Università Roma Tre

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) ratioanl argomentation 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;
(4) some forms of non-deductive reasoning
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. In addition, the module 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.

Module B: It will present basic facts, notions, and definitions of set theory, which are indispensible 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 resepct to the semantics of quantified classical logic. Russell paradox will also be introduced and discussed. The module will later with a brief overview of some forms of non-deductive reasoning (probabilistic reasoning, statistical reasoning, reasoning with explanatory hypotheses).

Main text:
Critical Thinking. Un’introduzione, a cura di D. Canale, R. Ciuni, A. Frigerio, G. Tuzet, Egea, Milano 2021
(Relativi al Modulo A: Capitoli 2 – 6. Relativi al Modulo B: Capitoli 7 – 10).