Gentzen-Type Methods for Bilattice Negation

A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (a conservative extension of CL) and CLS_cw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLS_cw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL.     

他连A都知道--认知逻辑EK1-EK3

首先,我们构造认知系统EK1-EK3,给出它们的一些证明论结果。其次,我们引入邻域语义,给出EK1-EK3的特征公理和规则的框架条件,证明EK1-EK3相对这些框架条件分别是框架可靠的。最后,我们证明EK1-EK3相对这些框架条件分别是框架完全的。     

Thomas Kuhn's Misunderstood Relation to Kripke-Putnam Essentialism

Kuhn's ‘taxonomic conception’ of natural kinds enables him to defend and re-specify the notion of incommensurability against the idea that it is reference, not meaning/use, that is overwhelmingly important. Kuhn's ghost still lacks any reason to believe that referentialist essentialism undercuts his central arguments in SSR – and indeed, any reason to believe that such essentialism is even coherent, considered as a doctrine about anything remotely resembling our actual science. The actual relation of Kuhn to Kripke-Putnam essentialism, is as follows: Kuhn decisively undermines it – drawing upon the inadequacies of such essentialism when faced with the failure of attempts to instantiate in history or contemporaneously its ‘thought-experiment’ – and leaves the field open instead for his own more ‘realistic’, deflationary way of thinking about the operation of ‘natural kinds’ in science. This revised version was published online in August 2006 with corrections to the Cover Date.     

Monadic Fuzzy Predicate Logics

Two variants of monadic fuzzy predicate logic are analyzed and compared with the full fuzzy predicate logic with respect to finite model property (properties) and arithmetical complexity of sets of tautologies, satisfiable formulas and of analogous notion restricted to finite models.     

The Basic Algebra of Game Equivalences

We give a complete axiomatization of the identities of the basic game algebra valid with respect to the abstract game board semantics. We also show that the additional conditions of termination and determinacy of game boards do not introduce new valid identities.En route we introduce a simple translation of game terms into plain modal logic and thus translate, while preserving validity both ways, game identities into modal formulae.The completeness proof is based on reduction of game terms to a certain minimal canonical form, by using only the axiomatic identities, and on showing that the equivalence of two minimal canonical terms can be established from these identities.     

Modal Incompleteness Revisited

In this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some L and some cannot be separated from by a suitably wide class of complete algebras — usually Kripke algebras. We are going to show that classical examples of incomplete logics, e.g., Fine logic, are not complete with respect to any class of complete BAOs. Even above Grz it is possible to find a continuum of such logics, which immediately implies the existence of a continuum of neighbourhood-incomplete Grz logics. Similar results can be proved for Löb logics. In addition, completely incomplete logics above Grz may be found uniformly as a result of failures of some admissible rule of a special kind.     

Reasoning about negligibility and proximity in the set of all hyperreals

We consider the binary relations of negligibility, comparability and proximity in the set of all hyperreals. Associating with negligibility, comparability and proximity the binary predicates N, C and P and the connectives $[N]$, $[C]$ and $[P]$, we consider a first-order theory based on these predicates and a modal logic based on these connectives. We investigate the axiomatization/completeness and the decidability/complexity of this first-order theory and this modal logic.