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Diamonds are a Philosopher's Best Friends. The Knowability Paradox and Modal Epistemic Relevance Logic (Extended Abstract)
 Journal of Philosophical Logic
, 2002
"... Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the ..."
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Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the antirealist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.
Intersection Type Systems and Logics Related to the Meyer–Routley System B +
, 2003
"... Abstract: Some, but not all, closed terms of the lambda calculus have types; these types are exactly the theorems of intuitionistic implicational logic. An extension of these simple (→) types to intersection (or →∧) types allows all closed lambda terms to have types. The corresponding → ∧ logic, rel ..."
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Abstract: Some, but not all, closed terms of the lambda calculus have types; these types are exactly the theorems of intuitionistic implicational logic. An extension of these simple (→) types to intersection (or →∧) types allows all closed lambda terms to have types. The corresponding → ∧ logic, related to the Meyer–Routley minimal logic B + (without ∨), is weaker than the → ∧ fragment of intuitionistic logic. In this paper we provide an introduction to the above work and also determine the →∧ logics that correspond to certain interesting subsystems of the full →∧ type theory. 1 Simple Typed Lambda Calculus In standard mathematical notation “f: α → β ” stands for “f is a function from α into β. ” If we interpret “: ” as “∈ ” we have the rule: f: α → β t: α f(t) : β This is one of the formation rules of typed lambda calculus, except that there we write ft instead of f(t). In λcalculus, λx.M represents the function f such that fx = M. This makes the following rule a natural one: [x: α] M: β λx.M: α → β We now set up the λterms and their types more formally.
Boolean Conservative Extension Results for some Modal Relevant Logics
, 2010
"... Abstract: This paper shows that a collection of modal relevant logics are conservatively extended by the addition of Boolean negation. 1 Dedication This paper is dedicated to the memory of Bob Meyer. Bob was our friend and colleague. We miss him greatly. 2 ..."
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Abstract: This paper shows that a collection of modal relevant logics are conservatively extended by the addition of Boolean negation. 1 Dedication This paper is dedicated to the memory of Bob Meyer. Bob was our friend and colleague. We miss him greatly. 2
Models for Substructural Arithmetics
, 2010
"... Abstract: This paper explores models for arithmetic in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (li ..."
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Abstract: This paper explores models for arithmetic in substructural logics. In the existing literature on substructural arithmetic, frame semantics for substructural logics are absent. We will start to fill in the picture in this paper by examining frame semantics for the substructural logics C (linear logic plus distribution), R (relevant logic) and CK (C plus weakening). The eventual goal is to find negation complete models for arithmetic in R. This paper is dedicated to my friend and mentor Professor Robert K. Meyer, who taught me the joys of Relevant Arithmetic and so much more. 1 substructural arithmetics Consider the Peano axioms for arithmetic. Identity 0 = 0 ∀x∀y(x = y → y = x) ∀x∀y∀z ( y = z → (x = y → x = z)) Successor ∀x∀y(x ′ = y ′ → x = y) ∀x∀y(x = y → x ′ = y ′) ∀x(0 ̸ = x ′) Addition ∀x(x + 0 = x) ∀x∀y ( x + y ′ = (x + y) ′)