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Jerzy Kalinowski’s Logic of Normative Sentences Revisited

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Abstract

The paper tackles two problems. The first one is to grasp the real meaning of Jerzy Kalinowski’s theory of normative sentences. His formal system K 1 is a simple logic formulated in a very limited language (negation is the only operator defined on actions). While presenting it Kalinowski formulated a few interesting philosophical remarks on norms and actions. He did not, however, possess the tools to formalise them fully. We propose a formulation of Kalinowski’s ideas with the use of a set-theoretical frame similar to the one presented by Krister Segerberg in his A Deontic Logic of Action. At the same time we enrich the language used by Kalinowski with more operators on actions (parallel execution and free choice) and present an adequate axiomatisation of the resulting system. That allows us to disclose some unrevealed aspects of Kalinowski’s theory. The most important one is a relation between acts which we call moral indiscernibility. Our second problem is a proper understanding of moral indiscernibility. We show how a repertoire of agent’s actions, defined with the use of simple observable elements of actions, can be filtrated by the relation of moral indiscernibility. That allows us to understand the consequences of Kalinowski’s claim that not doing something good is always bad.

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... We shall not enter into discussion about algebraical issues here. The interested Reader should consult section 2.3.1 in [24]. 8 Aquist's system F consists of PC laws, Modus Ponens, Substitution rule, Replacement in PC-theorems, and ten axioms: Three of them are mentioned explicitly: ...
... Axioms of the system consist of (16), (18), (20) and the following ones: (16), (24), (25) and (26) characterize relations between deontic concepts as defined in table 5. (18) and (27) characterize prohibition and (20) and (28) state the properties of obligation. ...
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Multivalued setting is quite natural for deontic action logic, where actions are usually treated as obligatory, neutral or forbidden. We apply the ideas of multivalued deontic logic to the phenomenon of a moral dilemma and, broader, to any situation where there are conflicting norms. We formalize three approaches towards normative conflicts. We present matrices for the systems and compare their tautologies. Finally, we present a sound and complete axiomatization of the systems.
... He declared that the systems were confluent but did not provide the proof. The adequacy of K 1 with respect to a model based on boolean algebra, corresponding to Kalinowski's intuitions presented in an informal way in [4], has recently been proved in [9]. In the present paper we prove the adequacy of the two Kalinowski's original approaches. ...
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Halldén, in [1], has recently pointed out that it is highly undesirable, in a system of sentential calculus, for there to exist two formulas α and β such that: (i) α and β contain no variable in common; (ii) neither α nor β is provable; (iii) α ∨ β is provable. We shall call a system unreasonable ( in the sense of Halldén ) if there exists a pair of formulas α and β having properties (i), (ii), and (iii). Halldén shows (in [1]) that the Lewis systems S 1 and S 3 are unreasonable in this sense; and that the same is true of any system which is between S 1 and S 3, as well as of every system which is stronger than S 3 but weaker than both S 4 and S 7. In the present note we shall show that this defect does not occur in S 4, nor in S 5, nor in any “quasi-normal” extension of S 5; we give an example, on the other hand, of an unreasonable system which lies between S 4 and S 5. When we speak, in what follows, of a system of modal logic , we shall mean a system having the same class of well-formed formulas as have the various Lewis calculi. Thus the well-formed formulas of a system of modal logic, when written in unabbreviated form, are just those formulas which can be built up from sentential variables by use of the binary connective ‘·’ (conjunction sign), and the two unary connectives ‘˜’ (negation sign) and ‘◇’ (possibility sign). We shall, however, also make use of some of the defined signs of Lewis.
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