We present a theoretical and practical approach to the modular natural deduction presentation of modal logics and their implementation in a logical framework. Our work treats a large and well-known class of modal logics (including K, D, T, B, S4, S4:2, KD45, S5) in a uniform way with respect to soundness and completeness for semantics, and faith-fulness and adequacy of the implementation. Moreo...

Proof-theory has traditionally been developed based on linguistic (symbolic) representations of logical proofs. Recently, however, logical reasoning based on diagrammatic or graphical representations has been investigated by logicians. Euler diagrams were introduced in the 18th century by Euler [1768]. But it is quite recent (more precisely, in the 1990s) that logicians started to study them fr...

We propose a new framework for representing logics, called LF + and based on the Ed-inburgh Logical Framework. The new framework allows us to give, apparently for the rst time, general deenitions which capture how well a logic has been represented. These deen-itions are possible since we are able to distinguish in a generic way that part of the LF + entailment which corresponds to the underlyin...

Modern software systems usually deal with several sorts (types) of data elements simultaneously. Some of these sorts, like integers, booleans, and so on, can be seen as having an immediate, direct nature and therefore are called visible, and they are contrasted with the others, like types of objects (in OO sense), which are called hidden sorts. A language used to specify such software system ha...

In less than a decade, Deduction with Constraints (DwC) has opened a new universe in computer science. DwC can be viewed from two perspectives: one related to the automated deduction framework 17, 19], the other to the development and usage of programming languages based on logic 7, 14]. Within the automated deduction framework, constraints on the generic data structure \terms" have become a po...

Structural synthesis of programs (SSP) is an approach to deductive synthesis of functional programs using types as speciications and based on the Curry-Howard correspondence and on an intensional treatment of the notion of type. The implemented programming environments employing SSP have been based on a fragment of intuitionistic propositional logic (simple type theory) and on a natural-deducti...

We introduce an abstract general notion of system of equations and rewrites between terms, called Term Equational Rewrite System (TERS), and develop a sound logical deduction system, called Term Equational Rewrite Logic (TERL), to reason about equality and rewriting. Further, we give an analysis of algebraic free constructions which together with an internal completeness result may be used to s...

We introduce an abstract general notion of system of equations between terms, called Term Equational System, and develop a sound logical deduction system, called Term Equational Logic, for equational reasoning. Further, we give an analysis of algebraic free constructions that together with an internal completeness result may be used to synthesise complete equational logics. Indeed, as an applic...

About a century ago, Ernst Mach argued that Archimedes’s deduction of the principle of the lever is invalid, since its premises contain the conclusion to be demonstrated. Subsequently, many scholars defended Archimedes, mostly on historical grounds, by raising objections to Mach’s reconstruction of Archimedes’s deduction. In the debate, the Italian philosopher and historian of science Giovanni ...

Deduction modulo is a way to express a theory using computation rules instead of axioms. We present in this paper an extension of deduction modulo, called Polarized deduction modulo, where some rules can only be used at positive occurrences, while others can only be used at negative ones. We show that all theories in propositional calculus can be expressed in this framework and that cuts can al...