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Multilanguage Hierarchical Logics (or: How We Can Do Without Modal Logics)
, 1994
"... 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 ..."
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Cited by 208 (51 self)
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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.
Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... 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 noth ..."
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Cited by 139 (25 self)
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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 BrouwerHeytingKolmogorov 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.
FirstOrder Logic of Proofs
, 2011
"... The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capa ..."
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Cited by 27 (11 self)
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The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the firstorder logic of proofs FOLP capable of realizing firstorder modal logic S4 and, therefore, the firstorder intuitionistic logic HPC. FOLP enjoys a natural provability interpretation; this provides a semantics of explicit proofs for firstorder S4 and HPC compliant with BrouwerHeytingKolmogorov requirements. FOLP opens the door to a general theory of firstorder justification.
On Epistemic Logic with Justification
 NATIONAL UNIVERSITY OF SINGAPORE
, 2005
"... 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 representat ..."
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Cited by 26 (9 self)
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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
Functionality in the Basic Logic of Proofs
, 1993
"... 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 ..."
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Cited by 22 (16 self)
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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.
Unified Semantics for Modality and lambdaterms via Proof Polynomials
"... 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 inte ..."
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Cited by 3 (1 self)
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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 systemindependent generalization of the CurryHoward 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...
Hierarchical MetaLogics for Belief and Provability: How We Can Do Without Modal Logics
, 1991
"... 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 a ..."
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Cited by 3 (3 self)
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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.
On propositional quantifiers in provability logic
 Notre Dame Journal of Formal Logic
, 1993
"... Abstract The first order theory of the Diagonalizable Algebra of Peano Arithmetic (DA(PA)) represents a natural fragment of provability logic with propositional quantifiers. We prove that the first order theory of the Ogenerated subalgebra of DA(PA) is decidable but not elementary recursive; the ..."
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Cited by 2 (2 self)
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Abstract The first order theory of the Diagonalizable Algebra of Peano Arithmetic (DA(PA)) represents a natural fragment of provability logic with propositional quantifiers. We prove that the first order theory of the Ogenerated subalgebra of DA(PA) is decidable but not elementary recursive; the same theory, enriched by a single free variable ranging over DA(PA), is already undecidable. This gives a negative answer to the question of the decidability of provability logics for recursive progressions of theories with quantifiers ranging over their ordinal notations. We also show that the first order theory of the free diagonalizable algebra on n independent generators is undecidable iff n Φ 0. / Introduction Gδdel was probably the first to consider the provability interpretation of modal logic: according to it the modality D is understood as the standard arithmetical Σipredicate Pr ( ) expressing provability in Peano arithmetic PA (cf. [15]). A complete axiomatization together with a decision procedure for the propositional modal logic ofprovability was given in Solovay [21]. On the other hand, it was shown
Operations on Proofs That Can Be Specified By Means of Modal Logic
"... 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 polynom ..."
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Cited by 2 (2 self)
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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.