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The l-operator^{*}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 propertyPhas the second-level propertyM". (For a concrete example, we might think of a cardinal number, e.g., 5; then '5(P)' says thatPhas cardinal number 5. Here, of course, 5 is regarded as a property of properties.) If we wish to assert that the property predicated ofaby the sentence 'PavQa' has the propertyM, we can do so with the help of the symbolism introduced earlier: for since 'PavQa' can also be written '(PvQ)a', the proposition named above can be formulated 'M(PvQ)'. [Example: we would read '5(PvQ)' as "the disjunction of propertiesPandQ(or: the union of classPandQ) has cardinal number 5".]

What we have just done in connection with 'PavQa'cannot be extended to more elaborate sentential compounds such as 'Pav(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

^{*}FromIntroduction to Symbolic Logic and its ApplicationsDover. 1958. pp. 129-136.

129 expression for the property predicated of individual

aby the sentential compound 'Pav(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 individualaby the sentence 'Pav(y)Rya'is to be designated by the predicate expression '(lx)(Pxv(y)Ryx)'.

An expression of the form '(lx)(...x...)' is called al-expression. In thel-expression '(lx)(...x...)' the portion written '(lx)' is an operator which we call thel-operator; and the portion written '(...x...)' is theoperandof thel-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 ofxsuch that ...x..." or the verbal expression "the class of thosexsuch that ...x..."; and the full expression '[(lx)(...x...)]a' is a sentence asserting that the individualahas the property(lx)(...x...).

The use of al-expression, e.g.; '(lx)(Pxv(y)Ryx)', would be superfluous if its purpose were merely to ascribe the property it designated to some individual, sayb. For this can be done simply by the sentence 'Pbv(y)Ryb', and the more complicated formulation '[(lx)(Pxv(y)Ryx)]b' can be dispensed with; both formulations say the same thing. Therefore our syntactical systemBcontains a primitive sentence scheme (it isP10in22a) that enables either one of the two sentences just named to be derived from the other; which is to say, we can find inBa sentence in the old symbolism, viz. 'Pbv(y)Ryb'that is synonymous with thel-expression, viz. '[(lx)(Pxv(y)Ryx)]b'. However, the old symbolism providesnoexpression that is synonymous with thel-expression itself. Hence the newl-expression is very useful if we wish to ascribe to the property designated by thisl-expression some property of the second level, for in this case thel-expression can serve as the argument-expression of the second level predicate expression.

The particular illustration of al-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. al-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 individualsxandyjust in case they satisfy the condition formulated in the operand. The formulation ofl-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 al-operator more than once.

While of great importance theoretically,l-expressions are relatively seldom used in languageC. The reason is that in languageCother forms of

130 expression (notably, functors) are often available for the construction of predicate expressions. E.g. the property predicated of

aby 'Pav($ y)Rya' can be designated 'Pavmem', hence in this case the less concise_{2}(R)l-expression '(lx)(Pxv($ 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 Pxv($ 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.

. Up to now we have dealt only withl-functor expressionsl-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 'afor '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 thel-expressions now under consideration are not predicate expressions, but functor expressions. (It should be noted that the primitive sentence scheme P10 of22astill serves for the transformation of our presentl-expressions.)

Examples. 1In accordance with the above, '[(lx)(prod (3, x))]a' is synonymous with 'prod (3, a)' and hence means "the triple ofa"; thus '(lx)(prod (3, x))' is a functor expression to be read "the triple of" or "the function whose value atxis 3x". From this example we see that anyl-functor expression '(lx)(...x...)' can be read "the function whose value atxis ...x...". --2. The one-place predicate expression '(lx) [($y)Rxy]' is read "the class of thosexsuch that there is somethingyto whichxbears the relation; henceR(in view of D18-1, and the fact that '(lx) [($y)Rxy]' means the same '($ y)Ray', i.e. 'mem'_{1}(R)a)it is clear that '(lx) [($y)Rxy]' is synonymous with 'mem'. Now suppose we let the_{1}(R)l-expression of this example be the operand of anotherl-expression, viz. '(lH)[(lx)[($y)Hxy]]'. This newl-expression is a functor expression; it is read "the function whose value atHis the class of thosexwhich bear the relationHto something" or "the function whose value atHis the class of first members ofH"; and hence it is synonymous with 'mem'. This last can be seen as follows: '[_{1}lH)[($y)Hxy]]](R)' is synonymous with '(lx) [$y)Rxy]', which in turn is synonymous with 'mem'; thus_{1}(R)(lH) '[($y)Hxy]]' is synonymous with 'mem'._{1}According to an earlier rule (

9a, (4)), thosebrackets can be omittedwhich 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 of9adoes not apply tol-expressions.)

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33b. Rule for theWhat is said below is a consequence of our explanation of the meaning ofl-operator.l-expressions. Suppose that immediately after alexpression whosel-operator containsnvariables there follows an argument-expression; then the whole complex is a full expression provided this argument-expression isn-place and the member in thekth place thereof (k= 1,...n) is of the same type as thekth variable in thel-operator. (The argument-expression referred to above is calledthe argument-expression belonging to the l-expression,orthe argument-expression belonging to the l-operator; thel-expression itself can, of course, be either a predicate or a functor expression.) If al-expression and its argument-expression together have this character, i.e. if the whole complex is a full expression, then thel-operator can be eliminated with the help of thel-rule given below. [So far as the syntactical systemBis concerned, thel-rule follows from the primitive schema P10 of22a. [So far as the semantical systemBis concerned, thisl-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 in25a.]

TheA full expression of the forml-rule.

[(ld(Z_{k1 },d_{k2},... d_{kn})_{i})](Z_{m1}, Z_{m2}, ...,Z_{mn}),

where Z_{i}is the operand of the l-operator, may be transformed into the expression Z_{k}which is obtained from Z_{i}by substituting in the later Z_{m1}for d_{k1 }, Z_{m2}for d_{k2 }... and Z_{mn}for d_{kn }.The transformation referred to in thisl-rule can be effected whether the displayedl-expression is an independent sentence or a part of another sentence. In view of the rule, al-operator can always be eliminated if there is an argument-expression belonging to it. If an expression consists of a single operand preceded by severall-operators and followed by several argument-expressions (each of these last is bracketed by itself; their number does not exceed the number ofl-operators), the first argument-expression belongs to the firstl-operator and can be eliminated with it; the second argument-expression belongs to the secondl-operator, and can be eliminated with it; and so on.

Example. By two applications of thel-rule (the second application involving two variables), the expression '(lxcan be transformed into '(_{1}) (lF, x_{2}_{3})(lH_{4}) (x_{1}...F_{2}...x_{3}...H_{4}...)(a_{1})(P_{2}, a_{3})'lH'._{4})(...a_{1}...P_{2}...a_{3}...H_{4}...)

Remarks. The use ofl-expressions requires careful attention tobrackets. According to our earlier stipulation (see the end of33a., it is permissible to write '(lx)(Px)a'instead of '[(lx)](a)'. On the other hand, brackets enclosing the operand of al-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

132 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 thel-rule into '...a...b...'. On the other hand, in view of our agreement about omission of brackets, '(lx)' is an abbreviation for '((ly)(...x...y...)lx)[(ly) (...x...y...)]' and hence is a functor expression; a full expression of it is e.g. '(lx)(ly)(a)', which by thel-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 thel-rule transforms first into '(ly) (...a...y...)(b)' and then into '...a...b...'.

Thel-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 thel-operator for functor expressions; he has given thel-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 section33b, 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.Home