MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of first order languages, each language containing names for the language below, and propose them as an alternative to modal logics. The motivations of our proposal are technical, epistemological and implementational. From a technical point of view, we prove, among other things, that the set of theorems of the most common modal logics can be embedded (under the obvious bijective mapping between a modal and a first order language) into that of the corresponding ML systems. Moreover, we show that ML systems have properties not holding for modal logics and argue that these properties are justified by our intuitions. This claim is motivated by the study of how ML systems can be used in the representation of beliefs (more generally, propositional attitudes) and provability, two areas where modal logics have been extensively used. Finally, from an implementation point of view, we argue that ML systems resemble closely the current practice in the computer representation of propositional attitudes and metatheoretic theorem proving.

In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #-calculus.

Abstract not found

This report describes the elimination of the injectivity restriction for functional arithmetical interpretations as used in the systems PF and PFM in the Basic Logic of Proofs. An appropriate axiom system PU in a language with operators "x is a proof of y" is defined and proved to be sound and complete with respect to all arithmetical interpretations based on functional proof predicates. Unification plays a major role in the formulation of the new axioms.

The true belief components of Plato's tripartite definition of knowledge as justified true belief are represented in formal epistemology by modal logic and its possible worlds semantics. At the same time, the justification component of Plato's definition did not have a formal representation. This

In this paper individual proofs are integrated into provability logic. Systems of axioms for a logic with operators "A is provable" and "p is a proof of A" are introduced, provided with Kripke semantics and decision procedure. Completeness theorems with respect to the arithmetical interpretation are proved. 1 Introduction In [1] and [2] proofs were incorporated into propositional logic by means of labeled modalities. The basic labeled modal logic contains the propositional logic enriched by unary operators 2 p i , i = 0; 1; 2; : : : . This language helps to provide a logical treatment of a rather general situation when we are interested not only to know that a certain statement A is valid, but also have to keep track on some evidences of its validness: 2 p A may stand for "p is a proof of A", "p is a program which computes A", "A has a proof of the complexity p", etc. The language of the provability logic ([3]) with the provability operator 2 only, where 2A stands for "A is provable...

MultiLanguage systems (ML systems) are formal systems allowing the use of multiple distinct logical languages. In this paper we introduce a class of ML systems which use a hierarchy of metatheories, each with a first order language containing names for the language below, and propose them as an alternative to modal logics. The motivations of our proposal are technical and epistemological. From a technical point of view, we prove, among other things, that modal logics can be embedded in the corresponding ML systems. Moreover, we show that ML systems have properties not holding for modal logics and argue that these properties are justified by our intuitions. We motivate our claim by studying how they can be used in the representation of beliefs (more generally, propositional attitudes) and provability, two areas where modal logics have been extensively used. 1

It is shown that the modal logic S4, simple -calculus and modal -calculus admit a realization in a very simple propositional logical system LP , which has an exact provability semantics. In LP both modality and -terms become objects of the same nature, namely, proof polynomials. The provability interpretation of modal -terms presented here may be regarded as a system-independent generalization of the Curry-Howard isomorphism of proofs and -terms. 1 Introduction The Logic of Proofs (LP , see Section 2) is a system in the propositional language with an extra basic proposition t : F for "t is a proof of F ". LP is supplied with a formal provability semantics, completeness theorems and decidability algorithms ([3], [4], [5]). In this paper it is shown that LP naturally encompasses -calculi corresponding to intuitionistic and modal logics, and combinatory logic. In addition, LP is strictly more expressive because it admits arbitrary combinations of ":" and propositional connectives. The id...

Explicit modal logic was first sketched by Gödel in [16] as the logic with the atoms "t is a proof of F". The complete axiomatization of the Logic of Proofs LP was found in [4] (see also [6],[7],[18]). In this paper we establish a sort of a functional completeness property of proof polynomials which constitute the system of proof terms in LP. Proof polynomials are built from variables and constants by three operations on proofs: "\Delta" (application), "!" (proof checker), and "+" (choice). Here constants stand for canonical proofs of "simple facts", namely instances of propositional axioms and axioms of LP in a given proof system. We show that every operation on proofs that (i) can be specified in a propositional modal language and (ii) is invariant with respect to the choice of a proof system is realized by a proof polynomial.

In 1933 Godel introduced a modal logic of provability (S4) and left open the problem of a formal provability semantics for this logic. Since then numerous attempts have been made to give an adequate provability semantics to Godel's provability logic with only partial success. In this paper we give the complete solution to this problem in the Logic of Proofs (LP). LP implements Godel's suggestion (1938) of replacing formulas "F is provable" by the propositions for explicit proofs "t is a proof of F" (t : F ). LP admits the reflection of explicit proofs t : F ! F thus circumventing restrictions imposed on the provability operator by Godel's second incompleteness theorem. LP formalizes the Kolmogorov calculus of problems and proves the Kolmogorov conjecture that intuitionistic logic coincides with the classical calculus of problems.