Order Sentences Research Articles

Amalgamation properties and interpolation theorems for equational theories

DEFINITION 0.1. Let Fx, F2, F be subsets of a boolean algebra B. Then the (F1, F2, F)-Separation Principle, written Sep(F~, F2, F) for short, is the following: whenever aEF1, bEF2 and a ^ b = O , there is c~F with a<.c and cAb=O. The (F1, F2, F)-Interpolation Principle, written Int(Ft , F2, F) for short, is the following: whenever a~F1, b~F2 and a<.b there is ceF such that a<~c<~b. Clearly Int (F~, /'2, / ') is just Sep (F l, F 2 , F) where F 2 = { b:b ~F2}, so that the two notions are coextensive. Which one to use is a matter of taste, convenience and history. In practice B is often the Lindenbaum algebra of formulas reduced modulo some theory: then we replace each equivalence class [~p] by its representative formula q~, the ordering [~0]<,. [O] of the algebra becomes a valid implication ~0-,% and O denotes falsity. We also abuse notation mildly by using F~, etc., for classes of formulas. Several standard examples of separation principles occur in Recursion Theory, Descriptive Set Theory and Model Theory: however, they are usually symmetric in the sense that F1 = F2, and strict in the sense that F~ ca F2 = F. In contrast, we shall often want to dispense with these assumptions, and this is why we phrase the definition with three parameters as opposed to the usual one. The best known interpolation theorem in Model Theory is probably Craig's Theorem: this can be summarised as In t (E 2, U 2, F) where F is the set of all first order sentences in a language L and E z [respectively U 2] is the set of existential [universal] second-order sentences over L. We shall, however, be interested only in first order formulas, of low quantifier complexity, and so we begin by looking at an interpolation theorem for such formulas, Herbrand's Theorem.

The research of Jaśkowski on decidability theory of first order sentences (I)

The research of Jaśkowski on decidability theory of first order sentences (I)

The Relationship between Syntax Discrimination and Sentence Imitation in Children

BOHANNON, JoHN NEL, III. The Relationship between Syntax Discrimination and Sentence Imitation in Children. CHILD DEVELOPMENT, 1975, 46, 444-451. 54 first-, second-, and fifthgrade children (9 boys and 9 girls per grade) were asked to imitate sentences that varied in terms of word order distortion (normal vs. scrambled syntax) and sentence length (5, 7, 9, 11, 13, and 15 words). Several months later the same children were asked to discriminate between the normal and scrambled sentences. Children who discriminated word order (i.e., made less than 5 errors) imitated normal word order sentences significantly better than nondiscriminator children. The effect was powerful enough to render the normal syntax imitations of first-grade discriminator children indistinguishable from the normal syntax imitations of fifth-grade children. The data suggest an important qualitative change in the manner in which 6-7-year-old children process structured language.

幼児における文法の習得

The possibility was examined that grammatical structure was acquired by “contextual generalization”Experiment 1 was the Verbal Group using sentences accompanied with particles “i”,“ga”,“to.”Experiment 2 was the Verbal Group using sentences not accompanied with particles.Experiment 3 was the Non-Verbal Gtoup.Each Experimental Group had eight subgroups: 2 (normal order sentence vs. reversed order sentence)×2 (two words sentence vs. four words sentence)×2 (five year old infants vs. four year old infants).Ss were 288 kindergarten infants. Half of them were five years old, and the others were four years old. They were divided into 24 subgroups.They were required to arrange picture cards.Results:(1) The Verbal Group was better than the Non- Verbal Group.(2) The Verbal Group using sentences accompanied with particles was better than the Verbal Group using sentences not accompanied with particles.(3) Five year old infants could do reversed order sentence as well as normal order sentence. But four year old infants could not do so.(4) Two words sentence was easier than four words sentence.(5) Five year old infants were better than four year old infants.

The model-companion of a class of structures

If Σ is the class of all fields and Σ* is the class of all algebraically closed fields, then it is well known that Σ* is characterized by the following properties:(i) Σ* is the class of models of some first order theory K*.(ii) If m1m2 are in Σ* and m1 ⊆ m2 then m1 ≺ m2 (m1 is an elementary substructure of m2, i.e. any first order sentence true in m1 is true in m2).(iii) If m1 is in Σ then there is a structure m2 in Σ* such that m1 ⊆ m2.If Σ is some other class of models of a first order theory K and a subclass Σ* of Σ exists satisfying (i)–(iii) then Σ* is uniquely determined and K* (which is unique up to logical equivalence) is called the model-companion of K. This notion is a generalization of the fundamental notion of model-completion introduced and extensively studied by A. Robinson [6], When the model-companion exists it provides the basis for a satisfactory treatment of the notion of an algebraically closed model of K.Recently A. Robinson has developed a more general formulation of the notion of “algebraically closed” structures in Σ, which is applicable to any inductive elementary class Σ of structures (by elementary we always mean ECΔ). Condition (i) must be weakened to(i′) Σ* is closed under elementary substructure (i.e. if m1 is in Σ* and m2 ≺ m1 then m2 is in Σ*).

Almost combinatorial Skolem functions

Let be a version of class set theory admitting urelemente, and with AC (= axiom of choice) replaced by AC0 (= axiom of choice for sets of finite sets), ω = nonnegative integers, and Δ = Dedekind cardinals. Let be an arbitrarily quantified positive first order sentence in functors for + and ·. Let ƒ0, … , ƒκ - 1 be function variables and the universal sentence obtained from by replacing existential quantifiers by the ƒ1 as Skolem functions.

Almost combinatorial Skolem functions

Let be a version of class set theory admitting urelemente, and with AC (= axiom of choice) replaced by AC0 (= axiom of choice for sets of finite sets), ω = nonnegative integers, and Δ = Dedekind cardinals. Let be an arbitrarily quantified positive first order sentence in functors for + and ·. Let ƒ0, … , ƒκ - 1 be function variables and the universal sentence obtained from by replacing existential quantifiers by the ƒ1 as Skolem functions.

Undecidability of first order sentences in the theory of free groupoids

Undecidability of first order sentences in the theory of free groupoids