Finite partially-ordered quantification

1970 ◽  
Vol 35 (4) ◽  
pp. 535-555 ◽  
Author(s):  
Wilbur John Walkoe
Keyword(s):  
Partial Order ◽  
Normal Forms ◽  
Second Order ◽  
Infinitary Logic ◽  
First Order ◽  
Partially Ordered ◽  
Order Language

In [3] Henkin made the observation that certain second-order existential formulas may be thought of as the Skolem normal forms of formulas of a language which is first-order in every respect except its incorporation of a form of partially-ordered quantification. One formulation of this sort of language is the closure of a first-order language under the formation rule that Qφ is a formula whenever φ is a formula and Q, which is to be thought of as a quantifier-prefix, is a system of partial order whose universe is a set of quantifiers. Although he introduced this idea in a discussion of infinitary logic, Henkin went on to discuss its application to finitary languages, and he concluded his discussion with a theorem of Ehrenfeucht that the incorporation of an extremely simple partially-ordered quantifier-prefix (the quantifiers ∀x, ∀y, ∃v, and ∃w, with the ordering {〈∀x, ∃v〉, 〈∀y, ∃w〉}) into any first-order language with identity gives a language capable of expressing the infinitary quantifier ∃zκ0x.

Infinitary logic and admissible sets

1969 ◽  
Vol 34 (2) ◽  
pp. 226-252 ◽  
Author(s):  
Jon Barwise
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Predicate Calculus ◽  
Generalized Quantifiers ◽  
Infinitary Logic ◽  
First Order ◽  
Admissible Sets ◽  
Classical Language ◽  
Second Order Logic ◽  
Order Predicate Calculus

In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is(I) strong enough to express interesting properties not expressible by the classical language, but(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.

scholarly journals Local Normal Forms for First-Order Logic with Applications to Games and Automata

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Thomas Schwentick ◽  
Klaus Barthelmann
Keyword(s):  
Normal Forms ◽  
Winning Strategy ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Relational Structures ◽  
Second Order Logic ◽  
Monadic Second Order Logic

International audience Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

scholarly journals הכל הבל in Ecclesiastes 1:2 and 12:8 – Descriptive metaphysics of properties as comparative-philosophical supplement

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Jaco Gericke
Keyword(s):  
Second Order ◽  
Property Theory ◽  
First Order ◽  
Conceptual Clarification ◽  
Order Language ◽  
The World ◽  
Original Contribution ◽  
The Way

In this article, a supplementary yet original contribution is made to the ongoing attempts at refining ways of comparative-philosophical conceptual clarification of Qohelet’s claim that הבל הכל in 1:2 (and 12:8). Adopting and adapting the latest analytic metaphysical concerns and categories for descriptive purposes only, a distinction is made between הבל as property of הכל and the properties of הבל in relation to הכל. Involving both correlation and contrast, the second-order language framework is hereby extended to a level of advanced nuance and specificity for restating the meaning of the book’s first-order language on its own terms, even if not in them.Contribution: By considering logical, ontological, mereological and typological aspects of property theory in dialogue with appearances of הכל and of הבל in Ecclesiastes 1:2 and 12:8 and in-between, a new way is presented in the quest to explain why things in the world of the text are the way they are, or why they are at all.

An interpretation of “finite” modal first-order languages in classical second-order languages

1976 ◽  
Vol 41 (2) ◽  
pp. 337-340
Author(s):  
Scott K. Lehmann
Keyword(s):  
Possible Worlds ◽  
Second Order ◽  
Universal Quantifier ◽  
Possible World ◽  
Simple Interpretation ◽  
First Order ◽  
Order Language ◽  
Finite Set ◽  
Modal Semantics ◽  
Individual Variables

This note describes a simple interpretation * of modal first-order languages K with but finitely many predicates in derived classical second-order languages L(K) such that if Γ is a set of K-formulae, Γ is satisfiable (according to Kripke's 55 semantics) iff Γ* is satisfiable (according to standard (or nonstandard) second-order semantics).The motivation for the interpretation is roughly as follows. Consider the “true” modal semantics, in which the relative possibility relation is universal. Here the necessity operator can be considered a universal quantifier over possible worlds. A possible world itself can be identified with an assignment of extensions to the predicates and of a range to the quantifiers; if the quantifiers are first relativized to an existence predicate, a possible world becomes simply an assignment of extensions to the predicates. Thus the necessity operator can be taken to be a universal quantifier over a class of assignments of extensions to the predicates. So if these predicates are regarded as naming functions from extensions to extensions, the necessity operator can be taken as a string of universal quantifiers over extensions.The alphabet of a “finite” modal first-order language K shall consist of a non-empty countable set Var of individual variables, a nonempty finite set Pred of predicates, the logical symbols ‘¬’ ‘∧’, and ‘∧’, and the operator ‘◊’. The formation rules of K generate the usual Polish notations as K-formulae. ‘ν’, ‘ν1’, … range over Var, ‘P’ over Pred, ‘A’ over K-formulae, and ‘Γ’ over sets of K-formulae.

