The Elimination of Atomic Cuts and the Semishortening Property for Gentzen's Sequent Calculus with Equality

We study various extensions of Gentzen's sequent calculus obtained by adding rules for equality. One of them is singled out as particularly natural and shown to satisfy full cut elimination, namely also atomic cuts can be eliminated. Furthermore we tell apart the extensions that satisfy full cut elimination from those that do not and establish a strengthened form of the nonlenghtening property of Lifschitz and Orevkov

Comparing theories: the dynamics of changing vocabulary. A case-study in relativity theory

There are several first-order logic (FOL) axiomatizations of special relativity theory in the literature, all looking essentially different but claiming to axiomatize the same physical theory. In this paper, we elaborate a comparison, in the framework of mathematical logic, between these FOL theories for special relativity. For this comparison, we use a version of mathematical definability theory in which new entities can also be defined besides new relations over already available entities. In particular, we build an interpretation of the reference-frame oriented theory SpecRel into the observationally oriented Signalling theory of James Ax. This interpretation provides SpecRel with an operational/experimental semantics. Then we make precise, "quantitative" comparisons between these two theories via using the notion of definitional equivalence. This is an application of logic to the philosophy of science and physics in the spirit of Johan van Benthem's work.Comment: 27 pages, 8 figures. To appear in Springer Book series Trends in Logi

Absorbing the structural rules in the sequent calculus with additional atomic rules

We show that if the structural rules are admissible over a set R of atomic rules, then they are admissible in the sequent calculus obtained by adding the rules in R to G3[mic]. Two applications to pure logic and to the sequent calculus with equality are presented.Comment: 16 page

The Subterm Property for the Sequent Calculus with Equality

By the sequent calculus with equality we mean Gentzen\u2019s systems LJ and LK, with the 00 \u21d2 and \u21d2 03 rules restricted to individual parameters, enriched by the reflexivity axioms \u21d2 t = t, for t an arbitrary term, and the multiple right congruence rule: \u393 \u21d2 F{x1/r1,...,xn/rn} \u393 \u21d2 r1 = s1 ...\u393 \u21d2 rn = sn \u393 \u21d2 F{x1/s1,...,xn/sn} for r1, . . . sn arbitrary terms. We show that for both systems, every deriv- able sequent has a cut-free derivation in which occur only terms and atomic formulae other than equalities, that are renamings of terms and atomic formulae occurring in the sequent itself. That is instrumental in the proof theoretic analysis of the logic of par- tial terms, where \u201dexistence\u201d, namely defineteness, of a term t is expressed by 03x(x = t), for x not occurring in t

Cut Elimination for Gentzen's Sequent Calculus with Equality and Logic of Partial Terms

We provide a natural formulation of the sequent calculus with equality and establish the cut elimination theorem. We also briefly comment on its applications to the logic of partial terms, when "existence" is formulated as equality with a (bound) variabl

A Note on the Sequent Calculi G3[mic]=

We show that the Replacement Rule of the sequent calculi G3[mic]=in [8] can be replaced by the simpler rule in which one of the principal formulae is not repeated in the premiss