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Andrzej Pietruszczak

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In introductory logic courses the authors often limit their considerations to the truthvalue operators. Then they write that conditionals and biconditionals of natural language (“if” and “if and only if”) may be represented as material implications and equivalences, respectively. Yet material implications are not suitable for conditionals. Lewis’ strict implications are much better for this purpose. Similarly, strict equivalences are better for representing biconditionals (than material equivalences). In this paper we prove that the methods from standard first courses in logic can be used for testing arguments with strict implications, strict equivalences and other operators which may represent connectives from natural language.

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Krzysztof Wójtowicz and Andrzej Pietruszczak

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In the article the problem of independence in mathematics is discussed. The status of the continuum hypothesis, large cardinal axioms and the axiom of constructablility is presented in some detail. The problem whether incompleteness is really relevant for ordinary mathematics and for empirical science is investigated. Another aim of the article is to give some arguments for the thesis that the problem of reliability and justification of new axioms is well-posed and worthy of attention. In my opinion, investigations concerning the status of independent sentences give insight into our understanding of mathematical concepts, of mathematical knowledge and of the role of mathematics in empirical science.

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Edited by Jacek Malinowski and Andrzej Pietruszczak

The aim of this book is to present essays centered upon the subjects of Formal Ontology and Logical Philosophy. The idea of investigating philosophical problems by means of logical methods was intensively promoted in Torun by the Department of Logic of Nicolaus Copernicus University during last decade. Another aim of this book is to present to the philosophical and logical audience the activities of the Torunian Department of Logic during this decade. The papers in this volume contain the results concerning Logic and Logical Philosophy, obtained within the confines of the projects initiated by the Department of Logic and other research projects in which the Torunian Department of Logic took part.
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Tomasz Jarmużek, Maciej Nowicki and Andrzej Pietruszczak

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The article presents a formalization of Anselm’s so-called Ontological Arguments from Proslogion. The main idea of our research is to stay to the original text as close as is possible. We show, against some common opinions, that

(i) the logic necessary for the formalization must be neither a purely sentential modal calculus, nor just non-modal first-order logic, but a modal first-order theory;

(ii) such logic cannot contain logical axiom ⌜A → ◊A⌝;

(iii) none of Anselm’s reasonings requires the assumptions that God is a consistent object or that existence of God is possible (in symbols “◊Eg”);

(iv) no such thing as the so-called Anselm’s Principle (in symbols “□(Eg → □Eg)”) is involved in any of the proofs;

Moreover we show a single line of reasoning underlying the whole Proslogion and allowing Anselm to deduce many theorems concerning God’s nature. Last but not least we study the possibility of proving the uniqueness of God within the outlined theory.