bluuebunny/crossref_metadata_2025 huggingface dataset:
Deliberative Democracy and Authority
Authors: Warren Mark E.
Published: May 2006

The topic of authority only rarely figures into theories of deliberative democracy, no doubt owing to the widely held view that authority is inherently undemocratic. But deliberative democrats need a concept of authoritative decision making, not least because the scale and complexity of contemporary societies radically limit the numbers of decisions that can be made by deliberatively democratic means. I argue for an inherently democratic conception of authority, in large part by examining and rejecting the view—held by radical democrats, conservatives, and most liberals—that authority involves a surrender of judgment by those subject to authority. In contrast, I develop the view that authority, particularly in posttraditional contexts, involves a limited suspension of judgment enabled by a context of democratic challenge and public accountability. An important point is that democratic authority supports robust deliberative decision making by enabling individuals to allocate their time, energy, and knowledge to the issues most significant to them.
Institutionalizing the Public Interest: The Defense of Deadlock and Beyond
Authors: Goodin Robert E.
Published: May 2006

Many bemoan divided government and the consequent deadlock of democracy. The logic ofThe Federalistarguably defends it, in ways heretofore unappreciated, by appeal to a least-common-denominator definition of the “public interest.” That quasi-Federalistlogic is explored, and alternative political structures are assessed against the public interest criterion to which it appeals. Another and more defensible notion of the public interest is introduced, and its very different political styles, institutions, and policies are adduced.
What To Do (and Not to Do) with Time-Series Cross-Section Data
Authors: Beck Nathaniel , Katz Jonathan N.
Published: May 2006

We examine some issues in the estimation of time-series cross-section models, calling into question the conclusions of many published studies, particularly in the field of comparative political economy. We show that the generalized least squares approach of Parks produces standard errors that lead to extreme overconfidence, often underestimating variability by 50% or more. We also provide an alternative estimator of the standard errors that is correct when the error structures show complications found in this type of model. Monte Carlo analysis shows that these “panel-corrected standard errors” perform well. The utility of our approach is demonstrated via a reanalysis of one “social democratic corporatist” model.
Issue Fluidity on the U.S. Supreme Court
Authors: McGuire Kevin T. , Palmer Barbara
Published: May 2006

In making decisions on the merits, the members of the U.S. Supreme Court are often willing to provide authoritative answers to questions that have not been asked and to disregard issues that the parties have presented. What accounts for these forms of issue fluidity? Analyzing data from the 1988 term of the Court, we find that issue transformation is quite common, occurring in roughly half of the cases on the plenary agenda. We propose models of both issue discovery and issue suppression that, while successful in explaining how the justices select issues, suggest that these two forms of fluidity result from largely different influences.
Avarice and Ambition in Congress: Representatives' Decisions to Run or Retire from the U.S. House
Authors: Hall Richard L. , Van Houweling Robert P.
Published: May 2006

Over two centuries ago, Adam Smith wrote of two passions that motivate those seeking public distinction: avarice and ambition. By investing these categories with appropriately concrete meaning, we develop a model of House members' career decisions. Like other individuals contemplating retirement options, politicians act with an eye to their financial interests, but not all financial interests are alike. The financial factor that matters most involves perennial considerations of post-retirement pension benefits, not fleeting opportunities to exploit ethically questionable sources of outside income. Second, we embed in the model a theory of intra-institutional ambition. Members impute value both to leadership positions they expect to retain and positions they expect to obtain. Majority members well-positioned to exert future legislative leverage are less likely to retire. Finally, several sources of electoral insecurity increase retirement probability. In the main, members may be reelection-seekers but will not pay any price to seek something they may not find.
Note on a recent set of postulates for the calculus of propositions
Authors: Huntington K. V.
Published: May 2006

In my American Academy paper of 1937 I gave a mathematical clarification and extension of C. I. Lewis's theory of strict implication using as the base (K, T, ×, ′, ≣). In that paper the element a⊰b (read “a hook b”) was defined by (a⊰b) = (ab′=cc′). The main purpose of the present paper is to show that this definition may be replaced by the simpler definition (a⊰b) = (ab≣a), provided Postulate 15 is replaced by two simpler postulates, 15a and 15b (see below). The paper may also serve to throw further light on the essential difference between Lewis's theory of “strict” implication and the more familiar theory of “material” implication. This paper and the earlier one, when read together, are entirely self-contained, all proofs (including proofs of independence) being given in full and no reference to earlier literature being required.

We begin by reviewing briefly the rôle which logical deduction plays in the writing of any scientific book.

Suppose you sit down to write. What do you actually do? In the last analysis, your activity as an author is a process of voluntary selection, restricted by certain inescapable rules. From an indefinite number of possible propositions which may occur to your imagination, you voluntarily select certain propositions which you accept as being worthy of being written down, one after another, in your book. Every proposition which occurs to you is either accepted, or rejected, or laid aside for further consideration.
A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis
Authors: Bernays Paul
Published: May 2006

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable.

