inductive and abductive reasoning; (4) securing an understanding of the function reasoning plays in rational discussion and the exchange of theses.

Objectives (1) – (4) are crucial since today, mainly due to the presence of social networks, our social interaction comes with an exchange of opinions that is increasingly more frequent and our connections with other agents are wider and wider. It has been acknowledged that the speed and frequency of these exchanges goes along with diminished reasoning skills, and this jeopardizes the understanding of problems of public interest on which our opinion is solicited.

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This course provides an introduction to:
(1) the role played by reasoning in rational interaction (discussions, exchanges of theses), in the solution of problems of logic and mathematics, and the consequence of a lack of adequate reasoning procedures in these areas;
(2) rational argumentation and the logical structure underlying valid arguments;
(3) a rigorous approach to deductive reasoning, based on the formal tools provided by propositional and quantified (deductive) logic.
The course also wishes to alert participants of the consequences of a lack of a rational course in the context of mass communication, information society, and online interaction, while developing the ability to correctly apply the basic rules of reasoning that are distinctive of deductive reasoning.

The course will apply, as far as possible, a `bottom-up' approach: from reasoning problems, to the tools required to solve them, to the theories in which such tools are defined, understood, and discussed. The course is divided into two modules:

Module A: It will approach and discuss the definition of an argument and of a good argument, the role played by arguments in our reactions to disagreement and in rational discussions, and the rational strategies for reacting to disagreement. It will then focus on deductive reasoning and on propositional logic in particular. In this context the course will introduce and discuss the basic rules of reasoning of propositional logic and it will discuss the notion of derivability, introduce the procedures for building a formal language, it will explore the semantics of propositional logic, the notions of logical consequence and validity, and the possible connections between derivability, logical consequence, and validity.

Module B: It will introduce the notion of a system of rules and that of an axiomatics system, together with the notions of soundness and completeness, and it will then focus on natural deduction and its soundness and completeness with respect to the semantics of classical propositional logic. It will then present basic facts, notions, and definitions of set theory, which are indispensable when it comes to an understanding of quantified logic. After that, the course will focus on quantified logic, by explaining the way in which quantified logic 'reads' predicates and quantifiers (expressions like 'Every' and 'Some'), it will introduce basic rules for reasoning with the quantifiers, and it will introduce the semantics of quantified logic. The course will then discuss soundness and completeness of natural deduction for quantified classical logic with respect to the semantics of quantified classical logic. Russell paradox will also be introduced and discussed.

The achievement of 12 CFU requires presenting the program of both modules; achievement of 6 CFU requires presenting the program of one of the two modules only.

Core Documentation

Francesco Berto. Logica. Da zero a Gödel, Laterza, Roma 2008.

Reference Bibliography

References: The following books help familiarizing more with some of the concepts introduced through the course and discussed in the textbook, while providing a more detailed framework for some specific topics. They also give a chance for an in-depth analysis of such topics: E.J. Lemmon. Elementi di Logica. Laterza, Rome 2021. (Particularly on natural deduction in Classical Logic, both propositional and quantified). Critical Thinking. Un’introduzione, a cura di D. Canale, R. Ciuni, A. Frigerio, G. Tuzet, Egea, Milano 2021. (Capitoli 2–3 e 5). (Particularly on the notions that are introduced in the first two classes, and with some references to the notions of derivability, truth table, introdction and elimination rules, and quantified logic). Mondadori M., D’Agostino M., Logica, Mondadori, Milano 1997. Chapters 1 – 4 and 8. (On a wide range of topics dealt with in the different classes)

Type of delivery of the course

Lectures. The lecturer will solicit the active participation of the students in given moments during the class. Whenever possible, the lecture will start from the discussion of problems, and this will require the participation of the students. Then, the lecture will present the reasoning tools that make a solution to the problems possible, while providing a framework of notions for them. The lectures will also approach aspect of the topics that are more abstract and theoretical in nature. Some lectures will have a prominently theoretical angle. In general, lectures aim at realizing, in general, a ‘bottom-up’ approach – in this case: an approach that goes from the problems, to the tools for their solution, to the theoretical frameworks in which these tools are described an understood – in particular, formal logic.

Type of evaluation

A written partial exam on module A (it will take place in the course hours). The partial exam will last 60 minutes. In addition, one written final exam in the regular exam dates. This will last 120 minutes. Success in the partial exam will release from questions on module A in the final exam. If a student does not pass or does not take part to the partial exam will have questions on the whole program The evaluation will assess: - acquisition of the reasoning abilities that are related to the kind of reasoning problems presented throughout the course; - understanding of the key notions introduced by the course; - the abilities of specifying and discussing such notions and their importance.