This book collects notes that were prepared for a university course taught in the Spring of 2018,... more This book collects notes that were prepared for a university course taught in the Spring of 2018, and delivered to an audience of students enrolled in the Master course in Logic, philosophy and history of science of the University of Florence. The goal of the course was to introduce students to some basic concepts from the area of research generally known as decision theory. This is done by focussing on the concept of ‘rational choice’, which is analyzed, methodologically speaking, by the means of the theory of games. To minimize prerequisites it was decided to restrict the attention to the theory of finite games in particular. The topics treated are vary, and belongs to both the theory of games ‘in normal form’ as well as that of games ‘in extensive form’, as they are usually referred to. The classical issues in both fields, such as the theory of ‘equilibria’ and the study of properties such as determinacy, are carefully discussed to make them clear to beginners and are addressed from a novel perspective which makes use of formal methods that are typical of researches connected with the study of logic.
Logic as it is intended in this volume, is the basis of human rational faculties: understanding, ... more Logic as it is intended in this volume, is the basis of human rational faculties: understanding, analyzing, reasoning. This book offers an overview of useful related topics, for each of which a theoretical introduction is coupled with a set of excercises to evaluate and improve the reader's skills.
The book is in italian only, and is freely available for download at the website of the publisher Firenze University Press (link is provided below).
Kurt Gödel è stato certamente il logico più influente del Novecento. I suoi lavori, visti come de... more Kurt Gödel è stato certamente il logico più influente del Novecento. I suoi lavori, visti come delle pietre miliari e come tali richiamati ancora oggi nella letteratura specialistica, investono aspetti fondamentali per la ricerca in campo matematico, quali la relazione tra verità e dimostrabilità o questioni relative alla natura e ai limiti della conoscenza matematica. Questo profilo nasce dal tentativo di correggere l’impressione che la grandezza del personaggio possa essere ricondotta a uno dei suoi molteplici contributi in particolare, come ad esempio il teorema di incompletezza per le teorie assiomatiche al quale egli deve gran parte della sua fama. Al contrario, il volume cerca di offrire una panoramica per quanto possibile esaustiva dell’opera logica e matematica di Gödel, così da valutarne meglio la portata.
Truth Meets Vagueness. Unifying the Semantic and the Soritical Paradoxes, 2023
Semantic and soritical paradoxes display remarkable family resemblances. For one thing, several n... more Semantic and soritical paradoxes display remarkable family resemblances. For one thing, several non-classical logics have been independently applied to both kinds of paradoxes. For another, revenge paradoxes and higher-order vagueness-among the most serious problems targeting solutions to semantic and soritical paradoxes-exhibit a rather similar dynamics. Some authors have taken these facts to suggest that truth and vagueness require a unified logical framework, or perhaps that the truth predicate is itself vague. However, a common core of semantic and soritical paradoxes has not been identified yet, and no explanation of their relationships has been provided. Here we aim at filling this lacuna, in the framework of many-valued logics. We provide a unified diagnosis of semantic and soritical paradoxes, identifying their source in a general form of indiscernibility. We then develop our diagnosis into a theory of paradoxicality, which formalizes both semantic and soritical paradoxes as arguments involving specific instances of our generalized indiscernibility principle, and correctly predicts which logics can non-trivially solve them.
At the commonsensical level of the manifest image, we seem to take for granted logical laws of al... more At the commonsensical level of the manifest image, we seem to take for granted logical laws of all sorts, including classical logic (CL) and naive principles of truth and predication (TP), which, however, generate logical paradoxes such as the liar, Russell's paradox and Curry's paradox. The formal logic of the scientific image comes to the rescue by proposing many competing formal systems that restore consistency, by sacrificing either principles of CL or principles of TP. We wish to explore a different path, which aims at saving both CL and TP, and deals with the paradoxes when they come to the fore, without swallowing contradictions or explosion. We consider the viability of Batens' Inconsistency-Adaptive Logic (IAL) to pursue this goal and we end up with a negative assessment. We then sketch an alternative proposal that incorporates IAL's distinction between provisional and final derivability.
Kurt Gödel (1906-1978) is undoubtedly the most influential logician of the 20th Century. Thanks t... more Kurt Gödel (1906-1978) is undoubtedly the most influential logician of the 20th Century. Thanks to him, we have some of the most notable results in field of foundational research, such as the completeness theorem for the first-order logical calculus, the two incompleteness theorems, the proof of consistency between Cantor’s hypothesis of the continuum and the axioms of set theory. He is also known for much disputed philosophical stances, such as his Platonism approach towards the existence and the nature of mathematical objects. His complex personality, along with his extraordinary scientific production, have spread his fame well beyond the field of logic.Kurt Gödel (1906-1978) è stato certamente il logico più influente del ‘900. A lui si devono alcuni dei risultati più noti nel campo delle ricerche fondazionali, come il teorema di completezza semantica per il calcolo logico del prim’ordine, i due teoremi di incompletezza, la dimostrazione di consistenza dell’ipotesi del continuo di...
