A Logic of Exceptions
As the title suggests this book discusses the subject of logic. It begins with the basic elements but it also introduces new elements such as a logic of exceptions. Hence the last part of the book requires a more advanced level. It also includes part of the exciting history of philosophy, mathematics and logic, that usually doesn’t appear in other logic texts.
The examples are written in Mathematica which is a language or system for doing mathematics on the computer, but you can still read and understand this book even if you are not familiar with Mathematica and you decide not to run the programs. In any case the basics are supplied.
The book is organised in three parts:
Part I – The basic propositional operators are introduced.
Part II – This part deals with the basic notions of inference: inferences, syllogisms, axiomatics and proof theory. It is first developed for propositional logic and then for predicate logic. For the first time, the existence of sentences that require some three-valued logic is discussed. (There are propositions that are either true or false (two-valued) and sentences that may be true, false or indeterminate.)
It also includes a short history of the Liar paradox , which is the root of many other paradoxes and its different solution approaches such as the theory of types, proof theory and three-valued logic. It analyses the problems which arise in the first two approaches, so that the author defends that only three-valued logic allows for a consistent development.
The historical review of the Liar shows how various cases propose the introduction of an exception as part of the proposed solution. Hence a logic of exceptions is developed.
Part III - This part deals with three-valued logic. First of all, the author discusses the need to introduce a third value different to true or false. Once again the basic operators are considered and its definitions modified because of this third value.
Finally three-valued logic is applied to the Liar paradox, which is now solved. Gödel’s theorems, and Brouwer’s intuitionism are revisited.