Of those who are approximately my contemporaries, Professor W. V.  Quine is one of the very few to whom I feel I owe the deepest of professional debts, the debt which is owed to someone from whom one has learned something very important about how philosophy should be done, and who has, in consequence, helped to shape one's own mode of thinking.  I hope that he will not think it inappropriate that my offering on this occasion should take the form not of a direct discussion of some part of Word and Object, but rather of an attempt to explore an alternative to one of his central positions, namely his advocacy of the idea of the general eliminability of singular terms, including names. I hope, also, that he will not be too shocked by my temerity in venturing into areas where my lack of expertise in formal logic is only too likely to be exposed. I have done my best to protect myself by consulting those who are in a position to advise me; they have suggested ideas for me to work on and have corrected some of my mistakes, but it would be too much to hope that none remain."  I. THE PROBLEM  It seems to me that there are certain quite natural inclinations which have an obvious bearing on the construction of a predicate calculus. They are as follows:  (I) To admit individual constants; that is to admit names or their representations.  To allow that sometimes a name, like "Pegasus", is not the name of any existent object; names are sometimes 'vacuous. In the light of (2), to allow individual constants to lack designata, so that sentences about Pegasus may be represented in the system. To regard Fa and ~ Fa as 'strong' contradictories; to suppose, that is, that one must be true and the other false in any conceivable state of the world. To hold that, if Pegasus does not exist, then "Pegasus does not fly" (or "It is not the case that Pegasus flies") will be true, while "Pegasus flies" will be false. To allow the inference rules U.I. and E.G. to hold generally, without special restriction, with respect to formulae containing individual constants. To admit the law of identity ((Vx) x=*) as a theorem. To suppose that, if is derivable from , then any statement represented by $ entails a corresponding statement represented by f. It is obviously difficult to accommodate all of these inclinations.  Given [by (7)] (x)x=x we can, given given derive first a =a by U.I. and then (3x)x=a by E.G. It is natural to take (3x)x=a as a representation of 'a exists'. So given (2) and (3), a representation of a false existential statement ("Pegasus exists') will be a theorem. Given (6), we may derive, by E.G., Sx) ~ Ex from ~ Fa. Given (3), this seemingly licenses an inference from "Pegasus does not fly" to "Something does not fly". But such an inference seems illegitimate if, by (5), "Pegasus does not fly" is true if Pegasus does not exist (as (2) allows). One should not be able, it seems, to assert that something does not fly on the basis of the truth of a statement to the effect that a certain admittedly non-existent object does not fly.  To meet such difficulties as these, various manoeuvres are available, which include the following:  To insist that a grammatically proper name N is only admissible as a substituend for an individual constant (is only classifiable as a name, in a certain appropriate sense of 'name') if N has a bearer. So "Pegasus" is eliminated as a substituend, and inclination (3) is rejected. To say that a statement of the form Fa, and again one of the form ~ Fa, presupposes the existence of an object named by a, and lacks a truth-value if there is no such object. [Inclinations (4) and (5) are rejected.] To exclude individual constants from the system, treating ordinary names as being reducible to definite descriptions. [Inclination (I) is rejected.] To hold that "Pegasus" does have a bearer, a bearer which has being though it does not exist, and to regard (3.x) Fx as entailing not theexistence but only the being of something which is F. [Inclination (2) is rejected.] To allow U.I. and E.G. only in conjunction with an additional premise, such as Ela, which represents a statement to the effect that a exists. [Inclination (6) is rejected.] To admit individual constants, to allow them to lack designata, and to retain normal U.I. and E.G.; but hold that inferences made in natural discourse in accordance with the inference-licences provided by the system are made subject to the 'marginal' (extra-systematic) assumption that all names which occur in the expression of such inferences have bearers. This amounts, I think, to the substitution of the concept of *entailing subject to assumption A' for the simple concept of entailment in one's account of the logical relation between the premises and the conclusions of such inferences. [Inclination (8) is rejected.]  I do not, in this paper, intend to discuss the merits or demerits of any of the proposals which I have just listed. Instead, I wish to investigate the possibility of adhering to all of the inclinations mentioned at the outset; of, after all, at least in a certain sense keeping everything. I should emphasize that I do not regard myself as committed to the suggestion which I shall endeavour to develop; my purpose is exploratory.  II. SYSTEM Q: OBJECTIVES  The suggestion with which I am concerned will involve the presentation and discussion of a first-order predicate calculus (which I shall call Q), the construction of which is based on a desire to achieve two goals:  (i) to distinguish two readings of the sentence "Pegasus does not fly" (and of other sentences containing the name "Pegasus" which do not explicitly involve any negation-device), and to provide a formal representation of these readings. The projected readings of "Pegasus does not fly" (S,) are such that on one of them an utterance of S, cannot be true, given that Pegasus does not exist and never has existed, while on the other an utterance of S, will be true just because Pegasus does not exist. ii) to allow the unqualified validity, on either reading, of a step from the assertion of S, to the assertion (suitably interpreted) of "Something [viz., Pegasus] does not fly" (S2).More fully, Q is designed to have the following properties.  