We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.
Concatenation as basis for a complete system of arithmetic
Authors: Löb M. H.
Published: May 2006

In [3] Myhill has constructed a complete system K which allows in it the development of a large and important section of classical mathematics. Completeness is achieved essentially by sacrificing universal quantification and introducing instead the proper ancestral as a primitive idea.

In the following we are presenting a system K0 which will be shown to be equivalent to K (i.e. the primitive operators of both systems are mutually definable in terms of one another). K0 is also complete and covers the same ground as K. K0, however, differs from K by the introduction of the limited universal quantifier instead of the proper ancestral and of concatenation instead of the ordered-pair function as primitive operators. By a further reduction K0 will be shown to be equivalent to the system K1 not containing the abstraction-operator and the class-membership relation.
On axiomatizability within a system
Authors: Craig William
Published: May 2006

Let C be the closure of a recursively enumerable set B under some relation R. Suppose there is a primitive recursive relation Q, such that Q is a symmetric subrelation of R (i.e. if Q(m, n), then Q(n, m) and R(m, n)), and such that, for each m ϵ B, Q(m, n) for infinitely many n. Then there exists a primitive recursive set A, such that C is the closure under R of A. For proof, note that , where f is a primitive recursive function which enumerates B, has the required properties. For each m ϵ B, there is an n ϵ A, such that Q(m, n) and hence Q(n, m); therefore the closure of A under Q, and hence that under R, includes B. Conversely, since Q is a subrelation of R, A is included in C. Finally, that A is primitive recursive follows from [2] p. 180.

This observation can be applied to many formal systems S, by letting R correspond to the relation of deducibility in S, so that R(m, n) if and only if m is the Gödel number of a formula of S, or of a sequence of formulas, from which, together with axioms of S, a formula with the Gödel number n can be obtained by applications of rules of inference of S.
Toward an inscriptional semantics
Authors: Martin R. M. , Woodger J. H.
Published: May 2006

Classical syntax and semantics make fundamental use of variables ranging over classes, relations, etc. In syntax, one speaks exclusively of such objects as symbols, formulae, terms, in short of what may be called (following Carnap) sign-designs. A sign-design is a class of similar concrete marks or sign-events or inscriptions or sequences of typographical characters. In semantics, in addition to sign-designs, one also speaks of the objects denoted or designated by the sign-designs called constants or of the objects which are values for the sign-designs called variables.

Interest in the possibility of a syntax and semantics based upon sign-events or inscriptions rather than upon sign-designs appears to have originated with Lesniewski and Tarski. Not until recently, however, in a paper by Goodman and Quine, has a purely inscriptional syntax been explicitly formulated. The discussion in that paper is confined to the syntax of one object language, but the methods developed are applicable to other systems as well and should suffice for many of the purposes of syntax.

In the present paper we make an attempt to construct a purely inscriptional semantics, in which, as with Goodman and Quine, sign-designs or classes of similar inscriptions in no way figure as values for variables. In rejecting such entities it might appear that the resulting semantics would be drastically curtailed. That the methods used here are of sufficient power to be of interest, however, is shown by constructing a definition of a semantical concept of truth for a given sample elementary object language L. The definition of a predicate for truth in L is given, and the predicate is shown to be adequate in essentially the sense due to Leśniewski, Kotarbiński, and Tarski.
On undecidable statements in enlarged systems of logic and the concept of truth
Authors: Tarski Alfred
Published: May 2006

It is my intention in this paper to add somewhat to the observations already made in my earlier publications on the existence of undecidable statements in systems of logic possessing rules of inference of a “non-finitary” (“non-constructive”) character (§§1–4).

I also wish to emphasize once more the part played by the concept of truth in relation to problems of this nature (§§5–8).

At the end of this paper I shall give a result which was not touched upon in my earlier publications. It seems to be of interest for the reason (among others) that it is an example of a result obtained by a fruitful combination of the method of constructing undecidable statements (due to K. Gödel) with the results obtained in the theory of truth.

1. Let us consider a formalized logical system L. Without giving a detailed description of the system we shall assume that it possesses the usual “finitary” (“constructive”) rules of inference, such as the rule of substitution and the rule of detachment (modus ponens), and that among the laws of the system are included all the postulates of the calculus of statements, and finally that the laws of the system suffice for the construction of the arithmetic of natural numbers. Moreover, the system L may be based upon the theory of types and so be the result of some formalization of Principia mathematica. It may also be a system which is independent of any theory of types and resembles Zermelo's set theory.
Note on a conjecture of Skolem
Authors: Post Emil L.
Published: May 2006

In his excellent review of four notes of Skolem on recursive functions of natural numbers Bernays states: “The question whether every relation y = f(x1,…, xn) with a recursive function ƒ is primitive recursive remains undecided.” Actually, the question is easily answered in the negative by a form of the familiar diagonal argument.

