The l-operator*by RUDOLF CARNAP
33a. The l operator. Let 'M' be a one-place predicate of the second level, i.e. designating a property of properties of individuals. Thus e.g. 'M(P)' might be rendered "the first-level property P has the second-level property M". (For a concrete example, we might think of a cardinal number, e.g., 5; then '5(P)' says that P has cardinal number 5. Here, of course, 5 is regarded as a property of properties.) If we wish to assert that the property predicated of a by the sentence 'Pa v Qa' has the property M, we can do so with the help of the symbolism introduced earlier: for since 'Pa v Qa' can also be written '(P v Q)a', the proposition named above can be formulated 'M(P v Q)'. [Example: we would read '5(P v Q)' as "the disjunction of properties P and Q (or: the union of class P and Q) has cardinal number 5".]
What we have just done in connection with 'Pa v Qa' cannot be extended to more elaborate sentential compounds such as 'Pa v (y)Rya'. The reason is that the symbolism available up to this point furnishes no predicate expressions for the properties predicated of an individual by most compound sentences about the individuals; e.g., we have no predicate
* From Introduction to Symbolic Logic and its Applications Dover. 1958. pp. 129-136.
expression for the property predicated of individual a by the sentential compound 'Pa v (y)Rya'. The operator sign 'l' now to be introduced will have the particular role of forming a predicate expression for any property ascribed to an individual by any sentence in language C. Thus it will appear in what follows that the property predicated of individual a by the sentence 'Pa v (y)Rya' is to be designated by the predicate expression '(lx)(Px v (y)Ryx)'.
An expression of the form '(lx)(...x...)' is called a l-expression. In the l-expression '(lx)(...x...)' the portion written '(lx)' is an operator which we call the l-operator; and the portion written '(...x...)' is the operand of the l-operator. Note therefore that the 'x' is bound in '(lx)(...x...)'. If '...x...' is a sentential formula, then '(lx)(...x...)' corresponds, say, to the verbal expression "the property of x such that ...x..." or the verbal expression "the class of those x such that ...x..."; and the full expression '[(lx)(...x...)]a' is a sentence asserting that the individual a has the property (lx)(...x...).
The use of a l-expression, e.g.; '(lx)(Px v (y)Ryx)', would be superfluous if its purpose were merely to ascribe the property it designated to some individual, say b. For this can be done simply by the sentence 'Pb v (y)Ryb', and the more complicated formulation '[(lx)(Px v (y)Ryx)]b' can be dispensed with; both formulations say the same thing. Therefore our syntactical system B contains a primitive sentence scheme (it is P10 in 22a) that enables either one of the two sentences just named to be derived from the other; which is to say, we can find in B a sentence in the old symbolism, viz. 'Pb v (y)Ryb' that is synonymous with the l-expression, viz. '[(lx)(Px v (y)Ryx)]b'. However, the old symbolism provides no expression that is synonymous with the l-expression itself. Hence the new l-expression is very useful if we wish to ascribe to the property designated by this l-expression some property of the second level, for in this case the l-expression can serve as the argument-expression of the second level predicate expression.
The particular illustration of a l-expression that appears above is a one-place predicate expression. In a similar way, l-operators with several variables can be used to construct many-place predicate expressions. E.g. a l-expression of the form '(lxy)(...x...y...)' whose operand '...x...y...' is a sentential formula with free variables 'x' and 'y' is to be recognized as a two-place predicate expression designating that relation which subsists between two individuals x and y just in case they satisfy the condition formulated in the operand. The formulation of l-predicate expressions with more than two argument-places and of arbitrary type is carried out in an analogous fashion. A variable must not occur in a l-operator more than once.
While of great importance theoretically, l-expressions are relatively seldom used in language C. The reason is that in language C other forms of
expression (notably, functors) are often available for the construction of predicate expressions. E.g. the property predicated of a by 'Pa v ($ y)Rya' can be designated 'Pa v mem2(R)', hence in this case the less concise l-expression '(lx)(Px v ($ y)Ryx' can be dispensed with. Again, it often happens that a discussion involves repeated reference to a certain property in a particular connection; in this event it may pay to introduce (by definition) a simple predicate for the property. Thus, reverting to our last example, we can introduce Q, say, by the definition 'Qx º Px v ($ y)Ryx' and thereafter render as 'M(Q)' the proposition contemplated about this property. As a general rule, l-expressions are of use only when there is no advantage either in defining predicates for the properties under consideration, or in defining functors which permit the designation of these properties by compound predicate expressions.
l-functor expressions. Up to now we have dealt only with l-expressions which are predicate expressions, i.e. l-expressions whose operands are expressions of arbitrary type in the type system. Here, as before, the full expression '[(lx)(...x...)]a' is synonymous with '...a...', i.e. with what results from substituting 'a for 'x' in the operand. But whereas formerly this full expression was a sentence, now the full expression is an expression of the type system. For this reason the l-expressions now under consideration are not predicate expressions, but functor expressions. (It should be noted that the primitive sentence scheme P10 of 22a still serves for the transformation of our present l-expressions.)
