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Outline of a Revised Formulation of the Logic of Sense and Denotation (Part I)Author(s): Alonzo ChurchReviewed work(s):Source: Noûs, Vol. 7, No. 1 (Mar., 1973), pp. 24-33Published by: Blackwell PublishingStable URL: http://www.jstor.org/stable/2216181 .Accessed: 13/02/2012 16:57

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24 NOUS

to the second one. However, I take every uch pair to be equivalent, ince in everycase he says that either of the two may be inferred rom the other" (Scott [11],p. 586, fn. 3). What Ockham does instead is to say that (3) can be inferred romone of the disjuncts of (3a), the point of which comment is presumably to dis-tinguish the supposition of 'man' in (3) from that of man' in (1). After ll, (3a)could have been got from 1) as well as from 3); but clearly 1) can't be inferredfrom This man is an animal.'

Outline of a RevisedFormulationof the Logic of Sense and

Denotation Part I)*ALONZO CHURCH

UNIVERSITY OF CALIFORNIA AT LOS ANGELES

In this paper I return o and reexamine my "A Formulationof the Logic of Sense and Denotation." For brevity shallgenerally not repeat passages and material from this latter paper[4] or its abstract [2], but shall merely refer to the paper (asA Formulation) r to the abstract, often n such a way that thereader must have at least [4] before him in order to follow whatis being said.

In fact A Formulation s unsound or faulty n many ways,some major and some minor, and though there has never beena detailed published account of the faults, believe that manyhave been aware of their existence. Frege's theory f meaning hasa very strong intuitive appeal when presented informally s aresolution of the difficulties which Frege considers about themeaning of identity tatements, bout denotationless names, andespecially about the logic of indirect discourse and of othercontexts nvolving what Frege calls the ungerade "oblique" or"indirect") occurrence f names. But in spite of this the conclusionhas no doubt often been drawn that a satisfactory reatment f

the intensional ogic demanded by Frege's theory, f it is possibleat all, is at least not possible along the general ines that proposed.I confess that I was at one time of this opinion myself, nd Ihave on several occasions advised others accordingly (to my



REVISEDFORMULATION OF THE LOGIC OFSENSE ANDDENOTATION 25

present regret). Nevertheless, now outline an attempt to set

matters ight.The principal flaw n the original axioms for Alternative 2)was first alled to my attention by A. F. Bausch not long afterthe publication of A Formulation. n effect, he discussion onpages 15-16 of A Formulation, eading up to the assumption thatb' is a concept f a function if and only f it is a characterizing

function f S, is incompatible with the principle of Alternative 2),that the senses of names A and B are the same if and only f theequation A B is logicallyvalid. Because this italicized) assump-tion is embodied in the axioms as Axioms

150Rnd

16a0, the resultis that the axioms lead, not quite to an inconsistency, ut to areduction to extensionality, n the sense that it is a theorem neach type ot hat wo concepts of the same thing re always dentical.

As a means of finding suitable amendment of the axiomsfor Alternative 2), I tried unsuccessfully n lectures at the Uni-versity of California, at Berkeley in 1960 and at Los Angelesin 1961, to construct satisfactory model along the lines of thenow familiar possible worlds" approach. Concerning the originsof this approach to the construction of models in intensionallogic, see Carnap [1] and Kaplan [9].1

To overcome the difficulties f 1960-1961, one thing thatseems to be necessary is to allow the occurrence of vacuousconcepts in the model more freely han was done at that time.For though it was known that such concepts are unavoidable(cf. footnotes 4, 18 of A Formulation), here was an attempt tominimize them that is now seen to have been ill judged.

However, a more important amendment s that the singleprimitive A is replaced by an infinite ist of primitive ymbolsAOAl A2, *- . Of these, Aois called the abstraction perator;and the subscript 0 may be omitted as an abbreviation, o thatA is written o stand for AO.Then the formation ule (2) (at thebottom of page 8 of A Formulation) s replaced by the following,where n = 0, 1, 2, * :

(2.) if xIn is a variable of type P and Ma is a well-formedformula f type a., then A.x3Ma, is a well-formed ormulahaving hetype xn/3

An occurrence of a variable x8 in a well-formed ormula s boundor free according as it is or is not an occurrence n a well-formedpart of the formula having the form A.x&M,).

That is, this change in the primitive basis of the system,



26 NOUS

which was originally ntended only for Alternative 0), is nowmade also for Alternative 2). And for Alternative 2) there s noother change in the list of primitive ymbols or in the formationrules, as these are given in A Formulation.

