Cut-Elimination and Quantification in Canonical Systems

Propositional canonical Gentzen-type systems, introduced in [1], are systems which in addition to the standard axioms and structural rules have only logical rules in which exactly one occurrence of a connective is introduced and no other connective is mentioned. [1] provides a constructive coherence criterion for the non-triviality of such systems and shows that a system of this kind admits cut-elimination iff it is coherent. The semantics of such systems is provided using two-valued non-deterministic matrices (2Nmatrices). [14] extends these results to systems with unary quantifiers of a very restricted form. In this paper we substantially extend the characterization of canonical systems to (n,k)-ary quantifiers, which bind k distinct variables and connect n formulas. We show that the coherence criterion remains constructive for such systems, and that for the case of k∈{0,1}: (i) a canonical system is coherent iff it has a strongly characteristic 2Nmatrix, and (ii) if a canonical system is coherent, then it admits cut-elimination.

We define the notions of a canonical inference rule and a canonical constructive system in the framework of strict single-conclusion Gentzen-type systems (or, equivalently, natural deduction systems), and develop a corresponding general non-deterministic Kripke-style semantics. We show that every constructive canonical system induces a class of non-deterministic Kripke-style frames, for which it is strongly sound and complete. This non-deterministic semantics is used to show that such a system always admits a strong form of the cut-elimination theorem, and for providing a decision procedure for such systems.

Canonical inference rules and canonical systems are defined in the framework of non-strict single-conclusion sequent systems, in which the succeedents of sequents can be empty. Important properties of this framework are investigated, and a general non-deterministic Kripke-style semantics is provided. This general semantics is then used to provide a constructive (and very natural), sufficient and necessary coherence criterion for the validity of the strong cut-elimination theorem in such a system. These results suggest new syntactic and semantic characterizations of basic constructive connectives.

Angiotensin Converting Enzyme (ACE) inhibition has been a promising avenue for anti-hypertensive drug development. Our study investigated the inhibitory potential of bioactive compounds derived from six medicinal plants (Allium sativum L., Zingiber officinale Roscoe, Acalypha godseffiana Mast., Moringa oleifera Lam., Vernonia amygdalina Delile, and Rauvolfia vomitoria Afzel.) against ACE using in silico methods. Thirty-one (31) bioactive compounds were screened while Ramipril, and Enalapril were employed as control drugs. 3D structures and canonical Simplified Molecular Input Line Entry System (SMILES) of the bioactive compounds and control drugs were obtained from the PubChem online server. Drug-likeness assessment of the bioactive compounds and protein-ligand docking of successful compounds were conducted using SwissADME online server and AutoDock Vina software. ADMET (absorption, distribution, metabolism, excretion, toxicity) analysis was also done to evaluate the suitability of the hit ligands for further drug development. Of the 31 compounds screened, 17 passed at least four of the five standard rules of drug-likeness determination, while the control drugs (Ramipril and Enalapril) failed one of the rules. Ajmaline, Apigenin, Quercetin, Cryptolepine, Luteolin, Hydroxyvernolide, Kaempferol and Vernodalol had higher binding energies of -9.6 kcal/mol, -8.7 kcal/mol, -8.5 kcal/mol, -8.4 kcal/mol, -8.4 kcal/mol, -8.3 kcal/mol, -8.3 kcal/mol and -7.8 kcal/mol, respectively than Ramipril and Enalapril (-7.6 kcal/mol, and -7.5 kcal/mol). The higher binding energies and the stability of their binding interactions denote these hit ligands as potential antihypertensive drugs targeting ACE. However, wet lab experimental investigation is necessary to validate the inhibitory activity of these compounds and elucidate their mechanisms of action.

Standard axioms for equality, or standard definitions of equality in secondorder logic or set theory, permit to infer versions limited to equalities of structural rules otherwise banned in substructural logics. In the presence of settheoretic extensionality and a rather natural form of separation, these limited structural rules yield the full structural rules in question. It is suggested that this may be prevented by restricting substitution for individual variables in the principle of replacement of equal terms. These restrictions are parallel to the restrictions for structural rules in substructural logics.

