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# The Calculi of Lambda Conversion. (AM-6)

ALONZO CHURCH
Pages: 77
Stable URL: http://www.jstor.org/stable/j.ctt1b9x12d
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1. Front Matter (pp. None)
3. Chapter I INTRODUCTORY (pp. 1-7)

1. THE CONCEPT OF A FUNCTION. Underlying the formal calculi which we shall develop is the concept of a function, as it appears in various branches of mathematics, either under that name or under one of the synonymous names, “operation” or “transformation.” The study of the general properties of functions, independently of their appearance in any particular mathematical (or other) domain, belongs to formal logic or lies on the boundary line between logic and mathematics. This study is the original motivation for the calculi — but they are so formulated that it is possible to abstract from the intended meaning and...

4. Chapter II LAMBDA-CONVERSION (pp. 8-27)

5. PRIMITIVE SYMBOLS, AND FORMULAS. We turn now to the development of a formal system, which we shall call the calculus of A-conversion, and which shall have as a possible interpretation or application the system of ideas about functions described in Chapter I.

The primitive symbols of this calculus are three symbols,

۸, (, ),

which we shall call improper symbols, and an infinite list of symbols,

a, b, c, ... , x, y, z, ā, Ƃ, ... , ẑ, ǟ, ... ,

which we shall call variables. The order in which the variables appear in this originally given infinite...

5. Chapter III LAMBDA-DEFINABILITY (pp. 28-42)

8. LAMBDA-DEFINABILITY OF FUNCTIONS OF POSITIVE INTEGERS. We define,

1 → λab.ab,

2 → λab.a(ab),

3 → λab.a(a(ab)),

and so on, each numeral (in the Arabic decimal notation) being introduced as an abbreviation for a corresponding formula of the indicated form. But where a numeral consists of more than one digit, a bar is used over it, in order to avoid confusion with other notations; thus,

П → λab.a(a(a(a(a(a(a(a(a(a(ab)))))))))),

but 11, without the bar, is an abbreviation for

(λab.ab) (Aab.ab).

In connection with these definitions an interpretation of the calculus of λ-conversion is contemplated under which each of the formulas...

6. Chapter IV COMBINATIONS, GÖDEL NUMBERS (pp. 43-57)

12. COMBINATIONS. If s is any set of well-formed formulas, the class of s-combinations is defined by the two following rules, a formula being an s-combination if and only if it is determined as such by these rules:

(1 ) Any formula of the set s, and any variable standing alone, is an s-combination.

(2) IfAandBare s-combinations,ABis an scombination.

In the cases in which we shall be interested the formulas of s will contain no free variables and will none of them be of the formAB. In such a case It is possible...

7. Chapter V THE CALCULI OF λ-K-CONVERSION AND λ-δ-CONVERSION (pp. 58-71)

17. THE CALCULUS OF λ-K-CONVERSION. The calculus of λ-K-converslon is obtained if a single change is made in the construction of the calculus of λ-conversion which appears in §§5,6: namely, in the definition of well-formed formula (§5) deleting the words “and contains at least one free occurrence of x” from the rule 3. The rules of conversion, I, II, III, in §6 remain unchanged, except that well-formed is understood in the new sense.

Typical of the difference between the calculi of λ-conversion and λ-K-conversion is the possibility of defining in the latter the constancy function,

K → λa(λba),

and the...

8. INDEX OF THE PRINCIPAL FORMULAS INTRODUCED BY DEFINITION (pp. 72-72)
9. BIBLIOGRAPHY (pp. 73-82)
10. CORRECTION AND ADDITIONS (pp. 82-82)