The calculi of lambda-conversion

The Calculi of Lambda Conversion by Alonzo Church (Paperback, 1941)
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1. The Concept of a Function

Rushmore as the value to every argument, while highest-mountain-in-the-Himalayas still assigns Mt. Everest to every object.

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In other words, highest-mountain-on-earth and highest-mountain-in-the-Himalayas are extensionally equivalent at the actual world but not intensionally equivalent. A function concept may now be called extensional if and only if it requires functions that are extensionally equivalent at the actual world to be identical. And a function concept may be classified as intensional if and only if it requires intensionally equivalent functions to be identical.

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Note that these classifications are conceptually different from the distinctions commonly used in the foundations of mathematics. On the terminology used in the foundations of mathematics, functions-as-sets are classified as extensional since they use the axiom of extensionality as their criterion of identity, and functions-as-rules are classified as intensional because they rely on the ostensibly intensional concept of a rule.

In the present possible-worlds terminology, function concepts are classified as extensional or intensional based of their behavior at possible-worlds. To see this, consider the following two functions:. These two functions are clearly extensionally equivalent: they assign the same value to the same input at the actual world. Moreover, given standard assumptions in possible worlds semantics, the two functions are also intensionally equivalent. If we assume that mathematical facts, like facts about addition and subtraction, are necessary in the sense that they are the same at every possible world, then we get that the two functions give the same value to the arguments at every possible world.

So, an intensional function concept would require the two functions to be identical. A function concept that allows for intensionally equivalent functions to be distinct is called hyperintensional. The point here was simply to clarify any conceptual confusions that might arise from different terminologies at play in philosophical discourse. On this, see section 9. On this, see section 5. Terms formed according to rule 2 are called application terms. Terms formed according to rule 3 are called abstraction terms.

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As is common when dealing with formal languages that have grouping symbols the left and right parenthesis, in our case , some parentheses will be omitted when it is safe to do so that is, when they can be reintroduced in only one sensible way. To avoid the tedium of always writing all needed parentheses, we adopt the following convention:. One can define, analogously, the notions of free and bound variable in the expected way, as follows. As is typical in other subjects where the concepts appear, such as first-order logic, one needs to be careful about the issue; a casual attitude about substitution can lead to syntactic difficulties.

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New paperback Quantity Available: 1. Salomon Bochner. Book Depository hard to find London, United Kingdom. Friend Reviews. Editorial team. Add co-authors Co-authors. Terms formed according to rule 2 are called application terms.

We now define substitution, and then lay down a convention that allows us to avoid such difficulties. Clause 3 tells us that substitution unconditionally distributes over applications. This explains clause 4. Such a substitution would yield a function different from the one intended. Such terms appear at first blush to be suspicious; one might suspect that using such terms could lead to inconsistency, and in any case one might find oneself reaching for a tool with which to forbid such terms. But no set can contain itself in this way, lest the axiom of foundation or regularity be violated.

Thus, from a set theoretical perspective such terms are clearly dubious. Below one can find a brief sketch of one such tool, type theory. Moreover, forbidding such terms, as in type theory, does not come for free e. Many more could be given and obviously there are infinitely many combinators , but the following have concise definitions and have proved their utility.

Thus, it is safe to write the function and the argument adjacent to one other. Recall our convention that we are to understand such officially illegal expressions by working from left to right, always putting parentheses around adjacent terms. Alternative notation. It is not necessary to include two kinds of grouping symbols parentheses and square brackets in the syntax. Parentheses or square brackets alone would obviously suffice. The two kinds of brackets are employed in this entry for the sake of readability. One sees this device used in earlier works of logic, such as Principia Mathematica , where the function of the symbol.

Some authors write abstraction terms without any device separating the bound variable from the body: such terms are crisply written as, e. But this practice is not flawless. Usually the names of variables are single letters, and theoretically this is clearly sufficient. But it seems unduly restrictive to forbid the practice of giving longer names to variables; indeed, such constructions arise naturally in computer programming languages. For the sake of uniformity, we will adopt the square bracket notation in this entry. Incidentally, this notation is used in Turing, Which notation to use for substitution seems to be a personal matter.

In this entry we use a linear notation, eschewing superscripts and subscripts. Frege observed, as we did above, that in the study of functions it is sufficient to focus on unary functions i.

Lambda calculus

The procedure of viewing a multiple-arity operation as a sequence of abstractions that yield an equivalent unary operation is called currying the operation. Perhaps it would be more historically accurate to call the operation fregeing , but there are often miscarriages of justice in the appellation of mathematical ideas. The proof that these two suffice to represent all functions is beyond the scope of this entry. For further discussion, see the entry on combinatory logic.

In the hands of Curry, Church, Kleene, and Rosser some of the pioneers in the subject the focus was on defining mathematical objects and carrying out logical reasoning inside the these new systems. This problem was shown to be undecidable. Initially one had results showing that certain terms were not interconvertible e. See below for more detailed discussion of consistency.

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Its relation to programming languages was also clarified.