# Multirole Logic (Extended Abstract)

We identify multirole logic as a new form of logic in which conjunction/disjunction is interpreted as an ultrafilter on the power set of some underlying set (of roles) and the notion of negation is generalized to endomorphisms on this underlying set. We formalize both multirole logic (MRL) and linear multirole logic (LMRL) as natural generalizations of classical logic (CL) and classical linear logic (CLL), respectively, and also present a filter-based interpretation for intuitionism in multirole logic. Among various meta-properties established for MRL and LMRL, we obtain one named multiparty cut-elimination stating that every cut involving one or more sequents (as a generalization of a (binary) cut involving exactly two sequents) can be eliminated, thus extending the celebrated result of cut-elimination by Gentzen.

## Similar Publications

Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. Read More

We study an infinite countable iteration of the natural product between ordinals. We present an "effective" way to compute this countable natural product; in the non trivial cases, the result depends only on the natural sum of the degrees of the factors, where the degree of a nonzero ordinal is the largest exponent in its Cantor normal form representation. Thus we are able to lift former results about infinitary sums to infinitary products. Read More

Recent results of Hindman, Leader and Strauss and of Fern\'andez-Bret\'on and Rinot showed that natural versions of Hindman's Theorem fail {\em for all} uncontable cardinals. On the other hand, Komj\'ath proved a result in the positive direction, showing that {\em there are} arbitrarily large abelian groups satisfying {\em some} Hindman-type property. In this note we show how a family of natural Hindman-type theorems for uncountable cardinals can be obtained by adapting some recent results of the author from their original countable setting. Read More

We show that the class of finite groups with n automorphisms has the amalgamation property for every n. Consequently, the automorphism group of Philip Hall's universal locally finite group has ample generics, that is, it admits comeager diagonal conjugacy classes in all dimensions. Read More

We are concerned with two separation theorems about analytic sets by Dyck and Preiss, the former involves the positively-defined subsets of the Cantor space and the latter the Borel-convex subsets of finite dimensional Banach spaces. We show by introducing the corresponding separation trees that both of these results admit a constructive proof. This enables us to give the uniform version of the separation theorems, and derive as corollaries the results, which are analogous to the fundamental fact "HYP is effectively bi-analytic" provided by the Souslin-Kleene Theorem. Read More

In Alm-Hirsch-Maddux (2016), relation algebras $\mathfrak{L}(q,n)$ were defined that generalize Roger Lyndon's relation algebras from projective lines, so that $\mathfrak{L}(q,0)$ is a Lyndon algebra. In that paper, it was shown that if $q>2304n^2+1$, $\mathfrak{L}(q,n)$ is representable, and if $q<2n$, $\mathfrak{L}(q,n)$ is not representable. In the present paper, we reduced this gap by proving that if $q\geq n(\log n)^{1+\varepsilon}$, $\mathfrak{L}(q,n)$ is representable. Read More

We further investigate a divisibility relation on the set $\beta N$ of ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a hierarchy in which a position of every ultrafilter depends on the set of prime ultrafilters it is divisible by. We also construct ultrafilters with many immediate successors in this hierarchy and find positions of products of ultrafilters. Read More

Since it was realized that the Curry-Howard isomorphism can be extended to the case of classical logic as well, several calculi has appeared as candidates for the encodings of proofs in classical logic. One of them was the $\lambda\mu$-calculus of Parigot ([19]). In this paper, following the reasoning of Xi presented for the $\l$-calculus ([27]), we give an upper bound for the lengths of reductions in the $\lambda\mu$-calculus extended with two more simplification rules: the $\rho$- and $\theta$-rules. Read More

These are notes of a series of talks about motivic integration I gave on the M\"unster Model Theory Month. Readers are assumed to have some basic knowledge of model theory and of valued fields. The notes are closest to the Cluckers-Loeser style of motivic integration, though mostly they are about doing p-adic integration uniformly in all $\mathbb{Q}_p$. Read More