Automated Synthesis of Tableau Calculi

This paper presents a method for synthesising sound and complete tableau calculi. Given a specification of the formal semantics of a logic, the method generates a set of tableau inference rules that can then be used to reason within the logic. The method guarantees that the generated rules form a calculus which is sound and constructively complete. If the logic can be shown to admit finite filtration with respect to a well-defined first-order semantics then adding a general blocking mechanism provides a terminating tableau calculus. The process of generating tableau rules can be completely automated and produces, together with the blocking mechanism, an automated procedure for generating tableau decision procedures. For illustration we show the workability of the approach for a description logic with transitive roles and propositional intuitionistic logic.Comment: 32 page

An Abstract Tableau Calculus for the Description Logic SHOI Using UnrestrictedBlocking and Rewriting

Abstract This paper presents an abstract tableau calculus for the description logic SHOI. SHOI is the extension of ALC with singleton concepts, role inverse, transitive roles and role inclusion axioms. The presented tableau calculus is inspired by a recently introduced tableau synthesis framework. Termination is achieved by a variation of the unrestricted blocking mechanism that immediately rewrites terms with respect to the conjectured equalities. This approach leads to reduced search space for decision procedures based on the calculus. We also discuss restrictions of the application of the blocking rule by means of additional side conditions and/or additional premises.

Addition of Sodium Bicarbonate to Irrigation Solution May Assist in Dissolution of Uric Acid Fragments During Ureteroscopy

Introduction: We hypothesized that adding sodium bicarbonate (bicarb) to normal saline (NS) irrigation during ureteroscopy in patients with uric acid (UA) nephrolithiasis may assist in dissolving small stone fragments produced during laser lithotripsy. In vitro testing was performed to determine whether dissolution of UA fragments could be accomplished within 1 hour. Materials and Methods: In total 100% UA renal calculi were fragmented, filtered, and separated by size. Fragment sizes were <0.5 mm and 0.5 to 1 mm. Similar amounts of stone material were agitated in solution at room temperature. Four solutions were tested (NS, NS +1 ampule bicarb/L, NS +2, NS +3). Both groups were filtered to remove solutions after fixed periods. Filtered specimens were dried and weighed. Fragment dissolution rates were calculated as percent removed per hour. Additional testing was performed to determine whether increasing the temperature of solution affected dissolution rates. Results: For fragments <0.5 mm, adding 2 or 3 bicarb ampules/L NS produced a dissolution rate averaging 91% ± 29% per hour. This rate averaged 226% faster than NS alone. With fragments 0.5 to 1 mm, addition of 2 or 3 bicarb ampules/L NS yielded a dissolution rate averaging 22% ± 7% per hour, which was nearly five times higher than NS alone. There was a trend for an increase in mean dissolution rate with higher temperature but this increase was not significant (p = 0.30). Conclusions: The addition of bicarbonate to NS more than doubles the dissolution rate of UA stone fragments and fragments less than 0.5 mm can be completely dissolved within 1 hour. Addition of bicarb to NS irrigation is a simple and inexpensive approach that may assist in the dissolution of UA fragments produced during ureteroscopic laser lithotripsy. Further studies are needed to determine whether a clinical benefit exists

Hypertableau Reasoning for Description Logics

We present a novel reasoning calculus for the description logic SHOIQ^+---a knowledge representation formalism with applications in areas such as the Semantic Web. Unnecessary nondeterminism and the construction of large models are two primary sources of inefficiency in the tableau-based reasoning calculi used in state-of-the-art reasoners. In order to reduce nondeterminism, we base our calculus on hypertableau and hyperresolution calculi, which we extend with a blocking condition to ensure termination. In order to reduce the size of the constructed models, we introduce anywhere pairwise blocking. We also present an improved nominal introduction rule that ensures termination in the presence of nominals, inverse roles, and number restrictions---a combination of DL constructs that has proven notoriously difficult to handle. Our implementation shows significant performance improvements over state-of-the-art reasoners on several well-known ontologies

From moral welfarism to technical non-welfarism : A step back to Bentham’s felicific calculus of its members

A focus on the information used in Bentham’s felicific calculus sheds new light on the contemporary debate in normative economics opposing non-welfarism to welfarism. As a utilitarian, Bentham is de facto welfarist on a moral sense. Unexpectedly, this study shows Bentham resorts to non-welfarist information in his calculus. Thus, technical non-welfarism is coherent with moral welfarism, and even, the former proves necessary not to betray utilitarian principles. To sustain this claim, we challenge a view opposing a “cardinal” to an “ordinal” calculus: these two calculi constitute different stages of a unique calculus; because of operational constraints, Bentham is bound to rely on proxies, hence on non-utility information.Bentham, individual utility, utility calculus, utilitarianism, welfarism, non-welfarism, social welfare, technical welfarism, moral welfarism.

Proof Theory of Finite-valued Logics

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics

Elimination of Cuts in First-order Finite-valued Logics

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information

From Logical Calculus to Logical Formality—What Kant Did with Euler’s Circles

John Venn has the “uneasy suspicion” that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kant’s “disastrous effect on logical method,” namely the “strictest preservation [of logic] from mathematical encroachment.” Kant’s actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Euler’s circles and comparing it with Euler’s own use. I do so in light of the developments in logical calculus from G. W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations