The diversity of quantifier prefixes

1973 ◽  
Vol 38 (1) ◽  
pp. 79-85 ◽  
Author(s):  
H. Jerome Keisler ◽  
Wilbur Walkoe
Keyword(s):  
Positive Integer ◽  
Analogous Result ◽  
Predicate Logic ◽  
Finite Sequence ◽  
Arithmetical Hierarchy ◽  
First Order ◽  
Free Variables ◽  
The Standard Model ◽  
Image Position ◽  
Identity Symbol

The Arithmetical Hierarchy Theorem of Kleene [1] states that in the complete theory of the standard model of arithmetic there is for each positive integer r a Σr0 formula which is not equivalent to any Πr0 formula, and a Πr0 formula which is not equivalent to any Πr0 formula. A Πr0 formula is a formula of the formwhere φ has only bounded quantifiers; Πr0 formulas are defined dually.The Linear Prefix Theorem in [3] is an analogous result for predicate logic. Consider the first order predicate logic L with identity symbol, countably many n-placed relation symbols for each n, and no constant or function symbols. A prefix is a finite sequenceof quantifier symbols ∃ and ∀, for example ∀∃∀∀∀∃. By a Q formula we mean a formula of L of the formwhere v1, …, vr are distinct variables and φ has no quantifiers. A sentence is a formula with no free variables. The Linear Prefix Theorem is as follows.Linear Prefix Theorem. Let Q and q be two different prefixes of the same length r. Then there is a Q sentence which is not logically equivalent to any q sentence.Moreover, for each s there is a Q formula with s free variables which is not logically equivalent to any q formula with s free variables.For example, there is an ∀∃∀∀∀∃ sentence which is not logically equivalent to any ∀∃∃∀∀∃ sentence, and vice versa. Recall that in arithmetic two consecutive ∃'s or ∀'s can be collapsed; for instance all ∀∃∀∀∀∃ and ∀∃∃∀∀∃ formulas are logically equivalent to Π40 formulas. But the Linear Prefix Theorem shows that in predicate logic the number of quantifiers in each block, as well as the number of blocks, counts.

An improved prenex normal form1

1962 ◽  
Vol 27 (3) ◽  
pp. 317-326 ◽  
Author(s):  
C. C. Chang ◽  
H. Jerome Keisler
Keyword(s):  
Positive Integer ◽  
Predicate Logic ◽  
Atomic Formula ◽  
First Order ◽  
Empty String ◽  
Image Position ◽  
First Order Predicate Logic

Let ℒ be the set of all formulas of a given first order predicate logic (with or without identity). For each positive integer n, let ℒn be the set of all formulas φ in ℒ logically equivalent to a formula of the form where Q is a (possibly empty) string of quantifiers, m is a positive integer, and each αij is either an atomic formula or the negation of an atomic formula.

On the formalisation of indirect discourse

1958 ◽  
Vol 23 (4) ◽  
pp. 417-419 ◽  
Author(s):  
R. L. Goodstein
Keyword(s):  
Predicate Logic ◽  
Logical Truth ◽  
Predicate Calculus ◽  
Inference Rules ◽  
First Order ◽  
Indirect Discourse ◽  
Image Position ◽  
Order Predicate Calculus

Mr. L. J. Cohen's interesting example of a logical truth of indirect discourse appears to be capable of a simple formalisation and proof in a variant of first order predicate calculus. His example has the form:If A says that anything which B says is false, and B says that something which A says is true, then something which A says is false and something which B says is true.Let ‘A says x’ be formalised by ‘A(x)’ and let assertions of truth and falsehood be formalised as in the following table.We treat both variables x and predicates A (x) as sentences and add to the familiar axioms and inference rules of predicate logic a rule permitting the inference of A(p) from (x)A(x), where p is a closed sentence.We have to prove that from

A theorem on deducibility for second-order functions

1939 ◽  
Vol 4 (2) ◽  
pp. 77-79 ◽  
Author(s):  
C. H. Langford
Keyword(s):  
Second Order ◽  
Order Function ◽  
Usual Definition ◽  
First Order ◽  
Free Variables ◽  
Usual Convention ◽  
Definition Of ◽  
Image Position

It is known that the usual definition of a dense series without extreme elements is complete with respect to first-order functions, in the sense that any first-order function on the base of a set of postulates defining such a series either is implied by the postulates or is inconsistent with them. It is here understood, in accordance with the usual convention, that when we speak of a function on the base , the function shall be such as to place restrictions only upon elements belonging to the class determined by f; or, more exactly, every variable with a universal prefix shall occur under the hypothesis that its values satisfy f, while every variable with an existential prefix shall have this condition categorically imposed upon it.Consider a set of postulates defining a dense series without extreme elements, and add to this set the condition of Dedekind section, to be formulated as follows. Let the conjunction of the three functions,be written H(ϕ), where the free variables f and g, being parameters throughout, are suppressed. This is the hypothesis of Dedekind's condition, and the conclusion iswhich may be written C(ϕ).

