Propositional dynamic logic for searching games with errors

We investigate some finitely-valued generalizations of propositional dynamic logic with tests. We start by introducing the (n+1)-valued Kripke models and a corresponding language based on a modal extension of {\L}ukasiewicz many-valued logic. We illustrate the definitions by providing a framework for an analysis of the R\'enyi - Ulam searching game with errors. Our main result is the axiomatization of the theory of the (n+1)-valued Kripke models. This result is obtained through filtration of the canonical model of the smallest (n+1)-valued propositional dynamic logic

Quantum Information Dynamics and Open World Science

One of the fundamental insights of quantum mechanics is that complete knowledge of the state of a quantum system is not possible. Such incomplete knowledge of a physical system is the norm rather than the exception. This is becoming increasingly apparent as we apply scientific methods to increasingly complex situations. Empirically intensive disciplines in the biological, human, and geosciences all operate in situations where valid conclusions must be drawn, but deductive completeness is impossible. This paper argues that such situations are emerging examples of {it Open World} Science. In this paradigm, scientific models are known to be acting with incomplete information. Open World models acknowledge their incompleteness, and respond positively when new information becomes available. Many methods for creating Open World models have been explored analytically in quantitative disciplines such as statistics, and the increasingly mature area of machine learning. This paper examines the role of quantum theory and quantum logic in the underpinnings of Open World models, examining the importance of structural features of such as non-commutativity, degrees of similarity, induction, and the impact of observation. Quantum mechanics is not a problem around the edges of classical theory, but is rather a secure bridgehead in the world of science to come

Poythress’s Trinitarian Logic: A Review Essay

Reviewed Title: Vern Sheridan Poythress. Logic: A God-Centered Approach to the Foundation of Western Thought. Crossway, 2013. 733 pp. ISBN: 978-1-4335-3229-0

Semantic Relevance

International audienceAbstract A clause C is syntactically relevant in some clause set N , if it occurs in every refutation of N . A clause C is syntactically semi-relevant, if it occurs in some refutation of N . While syntactic relevance coincides with satisfiability (if C is syntactically relevant then $$N\setminus \{C\}$$ N \ { C } is satisfiable), the semantic counterpart for syntactic semi-relevance was not known so far. Using the new notion of a conflict literal we show that for independent clause sets N a clause C is syntactically semi-relevant in the clause set N if and only if it adds to the number of conflict literals in N . A clause set is independent, if no clause out of the clause set is the consequence of different clauses from the clause set. Furthermore, we relate the notion of relevance to that of a minimally unsatisfiable subset (MUS) of some independent clause set N . In propositional logic, a clause C is relevant if it occurs in all MUSes of some clause set N and semi-relevant if it occurs in some MUS. For first-order logic the characterization needs to be refined with respect to ground instances of N and C

A general approach to define binders using matching logic

We propose a novel shallow embedding of binders using matching logic, where the binding behavior of object-level binders is obtained for free from the behavior of the built-in existential binder of matching logic. We show that binders in various logical systems such as lambda-calculus, System F, pi-calculus, pure type systems, etc., can be defined in matching logic. We show the correctness of our definitions by proving conservative extension theorems, which state that a sequent/judgment is provable in the original system if and only if it is provable in matching logic. An appealing aspect of our embedding of binders in matching logic is that it yields models to all binders, also for free. We show that models yielded by matching logic are deductively complete to the formal reasoning in the original systems. For lambda-calculus, we further show that the yielded models are representationally complete---a desired property that is not enjoyed by many existing lambda-calculus semantics.Ope

Iteration and Truth: A Fifth "Orientation of Thought"

This article offers a novel interpretation of Jacques Derrida's deconstructive thought in terms of model theory. Taking its cue from Paul Livingston's Politics of Logic, which interprets Derrida as a thinker of inconsistent totalities, the article argues that Livingston's description of Derrida is unable to accommodate certain consistency-driven aspects of Derrida's work. These aspects pertain to Derrida's notion of ”iterability”. The article demonstrates that the context-bound nature of iteration – the altering repetition of any discrete unit of meaning – and Derrida's possibilist view of context – that a context need not be part of the actual world to merit consideration – lead to the possibility of articulating iteration with the model-theoretical notion of truth. In model theory, truth is a relation between a sentence and the class of models in which the sentence is true. Arguing that the same holds for Derrida's iterations and contexts, the article, in presenting the first rigorous truth-definition internal to deconstructive thought, outlines a ”fifth orientation of thought” alongside the four orientations listed in Livingston's book: if, according to Livingston, one can relate the whole of being to the whole of thought in one of four different ways, the aspects of Derrida's work that do not fall within this schema call out for another possible orientation