1. MODELS FOR ALTERNATIVE (2). On this basis let us now goon to describe a class (or better, n approximate lass) of miniaturemodels which we propose for Alternative 2), and to illustrate bysetting up in detail two examples of such models.

These models are extensional n the sense that the intensionalnotion of concept is not presupposed but is provided for by anextensional construction hat uses the "possible worlds" of themodel. Otherwise they follow the general plan of the "intendedinterpretation" s this is described on pages 10-14, 16-17 ofA Formulation. But they are miniature n the sense that they useonly a small number of possible worlds and have only a smallnumber of individuals members of the type t).

Such miniature models may well be suggestive, and theymay enable us to control the rules of inference nd the axiomsin the sense of showing that (or amending them so that) theydo not lead to an inconsistency r a reduction to extensionalityor other undesired result. But any one of the miniature modelsor even all of them together may have some property hat doesnot accord with the intended nterpretation n general; and hencethey must always be used in conjunction with heuristic onsidera-tions and informal insights about the latter. Such appeal tointuition cannot be avoided by constructing more elaboratemodel or a larger variety of models, as it will still be needed inorder to decide which of various models that suggest themselvesare admissible. And so the models are to be regarded as an aidin formulating satisfactory ystem, rather than as providing afinal criterion.

We name the models by the number of individuals in eachdomain of individuals (in each possible world). For example,Model 2-2 makes use of two possible worlds, which we shallcall the principal world, or simply the world, and the antiworld,each having two individuals. This method of naming the modelsmay be inconvenient f we wish to allow an infinite number ofpossible worlds, or if we wish to allow two models which havethe same domains of individuals n each possible world but differin regard to the membership of other types. Nevertheless weshall use it at least temporarily. And to remove equivocacy we



REVISED FORMULATION OF THE LOGIC OF SENSE AND DENOTATION 27

specify that in Model 11-12- -ln there are n possible worlds, ofwhich the first world s taken as the principal world and the othersas antiworlds; hat n the ith world there are exactly i individualsas members of the type ; that the members of the type o areexactly t and f, the two truth-values; hat the membership f anytype op consists exactly of all functions rom the type P into thetype o; and that the membership f any simple type y, (i.e., anytype y, that is not a type ocj) consists of the maximum numberof concepts that is permissible under the guiding principle ofAlternative 2), including one and only one concept (0, 0, , 0)that is vacuous in all n worlds.

Heuristically, we think of the individuals as contingentthings, having contingent xistence, but of (e.g.) the truth-valuesas having necessary existence. Admittedly his heuristic notionmust not be pressed too far without more precise specification fit. But the axioms nd discussion n sections and 7 below may erveto give it at least some of the needed precision. And meanwhilewe,follow t in the models by taking lways t and f as the membersof the type o but allowing various different omains of individualsin different orlds.

When, however, two worlds in a particular model have thesame number of individuals, we shall by means of some arbitraryone-to-one correspondence dentify ach individual in one worldwith a corresponding ndividual in the other, and so treat thetwo domains of individuals as the same. Generally this willsimplify he description of the model and will not change theclass of closed formulas of type o which are validated by themodel (in the sense that they denote t in every world). Indeedit may be argued that relations f dentity r non-identity etweenindividuals in different worlds have no genuine significance,except by associating with each individual in one of the worldsa particular oncept of it (in that world) in some arbitrary way;2and hence for the particular purpose of setting up a model, theserelations of identity and non-identity between individuals indifferent orlds are a matter f ndifference nd are to be disposedof as convenient.

2. MODEL2-2. Nearly the simplest model that has more thanminor value for our purpose is Model 2-2, details of which wenow go on to give. As already stated, there are two worlds, whichwe call the (principal) world nd the antiworld. We shall speak ofa member .1of the type oci s being a concept f ,, nd an anticoncept



28 NOUS

of b.1,meaning hat t is a concept of a,, n the world and a conceptof b., in the antiworld. And we shall speak of the value andantivalue of any well-formed ormula closed or not) for a systemof values of any list of variables that ncludes all its free variables,meaning its value in the world and in the antiworld. But sincein Model 2-2 the membership of any type is the same in worldand antiworld, we speak simply of the members f the type, notneeding to distinguish members and antimembers.

That a,l is a concept of a,, implies that any closed formulawhich expresses acl as its sense denotes a,, (i.e., in the world).And that ac,l s an anticoncept f bc,mplies that any closed formulaexpressing .1 as its sense antidenotes bc. The sense of a closedformula s to be (unlike the denotation) always the same in bothworlds.

