It is shown that Friedman's problem, whether there exists a proper extension of first order logic satisfying the compactness and interpolation theorems, has extremely simple positive solutions if one considers extensions by generalized (finitary) propositional connectives. This does not solve, however, the problem of whether such extensions exist which are also closed under relativization of fo...

we present some model theoretic results for {l}ukasiewiczpredicate logic by using the methods of continuous model theorydeveloped by chang and keisler.we prove compactness theorem with respect to the class of allstructures taking values in the {l}ukasiewicz $texttt{bl}$-algebra.we also prove some appropriate preservation theorems concerning universal and inductive theories.finally, skolemizatio...

It is known that determining the satisfiability of n-valued ?ukasiewicz rules is NP-complete for n?4, as well as that it can be solved in time linear in the length of the formula in the Boolean case (when n=2). However, the complexity for n=3 is an open problem. In this paper we formally prove that the satisfiability problem for 3-valued ?ukasiewicz rules is NP-complete. Moreover, we also prove...

It is known that determining the satisfiability of n-valued ?ukasiewicz rules is NP-complete for n?4, as well as that it can be solved in time linear in the length of the formula in the Boolean case (when n=2). However, the complexity for n=3 is an open problem. In this paper we formally prove that the satisfiability problem for 3-valued ?ukasiewicz rules is NP-complete. Moreover, we also prove...

It is show11 that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives. including those proposed by Gabbay. are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases. the double negatio~l of such a connective is equivalent to a formula of intnitionistic calculus. Thus, under the excluded third law i...

We introduce two-dimensional logics based on \L{}ukasiewicz and G\"{o}del to formalize paraconsistent fuzzy reasoning. The are interpreted matrices, where the common underlying structure is bi-lattice (twisted) product of $[0,1]$ interval. first (resp.\ second) coordinate encodes positive negative) information one has about a statement. propose constraint tableaux that provide modular framework...

Proof-functional logical connectives allow reasoning about the structure of logical proofs, in this way giving to the latter the status of first-class objects. This is in contrast to classical truth-functional connectives where the meaning of a compound formula is dependent only on the truth value of its subformulas. In this paper we present a typed lambda calculus, enriched with strong product...

For two propositional fuzzy logics, we present analytic proof calculi, based on relational hypersequents. The logic considered first, called Mà L, is based on the finite ordinal sums of à Lukasiewicz t-norms. In addition to the usual connectives – the conjunction ̄, the implication →, and the constant 0 –, we use a further unary connective interpreted by the function associating with each truth...