Logical abilities are a ubiquitous ingredient in all those contexts that take into account soft s... more Logical abilities are a ubiquitous ingredient in all those contexts that take into account soft skills, argumentative skills or critical thinking. However, the relationship between logical models and the enhancement of these abilities is rarely explicitly considered. Two aspects of the issue are particularly critical in our opinion, namely: (i) the lack of statistically relevant data concerning these competences; (ii) the absence of reliable indices that might be used to detect and measure the possession of abilities underlying the aforementioned skills. This paper addresses both aspects of this topic by presenting the results of a research that we conducted between October and December 2020 on students enrolled in various degree courses at the University of Florence. The dataset has been collected by a three-stage initiative. We started from an entrance examination to assess the students' initial abilities. This test included ten questions, each regarding a specific aspect of l...
This book collects notes that were prepared for a university course taught in the Spring of 2018,... more This book collects notes that were prepared for a university course taught in the Spring of 2018, and delivered to an audience of students enrolled in the Master course in Logic, philosophy and history of science of the University of Florence. The goal of the course was to introduce students to some basic concepts from the area of research generally known as decision theory. This is done by focussing on the concept of ‘rational choice’, which is analyzed, methodologically speaking, by the means of the theory of games. To minimize prerequisites it was decided to restrict the attention to the theory of finite games in particular. The topics treated are vary, and belongs to both the theory of games ‘in normal form’ as well as that of games ‘in extensive form’, as they are usually referred to. The classical issues in both fields, such as the theory of ‘equilibria’ and the study of properties such as determinacy, are carefully discussed to make them clear to beginners and are addressed from a novel perspective which makes use of formal methods that are typical of researches connected with the study of logic.
Logic as it is intended in this volume, is the basis of human rational faculties: understanding, ... more Logic as it is intended in this volume, is the basis of human rational faculties: understanding, analyzing, reasoning. This book offers an overview of useful related topics, for each of which a theoretical introduction is coupled with a set of excercises to evaluate and improve the reader's skills.
The book is in italian only, and is freely available for download at the website of the publisher Firenze University Press (link is provided below).
Kurt Gödel è stato certamente il logico più influente del Novecento. I suoi lavori, visti come de... more Kurt Gödel è stato certamente il logico più influente del Novecento. I suoi lavori, visti come delle pietre miliari e come tali richiamati ancora oggi nella letteratura specialistica, investono aspetti fondamentali per la ricerca in campo matematico, quali la relazione tra verità e dimostrabilità o questioni relative alla natura e ai limiti della conoscenza matematica. Questo profilo nasce dal tentativo di correggere l’impressione che la grandezza del personaggio possa essere ricondotta a uno dei suoi molteplici contributi in particolare, come ad esempio il teorema di incompletezza per le teorie assiomatiche al quale egli deve gran parte della sua fama. Al contrario, il volume cerca di offrire una panoramica per quanto possibile esaustiva dell’opera logica e matematica di Gödel, così da valutarne meglio la portata.
Truth Meets Vagueness. Unifying the Semantic and the Soritical Paradoxes, 2023
Semantic and soritical paradoxes display remarkable family resemblances. For one thing, several n... more Semantic and soritical paradoxes display remarkable family resemblances. For one thing, several non-classical logics have been independently applied to both kinds of paradoxes. For another, revenge paradoxes and higher-order vagueness-among the most serious problems targeting solutions to semantic and soritical paradoxes-exhibit a rather similar dynamics. Some authors have taken these facts to suggest that truth and vagueness require a unified logical framework, or perhaps that the truth predicate is itself vague. However, a common core of semantic and soritical paradoxes has not been identified yet, and no explanation of their relationships has been provided. Here we aim at filling this lacuna, in the framework of many-valued logics. We provide a unified diagnosis of semantic and soritical paradoxes, identifying their source in a general form of indiscernibility. We then develop our diagnosis into a theory of paradoxicality, which formalizes both semantic and soritical paradoxes as arguments involving specific instances of our generalized indiscernibility principle, and correctly predicts which logics can non-trivially solve them.
At the commonsensical level of the manifest image, we seem to take for granted logical laws of al... more At the commonsensical level of the manifest image, we seem to take for granted logical laws of all sorts, including classical logic (CL) and naive principles of truth and predication (TP), which, however, generate logical paradoxes such as the liar, Russell's paradox and Curry's paradox. The formal logic of the scientific image comes to the rescue by proposing many competing formal systems that restore consistency, by sacrificing either principles of CL or principles of TP. We wish to explore a different path, which aims at saving both CL and TP, and deals with the paradoxes when they come to the fore, without swallowing contradictions or explosion. We consider the viability of Batens' Inconsistency-Adaptive Logic (IAL) to pursue this goal and we end up with a negative assessment. We then sketch an alternative proposal that incorporates IAL's distinction between provisional and final derivability.