We start with the ternary recursive relation R, referred to in the review, such that R(x, y, 0), R(x, y, 1), … is an enumeration of all binary primitive recursive relations.
A proof procedure for quantification theory
Authors: Quine W. V.
Published: May 2006

The purpose of this paper is to present and justify a simple proof procedure for quantification theory. The procedure will take the form of a method for proving a quantificational schema to be inconsistent, i.e., satisfiable in no non-empty universe. But it serves equally for proving validity, since we can show a schema valid by showing its negation inconsistent.

Method A, as I shall call it, will appear first, followed by a more practical adaptation which I shall call B. The soundness and completeness of A will be established, and the equivalence of A and B. Method A, as will appear, is not new.

The reader need be conversant with little more than the fairly conventional use (as in [8]) of such terms as ‘quantificational schema’, ‘interpretation’, ‘valid’, ‘consistent’, ‘prenex’, and my notation (as in [7]) of quasi-quotation.
Completeness in the theory of types
Authors: Henkin Leon
Published: May 2006

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.

For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem. This follows from results of Gödel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added.

By a valid formula of the second order calculus is meant one which expresses a true proposition whenever the individual variables are interpreted as ranging over an (arbitrary) domain of elements while the functional variables of degree n range over all sets of ordered n-tuples of individuals. Under this definition of validity, we must conclude from Gödel's results that the calculus is essentially incomplete.

It happens, however, that there is a wider class of models which furnish an interpretation for the symbolism of the calculus consistent with the usual axioms and formal rules of inference. Roughly, these models consist of an arbitrary domain of individuals, as before, but now an arbitrary class of sets of ordered n-tuples of individuals as the range for functional variables of degree n. If we redefine the notion of valid formula to mean one which expresses a true proposition with respect to every one of these models, we can then prove that the usual axiom system for the second order calculus is complete: a formula is valid if and only if it is a formal theorem.
The elimination of contextually defined predicates in a modal system
Authors: Marcus Ruth Barcan
Published: May 2006

In a recent note Bergmann states that “nonextensional languages may contain sentences from which contextually denned predicates are not eliminable.” A restatement of his argument is as follows. Suppose L is a functional calculus of fourth order. Included among the definitions occurring in L is

Let ϕ occur in A (where A represents a well-formed formula of L) without an argument. The elimination of ϕ from A in accordance with (1) requires

The proof of (2) depends on the principle of extensionality

Hence Bergmann's conclusion “no such general elimination rule can be constructed for nonextensional languages.”
The M-valued calculus of non-contradiction
Authors: Rose Alan
Published: May 2006

The 2-valued calculus of non-contradiction of Dexter has been extended to 3-valued logic. The methods used were, however, too complicated to be capable of generalisation to m-valued logics. The object of the present paper is to give an alternative method of generalising Dexter's work to m-valued logics with one designated truth-value. The rule of procedure is generalised in the same way as before, but the deductive completeness of the system is proved by applying results of Rosser and Turquette. The system has an infinite set of primitive functions, written n(P1, P2, …, Pr) (r = 1,2, …). With the notation of Post, n(P1, P2, …, Pr) has the same truth-value as ~(P1 & P2 & … & Pr). Thus n(P) is Post's primitive ~P, and we can define & by

We use n2(P1, P2, …, Pr) as an abbreviation for n(n(P1, P2, …, Pr)); similarly for higher powers of n. But if we set up the 1-1 correspondence of truth-values i ↔ m−i+1, then & corresponds to ∨ and ~m−1 corresponds to ~. Hence the functional completeness of our system follows from a theorem of Post.

We define the functions N(P), N(P, Q) by

Thus the truth-value of N(P) is undesignated if and only if the truth-value of P is designated, and the truth-value of N(P, Q) is undesignated if and only if the truth-values of P and Q are both designated.
Proof of a conjecture of Goodman
Authors: Fine N. J.
Published: May 2006

In Chapter X of The Structure of appearance the concept of linear array is given in terms of a two-place predicate “M” for matching. Also given in terms of “M” is the two-place predicate “B” for besideness. It is unnecessary to discuss here the formal definitions, since Goodman has proved that they are adequate to describe our presystematic ideas of linear array and besideness. Thus, we shall be concerned with a finite collection of elements with an assigned order satisfying:

This order is not uniquely defined by the relation M; either of the two end points may be taken as the ‘first.’ We shall represent the order in diagrams by the convention that x is to the left of y if x < y. For x ≠ y, we shall define S(x, y), the distance between x and y, as one more than the number of elements strictly between them, in the order <; also, S(x, x) = 0. Thus S(x, y) = 1 if and only if x is beside y. The order is connected with the reflexive and symmetric matching relation M by the conditions

(2) Bx, y ⊃ Mx, y,

(3) Mx, y▪x ≤; u ≤ ν ≤ y▪ ⊃ ▪Mu, ν.