On an Ackermann-type set theory

1973 ◽  
Vol 38 (3) ◽  
pp. 410-412
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Large Cardinal ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Individual Constant ◽  
Image Position ◽  
Order Predicate Calculus ◽  
Universal Closure

Ackermann's set theory A* is usually formulated in the first order predicate calculus with identity, ∈ for membership and V, an individual constant, for the class of all sets. We use small Greek letters to represent formulae which do not contain V and large Greek letters to represent any formulae. The axioms of A* are the universal closures ofwhere all free variables are shown in A4 and z does not occur in the Θ of A2.A+ is a generalisation of A* which Reinhardt introduced in [3] as an attempt to provide an elaboration of Ackermann's idea of “sharply delimited” collections. The language of A+ is that of A*'s augmented by a new constant V′, and its axioms are A1–A3, A5, V ⊆ V′ and the universal closure ofwhere all free variables are shown.Using a schema of indescribability, Reinhardt states in [3] that if ZF + ‘there exists a measurable cardinal’ is consistent then so is A+, and using [4] this result can be improved to a weaker large cardinal axiom. It seemed plausible that A+ was stronger than ZF, but our main result, which is contained in Theorem 5, shows that if ZF is consistent then so is A+, giving an improvement on the above results.

Extensional interpretations of modal logics

1966 ◽  
Vol 31 (1) ◽  
pp. 23-45 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Predicate Logic ◽  
Modal Logics ◽  
Predicate Calculus ◽  
First Order ◽  
Image Position ◽  
Order Predicate Calculus ◽  
First Order Predicate Logic

By ΡL we shall mean the first order predicate logic based on S4. More explicitly: Let Ρ0 stand for the first order predicate calculus. The formalisation of Ρ0 used in the present paper will be given later. ΡL is obtained from Ρ0 by adding the rules the propositional constant □ and

The foundations of Suslin logic

1975 ◽  
Vol 40 (4) ◽  
pp. 567-575 ◽  
Author(s):  
Erik Ellentuck
Keyword(s):  
Finite Sequence ◽  
Order Logic ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Finite Sequences ◽  
Image Position

Let L be a first order logic and the infinitary logic (as described in [K, p. 6] over L. Suslin logic is obtained from by adjoining new propositional operators and . Let f range over elements of ωω and n range over elements of ω. Seq is the set of all finite sequences of elements of ω. If θ: Seq → is a mapping into formulas of then and are formulas of LA. If is a structure in which we can interpret and h is an -assignment then we extend the notion of satisfaction from to by definingwhere f ∣ n is the finite sequence consisting of the first n values of f. We assume that has ω symbols for relations, functions, constants, and ω1 variables. θ is valid if θ ⊧ [h] for every h and is valid if -valid for every . We address ourselves to the problem of finding syntactical rules (or nearly so) which characterize validity .

Natural models and Ackermann-type set theories

1975 ◽  
Vol 40 (2) ◽  
pp. 151-158 ◽  
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Individual Constant ◽  
Image Position ◽  
Order Predicate Calculus

Our results concern the natural models of Ackermann-type set theories, but they can also be viewed as results about the definability of ordinals in certain sets.Ackermann's set theory A was introduced in [1] and it is now formulated in the first order predicate calculus with identity, using ∈ for membership and an individual constant V for the class of all sets. We use the letters ϕ, χ, θ, and χ to stand for formulae which do not contain V and capital Greek letters to stand for any formulae. Then, the axioms of A* are the universal closures ofwhere all the free variables are shown in A4 and z does not occur in the Θ of A2. A is the theory A* − A5.Most of our notation is standard (for instance, α, β, γ, δ, κ, λ, ξ are variables ranging over ordinals) and, in general, we follow the notation of [7]. When x ⊆ Rα, we use Df(Rα, x) for the set of those elements of Rα which are definable in 〈Rα, ∈〉, using a first order ∈-formula and parameters from x.We refer the reader to [7] for an outline of the results which are known about A, but we shall summarise those facts which are frequently used in this paper.