U.I. and E.G. will hold without restriction with respect to any formula & containing an individual constant « [Ф(c)]; no additional premise is to be required, and the steps licensed by U.I. and E.G. will not be subject to a marginal assumption or pretence that names occurring in such steps have bearers. For some (a), $ will be true on interpretations of Q which assign no designatum to a, and some such (a) will be theorems of Q. It will be possible, with respect to any @ (a), to decide on formal grounds whether or not its truth requires that a should have a designatun. It will be possible to find, in Q, a representation of sentences such as "Pegasus exists". There will be an extension of Q in which identity is represented. III. SCOPE  The double interpretation of S, may be informally clarified as follows: if S, is taken to say that Pegasus has the property of being something which does not fly, then S, is false (since it cannot be true that a nonexistent object has a property); but if S, is taken to deny that Pegasus has the property of being something which flies, then S, is true (for the reason given in explaining why, on the first interpretation, $, is false).  It seems to be natural to regard this distinction as a distinction between differing possible scopes of the name "Pegasus". In the case of connec-tives, scope-differences mirror the order in which the connectives are introduced in the building up of a formula [the application of formation rules; and the difference between the two interpretations of S, can be represented as the difference between regarding S, as being (i) the result of substituting "Pegasus" for "x" in "x does not fly" (negation having already been introduced), or ii) the result of denying the result of substituting "Pegasus" for "x" in "x flies" (the name being introduced before negation)J.  To deal with this distinction, and to preserve the unrestricted application of U.I. and E.G., Q incorporates the following features:  (1) Normal parentheses are replaced by numerical subscripts which are appended to logical constants and to quantifiers, and which indicatescope-precedence (the higher the subscript, the larger the scope). Subscripts are attached also to individual constants and to bound variables as scope-indicators. For convenience subscripts are also attached to predicate-constants and to propositional letters. There will be a distinction between  (a)  and  (b)  ~, F,a,.  (a) will represent the reading of S, in which S, is false if Pegasus does not exist; in (a) "a" has maximal scope. In (b) "a" has minimal scope, and the non-existence of a will be a sufficient condition for the truth of (b).  So (b) may be taken to represent the second reading of S,. To give further illustration of the working of the subscript notation, in the formula Faz→, Gava H,bs 'v' takes precedence over*→*, and while the scope of each occurrence of "a" is the atomic sub-formula containing that occurrence, the scope of "b" is the whole formula.  (2) The effect of extending scope-indicators to individual constants is to provide for a new formational operation, viz., the substitution of an individual constant for a free variable. The formation rules ensure that quantification takes place only after this new operation has been per-formed; bound variables will then retain the subscripts attaching to the individual constants which quantification eliminates. The following formational stages will be, for example, involved in the building of a simple quantificational formula:  There will be, then, a distinction between 3x4 ~ 2 Fixa, and  (a) will, in Q, be derivable from ~, F,a,, but not from ~ , F,az; (b) will be derivable directly (by E.G.) only from ~, Faz, though it will bederivable indirectly from ~, Fag. This distinction will be further dis-cussed.  (3) Though it was not essential to do so, I have in fact adapted a feature of the system set out in Mates' Elementary Logic; free variables do not occur in derivations, and U.I. always involves the replacement of one or more subscripted occurrences of a bound variable by one or more correspondingly subscripted occurrences of an individual constant.  Indeed, such expressions as Fix and G,xzV, are not formulae of Q (though to refer to them I shall define the expression "segment"). F,x and G,xy are formulae, but the sole function of free variables is to allow the introduction of an individual constant at different formational stages.  Faz→, G,agV 4 H, is admitted as a formula so that one may obtain from it a formula giving maximal scope to "b", viz., the formula  (4) Closed formulae of a predicate calculus may be looked upon in two different ways. The symbols of the system may be thought of as lexical items in an artificial language. Actual lexical entries (lexical rules) are provided only for the logical constants and quantifiers; on this view an atomic formula in a normal calculus, for example Fa, will be a categorical subject-predicate sentence in that language. Alternatively, formulae may be thought of as structures underlying, and exemplified by, sentences in a language (or in languages) the actual lexical items of which are left unidentified. On this view the formula Fav Gb will be a structure exemplified by a sub-class of the sentences which exemplify the structure Fa. The method of subscripting adopted in Q reflects the first of these approaches; in an atomic formula the subscripts on individual constants are always higher than that on the predicate-constant, in consonance with the fact that affirmative categorical subject-predicate sentences, like "Socrates is wise" or "Bellerophon rode Pegasus", imply the non-vacuousness of the names which they contain. Had I adopted the second approach, I should have had to allow not only F,az, etc., but also Fa,, etc., as formulae; I should have had to provide atomic formulae which would have substitution instances, e.g., F,a,→ G,b, in which the scope of the individual constants does not embrace the whole formula.  