As we have done in what was just said, we shall sometimesemploy ower-case talic letters or elements of the model, usuallywith subscript oindicate hetype. But as this threatens onfusionwith object-language variables, we shall (as at least a temporaryexpedient) use only the letters a, b, c, d and various lower-caseGreek letters for elements of the model, and then largely avoidusing a, b, c, d (with subscripts) as object-language variables.A few exceptions to this last may be made in cases in which noconfusion an arise.

In Model 2-2 the members of the type o are t and f, andthe members of the type t are 1 and 2. If ot s any simple type,the members of the type ociare all ordered pairs (a,,, bc,),wherea,, and bc, re any members of the type o; all ordered pairs (a,, 0),where a., is any member of the type o; all ordered pairs (0, bc,),where bc, s any member of the type o; and the ordered pair (0, 0).

For example, the members of the type ol are (t, t), (t, f), (f, t),(f, f), (t, 0), (f, 0), (0, t), (0, f), (0, 0); the members of the type t,are (1, 1), (1, 2), (2, 1), (2, 2), (1, 0), (2, 0), (0, 1), (0, 2), (0, 0); andthere are one hundred members of the type ?2, including e.g.((1, 2), (2, 2)), ((1, 2), (2, 0)), ((1, 2), (0, 0)), ((1, 2), 0), ((O, 0), (0, 0)),(0, (0, O)), (0, 0).

As already ndicated, he members of any type op are all thefunctions from the members of the type P into members of thetype oz.Thus there re four members f the type on, ixteen mem-bers of the type ott, ive hundred twelve members of the type ot, )As a recursive definition, et (0, O),1be the ordered pair (0, 0)if y, is a simple type, and let (0, 0),1,6s e the function lsA suchthat i1&91a91 (0, O)c,, or every member a,,1of the type Pi .



REVISEDFORMULATION OF THE LOGIC OF SENSE ANDDENOTATION 29

The concept relation and the anticoncept relation must nowboth be determined n the model. This will require anotherrecursive definition; s a first tep toward this we specify thatif y is a simple type, then (at, b,) is a concept of a, and an anti-concept of b,,; (a, , 0) is a concept of a, and is not an anticonceptof anything; 0, b,) is an anticoncept of b, and is not a conceptof anything; and (0, 0) is neither a concept nor an anticonceptof anything.

When y is not a simple type, t will be convenient o introducethe notations at,, b,), (a, , 0), (0, b,) in such a way that n analogueof the foregoing holds: (at,, b,) is a concept of a, and an anti-concept of b,,; (a, , 0) is a concept of a, and is not an anticonceptof anything; 0, b,) is an anticoncept of b, and is not a conceptof anything; and all remaining members of the type y, (not ofone of these three forms) are neither concepts nor anticonceptsof anything. Thus the notations at, b,), (at,, 0), (0, b,) will standfor ordered pairs only in case y is a simple type symbol; weprovide for the contrary ase by the following ecursivedefinition:

1. (Ke, ,C) is the function whose value for ny argument

of the form aq, b,) is (faRa, b whose value for any argumentof the form a,,, 0) is (40a; , 0), whose value for any argumentof the form 0, b,) is (0, b and whose value for all remainingarguments q, of type P, is (b, 0)aR1 (0, .

2. (f, , 0) is the function whose value for an argumentof either of the forms a, , b,) or (aq, 0) is (faRa, 0), and whosevalue for all remaining rguments of type P, is (0, 0),, .

3. (0, V'a)s the function whose value for an argumentof either of the forms a9 , bq) or (0, b,) is (0, Va1b), and whose

value for all remaining rguments of type P, is (0, 0),, .We say that the function 1 characterizes he function

Cie if, for every ,, and every concept a,9 of a, , 1,, ,9 is a conceptof 0,,,,a,, And we say that anticharacterizes ,3 if, for everya,, and every anticoncept a,9 of a,, +,,1a,9 is an anticoncept of0,,t93a;. From the foregoing definitions f the concept relationand of the anticoncept relation n the model it is clear that (inthe model) every concept of f, characterizes f, and everyanticoncept of 0, anticharacterizes ya; but the converses donot hold.