Condition (3) is what Goodman calls the weak mapping rule. All arrays considered here must satisfy (3). To avoid trivial cases, we shall assume that not all elements match. There is a strong mapping rule,

(4) Mx, y▪ ∼Mu, ν▪ ▪ ▪S(x, y) < S(u, ν),

which it is desirable, but not necessary, for arrays to satisfy. An array which does satisfy (4) is called regular.
On notation for ordinal numbers
Authors: Kleene S. C.
Published: May 2006

Consider a system of formal notations for ordinal numbers in the first and second number classes, with the following properties. Given a notation for an ordinal, it can be decided effectively whether the ordinal is zero, or the successor of an ordinal, or the limit of an increasing sequence of ordinals. In the second case, a notation for the preceding ordinal can be determined effectively. In the third case, notations for the ordinals of an increasing sequence of type ω with the given ordinal as limit can be determined effectively.

Are there systems of this sort which extend farthest into the second number class? When the conditions for the systems have been made precise, the question will be answered in the affirmative. There is an ordinal ω1in the second number class such that there are systems of notations of the sort described which extend to all ordinals less than ω1, but none in which ω1itself is assigned a notation.

1. An effective or constructive operation on the objects of an enumerable class is one for which a fixed set of instructions can be chosen such that, for each of the infinitely many objects (orn-tuples of objects), the operation can be completed by a finite process in accordance with the instructions. This notion is made exact by specifying the nature of the process and set of instructions. It appears possible to do so without loss of generality.
A homogeneous system for formal logic
Authors: Martin R. M.
Published: May 2006

Two more or less standard methods exist for the systematic, logical construction of classical mathematics, the so-called theory of types, due in the main to Russell, and the Zermelo axiomatic set theory. In systems based upon either of these, the connective of membership, “ε”, plays a fundamental role. Usually although not always it figures as a primitive or undefined symbol.

Following the familiar simplification of Russell's theory, let us mean by alogical typein the strict sense any one of the following: (i) the totality consisting exclusively of individuals, (ii) the totality consisting exclusively of classes whose members are exclusively individuals, (iii) the totality consisting exclusively of classes whose members are exclusively classes whose members in turn are exclusively individuals, and so on. Any entity from (ii) is said to beone type higherthan any entity from (i), any entity from (iii), one type higher than any entity from (ii), and so on. In systems based upon this simplified theory of types, the only significant atomic formulae involving “ε” are those asserting the membership of an entity in an entity one type higher. Thus any expression of the form “(x∈y)” is meaningless except where “y” denotes an entity of just one type higher than the type of the entity denoted by “x” It is by means of general type restrictions of this kind that the Russell and other paradoxes are avoided.
Independence of the primitive symbols of Lewis's calculi of propositions
Authors: Alban M. J.
Published: May 2006

In this paper it will be shown that the three primitive symbols used by Lewis are independent in each one of his systems S1–S6.

Lemma I. “·” is not definable in terms of “∼” and “◊” in any one of the systems S1–S6.

Proof. This is obvious, since no binary operation can be defined by unary operations alone.

Lemma II. “∼” is not definable in terms of “˙” and “◊” in any one of the systems S1–S5.
Recursive functions in basic logic
Authors: Fitch Frederic B.
Published: May 2006

1.1 The system K* of basic logic, as presented in a previous paper, will be shown to be formalizable in an alternative way according to which the rule [E],

is replaced by the rule [F],

1.2. General recursive functions will be shown to be definable in K* in a way that retains functional notation, so that the equation,

will be formalized in K* by the formula,

where ‘f’, ‘p1’, … ‘pn’ respectively denote φ, x1, …, xn, and where ‘≐’ plays the role of numerical equality. Partial recursive functions may be handled in a similar way. The rule [E] is not required for dealing with primitive recursive functions by this method.

1.3. An operator ‘G’ will be defined such that ‘[Ga ≐ p]’ is a theorem of K* if and only if ‘p’ denotes the Gödel number of ‘a’.

