Semantics for Dual and Symmetric Combinatory Calculi

We define dual and symmetric combinatory calculi (inequational and equational ones), and prove their consistency. Then, we introduce algebraic and set theoretical– relational and operational – semantics, and prove soundness and completeness. We analyze the relationship between these logics, and argue that inequational dual logics are the best suited to model computation.     

All Normal Extensions of S5-squared Are Finitely Axiomatizable

We prove that every normal extension of the bi-modal system S5² is finitely axiomatizable and that every proper normal extension has NP-complete satisfiability problem.Presented by Heinrich Wansing     

TPS: A hybrid automatic-interactive system for developing proofs

The theorem proving system Tps provides support for constructing proofs using a mix of automation and user interaction, and for manipulating and inspecting proofs. Its library facilities allow the user to store and organize work. Mathematical theorems can be expressed very naturally in Tps using higher-order logic. A number of proof representations are available in Tps, so proofs can be inspected from various perspectives.     

Standard Quantification Theory in the Analysis of English

Standard first-order logic plus quantifiers of all finite orders (SFOL) faces four well-known difficulties when used to characterize the behavior of certain English quantifier phrases. All four difficulties seem to stem from the typed structure of SFOL models. The typed structure of SFOL models is in turn a product of an asymmetry between the meaning of names and the meaning of predicates, the element-set asymmetry. In this paper we examine a class of models in which this asymmetry of meaning is removed. The models of this class permit definitions of the quantifiers which allow a desirable flexibility in fixing the domain of quantification. Certain SFOL type restrictions are thereby avoided. The resulting models of English validate all of the standard first-order logical truths and are free of the four deficiencies of SFOL models.     

Quantum Logic as a Fragment of Independence-Friendly Logic

The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation . Then in a Hilbert space turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann quantum logic can be interpreted by taking their disjunction to be ¬(A & B). Their logic can thus be mapped into a Boolean structure to which an additional operator has been added.     

Quantum Logic in Intuitionistic Perspective

In their seminal paper Birkhoff and von Neumann revealed the following dilemma:[ ] whereas for logicians the orthocomplementation properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic.In this paper we eliminate this dilemma, providing a way for maintaining both. Via the introduction of the "missing" disjunctions in the lattice of properties of a physical system while inheriting the meet as a conjunction we obtain a complete Heyting algebra of propositions on physical properties. In particular there is a bijective correspondence between property lattices and propositional lattices equipped with a so called operational resolution, an operation that exposes the properties on the level of the propositions. If the property lattice goes equipped with an orthocomplementation, then this bijective correspondence can be refined to one with propositional lattices equipped with an operational complementation, as such establishing the claim made above. Formally one rediscovers via physical and logical considerations as such respectively a specification and a refinement of the purely mathematical result by Bruns and Lakser (1970) on injective hulls of meet-semilattices. From our representation we can derive a truly intuitionistic functional implication on property lattices, as such confronting claims made in previous writings on the matter. We also make a detailed analysis of disjunctivity vs. distributivity and finitary vs. infinitary conjunctivity, we briefly review the Bruns-Lakser construction and indicate some questions which are left open.     

Possible-Worlds Semantics for Modal Notions Conceived as Predicates

If is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate , we tackle both problems. Given a frame <W,R> consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret at every world in such a way that A holds at a world wW if and only if A holds at every world vW such that wRv. The arithmetical vocabulary is interpreted by the standard model at every world. Several paradoxes (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the -inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic.