In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that ...

We propose probability logic as an appropriate standard of reference for evaluating human inferences. Probability logical accounts of nonmonotonic reasoning with system p, and conditional syllogisms (modus ponens, etc.) are explored. Furthermore, we present categorical syllogisms with intermediate quantifiers, like the “most . . . ” quantifier. While most of the paper is theoretical and intende...

In this paper we overview our recent work and outline our current line of research: First, we describe the WCSP framework and its related local consistencies. Second, the Max-SAT problem and some related inference rules including a new rule that is an extension of the classical Modus Ponens. Finally, we will show the relation between both ideas and how this link can help us to improve existent ...

In this paper we present a propositional logic programming language for reasoning under possibilistic uncertainty and represent­ ing vague knowledge. Formulas are repre­ sented by pairs (ip, a), where ip is a many­ valued proposition and a E [0, 1] is a lower bound on the belief on ip in terms of necessity measures. Belief states are modeled by pos­ sibility distributions on the set of all many...

We define a constructive model for ∆2-maps, that is, maps recursively definable from a map deciding the halting problem. Our model refines existing constructive interpretation for classical reasoning over one-quantifier formulas: it is compositional (Modus Ponens is interpreted as an application) and semantical (rather than translating classical proofs into intuitionistic ones, we define a math...

Redesign of a class structure requires transformation of the corresponding specification. In a previous paper we have shown how to use, what we call, interpretation functions for transformation of constraints. In this paper we study the way proof are transformed via such functions. We provide a sufficient condition for interpretation functions to preserve proofs using propositional logic with m...

In reporting on the theorem prover SCOTT (Slaney, SCOTT: A Semantically Guided Theorem Prover, Proc. IJCAI, 1993) we suggested semantic constraint as as an appropriate mechanism for guiding proof searches in propositional systems where the rule of inference is condensed detachment|a generalisation of Modus Ponens. Such constrained condensed detachment is closely analogous to semantic resolution...

Extended Abstract Representations of the lower and upper approximations of a set in the context of an approximation space as modal operators in the first order language of modal logics, are quite natural and widely familiar now to the rough-set community. According to the perception of an observer, objects of a universe (of discourse) are clustered. These are the basic information granules (or ...

We show that a set of three axioms, together with modus ponens, suffices to build up a formal system for the class R of all quadruples (L,6,T,J), in which (L,6) is a partially ordered set with a top element, and T is a triangular norm on (L,6) that has a residuated implication J. We show that this R is not a variety. 2004 Elsevier Inc. All rights reserved.

In [15],[16] Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated ε-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals ΦA of order type < ε0 which realize the Herbrand normal form A of A. Subsequently more perspic...