The second approach, however, could be accommodated with appropriate changes.(5) The significance of numerical subscripts is purely ordinal; so, for example, ~ Fa, and ~17 Fa, will be equivalent. More generally, any pair of "isomorphs" will be equivalent, and Q contains a rule providing for the interderivability of isomorphs. and & will be isomorphs iff (1) subscripts apart, @ and ( are identical, and (2) relations of magnitude (=, <,>) holding between any pair of subscripts in @ are preserved between the corresponding pair of subscripts in & [the subscripts in & mirror those in @ in respect of relative magnitudes].  Professor C. D. Parsons has suggested to me a notation in which I would avoid the necessity for such a rule, and has provided me with an axiom-set for a system embodying it which appears to be equivalent to Q (Mr. George Myro has made a similar proposal). The idea is to adopt the notation employed in Principia Mathematica for indicating the scope of definite descriptions. Instead of subscripts, normal parentheses are retained and the scope of an individual constant or bound variable is indicated by an occurrence of the constant or variable in square brackets, followed by parentheses which mark the scope boundaries. So the distinction between ~, F,a, and ~, Fa, is replaced by the distinction between ~[a] (Fa) and [a] (~Fa); and the distinction between Jxy~,F,x2 and 3xa~,F,x, is replaced by the distinction between  (x) (~ [x] (Ex)) and (3x) ([x] (~ Fx)). Parson's notation may well be found more perspicuous than mine, and it may be that I should have adopted it for the purposes of this paper, though I must confess to liking the obviousness of the link between subscripts and formation-rules.  The notion of scope may now be precisely defined for Q.  If y be a logical constant or quantifier occurring in a closed formula , the scope of an occurrence of y is the largest formula in @ which (a) contains the occurrence of n, (b) does not contain an occurrence a logical constant or quantifier bearing a higher subscript than that which attaches to the occurrence of „. If , be a term (individual constant or bound variable), the scope of , is the largest segment of @ which (a) contains the occurrence of n, (b) does not contain an occurrence of a logical constant bearing a higher subscript than that which attaches to the occurrence of n.  (3) A segment is a sequence of symbols which is either (a) a formula or (b) the result of substituting subscript-preserving occurrences ofvariables for one or more occurrences of individual constants in a formula.  We may now define the important related notion of "dominance". A term 0 dominates a segment @ ift @ falls within the scope of at least one of the occurrences, in , of 0. In other words, 0 dominates @ if at least one occurrence of 0 in @ bears a subscript higher than that attaching to any logical constant in @. Dominance is intimately connected with existential commitment, as will be explained.  IV. NATURAL DEDUCTION SYSTEM Q  A. Glossary  If "n" denotes a symbol of Q, "y." denotes the result of attaching, to that symbol, a subscript denoting n. "Ф(c, a)"= a formula @ containing occurrences of an individual constant a, each such occurrence being either an occurrence of aj, or of..., or of o'. [Similarly, if desired, for "$(ap,...w,)", where "o" ('omega') denotes a variable.] "Ф" ="a formula, the highest subscript within which denotes n". If 0, and 0, are terms (individual constants or bound variables). *(02/0,) -the result of replacing each occurrence of 0, in d by an occurrence of 0z, while preserving subscripts at substitution-points'.  • [The upper symbol indicates the substituend.]  B. Provisional Set of Rules for Q  1. Symbols  (a) Predicate-constants (*F", "fl"  ,... "G .)  (b) Individual constants ("a", 'a'*  '.... "b"  .( ...  (c) Variables (*x",  ... "y"...).  (d) Logical constants ("~"  , "&", "v", "→").  (e) Quantification-symbols (V, 9"). [A quantification-symbol followed by a subscripted variable is a quantifier.]  Numerical subscripts (denoting natural numbers). Propositional letters (*p", "q",....). 2. Formulae  A subscripted n-ary predicate constant followed by n unsubscripted variables; a subscripted propositional letter. If i is a formula, $(*,+m/∞) is a formula. If is a formula, Vo+Ф(∞/«) is a formula. [NB: Substitutions are to preserve subscripts.] If m is a formula, 3c, +Ф (∞/x) is a formula. [NB: Substitutions are to preserve subscripts.] If » is a formula, ~+m is a formula. If i-m and to-n are formulae, ф 8,4, ф. 4, ф→, 4 are for- mulae. is a formula only if it can be shown, by application of (1)-(6), that p is a formula. 3. Inference-Rules  (1) [Ass] Any formula may be assumed at any point.  ....中トリャー」&』～コース♥ローキーは。then  ,... ф*+~+ ф'.  「「ゆく。♥［m-n  (6 [v+] etnml)-*-nYa 0  (7) [v -] 1f(1) 4[-m)»Ф  (2) Хра- 17+ ф₴  ・がトら、  (3) ф°  then  (4) ф'  (8) [→ +, CP] If Ф(п-и]- 41  -…・・ロートスin-ns then o  (10) [V+] If v*  ,... w*F then v'.... v*+V@n+m $ (w/∞), provided  that a does not occur in '  (I1) [V-]V,Ф+ф(x∞), provided that Vo, is the scope of Va.  (*+) +30,+mV, where v is like except that, if a occurs in ф, at least one such occurrence is replaced in & by an occurrence of . (В-)Зо„Ф, x'... x*Hy if (a/0), x'... x*H/, provided that 3c,ф is the scope of 3o, that a does not occur in any of , x',.... x*, v. [NB. All substitutions referred to in (10)-(13) will preserve sub-scripts.]  