Henkin quantifiers and the definability of truth (original) (raw)

Henkin and function quantifiers

Annals of Pure and Applied Logic, 1989

introduced quantifiers which arise when usual (universal and existential) quantifiers are arranged in non-linear order. These quantifiers bre known as partially ordered quantifiers or Henkin quantifiers. Is. was observed already in [7] that languages with these quantifiers are stronger than first-order logic. A result of [ll] shows that even the weakest enkin quantifier has essentially the same expressive power as second-order logic. In this paper we introduce a simplification of Henkin quantifiers called function quantifiers. We show that function quantifiers are intimately connected with partially ordered quantifiers and suggest that they provide a natural framework for the study of partially ordered quantifiers and second-order definability in general. This paper continues work started in [9]. We are indebted to Alistair Lachlan and Lauri Hella for helpful discussions concerning preliminary versions of Section 8. We are also grateful to the referee for his or her remarks and suggestions. For basic notions concerning extensions of first-order logic and generalized uantifiers, the reader is referred to [ 11. Our logics 5? will mostly be extensions of 5$,,. obtained by adding geiazralized quantifiers in the sense of [12]. We recall the definition of a generalized quantifier from [ 121. Let K be a class of models of a relational similarity type r = (no, . . . , &). K is assumed to be closed under isomorphisms. The generalized quantifier Q asociated with K is defined as fQllows: Krynicki, J. V'n use 6e"' to denote the second-order logic with quantification over to denote the simplest non-trivial Henkin quantifier defined in [7]: true if for every x there exists u such that for every y there such that p. Another way of writing

First-Order Logic and Its Infinitary Quantifier Extensions over Countable Words

Fundamentals of Computation Theory, 2021

We contribute to the refined understanding of the languagelogic-algebra interplay in the context of first-order properties of countable words. We establish decidable algebraic characterizations of one variable fragment of FO as well as boolean closure of existential fragment of FO via a strengthening of Simon's theorem about piecewise testable languages. We propose a new extension of FO which admits infinitary quantifiers to reason about the inherent infinitary properties of countable words. We provide a very natural and hierarchical block-product based characterization of the new extension. We also explicate its role in view of other natural and classical logical systems such as WMSO and FO[cut]-an extension of FO where quantification over Dedekindcuts is allowed. We also rule out the possibility of a finite-basis for a block-product based characterization of these logical systems. Finally, we report simple but novel algebraic characterizations of one variable fragments of the hierarchies of the new proposed extension of FO.

A definability theorem for first order logic

1997

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a | M |= ϕ(a)} defined by some formula ϕ) is invariant under all automorphisms of M. The same is of course true for subsets of M n defined 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 definable 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 first section in which we review the basic definitions concerning Boolean valued models. The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. The construction of this space is closely related to the one in [1]. In fact, one of the results in that paper could be interpreted as a definability theorem for infinitary logic, using topological rather than Boolean valued models. * Both authors acknowledge support from the Netherlands Science Organisation (NWO).

On the construction of fully interpreted formal languages which posses their truth predicates

arXiv: Logic, 2015

We shall first construct by ordinary recursion method subsets to the set D of Gödel numbers of the sentences of a language L. That language is formed by the sentences of a fully interpreted formal language L, called an MA language, and sentences containing a monadic predicate letter T . From the class of the constructed subsets ofD we extract one set U by transfinite recursion method. Interpret those sentences whose Gödel numbers are in U as true, and their negations as false. These sentences together form an MA language. It is a sublanguage of L having L as its sublanguage, and T is its truth predicate. MSC: 00A30, 03B10, 47H04, 47H10, 68Q45

Characterizing second order logic with first order quantifiers

1979

In [l] and [Z] it is shoun that the language consisting of formulae of the form QM, where Q is a partially ordered qtiantifier prefix (Henkin prefix, abbreviated poq) and iVl is a quantifier-free matrix, is equal in expressive power to 2:(notation from ROGERS 131). Extending the language to allow the attachment of poq's to formulae as an additional formation rule (together with, say, A and i), yields A:(see [l]). This extension seems, howeirer. to destroy the natural character of the semantics of poq's which existed in the case QM.

