ABSTRACTS A1.1 Mathematical Logic On Implicational Connectives of Quantum Logics for Non-commutative Substructural Logics formulated Gentzen-style Natural Deduction

S A1.1 Mathematical Logic On Implicational Connectives of Quantum Logics for Non-commutative Substructural Logics formulated Gentzen-style Natural Deduction Takeshi Ueno, Food Science and Human Wellnes, Rakuno-Gakuen University, Ebetsu, JAPAN See below for abstract. A theory for systems of propositions referring to each other Denis Saveliev, Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, RUSSIAN FEDERATION As well-known, all classical paradoxes involve a kind of self-reference. A paradox without any self-reference was 
proposed by Yablo twenty years ago in [1] (for a subsequent discussion see [2]-[6]). This new paradox can be considered as an unfolding of the paradigmatic Liar Paradox: it consists of propositions indexed by natural numbers such that each of the propositions states “all propositions with greater indices are wrong”. Our purpose is to investigate arbitrary systems of propositions some of which state that some others are wrong, and to learn which of these systems are paradoxical and which are not. For this, we introduce a first-order theory of a language with one unary and one binary predicates, $T$ and $U$. Heuristically, variables mean propositions, $Tx$ means “$x$ is true”, and $Uxy$ means “$x$ states that $y$ is wrong”. As we show, the theory is $\Pi^{0} _2$ but not $\Sigma^{0}_2$. We study which model-theoretic operations preserve or do not preserve the theory, and provide a natural classification of its models. Furthermore, we say that a model $(X,U)$ is nonparadoxical iff it can be enriched to some model $(X,T,U)$ of this theory, and paradoxical otherwise. E.g. a model 
of the Liar Paradox consists of one reflexive point, a model of the Yablo Paradox is isomorphic to natural numbers with their usual ordering, and both are paradoxical. Generalizing these two instances, we note that any model with a transitive $U$ without maximal elements is paradoxical. On the other hand, any model with a well-founded $U^{-1}$ is not. We also examine other classes of relations, show that the paradoxicality (and hence nonparadoxicality) is a $\Delta^{1}_1$ but not elementary property, and provide a classification of nonparadoxical models. References [1] S. Yablo. Paradox without self-reference. Analysis 53:4 (1993), 251--252. 
[2] T. E. Forster. The Significance of Yablo's Paradox without Self-Reference. Manuscript, 1996.
 [3] G. Priest. Yablo's Paradox. Analysis 57:4 (1997), 236--242. 
[4] R. Sorensen. Yablo's Paradox and kindred infinite liars. Mind 107 (1998), 137--155. 
 [5] Jc. Beall. Is Yablo's paradox non-circular? Analysis 61:3 (2001), 176--187. 
[6] T. E. Forster. Yablo's Paradox and the Omitting Types Theorem for Propositional Languages. Manuscript, 2012. 
[7] J. Barwise, J. Etchemendi. The Liar: An Essay in Truth and Circularity. Oxford University Press, 1987. 
[8] J. Barwise, L. S. Moss. Vicious Circles. CSLI Lecture Notes 60, 1996. 
[9] C. C. Chang, H. J. Keisler. Model Theory. 3rd revised edition, NorthHolland, Amsterdam, 1990. 
[10] A. Gupta. Truth. In: L. Goble (ed.). The Blackwell Guide to Philosophical Logic. Blackwell, 2001. Logic and philosophy of trial and error mathematics: Dialectical and quasi-dialectical systems Luca San Mauro, Faculty of Humanities, Scuola Normale Superiore, Pisa, ITALY Formal systems represent mathematical theories in a somewhat static way, in which axioms of the represented theory have to be defined from the beginning, and no further modification is permitted. As is clear, this representation is not comprehensive of all aspects of real mathematical theories. In particular, these latter – as often argued, starting from the seminal work of Lakatos (see [3]) – are frequently the outcome of a much more dynamic process than the one captured by formal systems. For instance, in defining a new theory, axioms can be chosen through a trial and error process, instead of being initially selected. Dialectical systems, introduced by Roberto Magari in [4], are apt to characterize this dynamic feature of mathematical theories (see [2] for a similar, yet non equivalent, characterization). In this paper, we prove several results concerning dialectical systems and of the sets that they represent, called dialectical sets. In particular, we offer a degree theoretic characterization of dialectical sets. We prove that all dialectical sets are Turing equivalent to some computably enumerable set. Then, in order to better analyze the intended semantic of dialectical systems, we introduce a more general class of systems, that of quasi-dialectical systems. These are systems that naturally embeds a certain notion of “revision”. We prove that quasi-dialectical sets lie in the same Turing-degrees of dialectical sets, hence showing that they display the same computational power. Nonetheless, we conclude by proving that quasidialectical sets and dialectical sets are different, and by showing their respective place in the Ershov hierarchy (see [1]). References: [1] C. J. Ash and J. Knight. Computable Structures and the Hyperarithmetical Hierarchy. North-Holland Publishing Co., Amsterdam, 2000. [2] R. G. Jeroslow. Experimental logics and ?_02 theories. Journal of Philosophical Logic, 4(3):53–267, 1975. [3] I. Lakatos. Proofs and Refutations. Cambridge University Press, Cambridge, 1976. [4] R. Magari. Su certe teorie non enumerabili. Ann. Mat. Pura Appl. (4), XCVIII:119–152, 1974. Some general results on the translations between logics and theories Luiz Carlos Pereira, Philosophy, PUC-Rio/UERJ, Rio de Janeiro, BRAZIL In the late twenties and early thirties of last century several results were obtained connecting different logics and theories. These results assumed the form of translations/interpretations of one logic/theory into another logic/theory. A minimum requirement is that they preserve deducibility: Given two logics L1 and L2 and a translation T of L2 into L1, then S |-L2 A if and only if T[S] |-L1 T[A]. The aim of the present paper is to show the following results concerning translation between logics and theories: [1] Given two logics L1 and L2 and a translation of L2 into L1, then, given any intermediate logic L3 between L1 and L2, the same translation can be used to translate L2 into L3. It is also shown that this translation cannot be used to translate L3 into L1.
[2] In 1979, R. Statman showed a translation from Intuitionistic Propositional Logic into its implicational fragment. This reduction is polynomial and proves that Purely Implicational Minimal Logic is PSPACEcomplete. The methods that Statman used are based on prooftheory. The sub-formula principle for a Propositional Natural Deduction system NL for a logic L states that whenever α is provable from Γ in L, there is a derivation of α from a set of assumptions {δ1, . . . ,δk} ? Γ built up only with sub-formulas of α and/or {δ1, . . . , δk}. We show that any propositional logic L with a Natural Deduction system that satisfies the sub-formula principle has a translation into purely minimal implicational logic.
[3] The third result establishes that if T is a first order theory formulated in the language {~,&, →, } and T is atomically stable, then every theorem of T can be proved without the use of classical reasoning. On Implicational Connectives of Quantum Logics for Non-commutative Substructural Logics formulated Gentzen-style Natural Deduction Takeshi Ueno∗