Rules (I) (13) are not peculiar to Q, except insofar as they provide for the use of numerical subscripts as substitutes for parentheses. The role of term-subscripts has so far been ignored. The following three rules do not ignore the role of term-subscripts, and are special to Q.  (14) [Dom +]If(1) a dominates ,  (2) p,x,Rtw(a)0),  then  (3) ф, x). x'+* ((2, +m/c,),... (Фк+п/ок)) [m. п> 0].  [NB. v, thus altered, must remain a formula; for example, a must not acquire a subscript already attaching to a symbol other than x.]  (14) provides for the raising of subscripts on a in 4, including the case in which initially non-dominant a comes to dominate f.  [A subscript on an occurrence of a may always be lowered.]  (16) [Iso] If @ and y are isomorphs, @+v.  V. EXISTENCE  A. Closed Formulae Containing an Individual Constant &  (i) If a dominates @ then, for any interpretation Z, @ will be true on Z only if a is non-vacuous (only if Ta+exists? is true, where '+' is a  concatenation-symbol). If a does not dominate , it may still be the case that @ is true only if a is non-vacuous (for example if @="~, ~3 F,az" or ="FazV, G,az", though not if ="F,a→,G,az"). Whether or  not it is the case will be formally decidable. Let us abbreviate " is true only if a is non-vacuous" as "ф is E-committal for &". The conditions in which ф is E-committal for a can be specified recursively:  (1)  If a dominates , is E-committal for a.  (2)  If =~,~=-mV, and is E-committal for a, then @ is E-committal for a.  (3)  If =v&,x, and either or x is E-committal for a, then $ is E-committal for a.  (4)  If =v.x, and both y and & are E-committal for a, then ф is E-committal for a.  (5)  If =→x, and both ~_* and z are E-committal for a, then ф is E-committal for a [in being greater than the number denoted by any non-term-subscript in 4].  (6)  If =Vo, or 3o,v, and (B/∞) is E-committal for x, then ф is E-committal for a.  (ii) Since Fja, → ,F,a, is true whether or not "a" is vacuous, the truth of F,a,→, Fa, (in which "a" has become dominant) requires only that a exists, and so the latter formula may be taken as one representation of  "a exists". More generally, if (for some n) a is the only individual constant in » (x) and =→n-m then @ may be taken as a representation of Ta + exists?  B. 3-quantified Formulae  An 3-quantified formula 3o,ф will represent a claim that there exists an object which satisfied the condition specified in ¢ iff (a/∞) is E-com-mittal for o. To illustrate this point, compare (i)  3x4~, Fix, and  (ii)  ヨxュ～3Fix2.  Since ~, Fa, is E-committal for "a" (is true only if a exists) while  ~, F,a, is not E-committal for "a", (i) can, and ii) cannot, be read as a claim that there exists something which is not F. The idea which lies behind the treatment of quantification in Q is that while i) and ii) may be taken as representing different senses or different interpretations of  "something is not F" or of "there is something which is not F", these locutions must be distinguished from "there exists something which is not F"', which is represented only by (i). The degree of appeal which Q will have, as a model for natural discourse, will depend on one's willingness to distinguish, for example,  "There is something such that it is not the case that it flies" from "There is something such that it is something which does not fly", and to hold that (a) is justified by its being false that Pegasus flies, while (b) can be justified only by its being true of some actual object that it does not fly. This distinction will be further discussed in the next section.  Immediately, however, it must be made clear that to accept Q as a model for natural discourse is not to accept a Meinongian viewpoint; it is not to subscribe to the idea of a duality, or plurality, of 'modes of being'. Acceptance of Q as a model might be expected to lead one to hold that while some sentences of the form "Bertrand Russell _" will be interpretable in such a way as i) to be true, and (ii) to entail not merely "there is something which _  " but also "there exists something  which __", sentences of the form "Pegasus _  " will, if interpreted  so as to be true, entail only "there is something which _  -". But from  this it would be quite illegitimate to conclude that while Bertrand Russell both exists and is (or has being), Pegasus merely is (or has being). "Exists" has a licensed occurrence both in the form of expression "there exists something which " and in the form of expression "a exists"; "is" has a licensed occurrence in the form of expression "there is something which ___", but not in the form "a is". Q creates no ontological jungle.  VI. OBJECTION CONSIDERED  It would not be surprising if the combination of the admissibility, according to the natural interpretation of Q, of appropriate readings of  the inference-patterns  a does not exist a is not F  and (2)  a is (not) F  something is (not) F  have to be regarded as Q's most counter-intuitive feature.  Consider the following dialogue between A and B at a cocktail party:  Al) Is Marmaduke Bloggs here tonight?  B(1) Marmaduke Bloggs?  A(2) You know, the Merseyside stock-broker who last month climbed  Mt. Everest on hands and knees.  B(2) Oh! Well no, he isn't here.  A(3) How do you know he isn't here?  B(3) That Marmaduke Bloggs doesn't exist; he was invented by the  journalists.  A(4) So someone isn't at this party.  B(4) Didn't you hear me say that Marmaduke Bloggs does not exist?  A(5) I heard you quite distinctly; are you under the impression that you heard me say that there exists a person who isn't at this party?  B, in his remarks (3) and (4), seemingly accepts not only inference-pattern (I) but also inference-pattern (2).  