Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.

CHAPTER Addenda to Pure Combinatory Logic This chapter will treat various additions to, and modifications of, the subject matter of Chapters-7. ...

This essay explores the intersection of race science and plant taxonomy in the creation of evolutionary taxonomies (phylogenies) of populations of Zea mays, also known as maize or corn. Following recent work in the history and sociology of race, it analyzes maize taxonomy as technology. Through an analysis of successive attempts to classify diverse maize varieties, especially those originating in Mexico, it shows that taxonomy created possibilities for researchers to intervene in commercial agriculture, state development projects, biological conservation, and domestic (...) and international politics and policy. It further underscores that the modern science of maize taxonomy was distinct but never inseparable from assessments of maize’s human cultivators. Attending to particularities of this relationship is crucial, because it reveals the application of maize taxonomy as a technology for ordering human diversity and intervening in human lives, as well as managing the impressive diversity of Zea mays. (shrink)

This paper charts the history of the Rockefeller Foundation’s participation in the collection and long-term preservation of genetic diversity in crop plants from the 1940s through the 1970s. In the decades following the launch of its agricultural program in Mexico in 1943, the Rockefeller Foundation figured prominently in the creation of world collections of key economic crops. Through the efforts of its administrators and staff, the foundation subsequently parlayed this experience into a leadership role in international efforts to conserve so-called (...) plant genetic resources. Previous accounts of the Rockefeller Foundation’s interventions in international agricultural development have focused on the outcomes prioritized by foundation staff and administrators as they launched assistance programs and especially their characterization of the peoples and “problems” they encountered abroad. This paper highlights instead how foundation administrators and staff responded to a newly emergent international agricultural concern—the loss of crop genetic diversity. Charting the foundation’s responses to this concern, which developed only after agricultural modernization had begun and was understood to be produced by the successes of the foundation’s own agricultural assistance programs, allows for greater interrogation of how the foundation understood and projected its central position in international agricultural research activities by the 1970s. (shrink)

The Peanese convention for the use of dots as brackets has the disadvantage that it gives only an awkward method for representing chains of indefinite length, such as the compound implicationSuch chains occur frequently in logical investigations of a metatheoretic nature, and it is convenient to have a systematic method of abbreviating them. The most obvious method of doing this would be to leave the parentheses out entirely, and to understand that in such cases the implication sign or other operation (...) appearing on the extreme left is the most inclusive; but this method, which has been followed by Quine, is at variance with our ordinary algebraic usage, in which we write a − b + c − d for ((a − b) + c) − d, and consequently leads to confusion. It is desirable to have a modification of the Peanese convention which gives a simple method of representing such chain implications, and at the same time avoids this difficulty.Such a modification is easily obtained by generalizing somewhat a procedure of Church. The latter author uses only a single dot, which he writes on the right of an operator to signify a bracket extending from that point to the end of the formula (or parenthesized expression). If dots are used only on the right of operations this is all that can be desired. But the essential idea of this device can be extended to the case where dots are used also on the left, as follows: let us suppose that a group of dots on the right of an operation or prefix denotes the beginning of a bracket which extends to the right until it meets a group with an equal or larger number of dots on the left of an operation; and that the scope of a group of dots on the left of an operation shall extend to the left until it reaches a larger group of dots on the right of some operation. (shrink)

Lorenzen, in his book Einführung in die operative Logik und Mathematik has given a relatively precise form of syntactical system which he calls a calculus. The present paper deals with the relationship of Lorenzen's notion of calculus with the notion of formal system . It is shown that the obs of a formal system can be represented as the theses of a calculus of a certain type just when the calculus has a property called the tectonic property, and conditions are (...) given under which one form of system can be transformed into the other.ZusammenfassungLorenzen hat in seinem Buch, Einführung in die operative Logik und Mathematik, eine ziemlich genaue Auffassung eines syntaktischen Systems angegeben, die er Kalkül nennt. Die vorliegende Abhandlung betrifft die Verwandtschaft dieses Lorenzenschen Begriffs mit dem eines formalen Systems . Es wird bewiesen, dass die « Obs » eines formalen Systems sich als die Thesen eines passenden Kalküls genau dann repräsentieren lassen, wenn dieser Kalkül eine gewisse Eigenschaft, die « tectonic property » heisst, besitzt; die Bedingungen für die Überführung einer Art von System in die andere werden aufgezeigt.RésuméLorenzen, dans son livre Einführung in die operative Logik und Mathematik, a exposé une forme assez précise de systéme syntaxique qu'il a appelée un calcul. L'article actuel traite les relations entre cette idée et celle d'un système formel . L'auteur dèmontre que les « obs » d'un système formel peuvent se représenter comme les thèses d'un calcul d'une certaine espèce justement quand le calcul a une propriété que l'on appelle « the tectonic property »; et it énonce des conditions pour qu'un type de système soit transformable en l'autre. (shrink)