Display calculi and other modal calculi: a comparison

In this paper we introduce and compare four different syntactic methods for generating sequent calculi for the main systems of modal logic: the multiple sequents method, the higher-arity sequents method, the tree-hypersequents method and the display method. More precisely we show how the first three methods can all be translated in the fourth one. This result sheds new light on these generalisations of the sequent calculus and raises issues that will be examined in the last section.     

Some paraconsistent sentential calculi

In [8] Jakowski defined by means of an appropriate interpretation a paraconsistent calculusD ₂ . In [9] J. Kotas showed thatD ₂ is equivalent to the calculusM(S5) whose theses are exactly all formulasa such thatMa is a thesis ofS5. The papers [11], [7], [3], and [4] showed that interesting paraconsistent calculi could be obtained using modal systems other thanS5 and modalities other thanM. This paper generalises the above work. LetA be an arbitrary modality (i.e. string ofM's,L's and negation signs). Then theA-extension of a set of formulasX is {¦A X}}. Various properties ofA-extensions of normal modal systems are examined, including a problem of their axiomatizability     

Three prepositional calculi of probability

Attempts are made to transform the basis of elementary probability theory into the logical calculus.We obtain the propositional calculus NP by a naive approach. As rules of transformation, NP has rules of the classical propositional logic (for events), rules of the ukasiewicz logic ₀ (for probabilities) and axioms of probability theory, in the form of rules of inference. We prove equivalence of NP with a fragmentary probability theory, in which one may only add and subtract probabilities.The second calculus MP is a usual modal propositional calculus. It has the modal rules x x, x y x y, x x, x y (y x), (y x), in addition to the rules of classical propositional logic. One may read x as x is probable. Imbeddings of NP and of ₀ into MP are given.The third calculus P is a modal extension of ₀. It may be obtained by adding the rule ((xy)y) xy to the modal logic of quantum mechanics Q [5]. One may read x in P as x is observed. An imbedding of NP into P is given.     

Adjoint interpretations of sentential calculi

The aim of this paper is to give a general background and a uniform treatment of several notions of mutual interpretability. Sentential calculi are treated as preorders and logical invariants of adjoint situations, i.e. Galois connections are investigated. The class of all sentential calculi is treated as a quasiordered class.Some methods of the axiomatization of the M-counterparts of modal systems are based on particular adjoints. Also, invariants concerning adjoints for calculi with implication are pointed out. Finally, the notion of interpretability is generalized so that it may be applied to closure spaces as well.     

Quantum logical calculi and lattice structures

In a preceding paper [1] it was shown that quantum logic, given by the tableaux-calculus T _eff, is complete and consistent with respect to the dialogic foundation of logics. Since in formal dialogs the special property of the value-definiteness of propositions is not postulated, the calculus T _eff represents a calculus of effective (intuitionistic) quantum logic.Beginning with the tableaux-calculus the equivalence of T _eff to calculi which use more familiar figures such as sequents and implications can be investigated. In this paper we present a sequents-calculus of Gentzen-type and a propositional calculus of Brouwer-type which are shown to be equivalent to T _eff. The effective propositional calculus provides an interpretation for a lattice structure, called quasi-implicative lattice. If, in addition, the value-definiteness of quantum mechanical propositions is postulated, a propositional calculus is obtained which provides an interpretation for a quasi-modular orthocomplemented lattice which, as is well-known, has as a model the lattice of subspaces of a Hilbert space.     

Cut-free sequent calculi for some tense logics

We introduce certain enhanced systems of sequent calculi for tense logics, and prove their completeness with respect to Kripke-type semantics.Presented byMelvin Fitting.     

Degrees of maximality of Klukasiewicz-like sentential calculi

The paper is concerned with the problem of characterization of strengthenings of the so-called Lukasiewicz-like sentential calculi. The calculi under consideration are determined byn-valued Lukasiewicz matrices (n>2,n finite) with superdesignated logical values. In general. Lukasiewicz-like sentential calculi are not implicative in the sense of [7]. Despite of this fact, in our considerations we use matrices analogous toS-algebras of Rasiowa. The main result of the paper says that the degree of maximality of anyn-valued Lukasiewicz-like sentential calculus is finite and equal to the degree of maximality of the correspondingn-valued Lukasiewicz calculus. Allatum est die 15 Octobris 1976     

Completeness theorems for some intermediate predicate calculi

We give completeness results — with respect to Kripke's semantic — for the negation-free intermediate predicate calculi: (1) $$\begin{gathered} BD = positive predicate calculus PQ + B:(\alpha \to \beta )v(\beta \to \alpha ) \hfill \\ + D:\forall x\left( {a\left( x \right)v\beta } \right) \to \forall xav\beta \hfill \\ \end{gathered}$$ (2) $$T_n D = PQ + T_n :\left( {a_0 \to a_1 } \right)v \ldots v\left( {a_n \to a_{n + 1} } \right) + D\left( {n \geqslant 0} \right)$$ and the superintuitionistic predicate calculus: (3) $$B^1 DH_2^ \urcorner = BD + intuitionistic negation + H_2^ \urcorner : \urcorner \forall xa \to \exists x \urcorner a.$$ The central point is the completeness proof for (1), which is obtained modifying Klemke's construction [3]. For a general account on negation-free intermediate predicate calculi — see Casari-Minari [1]; for an algebraic treatment of some superintuitionistic predicate calculi involving schemasB andD — see Horn [4] and Görnemann [2].