The ludicrous aspects of this dialogue need to be accounted for. The obvious explanation is, of course, that the step on which B relies is at best dubious, while the step which A adds to it is patently illegitimate; if we accept pattern (I) we should not also accept pattern (2). But there is another possible explanation, namely that  (i given (P) "a does not exist and so a is not F" the putative conclusion from (P), "Something is not F" (C), is strictly speaking (on one reading) true, but  il) given that (P) is true there will be something wrong, odd, or misleading about saying or asserting (C).  In relation to this alternative explanation, there are two cases to con-  (a) that in which the utterer of (C) knows or thinks that a does not  exist, and advances (C) on the strength of this knowledge or belief; but the non-existence of a is not public knowledge, at least so far as the speaker's audience is concerned;  (b) that which differs from (a) in that all parties to the talk-exchange are aware, or think, that a does not exist. Case (a) will not, perhaps, present too great difficulties; if there is a sense of "Something is not F" such that for this to be true some real thing must fail to be F, the knowledge that in this sense something is not F will be much more useful than the knowledge that something is not F in the other (weaker) sense; and ceteris paribus one would suppose the more useful sense of (C) to be the more popular, and so, in the absence of counter-indications, to be the one employed by someone who utters (C). Which being the case, to utter  (C) on the strength of the non-existence of a will be misleading.  Case (b) is less easy for the alternative explanation to handle, and my dialogue was designed to be an example of case (b). There is a general consideration to be borne in mind, namely that it will be very unplausible to hold both that there exists a particular interpretation or sense of an expression E, and that to use E in this sense or interpretation is always to do something which is conversationally objectionable. So the alternative explanation will have (l) to say why such a case (b) example as that provided by the dialogue is conversationally objectionable, (2) to offer some examples, which should presumably be case (b) examples, in which the utterance of (C), bearing the putative weaker interpretation would be conversationally innocuous. These tasks might be attempted as follows.  (I) To say "Something is (not) such-and-such" might be expected to have one or other of two conversational purposes; either to show that it is possible (not) to be such-and-such, countering (perhaps in anticipation) the thesis that nothing is even (not) such-and-such, or to provide a prelude to the specification (perhaps after a query) of an item which is (not) such-and-such. A's remark (4) "So someone is not at this party" cannot have either of these purposes. First, M.B. has already been agreed by A and B not to exist, and so cannot provide a counter-example to any envisaged thesis that every member of a certain set (c.g. leading local business men) is at the party. M.B., being non-existent, is not a member of any set. Second, it is clear that A's remark (4) was advanced on the strength of the belief that M.B. does not exist; so whatever specification is relevant has already been given.  (2) The following example might provide a conversationally innocuous use of (C) bearing the weaker interpretation. The cocktail party is a special one given by the Merseyside Geographical Society for its members in honour of M.B., who was at the last meeting elected a member as a recognition of his reputed exploit. A and B have been, before the party, discussing those who are expected to attend it; C has been listening, and is in the know about M.B.  C Well, someone won't be at this party  A, B Who?  C Marmaduke Bloggs  A, B But it's in his honour  C That's as may be, but he doesn't exist; he was invented by the  journalists.  Here C makes his initial remark (bearing putative weak interpretation), intending to cite M.B. in specification and to disclose his non-existence.  It should be made clear that I am not trying to prove the existence or admissibility of a weaker interpretation for (C); I am merely trying to show that the prima facie case for it is strong enough to make investigation worth-while; if the matter is worth investigation, then the formulation of Q is one direction in which such investigation should proceed, in order to see whether a systematic formal representation of such a reading of "Something is (not) F" can be constructed.  As a further consideration in favour of the acceptability of the weaker interpretation of "Something is (not) F", let me present the following  "slide":  To say "M.B. is at this party" would be to say something which is not true. To say "It is not true that M.B. is at this party" would be to say something which is true. To say "M.B. is not at this party" would be to say something which is true. M.B. is not at this party. M.B. can be truly said not to be at this party. Someone (viz. M.B.) can be truly said not to be at this party. Someone is not at this party (viz. M.B.).It seems to me plausible to suppose that remark (I) could have been uttered with truth and propriety, though with some inelegance, by B in the circumstances of the first dialogue. It also seems to me that there is sufficient difficulty in drawing a line before any one of remarks (2) to (7), and claiming that to make that remark would be to make an illegitimate transition from its legitimate predecessor, for it to be worth considering whether one should not, given the non-existence of M.B., accept all seven as being (strictly speaking) true. Slides are dangerous instruments of proof, but it may be legitimate to use them to back up a theoretical proposal. VIL. IDENTITY  So far as I can see, there will be no difficulty in formulating a system Q', as an extension of Q which includes an identity theory. In a classical second-order predicate calculus one would expect to find that the formula (VF) (Fa→Fb) (or the formula (VF) (Fa-›Fb)) is