On Constructive Connectives and 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

Minimal Paradefinite Logics for Reasoning with Incompleteness and Inconsistency

Paradefinite (`beyond the definite\u27) logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for defining paradefinite logics, consisting of four-valued matrices, and study the better accepted logics that are induced by these matrices

Applicable Mathematics in a Minimal Computational Theory of Sets

In previous papers on this project a general static logical framework for formalizing and mechanizing set theories of different strength was suggested, and the power of some predicatively acceptable theories in that framework was explored. In this work we first improve that framework by enriching it with means for coherently extending by definitions its theories, without destroying its static nature or violating any of the principles on which it is based. Then we turn to investigate within the enriched framework the power of the minimal (predicatively acceptable) theory in it that proves the existence of infinite sets. We show that that theory is a computational theory, in the sense that every element of its minimal transitive model is denoted by some of its closed terms. (That model happens to be the second universe in Jensen's hierarchy.) Then we show that already this minimal theory suffices for developing very large portions (if not all) of scientifically applicable mathematics. This requires treating the collection of real numbers as a proper class, that is: a unary predicate which can be introduced in the theory by the static extension method described in the first part of the paper

Safety, Absoluteness, and Computability

The semantic notion of dependent safety is a common generalization of the notion of absoluteness used in set theory and the notion of domain independence used in database theory for characterizing safe queries. This notion has been used in previous works to provide a unified theory of constructions and operations as they are used in different branches of mathematics and computer science, including set theory, computability theory, and database theory. In this paper we provide a complete syntactic characterization of general first-order dependent safety. We also show that this syntactic safety relation can be used for characterizing the set of strictly decidable relations on the natural numbers, as well as for characterizing rudimentary set theory and absoluteness of formulas within it

Non-deterministic Multi-valued Matrices for First-Order Logics of Formal Inconsistency

Paraconsistent logic is the study of contradictory yet non-trivial theories. One of the best-known approaches to designing useful paraconsistent logics is da Costa’s ap-proach, which has led to the family of Logics of Formal Inconsistency (LFIs), where the notion of inconsistency is expressed at the object level. In this paper we use non-deterministic matrices, a generalization of standard multi-valued matrices, to provide simple and modular finite-valued semantics for a large family of first-order LFIs. The modular approach provides new insights into the semantic role of each of the studied axioms and the dependencies be-tween them. We also prove the effectiveness of our seman-tics, a crucial property for constructing counterexamples, and apply it to show a non-trivial proof-theoretical prop-erty of the studied LFIs. 1

What is a Paraconsistent Logic?

Paraconsistent logics are logical systems that reject the classical principle, usually dubbed Explosion, that a contradiction implies everything. However, the received view about paraconsistency focuses only the inferential version of Explosion, which is concerned with formulae, thereby overlooking other possible accounts. In this paper, we propose to focus, additionally, on a meta-inferential version of Explosion, i.e. which is concerned with inferences or sequents. In doing so, we will offer a new characterization of paraconsistency by means of which a logic is paraconsistent if it invalidates either the inferential or the meta-inferential notion of Explosion. We show the non-triviality of this criterion by discussing a number of logics. On the one hand, logics which validate and invalidate both versions of Explosion, such as classical logic and Asenjo–Priest’s 3-valued logic LP. On the other hand, logics which validate one version of Explosion but not the other, such as the substructural logics TS and ST, introduced by Malinowski and Cobreros, Egré, Ripley and van Rooij, which are obtained via Malinowski’s and Frankowski’s q- and p-matrices, respectively

Photosystem II -- Mediated cyclic photophosphorylation

DCMU-sensitive synthesis of ATP can be shown to continue in KCN-treated chloroplasts after cessation of O2 evolution. The catalyst for this reaction, -phenylenediamine, also stimulates synthesis of ATP in NH2OH-treated chloroplasts, but at much higher rates. This ATP synthesis can be observed in the presence of the quinone antagonist dibromothymoquinone, and under the appropriate conditions it is completely sensitive to DCMU. Since neither uptake nor evolution of O2 can be observed during illumination, these results are interpreted as evidence for catalysis of cyclic photophosphorylation by photosystem II.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/21823/1/0000224.pd