This paper explores proof-theoretic aspects of hybrid type-logical grammars, a logic combining Lambek grammars with lambda grammars. We prove some basic properties of the calculus, such as normalisation and the subformula property and also present both a sequent and a proof net calculus for hybrid type-logical grammars. In addition to clarifying the logical foundations of hybrid type-logical grammars, the current study opens the way to variants and extensions of the original system, including but not limited to a non-associative version and a multimodal version incorporating structural rules and unary modes.

In this paper we generalize the existing tableau methods for modal logics. First of all, while usual modal tableaux are based on trees, our basic structures are rooted directed acyclic graphs (RDAG). This allows natural tableau rules for some modal logics that are difficult to capture in the usual way (such as those having an accessibility relation that is dense or confluent). Second, tableau rules rewrite patterns, which are (schemas of) parts of a RDAG. A particular case of these rules are the single-step rules recently proposed by Massacci. This allows in particular tableau rule presentations for K5, KD5, K45, KD45, and S5 that respect the subformula property. Third, we divide modal tableau rules into propagation rules and structural rules. Structural rules construct new edges and nodes (without adding formulas to nodes), while propagation rules add formulas to nodes. This distinction allows to prove completeness in a modular way.

A normalization procedure is given for classical natural deduction with the standard rule of indirect proof applied to arbitrary formulas. For normal derivability and the subformula property, it is sufficient to permute down instances of indirect proof whenever they have been used for concluding a major premiss of an elimination rule. The result applies even to natural deduction for classical modal logic.

We explore the consequences, for logical system-building, of taking seriously (i) the aim of having irredundant rules of inference, and (ii) a preference for proofs of stronger results over proofs of weaker ones. This leads one to reconsider the structural rules of REFLEXIVITY, THINNING, and CUT. REFLEXIVITY survives in the minimally necessary form φ:φ. Proofs have to get started. CUT is subject to a CUT-elimination theorem, to the effect that one can always make do without applications of CUT. So CUT is redundant, and should not be a rule of the system. CUT-elimination, however, in the context of the usual forms of logical rules, requires the presence, in the system, of THINNING. But THINNING, it turns out, is not really necessary. Provided only that one liberalizes the statement of certain logical rules in an appropriate way, one can make do without CUT or THINNING. From the methodological point of view of this study, the logical rules ought to be framed in this newly liberalized form. These liberalized logical rules determine the system of core logic. Given any intuitionistic Gentzen proof of Δ:φ, one can determine from it a Core proof of some subsequent of Δ:φ. Given any classical Gentzen proof of Δ:φ, one can determine from it a classical Core proof of some subsequent of Δ:φ. In both cases the Core proof is of a result at least as strong as that of the Gentzen proof; and the only structural rule used is φ:φ.

An inelastic beam-like model for nonlinear dynamic analyses of multi-storey buildings

We introduce a sequent system which is Gentzen algebraisable with orthomodular lattices as equivalent algebraic semantics, and therefore can be viewed as a calculus for orthomodular quantum logic. Its sequents are pairs of non-associative structures, formed via a structural connective whose algebraic interpretation is the Sasaki product on the left-hand side and its De Morgan dual on the right-hand side. It is a substructural calculus, because some of the standard structural sequent rules are restricted—by lifting all such restrictions, one recovers a calculus for classical logic.

Sequent calculi for applied logics have nonstandard logical inference rules and various axioms, cut is essential in them. The following constraints on derivations in these calculi do not compromise completeness. Contraction rules follow cut and logical rules. Weakening rules precede logical rules. Consecutive cut rules are ordered so that larger formulas are cut first. Consecutive standard logical rules are ordered so that their principal formulas are in an increasing order.

18 - Application Strategies for Primary Structure-Activity Relationship Exploration

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa's school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the "core" of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency.

10.0 Logical Structure and Logical Rules