1.4. In reformulating K* we assume ‘o0’, ‘o1,’ ‘o2’, …, have been so chosen that we can determine effectively whether or not a given U-expression is the mth member of the above series. The revised rules for K* are then as follows. (Double-arrow forms of these rules are derivable, except in the case of rule [V].)
On the calculus of relations
Authors: Tarski Alfred
Published: May 2006

The logical theory which is called thecalculus of (binary) relations, and which will constitute the subject of this paper, has had a strange and rather capricious line of historical development. Although some scattered remarks regarding the concept of relations are to be found already in the writings of medieval logicians, it is only within the last hundred years that this topic has become the subject of systematic investigation. The first beginnings of the contemporary theory of relations are to be found in the writings of A. De Morgan, who carried out extensive investigations in this domain in the fifties of the Nineteenth Century. De Morgan clearly realized the inadequacy of traditional logic for the expression and justification, not merely of the more intricate arguments of mathematics and the sciences, but even of simple arguments occurring in every-day life; witness his famous aphorism, that all the logic of Aristotle does not permit us, from the fact that a horse is an animal, to conclude that the head of a horse is the head of an animal. In his effort to break the bonds of traditional logic and to expand the limits of logical inquiry, he directed his attention to the general concept of relations and fully recognized its significance. Nevertheless, De Morgan cannot be regarded as the creator of the modern theory of relations, since he did not possess an adequate apparatus for treating the subject in which he was interested, and was apparently unable to create such an apparatus. His investigations on relations show a lack of clarity and rigor which perhaps accounts for the neglect into which they fell in the following years.
A generalization of the antilogism
Authors: Dotterer Ray H.
Published: May 2006

The Ladd-Franklin “antilogism” provides a very convenient procedure for testing syllogistic arguments, especially when they are expressed algebraically. It is well known, however, that the antilogism test indicates the invalidity of certain moods which the traditional logic regards as valid. These are the moods in which a particular conclusion is inferred from two universal premises. It is the aim of this paper to show that the antilogism can be generalized in such a way as to cover these moods, and also to be applicable to certain forms of “immediate inference” and “sorites,” as these designations have been traditionally employed.

We shall find that when generalized in this fashion there are two principal kinds of antilogisms: (1) those which include an inequation of the form ab≠0, and (2) those which include an inequation of the form a≠0.

Definition. An antilogism is a group of propositions so constructed that the contradictory of any one of them is implied by the logical product of the others.
A generalization of the concept of ω-consistency
Authors: Henkin Leon
Published: May 2006

In this paper we consider certain formal properties of deductive systems which, in special cases, reduce to the property of ω-consistency; and we then seek to understand the significance of these properties by relating them to the use of models in providing interpretations of the deductive systems.

The notion of ω-consistency arises in connection with deductive systems of arithmetic. For definiteness, let us suppose that the system is a functional calculus whose domain of individuals is construed as the set of natural numbers, and that the system possesses individual constants ν0, ν1, ν2, … such that νi functions as a name for the number i. Such a system is called ω-consistent, if there is no well-formed formula A(x) (in which x is the only free variable) such that A(ν0), A(ν1), A(ν2), … and ∼(x)A(x) are all formal theorems of the system, where A(νi) is the formula resulting from A(x) by substituting the constant νi for each free occurrence of the individual variable x.

Now consider an arbitrary applied functional calculus F, and let Γ be any non-empty set of its individual constants. In imitation of the definition of ω-consistency, we may say that the system F is Γ-consistent, if it contains no formula A(x) (in which x is the only free variable) such that ⊦ A (α) for every constant α in Γ, and also ⊦ ∼(x)A(x) (where an occurrence of “⊦” indicates that the formula which it precedes is a formal theorem). We easily see that the condition of Γ-consistency is equivalent to the condition that the system F contain no formula B(x) such that ⊦ ∼ B(α) for each α in Γ, and also ⊦ (∃x)B(x).
Modalities in the Survey system of strict implication
Authors: Parry William Tuthill
Published: May 2006

Professor C. I. Lewis, in Lewis and Langford's Symbolic logic, designates the system (S2) determined by the postulates used in Chapter VI—namely, 11.1–7 (B1–7) and

as the system of strict implication. For certain reasons, he prefers it to either the earlier system (S3) determined by the stronger set of postulates of his Survey of symbolic logic, as emended, namely, A1–7 and

or the system (S1) determined by the weaker set of postulates B1–7 or A1–7.

But Lewis and others, following O. Becker, have also given consideration to systems which contain some additional principle effecting the reduction of complex modalities to simpler ones. Notable are the system (S4) determined by B1–7 plus

which includes (is stronger than) S3; and the system (S5) determined by B1–7 plus

which includes S4.

It seems worth while to investigate further the system of the Survey (emended), S3, which is intermediate between S2 and S4. This is the purpose of the present paper. We first prove some additional theorems in S2 and S3. These enable us to reduce all the complex modalities in S3 to a finite number, viz. 42; and it is shown that no further reduction is possible. Finally, several systems which include S3 are considered.
A new approach to semantics – Part II
Authors: Kemeny John G.
Published: May 2006

I will give precise versions of the foregoing verbal definitions, by formalizing them within a formal metalanguage.