scholarly journals Predicate functors revisited

1981 ◽  
Vol 46 (3) ◽  
pp. 649-652 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Set Theory ◽  
Predicate Logic ◽  
Combinatory Logic ◽  
Homogeneous Case ◽  
Quantification Theory ◽  
Present Scheme ◽  
First Order ◽  
Image Position ◽  
First Order Predicate Logic

Quantification theory, or first-order predicate logic, can be formulated in terms purely of predicate letters and a few predicate functors which attach to predicates to form further predicates. Apart from the predicate letters, which are schematic, there are no variables. On this score the plan is reminiscent of the combinatory logic of Schönfinkel and Curry. Theirs, however, had the whole of higher set theory as its domain; the present scheme stays within the bounds of predicate logic.In 1960 I published an apparatus to this effect, and an improved version in 1971. In both versions I assumed two inversion functors, major and minor; also a cropping functor and the obvious complement functor. The effects of these functors, when applied to an n-place predicate, are as follows:The variables here are explanatory only and no part of the final notation. Ultimately the predicate letters need exponents showing the number of places, but I omit them in these pages.A further functor-to continue now with the 1971 version-was padding:Finally there was a zero-place predicate functor, which is to say simply a constant predicate, namely the predicate ‘I’ of identity, and there was a two-place predicate functor ‘∩’ of intersection. The intersection ‘F ∩ G’ received a generalized interpretation, allowing ‘F’ and ‘G’ to be predicates with unlike numbers of places. However, Steven T. Kuhn has lately shown me that the generalization is unnecessary and reducible to the homogeneous case.

The nonconstructive content of sentences of arithmetic

1978 ◽  
Vol 43 (3) ◽  
pp. 497-501
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Natural Number ◽  
Recursive Function ◽  
Function Theory ◽  
First Order ◽  
The Standard Model ◽  
Image Position ◽  
The Relationship ◽  
Excluded Middle ◽  
Degrees Of Unsolvability ◽  
Intuitionistic Arithmetic

This note is concerned with the old topic, initiated by Kleene, of the connections between recursive function theory and provability in intuitionistic arithmetic. More specifically, we are interested in the relationship between the hierarchy of degrees of unsolvability and the interdeducibility of cases of excluded middle. The work described below was motivated by a counterexample, to be given presently, which shows that that relationship is more complicated than one might suppose.Let HA be first-order intuitionistic arithmetic. Let the symbol ⊢ mean derivability in HA. For each natural number n, let n¯ be the corresponding numeral. Let Ω be the standard model of arithmetic. Say that a sentence ϕ is true iff Ω⊨ ϕ. Now suppose ϕ(x) and Ψ(x) are formulas with only the variable x free. SupposeThen it is natural to conjecture that {n∣Ω⊨Ψ(n¯)} is recursive in {n∣Ω⊨ϕ(n¯)}.However, this conjecture is false. Consider the formula is a formalization of Kleene's T-predicate.

scholarly journals A NEW DP-MINIMAL EXPANSION OF THE INTEGERS

2019 ◽  
Vol 84 (02) ◽  
pp. 632-663 ◽  
Author(s):  
ERAN ALOUF ◽  
CHRISTIAN D’ELBÉE
Keyword(s):  
Analogous Result ◽  
Abelian Groups ◽  
Quantifier Elimination ◽  
Alternative Proof ◽  
Order Structure ◽  
First Order ◽  
Adic Valuation ◽  
Image Position ◽  
Natural Expansion

AbstractWe consider the structure $({\Bbb Z}, + ,0,|_{p_1 } , \ldots ,|_{p_n } )$, where $x|_p y$ means $v_p \left( x \right) \leqslant v_p \left( y \right)$ and vp is the p-adic valuation. We prove that this structure has quantifier elimination in a natural expansion of the language of abelian groups, and that it has dp-rank n. In addition, we prove that a first order structure with universe ${\Bbb Z}$ which is an expansion of $({\Bbb Z}, + ,0)$ and a reduct of $({\Bbb Z}, + ,0,|_p )$ must be interdefinable with one of them. We also give an alternative proof for Conant’s analogous result about $({\Bbb Z}, + ,0, < )$.

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