a definitional sub-stituend for, or at least is equivalent to, the formula a =b. Now in Q the sequence Fa→Fb will be incomplete, since subscripts are lacking, and there will be two significantly different ways of introducing subscripts, (i F,as→2F,be and (ii) Faz→, F,b,. In (i "a" and "b" are dominant,  and the existence of a and of b is implied; in ii) this is not the case. This difference of subscripting will reappear within a second-order predicate calculus which is an extension of Q; we shall find both (i) (a) VF,F,a,→, F,b, and (ii) (a) VE,F,a2→4 F,b,. If we introduce the symbol  * into Q, we shall also find iii) VF, F,a,,F,ba and (iv) VF,F,a,**F,b,. We may now ask whether we want to link the identity of a and b with the truth of (iii) or with the truth of (iv), or with both. If identity is linked with (iii) then any affirmative identity-formula involving a vacuous individual constant will be false; if identity is linked with (iv) any affirmative identity formula involving two vacuous individual  constants will be true.  A natural course in this situation seems to be to admit to Q' two types of identity formula, one linked with (iii) and one with (iv), particularly if one is willing to allow two interpretations of (for example) the sentence  "Pegasus is identical with Pegasus", on one of which the sentence is false because Pegasus does not exist, and on the other of which the sentence is true because Pegasus does not exist (just as "Pegasus is identical withBellerophon" will be true because neither Pegasus nor Bellerophon exist). We cannot mark this distinction in Q simply by introducing two different identity-signs, and distinguishing between (say) a,=,b, and a,=, b3. Since in both these formulae "a" and "b" are dominant, the formulae will be true only if a and b exist. Just as the difference between (iii) and (iv) lies in whether "a" and "b" are dominant or non-dominant, so must the difference between the two classes of identity formulae which we are endeavouring to express in Q'. So Q' must contain both such formulae as az=,b, (strong' identity formulae) and such formulae as aj=,b2 ('weak' identity formulae). To allow individual constants to be non-dominant in a formula which is not molecular will be a temporary departure from the practice so far adopted in Q; but in view of the possibility of eventually defining "=" in a second-order calculus which is an extension of Q one may perhaps regard this departure as justified.  Q' then might add to Q  one new symbol, "="; two new formation rules; (1)  ' =,? is a formula,  (2)  If aj+ =, Bj+, is a formula, &,+ =-Bj+, is a formula, where m> j+k and m> j+ 1.  (c) two new inference-rules  (I)  (2)  A-Vo,+,C0,-,-,0,-, [a weak identity law],  a, - Be. ф+ф(Ba). [There is substitutivity both on strong and on weak identity.]  I hope that these additions would be adequate, though I have not taken steps to assure myself that they are. I might add that to develop a representation of an interesting weak notion of identity, one such that Pegasus will be identical with Pegasus but not with Bellerophon, I think that one would need a system within which such psychological notions as "it is believed that" were represented.  VIII. SEMANTICS FOR Q  The task of providing a semantics for Q might, I think, be discharged inmore than one way; the procedure which I shall suggest will, I hope, continue the following features: (a) it will be reasonably intuitive, (b) it will not contravene the philosophical ideas underlying the construction of Q by, for example, invoking imaginary or non-real entities, (c) it will offer reasonable prospects for the provision of proofs of the soundness and completeness of Q (though I must defer the discussion of these prospects to another occasion).  A. Interpretation  The provision of an interpretation Z for Q will involve the following steps:  The specification of a non-empty domain D, within which two sub-domains are to be distinguished: the special sub-domain (which may be empty), the elements of which will be each unit set in D whose element is also in D; and the residual sub-domain, consisting of all elements of D which do not belong to the special sub-domain. The assignment of each propositional letter either to 1 or to 0. The assignment of each -ary predicate constant y to a set (the E-set of y) of ordered n-tuples, each of which has, as its elements, elements of D. An E-set may be empty. The assignment of each individual constant a to a single clement of D (the correlatum of a). If the correlatum of a belongs to the special sub-domain, it will be a unit-set whose element is also in D, and that element will be the designatum of a. If the correlatum of a is not in the special sub-domain, then & will have no designatum. [I have in mind a special case of the fulfilment of step (4), in which every individual constant has as its correlatum either an element of the special sub-domain or the null-set. Such a method of assignment seems particularly intuitive.] If an individual constant a is, in Z, assigned to a correlatum belonging to the special sub-domain, I shall say that the assignment of a is efficient. If, in Z, all individual constants are efficiently assigned, I shall say that Z is an efficient interpretation of Q  It will be noted that, as I envisage them, interpretations of Q will be of a non-standard type, in that a distinction is made between the correlation of an individual constant and its description. All individual constants are given correlata, but only those which on a given interpretation are non-vacuous have, on that interpretation, designata. Interpretations of this kind may be called Q-type interpretations.I shall use the expressions "Corr (1)" and "Corr (O)" as abbreviations, respectively, for "correlated with 1" and "correlated with 0"  *. By "atomic  formula" I shall mean a formula consisting of a subscripted n-ary predicate constant followed by a subscripted individual constant.  