Now to complete the account of Model 2-2 we turn to whatwe shall call the model-semantics. euristically his s the semanticsof the object language as it would be if Model 2-2 were the true



30 NOUS

model. But we observe that there s a sense in which the actuallyintended meaning of the object language is not given by themodel- emantics of any one model, even the supposed truemodel, as the object language is designed for users who do notknow the true model or know it only in part.3

We state only the semantical rules that determine he valueand the antivalue of any well-formed formula for a system ofvalues of ts free variables. t is understood hat value of a variablemust alwaysbe a member of the type to which the variable belongs(as indicated by its subscript) nd that the value and the antivalueof a well-formed ormula must always be members of the typeto which the formula belongs. Moreover, the value (the antivalue)of a well-formed ormula, for a given system of values of anylist of variables that includes all its free variables, is the sameas its value (its antivalue) for the system of values that resultsby deleting from he list all the variables that are not free variablesof the given formula. The denotation and the antidenotation fa closed formula re then the same as its value and its antivaluefor any system of values of any list of variables (not excludingthe empty list of variables). The sense of a closed formula oftype ox s (a. , ba), where a, is its denotation and ba is its anti-denotation.4 And the sense-value of a well-formed ormula for agiven ystem f sense-values of ts free variables may be determinedanalogously.5

The denotation and the antidenotation of the primitiveconstant COOOs the function oo,such that coooaobos f if aO is tand bo s f, and coooaobos t if either Ois f or bo s t. If the denotationand the antidenotation f C0o0? is Co 0 , the denotation nd theantidenotation f C000 is (CO00 Co0000).

The denotation nd the antidenotation f the constant HO(O.)is the function o(oa) whose value ro(0ob000, ith any member 0.of the type cxas argument, s t if )o. is the function hosevalue?Oba.is t for all arguments a, and is f in all remaining cases.If the denotation and the antidenotation f HI(-I ) is von(o'.%)

the denotation and antidenotation of 170n+(onlan%l) is (7on(Onan)

'TOn(001X)).

The denotation f the constant AO. is the function o.0asuchthat SO.10aa0.1s t if a01 s a concept of a,, and is f in the contrarycase. The antidenotation f 2A10is the function S' such thatS'a laa is t if a01 s an anticoncept f a, and is f in the contrarycase. The denotation nd the antidenotation f 010a2alis (8oa a ) ?31?s).If the denotation nd antidenotation f 21 a +la is 80 a +1? (where



REVISEDFORMULATION OF THE LOGIC OF SENSEAND DENOTATION 31

n is not 0), the denotation and antidenotation f i, 1+2+1 iS

For the principal member f a type we make a definition yrecursion as follows. The principal member of the type o is t.The principal member of the type is 1. In the case of any othersimple type the principal member s (0, 0). The principal memberof the type oj is the function ,x,whose value 0f3a13s, for everymember a,3of the type 3, the principal member of the type oz.

For those type ymbols for which 1,(oq)s a primitive onstant,the denotation and antidenotation of the constant t,9(0,1)s thefunction 610,3(O)hose value 013(0,g)f013,or any member O, of thetype op, s the member

,3f the type P such that Oo,3a,3s t, provided

there is a unique such member a,3of the type 3, and is in thecontrary ase the principal member of the type P. If the denotationand antidenotation of ?13l(e) is 613n(Oe) , the denotation andantidenotation f te+ (On+13n 1) is (f61(o3) X 1n(0n13)).

If a formula consists of a variable x,x tanding lone, its valuefor the value ao of x,x s ao .

For any system of values of the free variables of F0,9A,9ifthe value of Fa, is O,,, nd the value of A,, s a,3, the value of Fo,,,Agis S&13a,;nd if the antivalue of

F,,,is 3&<xand the antivalue of

A,9 s b13, he antivalue of FoA13 s 0,rbrFor a given system of values of its free variables, the value

of Ax,1M, s the function ,x, uch that, for any member a13 f thetype P, the function-value b,,3a,3s the same as the value of Mafor the value a,3of x,1 nd for the given system of values of theremaining reevariables; and the antivalue of Ax,1M,s the functionke such that, for any member a,3 f the type P, the function-valueVJ9f3a13s the same as the antivalue of Ma for the value a13 f x,and for the given system f values of the remaining reevariables.

We now introduce the notion of the n-fold onceptualizationf4% f a function b,. The definition s by recursion s follows.+b%s the same as sb. If + characterizes fr and anti-characterizes 0, then kn+13n is (orn"n , 7 Ifcharacterizes f, and does not anticharacterize ny function,then + iS (o'n , 0). If e, anticharacterizes0abq, nddoes not characterize ny function, hen j n+1, +l is (0, 00 If

neither characterizes nor anticharacterizes ny function,

Then, if Oo, 6 characterizes , we have that 1 isa concept of 4,0, and that Om+' is a concept of Om if m

,n.

Similarly, f O anticharacterizes i we have that 0 1

3



32 NOUS

is an anticoncept f , and that ?,+l 1 is an anticoncept fM if m < n.

For a given system f values of the free variables, f the valueand the antivalue of Ax,Mon are ann nd Oan respectively, hevalue of A,x"Ma s sbNnnnd the antivalue s zb,.

This now completes he statement f the details of Model 2-2,including the model-semantics.

