Encoding many-valued logic in $\lambda$-calculus

We will extend the well-known Church encoding of Boolean logic into $\lambda$-calculus to an encoding of McCarthy's $3$-valued logic into a suitable infinitary extension of $\lambda$-calculus that identifies all unsolvables by $\bot$, where $\bot$ is a fresh constant. This encoding refines to $n$-valued logic for $n\in\{4,5\}$. Such encodings also exist for Church's original $\lambda\mathbf{I}$-calculus. By way of motivation we consider Russell's paradox, exploiting the fact that the same encoding allows us also to calculate truth values of infinite closed propositions in this infinitary setting.

Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I)

In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.

Russell’s Paradox: a historical study about the paradox in Frege’s theories

For over twenty years, Frege tried to find the foundations of arithmetic through logic, and by doing this, he attempted to establish the truth and certainty of the knowledge. However, when he believed his work wasdone, Bertrand Russell sent him a letter pointing out a paradox, known as Russell‟s paradox. It is often considered that Russell identified the paradox in Frege‟s theories. However, as shown in this paper, Russell, Frege and also George Cantor contributedsignificantly to the identification of the paradox. In 1902, Russell encouraged Frege to reconsider a portion of his work based in a paradox built from Cantor‟s theories. Previously, in 1885, Cantor had warned Frege about taking extensions of concepts in the construction of his system. With these considerations, Frege managed to identify the precise law and definitions that allowed the generation of the paradox in his system. The objective of this paper is to present a historical reconstruction of the paradox in Frege‟s publications and discuss it considering the correspondences exchanged between him and Russell. We shall take special attention to the role played by each of these mathematicians in the identification of the paradox and its developments. We also will show how Frege anticipated the solutions and new theories that would arise when dealing with logico-mathematical paradoxes, including but not limited to Russell‟s paradox.

Russell's Paradox: A Historical Study about the Paradox in Frege's Theories

For over twenty years, Frege tried to find the foundations of arithmetic through logic, and by doing this, he attempted to establish the truth and certainty of the knowledge. However, when he believed his work was done, Bertrand Russell sent him a letter pointing out a paradox, known as Russell’s paradox. It is often considered that Russell identified the paradox in Frege’s theories. However, as shown in this paper, Russell, Frege and also George Cantor contributed significantly to the identification of the paradox. In 1902, Russell encouraged Frege to reconsider a portion of his work based in a paradox built from Cantor’s theories. Previously, in 1885, Cantor had warned Frege about taking extensions of concepts in the construction of his system. With these considerations, Frege managed to identify the precise law and definitions that allowed the generation of the paradox in his system. The objective of this paper is to present a historical reconstruction of the paradox in Frege’s publications and discuss it considering the correspondences exchanged between him and Russell. We shall take special attention to the role played by each of these mathematicians in the identification of the paradox and its developments. We also will show how Frege anticipated the solutions and new theories that would arise when dealing with logico-mathematical paradoxes, including but not limited to Russell’s paradox.

Founded semantics and constraint semantics of logic rules

Logic rules and inference are fundamental in computer science and have been studied extensively. However, prior semantics of logic languages can have subtle implications and can disagree significantly, on even very simple programs, including in attempting to solve the well-known Russell’s paradox. These semantics are often non-intuitive and hard-to-understand when unrestricted negation is used in recursion. This paper describes a simple new semantics for logic rules, founded semantics, and its straightforward extension to another simple new semantics, constraint semantics, that unify the core of different prior semantics. The new semantics support unrestricted negation, as well as unrestricted existential and universal quantifications. They are uniquely expressive and intuitive by allowing assumptions about the predicates, rules and reasoning to be specified explicitly, as simple and precise binary choices. They are completely declarative and relate cleanly to prior semantics. In addition, founded semantics can be computed in linear time in the size of the ground program.

Second-Order Abstraction Before and After Russell’s Paradox

In this article, I discuss certain aspects of Frege’s paradigms of second-order abstraction principles, Hume’s Principle and Basic Law V, with special emphasis on the latter. I begin by arguing that, contrary to a widespread view, Frege did not express any dissatisfaction with Basic Law V before 1902. In particular, he did not raise any doubt about its assumed logical nature. I then show why Frege nonetheless fails to justify Basic Law V as a primitive logical truth along the lines of the semantic justification that he provides for the other axioms of his system. In subsequent sections, I argue (a) that Frege could not have chosen Hume’s Principle as a logical axiom, neither before 1902 nor after 1902; (b) that even if in the wake of Russell’s Paradox Frege had accepted Hume’s Principle as a logical axiom, such an axiom could not have replaced Basic Law V which was designed to introduce logical objects of a fundamental and irreducible kind and to afford us the right cognitive access to them; (c) that Frege most likely held that the two sides of Basic Law V express different thoughts; (d) that for Basic Law V or for any other Fregean abstraction principle that is laid down as an axiom of a theory, the case in which both real epistemic value and self-evidence are given their due is ruled out. I make a proposal as to how Frege might have escaped this epistemic dilemma.

The axiom of selection resolves Russell’s paradox

We show that the axiom of selection solves Russell's paradox.