ABSTRACT. Ever since the compilers of Euclid's Elements gave the "definitions" that "a point is that which has no part" and "a line is breadthless length", philosophers and mathematicians have worried that the the basic concepts of geometry are too abstract and too idealized. In the 20th century writers such as Husserl, Lesniewski, Whitehead, Tarski, Blumenthal, and von Neumann have proposed "pointless" approaches.

A problem more recent authors have emphasized it that there are difficulties in having a rich theory of a part-whole relationship without atoms and providing both size and geometric dimension as part of the theory.

A solution will be proposed using the Boolean algebra of measurable sets modulo null sets along with relations derived from the group of rigid motions in Euclidean n-space.

(This is a preliminary report on on-going joint work with Tamar Lando, Columbia University.)

ABSTRACT. We introduce proof systems and semantics for two paraconsistent extensions of the system $\mathbf{T}$ of Anderson and and Belnap, and prove strong soundness, completeness, and decidability for both. The semantics of both systems is based on excluding just one element from the set of designated values. One of the systems has the variable sharing property, and so it is a relevant logic. The other is an extension of the first that may be viewed as a semi-relevant dual of {\L}ukasiewicz Logic.

ABSTRACT. For involutive FL$_e$-algebras, several construction and decomposition theorems will be presented: connected co-rotations, connected and disconnected co-rotation-annihilations, pistil-petal construction and decomposition, and involutive ordinals sums. The constructions provide us with a huge set of examples of negative or zero rank involutive FL$_e$-algebras.

ABSTRACT. Our main issue was to understand the connection between Lukasiewicz logic with product and the Pierce-Birkhoff conjecture, and to express it in a mathematical way. To do this we define the class of fMV-algebras, which are MV-algebras endowed with both an internal binary product and a scalar product with scalars from [0,1]. The proper quasi-variety generated by [0,1], with both products interpreted as the real product, provides the desired framework: the normal form theorem of its corresponding logical system can be seen as a local version of the Pierce-Birkhoff conjecture. We survey the theory of MV-algebras with product (PMV-algebras, Riesz MV-algebras, fMV-algebras) with a special focus on the normal form theorems and their connection with the Pierce-Birkhoff conjecture.

ABSTRACT. Our goal is to establish functorial adjunctions between the category of MV-algebras and the categories of structures obtained by adding product operations, i.e. Riesz MV-algebras, PMV-algebras and fMV-algebras. We succed to do this for the corresponding subclasses of semisimple structures. Our main tool is the semisimple tensor product defined by Mundici. As consequence we prove the amalgamation property for unital and semisimple fMV-algebras. On our way we also prove that the MV-algebraic tensor product commutes, via Mundici's $\Gamma$ functor, with the tensor product of abelian lattice-ordered groups defined by Martinez.

ABSTRACT. The paper is devoted to the logic Lp of perfect MV-algebras. It is shown that a) m-generated perfect MV-algebra is projective if and only if it is finitely presented; b) there exists a one-to-one correspondence between projective formulas of Lp with m-variables and the m-generated projective subalgebras of the m-generated free algebras of the variety generated by perfect MV-algebras; c) Lp is structurally complete.

ABSTRACT. In this work, we provide bases for admissible rules for certain fragments of RMt, the logic R-mingle extended with a constant t.

ABSTRACT. The admissible rules of a logic are exactly those rules under which its set of theorems is closed. In this talk we will consider a class of intermediate logics, known as the subframe logics, and explore some of the structure of their admissible rules. We will show a particular scheme of admissible rules to be admissible in all subframe logics. Using this scheme we provide a complete description of the admissible rules of the intermediate logic BD2.

ABSTRACT. Almost Structural Completeness is proved and the form of admissible rules is found for some first-order modal logics extending S4.3. Bases for admissible rules are also investigated.

ABSTRACT. We consider intermediate predicate logics defined by fixed well-ordered (or dually well-ordered) linear Kripke frames with constant domains where the order-type of the well-order is strictly smaller than ω^ω. We show that two such logics of different order-type are separated by a first-order sentence using only one monadic predicate symbol. Previous results by Minari, Takano and Ono, as well as the second author, obtained the same separation but relied on the use of predicate symbols of unbounded arity.

ABSTRACT. Our goal is to introduce a framework for working with generalized multiple-conclusion rules (in propositional logics) containing asserted and rejected propositions. We construct a meta-logic that gives the syntactic means for manipulations with such type of rules. And we discuss the different flavors of the notion of admissibility that arises due to presence of multiple conclusions.