A multiple quantifier is a quantifier having inference rules that introduce or eliminate arbitrary number of quantifiers by one inference. This paper introduces the lambda calculus with negation, conjunction, and multiple existential quantifiers, and the lambda calculus with implication and multiple universal quantifiers. Their type checking and type inference are proved to be undecidable. This...

We refine the constructions of Ferrante-Rackoff and Solovay on iterated definitions in first-order logic and their expressibility in with polynomial size formulas. These constructions introduce additional quantifiers; however, we show that these extra quantifiers range over only finite sets. We prove optimal upper and lower bounds on the quantifier complexity of polynomial size formulas obtaine...

We show that for each n and m, there is an existential first order sentence which is NOT logically equivalent to a sentence of quantifier rank at most m in infinitary logic augmented with all generalized quantifiers of arity at most n. We use this to show the strictness of the quantifier rank hierarchies for various logics ranging from existential (or universal) fragments of first order logic t...

We shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide an easy alternative to nonstandard methods of proving existence theorems in probability theory, wh...

We obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like ∃≥3/4y, which says that “for at least 3/4 of all y”. These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are: 1. We deal with the quantifi...

Term algebras have wide applicability in computer science. Unfortunately, the decision problem for term algebras has a nonelementary lower bound, which makes the theory and any extension of it intractable in practice. However, it is often more appropriate to consider the bounded class, in which formulae can have arbitrarily long sequences of quantifiers but the quantifier alternation depth is b...

Logic programming has traditionally lacked devices for expressing iterative tasks. To overcome this problem, this paper proposes iterative goal formulas of the form ∧LxG where G is a goal, x is a variable, and L is a list. ∧Lx is called a parallel bounded quantifier. These goals allow us to specify the following task: iterate G with x ranging over all the elements of L. keywords: for-loop, iter...

Let V be a set of number-theoretical functions. We define notion absolute -realizability for predicate formulas and sequents in such way that the indices functions are used interpreting implication universal quantifier. In this article, we prove Basic Predicate Calculus is sound with respect to semantics if satisfies some natural conditions.

0. Introduction 2 The impact of logic in Banach space theory 2 The case of model theory 2 Model theory for structures of functional analysis 3 Two famous applications 4 A note on the exposition 4 1. Preliminaries: Banach Space Models 5 Banach space structures and Banach space ultrapowers 5 Positive bounded formulas 7 Approximate satisfaction 8 (1 + )-isomorphism and (1 + )-equivalence of struct...