Examples. 1 In accordance with the above, '[(lx)(prod (3, x))]a' is synonymous with 'prod (3, a)' and hence means "the triple of a"; thus '(lx)(prod (3, x))' is a functor expression to be read "the triple of" or "the function whose value at x is 3x". From this example we see that any l-functor expression '(lx)(...x...)' can be read "the function whose value at x is ...x...". -- 2. The one-place predicate expression '(l x) [ ($ y)Rxy]' is read "the class of those x such that there is something y to which x bears the relation R; hence (in view of D18-1, and the fact that '(l x) [ ($ y)Rxy]' means the same '($ y)Ray', i.e. 'mem1(R)a') it is clear that '(l x) [ ($ y)Rxy]' is synonymous with 'mem1(R)'. Now suppose we let the l-expression of this example be the operand of another l-expression, viz. '(lH)[(lx) [($y)Hxy]]'. This new l-expression is a functor expression; it is read "the function whose value at H is the class of those x which bear the relation H to something" or "the function whose value at H is the class of first members of H"; and hence it is synonymous with 'mem1'. This last can be seen as follows: '[lH) [($y)Hxy]]](R)' is synonymous with '(l x) [ $ y)Rxy]', which in turn is synonymous with 'mem1(R)'; thus (lH) '[($y)Hxy]]' is synonymous with 'mem1'.
According to an earlier rule (9a, (4)), those brackets can be omitted which immediately enclose an expression consisting of an operator and the operand belonging thereto. This rule permits us to omit e.g. all the square brackets from the illustrative expressions given above; thus '(lH) (lx)($y) (Hxy)(R)' can be written in place of '[lH) [(lx)[($ y)(Hxy)]]](R)'. (It should be observed that Rule 5 of 9a does not apply to l-expressions.)
33b. Rule for the l-operator. What is said below is a consequence of our explanation of the meaning of l-expressions. Suppose that immediately after a l expression whose l-operator contains n variables there follows an argument-expression; then the whole complex is a full expression provided this argument-expression is n-place and the member in the kth place thereof (k = 1,...n) is of the same type as the kth variable in the l-operator. (The argument-expression referred to above is called the argument-expression belonging to the l-expression, or the argument-expression belonging to the l-operator; the l-expression itself can, of course, be either a predicate or a functor expression.) If a l-expression and its argument-expression together have this character, i.e. if the whole complex is a full expression, then the l-operator can be eliminated with the help of the l-rule given below. [So far as the syntactical system B is concerned, the l-rule follows from the primitive schema P10 of 22a. [So far as the semantical system B is concerned, this l-rule always produces from a given expression a second that is L-interchangeable with the first; this follows from the fact that sentences of the form P10 are L-true on the basis of the evaluation rules given in 25a.]
The l-rule. A full expression of the form
[(ldk1 ,dk2,... dkn) (Zi)](Z m1, Z m2, ..., Z mn),
where Zi is the operand of the l-operator, may be transformed into the expression Zk which is obtained from Zi by substituting in the later Zm1 for dk1 , Z m2 for dk2 ... and Z mn for dkn .The transformation referred to in this l-rule can be effected whether the displayed l-expression is an independent sentence or a part of another sentence. In view of the rule, a l-operator can always be eliminated if there is an argument-expression belonging to it. If an expression consists of a single operand preceded by several l-operators and followed by several argument-expressions (each of these last is bracketed by itself; their number does not exceed the number of l-operators), the first argument-expression belongs to the first l-operator and can be eliminated with it; the second argument-expression belongs to the second l-operator, and can be eliminated with it; and so on.
Example. By two applications of the l-rule (the second application involving two variables), the expression '(lx1) (lF2, x3)( lH4) (x1...F2...x3...H4...)(a1)(P2, a3)' can be transformed into '(lH4)(...a1...P2...a3 ...H4...)'.
Remarks. The use of l-expressions requires careful attention to brackets. According to our earlier stipulation (see the end of 33a., it is permissible to write '(lx)(Px)a' instead of '[(lx)](a)'. On the other hand, brackets enclosing the operand of a l-operator (e.g. those around 'Px' in the expression just given) are generally not to be omitted; they may be omitted only if some other rule permits. Thus '(lx)(...x...)(a)' is to be regarded as an abbreviation for '[(lx)(...x...)](a)', but not for '(lx)(...x...)(a)'. In other words: a
predicate expressions or functor expression which stands between an operator and an argument-expression belongs to the l-operator.
Again, the difference between '(lx,y)' and '(lx)(ly))' should be noticed. Suppose '...x...y...' is a sentential formula. Then '(l x,y)(...x...y...)' is a predicate expression; and '(lx,y) (...x...y...)(a,b)' can be transformed by the l-rule into '...a...b...'. On the other hand, in view of our agreement about omission of brackets, '(lx)(ly)(...x...y...)' is an abbreviation for '(lx)[(ly) (...x...y...)]' and hence is a functor expression; a full expression of it is e.g. '(lx)(ly)(a)', which by the l-rule may be transformed into '(ly)(...a...y...)' and so recognized as a predicate expression. Using this predicate expression, let us form the full sentence '(lx)(ly)(...x...y...)(a)(b)'. This sentence is an abbreviation for '[(lx)[(ly)(...x...y...) ]](a)(b)', which by two successive applications of the l-rule transforms first into '(ly) (...a...y...)(b)' and then into '...a...b...'.
The l-predicate expression are entirely analogous to the class expressions of [P.M.]. Here, however, they are genuine predicate expressions, and are used exactly like predicates. Thus e.g. '(lx)(Px)' and 'P' are interchangeable in any context whatsoever. Concerning the line of development which led to this identification of predicate expressions and class expressions, see [Syntax] ¶ 37, ¶ 38. This development was initiated by Russell (see [P.M.], Introduction to vol. 1, 2nd ed., and Chap. VI). -- Church was the first to use the l-operator for functor expressions; he has given the l-operator a central role in his system ("The calculi of lambda-conversion", Annals of Math. Studies, No. 6, Princeton, 1941.
With the background provided by the present section 33b, we can state the following theorem.
T33-1. The following sentential formulas are L-true:+a. (lx)(Fx) = F.
b. (lx)(Fx)(y) º Fy
c. (lx,y)(Hxy) = H.
d. (lx,y)(Hxy)(u,v) º Huv.