A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points

Following F. William Lawvere, we show that many self-referential paradoxes, incompleteness theorems and fixed point theorems fall out of the same simple scheme. We demonstrate these similarities by showing how this simple scheme encompasses the semantic paradoxes, and how they arise as diagonal arguments and fixed point theorems in logic, computability theory, complexity theory and formal language theory

A Recipe for Paradox

In this paper, we provide a recipe that not only captures the common structure of semantic paradoxes but also captures our intuitions regarding the relations between these paradoxes. Before we unveil our recipe, we first talk about a well-known schema introduced by Graham Priest, namely, the Inclosure Schema. Without rehashing previous arguments against the Inclosure Schema, we contribute different arguments for the same concern that the Inclosure Schema bundles together the wrong paradoxes. That is, we will provide further arguments on why the Inclosure Schema is both too narrow and too broad. We then spell out our recipe. The recipe shows that all of the following paradoxes share the same structure: The Liar, Curry's paradox, Validity Curry, Provability Liar, Provability Curry, Knower's paradox, Knower's Curry, Grelling-Nelson's paradox, Russell's paradox in terms of extensions, alternative Liar and alternative Curry, and hitherto unexplored paradoxes. We conclude the paper by stating the lessons that we can learn from the recipe, and what kind of solutions the recipe suggests if we want to adhere to the Principle of Uniform Solution

Osservazioni su autoriferimento e verità

The present essay deals with the fundamental role of self-referential notions in contemporary logic. As a special case study, we survey recent ideas and results in formal semantics

Abstract logical structuralism

Structuralism has recently moved center stage in philosophy of mathematics. One of the issues discussed is the underlying logic of mathematical structuralism. In this paper, I want to look at the dual question, namely the underlying structures of logic. Indeed, from a mathematical structuralist standpoint, it makes perfect sense to try to identify the abstract structures underlying logic. We claim that one answer to this question is provided by categorical logic. In fact, we claim that the latter can be seen—and probably should be seen—as being a structuralist approach to logic and it is from this angle that categorical logic is best understood

Categories with New Foundations

While the interaction between set theory and category theory has been studied extensively, the set theories considered have remained almost entirely within the Zermelo family. Quine’s New Foundations has received limited attention, despite being the one-sorted version of a theory mentioned as a possible foundation for Category Theory by Mac Lane and Eilenberg in their seminal paper on the subject. The lack of attention given to NF is not without justification. The category of NF sets is not cartesian closed and the failure of choice is a theorem of NF. But those results should not obscure the aspects of NF that have foundational appeal, nor the value of studying category theory in the context of a universal set. The present research is not intended to “advocate” for the use of NF as a practical foundation for category theory. Instead, the work presents a broad survey of the interaction between the set theory and category theory of NF, examining the relationship in both directions. The abstract structure, of which both type restriction (in the category of NF sets) and size restriction (in the category of all categories) are specific cases, appears to be the study of relative algebra. In a number of cases, the existence of relative algebraic structures in NF can be proven more generally for a class of relative adjoints, (pseudo)monads, etc. Thus, where it seems appropriate to do so, this thesis seeks to contribute to the broader study of relative algebra