Universal Logic

We develop a basic theory of logical systems. Our aim is to provide elementary concepts and methods used for the study of propositional logics. We focus on the notion of completeness in both its aspects: global and local. “Global” means here reference to the set all correct and reliable schemata of argumentation. This approach impels such formal variants of the notion as Post-completeness or structural completeness. In the “local” view we have to do with the notion of truth relative to given semantices such as logical matrices or Kripke models.

We keep to Hilbert-style formalization of logical system. The basic concept is that of consequence operator due to Alfred Tarski. In the first part of our tutorial, we introduce concept and results central for the further study of the lattice of consequence operators. Logical matrices, and their variants, are discussed in the second part. They are regarded as propositional semantices and used for characterization of logical systems. Then we concentrate on different variants of the notion of completeness. Some methods and results used in the study of propositional systems are developed and briefly discussed there.


Consequence operators Piotr Wojtylak Institute of Mathematics and Informatics Silesian University - Poland
	We develop a basic theory of logical systems. Our aim is to provide elementary concepts and methods used for the study of propositional logics. We focus on the notion of completeness in both its aspects: global and local. “Global” means here reference to the set all correct and reliable schemata of argumentation. This approach impels such formal variants of the notion as Post-completeness or structural completeness. In the “local” view we have to do with the notion of truth relative to given semantices such as logical matrices or Kripke models. We keep to Hilbert-style formalization of logical system. The basic concept is that of consequence operator due to Alfred Tarski. In the first part of our tutorial, we introduce concept and results central for the further study of the lattice of consequence operators. Logical matrices, and their variants, are discussed in the second part. They are regarded as propositional semantices and used for characterization of logical systems. Then we concentrate on different variants of the notion of completeness. Some methods and results used in the study of propositional systems are developed and briefly discussed there.
		References A.Tarski, Fundamentale Begriffe der Methodologie der deductiven Wissenschaften I, Monatshefte für Mathematik und Physik, 37 (1930), 361-404. J. Łoś, R.Suszko, Remarks on sentential logics, Indagationes Mathematicae 20(1958), pp. 177-183. W.A. Pogorzelski, P. Wojtylak , Elements of the theory of completeness in Propositional Logic, Silesian University 1982. R. Wójcicki, Theory of logical calculi, Kluwer Academic Publishers 1988.