Identity, many-valuedness and referentiality

In the paper * we discuss a distinctive versatility of the non-Fregean approach to the sentential identity. We present many-valued and referential counterparts of the systems of SCI, the sentential calculus with identity, including Suszko’s logical valuation programme as applied to many-valued logics. The similarity of different constructions: many-valued, referential and mixed, leads us to the conviction of the universality of the non-Fregean paradigm of sentential identity as distinguished from the equivalence, cf. [9]

A partial squeezing theorem for a particular class of many-valued logics

The problem to be studied for this thesis was that of whether the usual statement calculus is a suitable formal system for every many-valued logic in a particular collection of logics. The logics in question are those that fall between the usual two-valued logic and a modified form of the Lukasiewicz-Tarski three-valued logic. Since this betweenness relationship was an original concept and appeared nowhere in the literature, the first goal in the research plan was to define this relationship precisely. Preliminary concepts included truth value mapping and forgivingness of logics, concepts that, like betweenness, are original to this paper and that facilitate the comparison of many-valued logics. After betweenness was defined, the next stage of the research would be to investigate the logics between the classical logic and the modified Lg and to see for which of these logics the statement calculus is suitable. This would involve direct calculations with truth tables as well as the use of any published results on the axiomatization and also the comparison of many-valued logics. Because of the scarcity of work or literature on the problem of comparing many-valued logics, direct calculation turned out to be the most effective method of research. The introduction of a device called a truth class table proved to be invaluable. Such a table allows the logician to work with sets of truth values instead of with individual truth values themselves. Truth class tables were calculated that are characteristic of the logics under consideration. It was discovered that the usual statement calculus is not suitable for every logic between the two-valued logic and the modified L3. The main result of the thesis is a theorem relating sufficient conditions under which a many-valued logic will have the usual statement calculus for a suitable formal system. It is not yet known whether these conditions are necessary as well as sufficient. Concluding remarks demonstrate, as a corollary to the main result, that for any integer n there exist n-valued logics for which the usual statement calculus is suitable

A map of dependencies among three-valued logics

International audienceThree-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value

Proof Theory of Finite-valued Logics

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics

Plausible reasoning for the problems of cognitive sociology

The plausible reasoning class (called the JSM-reasoning in honour of John Stuart Mill) is described. It implements interaction of three forms of non-deductive procedures induction, analogy and abduction. Empirical induction in the JSM-reasoning is the basis for generation of hypotheses on causal relations (determinants of social behaviour). Inference by analogy means that predictions about previously unknown properties of objects (individual’s behaviour) are inferred from causal relations. Abductive inference is performed to check on the explanatory adequacy of generated hypotheses. To recognize rationality of respondents’ opinion deductive inference is used. Plausible reasoning, semantics of argumentation logic and deductive recognition of opinion rationality represent logical tool for cognitive sociology problems

The Logic of Internal Rational Agent

In this paper, we introduce a new four-valued logic which may be viewed as a variation on the theme of Kubyshkina and Zaitsev's Logic of Rational Agent \textbf{LRA} \cite{LRA}. We call our logic $\bf LIRA$ (Logic of Internal Rational Agency). In contrast to \textbf{LRA}, it has three designated values instead of one and a different interpretation of truth values, the same as in Zaitsev and Shramko's bi-facial truth logic \cite{ZS}. This logic may be useful in a situation when according to an agent's point of view (i.e. internal point of view) her/his reasoning is rational, while from the external one it might be not the case. One may use \textbf{LIRA}, if one wants to reconstruct an agent's way of thinking, compare it with respect to the real state of affairs, and understand why an agent thought in this or that way. Moreover, we discuss Kubyshkina and Zaitsev's necessity and possibility operators for \textbf{LRA} definable by means of four-valued Kripke-style semantics and show that, due to two negations (as well as their combination) of \textbf{LRA}, two more possibility operators for \textbf{LRA} can be defined. Then we slightly modify all these modalities to be appropriate for $\bf LIRA$. Finally, we formalize all the truth-functional $n$-ary extensions of the negation fragment of $\bf LIRA$ (including $\bf LIRA$ itself) as well as their basic modal extension via linear-type natural deduction systems