In order to avoid the necessity of formulating a meta-language, I will make use of VI 112, (i.e., Gödel's The consistency of the continuum hypothesis). Actually, the system there developed is unnecessarily strong for the limited purpose of this illustration. It will suffice to choose for the system of VI 112 without axioms C4 and E (axioms of replacement and choice); this system will be the logical basis of , and to this I will add some set-constants and some meaning postulates. The logical basis is roughly as strong as Zermelo's system with an axiom of infinity.

Within I will set up analogues of the previous definitions, applicable to a wide variety of object-languages. For the sake of concreteness it will be convenient to think of as having Zermelo set theory, without the axiom of infinity, as its logical basis – but it must be kept in mind that this is not an essential feature of the forthcoming definitions.

Our is certainly strong enough to have all primitive recursive functions calculable within it. This supplies us with the needed tool for talking about , within , by introducing a number of primitive recursive functions, properties, and relations describing the structure of . The exact definitions of these will depend on the exact form of , but for the forthcoming definitions it will suffice to know that these functions, properties, and relations are representable in for the given . This makes the resulting definitions most useful, since they can be applied to any of the type described.
Arithmetic with creative definitions by induction
Authors: Myhill John
Published: May 2006

The purpose of this paper is to present a system of arithmetic stronger than those usually employed, and to prove some syntactical theorems concerning it.

We presuppose the lower functional calculus with identity and functions, and we start with three of Peano's axioms.

The other two (0 ϵ N and x ϵ N .⊃. x′ ϵ N) we do not need since our variables are anyhow restricted to natural numbers. Sometimes in the interest of a uniform notation for functions, we write Sx instead of x′.

Next we have two axioms for μ (the smallest number such that) as follows.

A third axiom for μ must wait until we have defined ≤.

Now we introduce the central feature of the system, the following rule of definition.

RD. Let Φ be a previously unused symbol. Then we can introduce it by a pair of definitions of the following form (n ≥ 0),

where F(x1, …, xn) is a wff in which no symbol occurs which was not previously defined (in particular, not Φ), and in which no free variables occur other than x1, …, xn (and possibly not all of these), and G(x1, …, xn, y) is a wff in which no free variables occur other than x1, …, xn, y (and possibly not all of these), and in which no symbol occurs which was not previously defined, except that Φ may occur but only if its last argument is y.
A note on nominalism and recursive functions
Authors: Martin R. M.
Published: May 2006

The purpose of this note is (i) to point out an important similarity between the nominalistic system discussed by Quine in his recent paper On universals and the system of logic (the system н) developed by the author in A homogeneous system for formal logic, (ii) to offer certain corrections to the latter, and (iii) to show that that system (н) is adequate for the general theory of ancestrale and for the definition of any general recursive function of natural numbers.

Nominalism as a thesis in the philosophy of science, according to Quine, is the view that it is possible to construct a language adequate for the purposes of science, which in no wise admits classes, properties, relations, or other abstract objects as values for variables.
Extensions of some theorems of Gödel and Church
Authors: Rosser Barkley
Published: May 2006

We shall say that a logic is “simply consistent” if there is no formula A such that both A and ∼ A are provable. “ω-consistent” will be used in the sense of Gödel. “General recursive” and “primitive recursive” will be used in the sense of Kleene, so that what Gödel calls “rekursiv” will be called “primitive recursive.” By an “Entscheidungsverfahren” will be meant a general recursive function ϕ(n) such that, if n is the Gödel number of a provable formula, ϕ(n) = 0 and, if n is not the Gödel number of a provable formula, ϕ(n) = 1. In specifying that ϕ must be general recursive we are following Church in identifying “general recursiveness” and “effective calculability.”

First, a modification is made in Gödel's proofs of his theorems, Satz VI (Gödel, p. 187—this is the theorem which states that ω-consistency implies the existence of undecidable propositions) and Satz XI (Gödel, p. 196—this is the theorem which states that simple consistency implies that the formula which states simple consistency is not provable). The modifications of the proofs make these theorems hold for a much more general class of logics. Then, by sacrificing some generality, it is proved that simple consistency implies the existence of undecidable propositions (a strengthening of Gödel's Satz VI and Kleene's Theorem XIII) and that simple consistency implies the non-existence of an Entscheidungsverfahren (a strengthening of the result in the last paragraph of Church).
Finite combinatory processes—formulation
Authors: Post Emil L.
Published: May 2006

The present formulation should prove significant in the development of symbolic logic along the lines of Gödel's theorem on the incompleteness of symbolic logics and Church's results concerning absolutely unsolvable problems.

We have in mind a general problem consisting of a class of specific problems. A solution of the general problem will then be one which furnishes an answer to each specific problem.

In the following formulation of such a solution two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions which will both direct operations in the symbol space and determine the order in which those directions are to be applied.

In the present formulation the symbol space is to consist of a two way infinite sequence of spaces or boxes, i.e., ordinally similar to the series of integers …, − 3, − 2, − 1, 0, 1, 2, 3, …. The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time. And apart from the presence of the worker, a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.
Constructibility as a criterion for existence
Authors: Rosser Barkley
Published: May 2006

Problems of construction have engaged the attention of mathematicians from the earliest times. The three famous unsolved problems of the Greeks were problems of construction. Apparently the attitude of the Greeks toward these problems was that, if a geometric entity exists, it must be constructible. For example they believed that given an arbitrary angle, there is another one equal to a third of the first. Hence there must be a way of constructing the second angle, that is, a way of trisecting the first angle. This attitude is very similar in formulation but vastly different in content from the attitude of the modern intuitionists, who say that a mathematical entity does not exist unless it is constructible. Had Brouwer been a Greek, he would probably have declared that, given an angle, it is non-sense to speak of another angle equal to a third of it until such a second angle has been constructed.

However, it seems scarcely necessary to remind the reader that, for the Greeks, constructibility meant a special kind of constructibility, namely, constructibility by ruler and compass. Other methods of constructing geometric entities are known, and by use of them, the problems of the Greeks can be solved. More than that, it is now known that ruler and compass constructions alone do not suffice to solve any one of the three famous problems of the Greeks. Therefore, we see that the Greeks erred in restricting themselves to ruler and compass constructions, even though they may have felt justified in making these restrictions, due to considerations of mathematical elegance, or to ignorance of other methods.
Universally free logic and standard quantification theory
Authors: Meyer Robert K. , Lambert Karel
Published: May 2006

Interest has steadily increased among logicians and philosophers in versions of quantification theory which meet the following criteria: (1) no existence assumptions are made with respect to individual constants, and (2) theorems are valid in every domain including the empty domain. Logics meeting the former of these criteria are called free logics by Lambert and have been investigated in a series of papers by him and by van Fraassen, and by Leblanc and Thomason.1

Although it is natural to impose (2) in the presence of (1), the criteria are independent.2 Hence we baptize logics which meet both criteria universally free.
A minimal canonically complete m-valued proper logic for each m
Authors: Call Richard L.
Published: May 2006

In [1], William H. Jobe has shown that a certain three-valued proper logic is canonically complete, i.e., each first order J is expressible by a formula that is in M-normal form. The first order operator Jk(p)1 represents the truth table of a possible formula which has the truth value k in the ith row and the truth value 1 in all other rows. “The M-normal form of a given formula is another formula which has the same truth table as the given formula but which contains no operation symbols other than those of conjunction, disjunction, and undefined unary operations, with no nonunary operation, i.e., conjunction or alternation, included in the scope of an operation which is unary, and no alternation included in the scope of a conjunction.” Here it is shown that it is possible to construct a canonically complete proper logic for each choice of M which contains, in addition to alternation and conjunction, exactly two unary operations. Moreover, it is further shown that at least two unary operations are necessary in order for such a logic to be canonically complete.
Venn-type diagrams for arguments ofnterms
Authors: Anderson Daniel E. , Cleaver Frank L.
Published: May 2006

The attempt to find usable diagrams fornterms of the sort devised by John Venn seems to have originated with Venn himself, who published diagrams for up to five classes (the fifth class, however, was shaped like a doughnut, and contained an area outside itself — like the hole in the doughnut). Venn then suggested that “if we wanted to use a diagram forsixterms (x, y, z, w, v, u) the best plan would probably be to taketwofive-term figures, one for theupart and one for the non-upart of all the other combinations …” Such a method would, as Venn admits, be somewhat confusing to the eye, and thus fail to fulfil one of the major purposes of the diagram; nevertheless, it is a method which could be extended to covernclasses.

Venn also suggests (but without illustration) another method for drawing diagrams fornterms: “… the rule of formation would be very simple. We should merely have to begin by drawing any closed figure, and then proceed to draw others subject to the one condition that each intersect once, and once only, all the existing subdivisions produced by those which had gone before.” Then, in a rather surprising footnote, he adds: “It will be found that when we adhere to continuous figures, instead of the discontinuous five-term figure given above, there is a tendency for the resultant outlines thus successively drawn to assume a comb-like shape after the first four or five. If we begin by circles or other rounded figures the teeth are curved, if by parallelograms then they are straight.
Limit ultraproducts
Authors: Keisler H. Jerome
Published: May 2006

This paper is a sequel to our earlier paper, “Limit Ultrapowers”, [6]. In that paper we introduced the limit ultrapower construction and proved that is isomorphic to a limit ultrapower of if and only if every PCΔ class which contains also contains . In Section 1 of this paper we introduce the more general limit ultraproduct construction, and in Section 2 we prove that, for any class K of relational systems, a relational system is isomorphic to a limit ultraproduct of members of K if and only if every PCΔ class which includes K also contains . As a consequence, the property of K being an intersection of PCΔ classes is characterized purely set-theoretically by the property of K being closed under isomorphisms and limit ultraproducts.