Model-completeness in a first order language with a generalized quantifier

Pacific Journal of Mathematics, 1975

The concept of Model-Completeness is defined in a first order language with a generalized quantifier. A necessary and sufficient condition is given for that Model-Completeness and its relation to categoricity is discussed. Some results of this paper were obtained in the author's thesis [12] and were announced in [11], They, together with other results of [12] were improved independently by the author and by S. Shelah. A suggestion of S. Shelah made some proofs simpler and due to it, better results were obtained in Theorem 1.5. The author wishes to thank S. Shelah for his remarks. Let L be a first order language with equality and let L(Q) be the language obtained from L by adding a new quantifier Q. Let α, β denote infinite cardinals. We define α-satisfaction for L(Q) by interpreting Q as "there exist at least a elements". If a sentence φ of L(Q) is a-satisfied in a model 91 for L we write 9ll= α φ and we say that 91 is an a-model for φ. Let 31,93 be two models for L, |9l|^α and 91 C 93. Write 21 < a 93 if for every n, every formula φ{x u-,x n) in L(Q) and every a u-,a n in 9Ϊ: 9lM>[α,, ,αj iff 93h α φ[α,, ,α n ]. Let T be an ordinary first order theory (namely a theory in L) that has infinite models. Define T(Q) = TU{Qx[x =x]}. Call T a-model-complete if for every 21,93 which are α-models for T(Q) and 91 C 93 also 9l< α 93. A necessary and sufficient condition for T to be a-model-complete for a > H o is given in section 1. Let T be as before. Define T(a) = {φ: φ is a sentence in L(Q) and for every 91, if 9lh α T(Q) then 9ίh α φ}. Call T α-complete if for every sentence φ in L(Q) either φ G T(a) ory φ G Γ(α). In §2, it is shown that if T is categorical in one uncountable power, it is incomplete and for every a^H*: T(a)=T(Ho). If T is also modelcomplete (in the usual sense) then it is α-model-complete for every α^No and T(a) is decidable provided T is axiomatic. 1. ci-Model-Completeness. DEFINITION 1.1. Let φ(x,x u ,jc m) be a formula in L such that x, jt,, ,jc m are exactly all its free variables. Let 91 be a model for L and let α,, ,α m be elements in 91. Define: 265

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.

Unifying some modifications of the Henkin construction

Notre Dame Journal of Formal Logic, 1992

This paper is a continuation of the work of Leblanc, Roeper, Thau, and Weaver, which modified the Henkin construction to yield various necessary and sufficient conditions for extending a consistent set of sentences in a countable first order language to a maximally consistent and ω-complete set in that language. In this paper the theory of abstract deducibility relations introduced by Goldblatt is extended to provide an abstract setting for these and related results. Modifications of Henkin construction are replaced by Goldblatt's Countable Henkin Principle to yield abstract forms of the ω-completeness theorem, the soundness and completeness of ω-logic, the theorem to the effect that ω-logic is a conservative extension of standard logic for ωcomplete sets, and the theorem that all ω-complete sets are ω-consistent. These abstract results specialize to yield the corresponding "concrete" ones.

Degrees of logics with Henkin quantifiers in poor vocabularies

Archive for Mathematical Logic, 2004

We investigate some logics with Henkin quantifiers. For a given logic L, we consider questions of the form: what is the degree of the set of L-tautologies in a poor vocabulary (monadic or empty)? We prove that the set of tautologies of the logic with all Henkin quantifiers in empty vocabulary L * ∅ is of degree 0 . We show that the same holds also for some weaker logics like L ∅ (H ω ) and L ∅ (E ω ).