
Authors:
Paulo A.S.VELOSO
Title:On some misconceptions about ultrafilter logic
Pages:112
File:bibtex
Abstract ( + )
We examine some apparent misconceptions about ultrafilter logic, an extension of classical firstorder logic by a generalized quantifier, whose intended interpretation is 'almost all'. These arise from some alleged objections against the usage of ultrafilters for capturing the intended meaning of 'almost universal' assertions. The two alleged objections on which we focus concern degree of arbitrariness and trivialization in finite situations. We argue that these two apparent objections can be overcome by a proper understanding of the roles of models and theories.

Authors:
Vladimir L.VASYUKOV
Title:Implicative logics, sequential deductive systems and exponential multicategories
Pages:1325
File:bibtex
Abstract ( + )
In the paper Lambektype sequential implicative deductive systems and exponential multicategories are considered. The cut elimination theorem for those is proven. A sequential implicative deductive metasystems and dual deductive systems and categories are built and some metamathematical results are obtained.

Authors:
Tomasz KOWALSKI
Title:A remark on quasivarieties of modal algebras
Pages:2730
File:bibtex
Abstract ( + )
[Abstract]

Authors:
Ewa GRACZYNSKA
Title:Proofs of tree identities
Pages:3141
File:bibtex
Abstract ( + )
We present special identities via tree operads. Among them there are: normal and regular, left and rightmost identities. We explain their role in universal algebra and equational logic. We also add some generalizations of theorems connected with the problem of representing algebras from an Ashift of a variety, for a given algebra A, i.e. from the variety defined by all the Aidentities of a given variety and with the lattice of all subvarieties of a given variety etc. One may apply our observations to a suitable extension of varieties by a (twoelement) algebra. We formalize the notion of a graphical proof.

Authors:
George TOURLAKIS
Title:A basic formal equational predicate logic  Part I
Pages:4356
File:bibtex
Abstract ( + )
We present, in two parts, the details of a formalization of Equational Predicate Logic based on a 'propositional' version of the Leibniz rule (Propositional Strong Leibniz, or PSL), and Equanimity (EQN). Both rules are 'strong', that is, they are applicable to arbitrary premises (not just to absolute theorems). We show (Part II) that a strong 'nocapture' Leibniz, and a weak 'fullcapture' version are derived rules (both access the interior of quantifier scopes). We also derive general rules MON (monotonicity) and AMON (antimonotonicity), which allow us to 'calculate' appropriate conclusions  C[p\A] > C[p\B] or  C[p\A] < C[p\B] from the assumption  A > B"[p\...]" denoting fullcapture substitution. We show that these rules are 'as strong as possible' for our logic. Finally, we show that our logic is sound and complete.

Authors:
Janusz CIUCIURA
Title:Intuitionistic discursive system (IDS)
Pages:5762
File:bibtex
Abstract ( + )
In the late forties, Stanislaw Jaskowski presented a calculus called by him discursive sentential calculus, D2. He provided a definition of it by an interpretation in the language of S5 of Lewis. A number of publications have followed with the course of time. The aim of this paper is to introduce a new axiomatization for the discursive system suggested by some features of intuitionism.

Authors:
Anna GOMOLINSKA and Justyna K.MYSLINSKA
Title:A note on relationship between the consequence operations Cn' and dCn
Pages:6367
File:bibtex
Abstract ( + )
The consequence operation Cn' was defined for and applied in formalisation of reasoning on rejected information in the logic of acceptance and rejection. We prove that Cn' and the dual counterpart dCn of the classical consequence Cn are equal. Thanks to the previous results, the relationship between Cn' and the consequence operation Cn^{1} obtains immediately.