Semantic Criteria of Correct Formalization

This paper compares several models of formalization. It articulates criteria of correct formalization and identifies their problems. All of the discussed criteria are so called “semantic” criteria, which refer to the interpretation of logical formulas. However, as will be shown, different versions of an implicitly applied or explicitly stated criterion of correctness depend on different understandings of “interpretation” in this context

Transformation of propositional calculus statements into integer and mixed integer programs: An approach towards automatic reformulation

A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Progamming (ILP) formulation Mixed Integer Programming (MIP) formulation is presented. An ILP stated as a system of linear constraints involving integer variables and an objective function, provides a powerful representation of decision problems through a tightly interrelated closed system of choices. It supports direct representation of logical (Boolean or prepositional calculus) expressions. Binary variables (hereafter called logical variables) are first introduced and methods of logically connecting these to other variables are then presented. Simple constraints can be combined to construct logical relationships and the methods of formulating these are discussed. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. These reformulation procedures are illustrated by two examples. A scheme of implementation.ithin an LP modelling system is outlined

On Affine Logic and {\L}ukasiewicz Logic

The multi-valued logic of {\L}ukasiewicz is a substructural logic that has been widely studied and has many interesting properties. It is classical, in the sense that it admits the axiom schema of double negation, [DNE]. However, our understanding of {\L}ukasiewicz logic can be improved by separating its classical and intuitionistic aspects. The intuitionistic aspect of {\L}ukasiewicz logic is captured in an axiom schema, [CWC], which asserts the commutativity of a weak form of conjunction. This is equivalent to a very restricted form of contraction. We show how {\L}ukasiewicz Logic can be viewed both as an extension of classical affine logic with [CWC], or as an extension of what we call \emph{intuitionistic} {\L}ukasiewicz logic with [DNE], intuitionistic {\L}ukasiewicz logic being the extension of intuitionistic affine logic by the schema [CWC]. At first glance, intuitionistic affine logic seems very weak, but, in fact, [CWC] is surprisingly powerful, implying results such as intuitionistic analogues of De Morgan's laws. However the proofs can be very intricate. We present these results using derived connectives to clarify and motivate the proofs and give several applications. We give an analysis of the applicability to these logics of the well-known methods that use negation to translate classical logic into intuitionistic logic. The usual proofs of correctness for these translations make much use of contraction. Nonetheless, we show that all the usual negative translations are already correct for intuitionistic {\L}ukasiewicz logic, where only the limited amount of contraction given by [CWC] is allowed. This is in contrast with affine logic for which we show, by appeal to results on semantics proved in a companion paper, that both the Gentzen and the Glivenko translations fail.Comment: 28 page

A Troubled Transition: From President Morgan to President Waugh

Dickinson College\u27s twentieth-century journey has been marked primarily, though not entirely, by gains: increases in numbers of students and faculty, advances in the quality of the program offered, and a general broadening of opportunities for those enrolled in this program. Specific advances have been identified with particular presidential administrations, and have been gracefully limned by Charles Coleman Sellers\u27s general history of the college. For those interested in the academic policies of Dickinson College in this century, one administration stands out for the potential it embodied, but did not realize: the administration, in the early thirties, of Karl Tinsley Waugh. Waugh\u27s brief tenure at Dickinson offers a case study in the kinds of tensions and frustrations which can spring from any effort to orchestrate change, and it is presented here as a vignette of Dickinson history. Because of its brevity, Waugh\u27s administration was not a landmark in Dickinson history. But it might have been, and deserves on that account to be better known. To understand President Karl Tinsley Waugh and his travails, however, it is essential first to introduce his immediate predecessor and ultimately his nemesis, James Henry Morgan, and to place each of their presidencies, as well as their personalities, in the context of the other. [excerpt

A Type-Directed Negation Elimination

In the modal mu-calculus, a formula is well-formed if each recursive variable occurs underneath an even number of negations. By means of De Morgan's laws, it is easy to transform any well-formed formula into an equivalent formula without negations -- its negation normal form. Moreover, if the formula is of size n, its negation normal form of is of the same size O(n). The full modal mu-calculus and the negation normal form fragment are thus equally expressive and concise. In this paper we extend this result to the higher-order modal fixed point logic (HFL), an extension of the modal mu-calculus with higher-order recursive predicate transformers. We present a procedure that converts a formula into an equivalent formula without negations of quadratic size in the worst case and of linear size when the number of variables of the formula is fixed.Comment: In Proceedings FICS 2015, arXiv:1509.0282