In Section 3 we apply limit ultraproducts to obtain model-theoretic conditions equivalent to the set-theoretic condition that every α-complete ultrafilter is γ+-complete. The first result, Theorem 3.7, was announced in the abstract [8], and it is also closely related to a result which was stated without proof in [10], namely Theorem 2 of that paper.

In Sections 4 and 5 we apply our results in order to improve a theorem of Craig in [2]. Craig considered the logic L(Q), where Q is a set of cardinals, obtained from ordinary first order logic by adding for each α ϵ Q the quantifier “there exist at least α”.
On theories categorical in their own power
Authors: Keisler H. Jerome
Published: May 2006

A theory T is said to be categorical in power κ iff T has a model of power κ and any two models of power κ are isomorphic.

It was conjectured by Morley [4] that if T is a theory in a language with κ > ω symbols and T is categorical in power κ, then T has a model of power < κ. The aim of this paper is to prove the following theorem.

Theorem A. Let κ be a regular cardinal such that ω < κ < 2ω. Let T be a theory in a language with κ symbols such that T is categorical in power κ. Then:

(a) T has a model of power < κ.

(b) T is categorical in all powers μ ≥ κ.
Some contrasts between degrees and the arithmetical hierarchy
Authors: Manaster Alfred B.
Published: May 2006

The nonrecursiveness of an arithmetical set can be measured in at least two ways. One way is to consider the level of the set in the arithmetical hierarchy. Another is to consider the degree of unsolvability of the set; more precisely to look at the set of degrees recursive in the degree of the set. The Hierarchy Theorem [1] indicates a correspondence between these two ways of considering the nonrecursiveness of an arithmetical set. The results below indicate some of the contrasts between these two points of view. (It should perhaps be mentioned that, although the results are stated in terms of degrees and levels in the arithmetical hierarchy, they can also be stated, in view of the Hierarchy Theorem, strictly in terms of degrees.)
A note on computable real fields
Authors: Madison E. W.
Published: May 2006

It is well known that every field (formally, real field ) has an algebraic closure (real-closure ). This is to say is an algebraic extension of which is algebraically closed (real-closed). Of course, certain properties of carry over to . In particular, M. O. Rabin has proved in [3] that the algebraic closure—which is of course unique up to isomorphism—of a computable field is computable. The purpose of this note is to establish an analogue of Rabin's theorem for formally real fields. It is clear that a direct analogue can be formulated only in the case of ordered fields, for otherwise there may be many (nonisomorphic) such .
Limiting recursion
Authors: Gold E. Mark
Published: May 2006

A class of problems is called decidable if there is an algorithm which will give the answer to any problem of the class after a finite length of time. The purpose of this paper is to discuss the classes of problems that can be solved by infinitely long decision procedures in the following sense: An algorithm is given which, for any problem of the class, generates an infinitely long sequence of guesses. The problem will be said to be solved in the limit if, after some finite point in the sequence, all the guesses are correct and the same (in case there is more than one correct answer). Functions, sets, and functionals which are decidable by such infinite algorithms will be calledlimiting recursive. These, together with classes of objects which can beidentified in the limit, are the subjects of this report.

Without qualification,setwill mean set of numbers;functionwill mean number-theoretic function of 1 variable, possibly partial;functionalswill take numerical values and have any number of numerical and/or function variables, the latter ranging solely over total functions of 1 variable. Thus a function is a special case of a functional,xwill invariably stand for a numerical variable;φfor a function variable;gfor a guess (a number);nfor the numerical variable which indexes the guesses, referred to as thetime.
Algebraic semantics for modal logics I
Authors: Lemmon E. J.
Published: May 2006

Modal logic received its modern impetus from the work of Lewis and Langford [10]. In recent years, however, their axiomatic approach, aided by somewhat ad hoc matrices for distinguishing different modal systems, has been supplemented by other techniques. Two of the most profound of these were, first, the algebraic methods employed by McKinsey and Tarski (see [11] and [12]) and, second, the semantic method of Kripke (see [5] and [6]); and there have been others. The aim of the present series of papers is to afford a synthesis of these methods. Thus, though new results are given, the interest lies rather in revealing interconnexions between familiar results and in providing a general framework for future research. In general, we show that semantic completeness results of the Kripke kind can be deduced from the algebraic results by means of one central theorem (Theorem 21).