Ordering finite variable types with generalized quantifiers (original) (raw)

The hierarchy theorem for generalized quantifiers

The Journal of Symbolic Logic, 1996

The concept of a generalized quanti er of a given similarity type was de ned in Lin66]. Our main result says that on nite structures di erent similarity types give rise to di erent classes of generalized quanti ers. More exactly, for every similarity type t there is a generalized quanti er of type t which is not de nable in the extension of rst order logic by all generalized quanti ers of type smaller than t. This was proved for unary similarity types by Per Lindstr om Wes] with a counting argument. We extend his method to arbitrary similarity types.

On orderings of the family of all logics

Archive for Mathematical Logic, 1980

The purpose of this paper is to examine the structural complexity of the sublogic relation between abstract logics. Let ~(") denote n 'h order logic. Then cp(,) is a proper subtogic of A °("÷ 1~ for each n < co, and we have the chain ~(1)<~(2)< ... <~cp(,)< .... The question naturally arises, what other kinds of chains or partial orderings we can have among sufficiently regular abstract logics. Can one have a family of logics ordered in the order type of the rationals (or perhaps the reals)? In Chapter 2 we prove that any distributive lattice, and therefore any partial ordering, can be embedded into the sublogic relation among what we call normal logics. Chapter 3 is devoted to a sublogic relation defined using PC-classes rather than elementary classes. In the last chapter we restrict ourselves to the very special logics of the form ~(Q1,-, Q,)-The situation becomes more problematic but we can still prove that any countable partial ordering is embeddable into the sublogic relation of these logics. We confine ourselves to single-sorted structures, but many of the results are equally true of many-sorted structures.

A de nability theorem for rst order logic

1997

In this paper we will present a de nability theorem for rst order logic This theorem is very easy to state and its proof only uses elementary tools To explain the theorem let us rst observe that if M is a model of a theory T in a language L then clearly any de nable subset S M i e a subset S fa j M j a g de ned by some formula is invariant under all automorphisms of M The same is of course true for subsets of M n de ned by formulas with n free variables Our theorem states that if one allows Boolean valued models the converse holds More precisely for any theory T we will construct a Boolean valued model M in which precisely the T provable formulas hold and in which every Boolean valued subset which is invariant under all automorphisms of M is de nable by a formula of L Our presentation is entirely selfcontained and only requires familiarity with the most elementary properties of model theory In particular we have added a rst section in which we review the basic de nitions concerning ...

Normalizable linear orders and generic computations in finite models

Archive for Mathematical Logic, 1999

Numerous results about capturing complexity classes of queries by means of logical languages work for ordered structures only, and deal with non-generic, or order-dependent, queries. Recent attempts to improve the situation by characterizing wide classes of finite models where linear order is definable by certain simple means have not been very promising, as certain commonly believed conjectures were recently refuted (Dawar's Conjecture). We take on another approach that has to do with normalization of a given order (rather than with defining a linear order from scratch). To this end, we show that normalizability of linear order is a strictly weaker condition than definability (say, in the least fixpoint logic), and still allows for extending Immerman-Vardi-style results to generic queries. It seems to be the weakest such condition. We then conjecture that linear order is normalizable in the least fixpoint logic for any finitely axiomatizable class of rigid structures. Truth of this conjecture, which is a strengthened version of Stolboushkin's conjecture, would have the same practical implications as Dawar's Conjecture. Finally, we suggest a series of reductions of the two conjectures to specialized classes of graphs, which we believe should simplify further work.

Reducibility orderings: Theories, definability and automorphisms

Annals of Mathematical Logic, 1980

introduced the notion of degree of unsolvability and the partial ordering ~ i on ~T, the set of such Tt, ring degrees, induced by Turing reducibility (Turing 13711, His paper with Kleene [ 1,4] contains the first serious analysis of this structure (~'r, ~x). They prove, for example, that all coantable partial orderings can be embedded in (ar, ~<~-). These embeddings show that the existential (it;st order~ theory of (~-r, ~r) is decidable, Next Spector [35], in a paper arising from Kleene's 1953 seminar, made an important inroad on the two quantifie, (i.e., VzI~ theory by showing that there is a minimal (Turing) degree. Sacks [31] extended these results and set forth some important conjectures on embeddings a~nd initial segments of ~'r. In particular he points out that one can prove the undecidability of the theory of (£ar, ~<v) by such results. This work inspired many papers by others eszabli,,hing better and better initial segment results. One milesto;~e was kachlan which showed that every countable distributive lattice can bc embedded as an initial segment of the Turing degrees. As the theory of distributive lattices was known to be undecidable, this sufficed to verify Sacks' conjezture that so is the theory of (@r, <~a-). (In fact it would have sufficed to embed all f-~nitc distributive lattices as was pointed out by Thomason [36] for hyperdegrees.) Two directions in which such results can be sharpened immediately come to mind, One is, where does the undecidability first arise in terms of quantifier complexity. The second is just how complicated is the full theory of (~, ~). (qhe results of Kleene and Post [ 14] showed only that the 3-theory was decidable while the coding of distributive lattices only showed that the full theory ha', degree at least 0'.) Further progress required further structural results. For the first question Lerman [20] supplied an essential ingredient by settling the full conjecture from Sacks , He showed that every finite lattice is embeddable ~, an initial segment of fib-. This can be combined with Kleene and Pc, st [14] to decide the 'q::l theory of

Rank Hierarchies for Generalized Quantifiers

Journal of Logic and Computation, 2011

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 to infinitary logics augmented with arbitrary classes of generalized quantifiers of bounded arity.

Partially-Ordered (Branching) Generalized Quantifiers: A General Definition

Journal of Philosophical Logic 26: 1-43., 1997

Following Henkin's discovery of partially-ordered (branching) quantification (POQ) with standard quantifiers in 1959, philosophers of language have attempted to extend his definition to POQ with generalized quantifiers. In this paper I propose a general definition of POQ with 1-place generalized quantifiers of the simplest kind: namely, predicative, or "cardinality" quantifiers, e.g., "most", "few", "finitely many", "exactly α", where α is any cardinal, etc. The definition is obtained in a series of generalizations, extending the original, Henkin definition first to a general definition of monotone-increasing (M↑) POQ and then to a general definition of generalized POQ, regardless of monotonicity. The extension is based on (i) Barwise's 1979 analysis of the basic case of M↑ POQ and (ii) my 1990 analysis of the basic case of generalized POQ. POQ is a non-compositional 1st-order structure, hence the problem of extending the definition of the basic case to a general definition is not trivial. The paper concludes with a sample of applications to natural and mathematical languages.

On the Deductive System of the Order of an Equationally Orderable Quasivariety

Studia Logica, 2016

We consider the equationally orderable quasivarieties and associate with them deductive systems defined using the order. The method of definition of these deductive systems encompasses the definition of logics preserving degrees of truth we find in the research areas of substructural logics and mathematical fuzzy logic. We prove several general results, for example that the deductive systems so defined are finitary and that the ones associated with equationally orderable varieties are congruential.

The hierarchy theorem for second order generalized quantifiers

Journal of Symbolic Logic, 2006

We study definability of second-order generalized quantifiers. We show that the question whether a second-order generalized quantifier Q 1 is definable in terms of another quantifier Q 2 , the base logic being monadic second-order logic, reduces to the question if a quantifier Q 1 is definable in FO(Q 2 , <, +, ×) for certain first-order quantifiers Q 1 and Q 2 . We use our characterization to show new definability and nondefinability results for second-order generalized quantifiers. In particular, we show that the monadic second-order majority quantifier Most 1 is not definable in second-order logic.