Kurt Gödel (1906-1978) is undoubtedly the most influential logician of the 20th Century. Thanks t... more Kurt Gödel (1906-1978) is undoubtedly the most influential logician of the 20th Century. Thanks to him, we have some of the most notable results in field of foundational research, such as the completeness theorem for the first-order logical calculus, the two incompleteness theorems, the proof of consistency between Cantor’s hypothesis of the continuum and the axioms of set theory. He is also known for much disputed philosophical stances, such as his Platonism approach towards the existence and the nature of mathematical objects. His complex personality, along with his extraordinary scientific production, have spread his fame well beyond the field of logic.Kurt Gödel (1906-1978) è stato certamente il logico più influente del ‘900. A lui si devono alcuni dei risultati più noti nel campo delle ricerche fondazionali, come il teorema di completezza semantica per il calcolo logico del prim’ordine, i due teoremi di incompletezza, la dimostrazione di consistenza dell’ipotesi del continuo di...
Logical abilities are a ubiquitous ingredient in all those contexts that take into account soft s... more Logical abilities are a ubiquitous ingredient in all those contexts that take into account soft skills, argumentative skills or critical thinking. However, the relationship between logical models and the enhancement of these abilities is rarely explicitly considered. Two aspects of the issue are particularly critical in our opinion, namely: (i) the lack of statistically relevant data concerning these competences; (ii) the absence of reliable indices that might be used to detect and measure the possession of abilities underlying the aforementioned skills. This paper addresses both aspects of this topic by presenting the results of a research that we conducted between October and December 2020 on students enrolled in various degree courses at the University of Florence. The dataset has been collected by a three-stage initiative. We started from an entrance examination to assess the students' initial abilities. This test included ten questions, each regarding a specific aspect of l...
Mathesis Universalis, Computability and Proof, 2019
Definitions are important tools for our attempt to construct an intelligible imagine of reality. ... more Definitions are important tools for our attempt to construct an intelligible imagine of reality. Regarded as such, there are interesting epistemological clues about them to consider: What does a legitimate definition look like? Or: Is there a privileged form that definitions should have? How do we find good definitions? Or: How do we know that a given definition is a good one? Traces of the debate on these, and similar other questions are ubiquitous in the history of philosophy. The aim of this note, however, is not to give a historical account on this matter. Rather, on the basis of some very recent work of a proof–theoretical nature, I plan to address those issues in the context of a discussion concerning circular definitions. A definition is circular in the sense of this paper if the very concept that one is defining is used in the condition defining it. Despite their peculiar character, circular definitions are neither rare, nor easy to dispense with: it turns out that they affe...
Paradoxes have represented an area of research where the interplay between mathematical methods o... more Paradoxes have represented an area of research where the interplay between mathematical methods of investigation and conceptual analysis proved to be particularly fruitful. As is well known, the discovery of paradoxes at the end of the nineteenth century occasioned the debate on the foundations of mathematics, which in turn was pivotal for the development of contemporary logic. However, rather than remaining a phenomenon confined in this area, paradoxes also arose in other contexts and became a matter of dispute which was also important for domains such as philosophy of logic and language, epistemology, decision theory, computer science.The book by Roy T. Cook under review aims at illustrating the richness of the debate on paradoxes, as well as suggesting a unifying view of them. This is done by taking into account a representative number and variety of paradoxical arguments which the author explains and discusses in detail. Cook’s initial remarks are intended as a guide for the inexpe
The revision theory of truth [2] is a semantical tool devised to solve the problems of formal tru... more The revision theory of truth [2] is a semantical tool devised to solve the problems of formal truth. More recently, Gupta [1] has proposed a definition of strategic rationality that makes use only of the finite part of the revision theory. In this talk I plan to explain the basic features of the revision-theoretic approach to circularity, compare the properties of the full transfinite theory wiTh those of the finite one, and illustrate Gupta's recent proposal. I will then present an attempt to extend the scope of the latter, analyze issues triggered by it and propose a solution which is due to a joint work with Giacomo Sillari (LUISS, Rome).
[1] Anil Gupta. On circular concepts. In A. Chapuis and A. Gupta, editors, Circularity, Definition, and Truth, pages 123–153. Indian Council of Philosohical Research, 2000.
[2] Anil Gupta and Nuel Belnap. The Revision Theory of Truth. MIT Press, 1993.
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BOOKS by Riccardo Bruni
The book is in italian only, and is freely available for download at the website of the publisher Firenze University Press (link is provided below).
Papers by Riccardo Bruni
The book is in italian only, and is freely available for download at the website of the publisher Firenze University Press (link is provided below).
circularity, compare the properties of the full transfinite theory wiTh those of the finite one, and illustrate Gupta's recent proposal. I will then present an attempt to extend the scope of the latter, analyze issues triggered by it and propose a solution which is due to a joint work with Giacomo Sillari (LUISS, Rome).
[1] Anil Gupta. On circular concepts. In A. Chapuis and A. Gupta, editors, Circularity, Definition, and Truth, pages 123–153. Indian Council of Philosohical Research, 2000.
[2] Anil Gupta and Nuel Belnap. The Revision Theory of Truth. MIT Press, 1993.