Polynomial rings and weak second-order logic

1985 ◽  
Vol 50 (4) ◽  
pp. 953-972 ◽  
Author(s):  
Anne Bauval
Keyword(s):  
Zero Divisor ◽  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
Polynomial Rings ◽  
Commutative Field ◽  
First Order ◽  
First Order Theory ◽  
Order Language ◽  
Second Order Logic

This article is a rewriting of my Ph.D. Thesis, supervised by Professor G. Sabbagh, and incorporates a suggestion from Professor B. Poizat. My main result can be crudely summarized (but see below for detailed statements) by the equality: first-order theory of F[Xi]i∈I = weak second-order theory of F.§I.1. Conventions. The letter F will always denote a commutative field, and I a nonempty set. A field or a ring (A; +, ·) will often be written A for short. We shall use symbols which are definable in all our models, and in the structure of natural numbers (N; +, ·):— the constant 0, defined by the formula Z(x): ∀y (x + y = y);— the constant 1, defined by the formula U(x): ∀y (x · y = y);— the operation ∹ x − y = z ↔ x = y + z;— the relation of division: x ∣ y ↔ ∃ z(x · z = y).A domain is a commutative ring with unity and without any zero divisor.By “… → …” we mean “… is definable in …, uniformly in any model M of L”.All our constructions will be uniform, unless otherwise mentioned.§I.2. Weak second-order models and languages. First of all, we have to define the models Pf(M), Sf(M), Sf′(M) and HF(M) associated to a model M = {A; ℐ) of a first-order language L [CK, pp. 18–20]. Let L1 be the extension of L obtained by adjunction of a second list of variables (denoted by capital letters), and of a membership symbol ∈. Pf(M) is the model (A, Pf(A); ℐ, ∈) of L1, (where Pf(A) is the set of finite subsets of A. Let L2 be the extension of L obtained by adjunction of a second list of variables, a membership symbol ∈, and a concatenation symbol ◠.

scholarly journals A dichotomy in classifying quantifiers for finite models

2005 ◽  
Vol 70 (4) ◽  
pp. 1297-1324
Author(s):  
Saharon Shelah ◽  
Mor Doron
Keyword(s):  
Partial Order ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Partial Order ◽  
Existential Quantifier ◽  
First Order ◽  
Finite Models ◽  
One To One

AbstractWe consider a family of finite universes. The second order existential quantifier Qℜ means for each U Є quantifying over a set of n(ℜ)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Qℜ, either Qℜ is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Qℜ) (first order logic plus the quantifier Qℜ) is undecidable.

scholarly journals Applying the method of normal forms to second-order nonlinear vibration problems

2010 ◽  
Vol 467 (2128) ◽  
pp. 1141-1163 ◽  
Author(s):  
Simon A. Neild ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Nonlinear Vibration ◽  
Normal Forms ◽  
Equations Of Motion ◽  
Second Order ◽  
Form Analysis ◽  
First Order ◽  
Invariance Properties ◽  
Vibration Problems ◽  
Second Order Equations

Vibration problems are naturally formulated with second-order equations of motion. When the vibration problem is nonlinear in nature, using normal form analysis currently requires that the second-order equations of motion be put into first-order form. In this paper, we demonstrate that normal form analysis can be carried out on the second-order equations of motion. In addition, for forced, damped, nonlinear vibration problems, we show that the invariance properties of the first- and second-order transforms differ. As a result, using the second-order approach leads to a simplified formulation for forced, damped, nonlinear vibration problems.

scholarly journals A Note on Identity and Higher Order Quantification

2009 ◽  
Vol 7 ◽  
Author(s):  
Rafal Urbaniak
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Identity Relation ◽  
First Order ◽  
Order Language ◽  
Order Set

It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be defined in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantifiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics (where the variables range over all subsets of the domain) in which the identity relation is not definable. The point is that the definability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed. This paper is a follow-up to (Urbaniak 2006), where it has been proven that no actual axiomatization of Leśniewski’s Ontology determines the standard semantics for the epsilon connective.

scholarly journals Betweenness of partial orders

2020 ◽  
Vol 54 ◽  
pp. 7
Author(s):  
Bruno Courcelle
Keyword(s):  
Partial Order ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Second Order ◽  
Partial Orders ◽  
First Order ◽  
Betweenness Relations

We construct a monadic second-order sentence that characterizes the ternary relations that are the betweenness relations of finite or infinite partial orders. We prove that no first-order sentence can do that. We characterize the partial orders that can be reconstructed from their betweenness relations. We propose a polynomial time algorithm that tests if a finite relation is the betweenness of a partial order.

Forcing and reducibilities. III. Forcing in fragments of set theory

1983 ◽  
Vol 48 (4) ◽  
pp. 1013-1034
Author(s):  
Piergiorgio Odifreddi
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
Second Order ◽  
Analytic Hierarchy ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Language ◽  
Order Arithmetic ◽  
Definition Of ◽  
Generic Set

We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:A0 = {X ⊆ ω: X is arithmetic},Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},Aλ = ⋃α<λAα (λ limit),RA = ⋃αAα.We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:L0 = ⊘,Lα+1 = {X: X is (first-order) definable over Lα},Lλ = ⋃α<λLα (λ limit),L = ⋃αLα.

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