We shall rely on Model 2-2 as our principal test of proposedaxioms, and as a means of establishing bout any system f axiomswe are thus led to that it is consistent nd does not reduce toextensionality. he value of other models s to supplement ntuitionin enabling us to reject axioms which may be validated by theparticular model but seem to be in some degree contrary o theoriginal intention or otherwise doubtful. And as an illustrationwe go on to outline one additional model.

REFERENCES

[1] Carnap, Rudolf, "Replies and Systematic Expositions," The Philosophy ofRudolf Carnap (La Salle, Illinois: 1963): 859-1013; see pp. 889-900, and alsopp. 1045, 1052 of the same book.

[2] Church, Alonzo, "A Formulation of the Logic of Sense and Denotation,"abstract, The Journal of Symbolic ogic 11 (1946): 31.

[3] , "On Carnap's Analysis of Statements of Assertion and Belief,"Analysis 10 (1950): 97-99.

[4] , "A Formulation of the Logic of Sense and Denotation," Structure,Method, nd Meaning, Essays in Honor of Henry M. Sheffer New York: 1951):3-24.

[5] , "The Need for Abstract Entities n Semantic Analysis," Proceedings fthe American Academy of Arts and Sciences 80 (1951): 100-112.

[6] , "Intepjional Isomorphism and Identity of Belief," PhilosophicalStudies 5 (19545: 65-73.

[7] , Introduction o Mathematical Logic, Vol. I (Princeton: 1956).[8] Hailperin, Theodore, "Quantification Theory and Empty Individual-

Domains," The Journal of Symbolic ogic 18 (1953): 197-200.[9] Kaplan, David, Review of Saul Kripke's "Semantical Analysis of Modal

Logic I," The Journal of Symbolic Logic 31 (1966): 120-122.

NOTES

* Part II will be forthcoming n a future ssue of NO1S.'David Kaplan's unpublished dissertation University of California at Los

Angeles, 1964), should also be mentioned as an early study of the question offormalization f the ogic of sense and denotation nd of the use, in this connection,of models along the lines which had been suggested by Carnap.

2 That is, with every ndividual a, in one of the worlds let a concept aL beassociated in such a way that aq is in this first world a concept of a,. Then anindividual d, in the second world is identified with the individual a, in the first



REVISEDFORMULATION OF THE LOGIC OF SENSE ANDDENOTATION 33

world if and only f aq. is in the second world a concept of d. Depending on howthe concepts of the individuals n the first world are chosen, this may sometimes

result in identifying wo different ndividuals in the first world with the sameindividual in the second world. We regard this as not excluded. But our pointhere is a different ne, that there s no basis at all for deciding relations of identityand non-identity etween individuals in different orlds, except with respect tosome choice of concepts of them.

3 Thus the users may have a concept, and extend the language by introducinga name that expresses this concept as its sense, and yet in some sense they maynot know what the concept is a concept of or what the name denotes; e.g. theymay not be able to determine he identity r non-identity f the denotation withother members of the appropriate type which they regard as previously known.On this ground t may be argued that the sense of a name is prior to its denotation,and that the construction f the model misleads by reversing his order, using the

members of (say) the type t as a means of obtaining the concepts of them in thetype tl. This may be admitted, nd yet the models serve their purpose.The reality to which the artificial models correspond s of course that our

ignorance and varying degrees of uncertainty bout details of this world force usto think of ourselves as being in some unknown one of a large number of partlycomprehended possible worlds.

4 Because the object language is designed to express the purely logical andnothing lse (as far s this s possible) and to have only necessary ruths s theorems,we do not admit nto t names which ack either denotation r an antidenotation.If we extend the language in order to be able to express contingent matters, ndadd postulates that constitute theory about the real but contingent world, westill (following Frege) confine ourselves to names which, according to the theory,

have a denotation. Indeed, the validity of each of the rules of inference II andIV requires this.) But in this case there will at least arise names that in somemodels lack an antidenotation n some of the antiworlds. The statement n thetext must then be supplemented appropriately; e.g., in a two-world model, if aclosed formula a name) has a. as its denotation but lacks an antidenotation, tssense is (a. , O).

(A modified ormulation, ith modified ules of nference, o allow denotation-less as well as antidenotationless ames may indeed be worth working out, but itis a complication which we prefer o avoid in the present treatment.)

5 See the discussion of the notion of sense-value in [5]. We observe that ifthe sense-value assigned to one of the variables is not a concept of anything notan anticoncept of anything), then the sense-value of the entire formula must

correspondingly ot be a concept of anything an anticoncept of anything).

church logic sense denotation 1

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