I shall, initially, in defining "Corr(1) on Z" ignore quantificational formulae.  If ф is atomic, @ is CorrI) on Z iff i) each individual constant in has in Z a designatum (i.e. its correlatum is a unit set in D whose element is also in D), and ii) the designata of the individual constants in , taken in the order in which the individual constants which designate them occur in , form an ordered n-tuple which is in the E-set assigned in Z to the predicate constant in ф. If no individual constant dominates , is Corr(1) on Z ifl (i If =~,V, y is Corr(0) on Z; (ii)  If =v&,x. v and z are each Corr(1) on Z;  (ili)  If ф=wv. X, either or y is Corr(1) on Z;  (iv)  If =/→,x, either is Corr(0) on Z or x is Corr(1) on Z.  If (x) is a closed formula in which & is non-dominant, and if is like « except that & dominates $, then is Corr(1) on Z iff i) v is Corr(1) on Z and (ii) a is efficiently assigned in Z. If a closed formula is not Corr(1) on Z, then it is Corr(0) on Z. To provide for quantificational formulae, some further notions are required.  An interpretation Z' is an i.c.-variant of Z iff Z' differs from Z (if at all) only in that, for at least one individual constant a, the correlatum of a in Z' is different from the correlatum of a in Z. Z' is an efficiency-preserving i.c.-variant of Z iff Z' is an i.c.-variant of Z and, for any a, if a is efficiently assigned in Z a is also efficiently assigned in Z'. Z' is an efficiency-quota-preserving i.c.-variant of Z iff Z' is an i.c.-variant of Z and the number of individual constants efficiently assigned in Z' is not less than the number efficiently assigned in Z.' Let us approach the treatment of quantificational formulae by considering the 3-quantifier. Suppose that, closely following Mates's procedure in Elementary Logic, we stipulate that Jw,ф is CorrI) on Z iff $ (a'/∞)is Corr (1) on at least one i.c.-variant of Z, where a is the first individual constant in Q. (We assume that the individual constants of Q can be ordered, and that some principle of ordering has been selected). In other words, 3w,ф will be Corr(I) on Z iff, without altering the assignment in Z of any predicate constant, there is some way of assigning &' so that ф (a/∞) is Corr(l) on that assignment. Let us also suppose that we shall define validity in Q by stipulating that @ is valid in Q iff, for any interpretation Z, ф is Corr(1) on Z.  We are now faced with a problem. Consider the "weak existential" formula 3x2~, F,x,. If we proceed as we have just suggested, we shall be forced to admit this formula as valid; if "a" is the first individual constant in Q, we have only to provide a non-efficient assignment for  "a" to ensure that on that assignment ~, Fa, is Corr(1); for any interpretation Z, some i.c.-variant of Z will provide such an assignment for "a", and so 3x4~3 F,x2 will be CorrI) on Z. But do we want to have to admit this formula as valid? First, if it is valid then I am reasonably sure that Q, as it stands, is incomplete, for I see no way in which this formula can be proved. Second, if in so far as we are inclined to regard the natural language counterparts of valid formulae as expressing conceptual truths, we shall have to say that e.g. "Someone won't be at this party", if given the 'weak' interpretation which it was supposed to bear in the conversations imagined in Section VI, will express a conceptual truth; while my argument in that section does not demand that the sentence in question express an exciting truth, I am not sure that I welcome quite the degree of triviality which is now threatened.  It is possible, however, to avoid the admission of 3x,~,F,x2 as a valid formula by adopting a slightly different semantical rule for the  3-quantifier. We stipulate that 3o,$ is Corr(I) on Z iff @ (c'/co) is Corr(I) on at least one efficiency-preserving i.c.-variant of Z. Some interpretations of Q will be efficient interpretations, in which "a" will be efficiently assigned; and in any efficiency-preserving i.c.-variant of such an interpretation "a" will remain efficiently assigned; moreover among these efficient interpretations there will be some in which the E-set assigned to  "F" contains (to speak with a slight looseness) the member of each unit-set belonging to the special sub-domain. For any efficient interpretation in which "p" is thus assigned, F,a, will be Corr(1), and ~ , F,a, will be Corr(0), on all efficiency-preserving i.c. -variants.So 3x4~gF,xz will not be Corr(1) on all interpretations, i.e. will not be valid.  A similar result may be achieved by using the notion of an efficiency-quota-preserving i.c.-variant instead of that of an efficiency-preserving i.c.-variant; and the use of the former notion must be preferred for the following reason. Suppose  that we use the latter notion;  (ii)  (iii)  that "a?" is non-efficiently assigned in Z;  that "a" is the first individual constant, and is efficiently assigned in Z;  (iv)  that "F" includes in its extension the member of each unit-set in the special sub-domain.  Then ~, Faz is Corr(1) on Z, and so (by E.G.) 3x2~, Fix, is Corr(1) on Z. But "a" is efficiently assigned in Z, so ~g F,a, is Corr(0) on every efficiency-preserving i.c.-variant of Z (since "F" includes in its extension every designable object). So x~, F,*z is Corr(0) on Z.  This contradiction is avoided if we use the notion of efficiency-quota-preserving i.c.-variant, since such a variant of Z may provide a non-efficient assignment for an individual constant which is efficiently assigned in Z itself; and so 3xz~, F,x, may be Corr(I) on Z even though "a" is efficiently assigned in Z.  So I add to the definition of "Cort(I) on Z", the following clauses:  (5) If =Vo,k, is CorrI) on Z, iff V(a'/a) is Corr(1) on every  efficiency-quota-preserving i.c.-variant of Z.  (6) If ф =3o,/, is Corr(1) on Z iff y (x'/c) is Corr(1) on at least  one efficiency-quota-preserving i.c.-variant of Z.  [In each clause, "a is to be taken as denoting the first individual constant in Q.]  Validity may be defined as follows:  ф is valid in Q iff, for any interpretation Z, ф is Corr(1) on Z.  Finally, we may, if we like, say that p is true on Z iff p is CorrI) on Z.  IX. NAMES AND DESCRIPTIONS  It might be objected that, in setting up Q in such a way as to allow for the representation of vacuous names, I have ensured the abandonment, at least in spirit, of one of the desiderata which I have had in mind; for(it might be suggested) if Q is extended so as to include a Theory of Descriptions, its individual constants will be seen to be indistinguishable, both syntactically and semantically, from unanalysed definite descrip-tions; they will be related to representations of descriptions in very much the same way as propositional letters are related to formulae, having lost the feature which is needed to distinguish them from representations of descriptions, namely that of being interpretable only by the assignment of a designatum.  I do not propose to prolong this paper by including the actual presentation of an extension of Q which includes the representation of descrip-tions, but I hope to be able to say enough about how I envisage such an extension to make it clear that there will be a formal difference between the individual constants of Q and definite descriptions. It is a familiar fact that there are at least two ways in which a notation for representing definite descriptions may be developed within a classical system; one may represent "The haberdasher of Mr. Spurgeon is bald" either by (1) G(1x. Ex) or by (2) (9x. Fx) Gx; one may, that is, treat "ix. Fx" either as a term or as being analogous to a (restricted) quantifier. The first method does not allow for the representation of scope-differences, so a general decision will have to be taken with regard to the scope of definite de-scriptions, for example that they are to have maximal scope. The second method does provide for scope-distinctions; there will be a distinction between, for example, (ix. Fx) ~ Gx and ~(1x. Fx) Gx. The apparatus of Q, however, will allow us, if we wish, to combine the first method, that of representing definite descriptions by terms, with the representation of differences of scope; we can, if we like, distinguish between c.g., ~,G,ax,F,x, and ~,G,1xgF,xz, and ensure that from the first formula we may, and from the second we may not, derive E!, 1x, F,*2. We might, alternatively, treat descriptions as syntactically analogous to restricted quantifiers, if we so desire. Let us assume (arbitrarily) that the first method is adopted, the scope-boundaries of a descriptive term being, in each direction, the first operator with a higher subscript than that borne by the iota-operator or the first sentential boundary, whichever is nearer.  Let us further assume (perhaps no less arbitrarily) that the iota-operator is introduced as a defined expression, so that such a formula as  nitional substitution for the right-hand side of the formulaG, xgF,x2→4G,x,F,x2, together with applications of the rules for subscript-adjustment.  Now, as I envisage the appropriate extension of Q, the formal difference between individual constants and descriptive terms will lie in there being a legitimate step (by E. G.) from a formula containing a non-dominant individual constant to the related "weak' existential form, e.g.. from ~, Faz to 3x4~, F,x2, while there will, for example, be no analogous step from ~ G, 1x, Fxz to 3x4~, G,x2. Such a distinction between individual constants and descriptive terms seems to me to have, at least prima facie, a basis in intuition; I have at least some inclination to say that, if Mr. Spurgeon has no haberdasher, then it would be true (though no doubt conversationally odd) to say "It is not the case that Mr. Spurgeon's haberdasher is bald" (S), even though no one has even suggested or imagined that Mr. Spurgeon has a haberdasher; even though, that is, there is no answer to the question who Mr. Spurgeon's haberdasher is or has been supposed to be, or to the question whom the speaker means by the phrase "Mr. Spurgeon's haberdasher." If that inclination is admissible, then it will naturally be accompanied by a reluctance to allow a step from S to "Someone is not bald" (S,) even when S, is given its 'weak' interpretation. I have, however, already suggested that an utterance of the sentence "It is not the case that Mr. Spurgeon is bald" (S') is not assessable for truth or falsity unless something can be said about who Mr. Spurgeon is or is supposed to be; in which case the step from S' to S, (weakly interpreted) seems less un-justifiable.  I can, nevertheless, conceive of this argument's failing to produce conviction. The following reply might be made: "If one is given the truth of S, on the basis of there being no one who is haberdasher to Mr. Spur-geon, all one has to do is first to introduce a name, say 'Bill', laying down that 'Bill' is to designate whoever is haberdasher to Mr. Spurgeon, then to state (truly) that it is not the case that Bill is bald (since there is no such person), and finally to draw the conclusion (now legitimate) that someone is not bald (on the 'weak' reading of that sentence). If only a stroke of the pen, so to speak, is required to legitimize the step from S to S, (weakly interpreted), why not legitimize the step directly, in which case the formal distinction in Q" between individual constants and descriptive terms must either disappear or else become wholly arbitrary?"A full treatment of this reply would, I suspect, be possible only within the framework of a discussion of reference too elaborate for the present occasion; I can hope only to give an indication of one of the directions in which I should have some inclination to proceed. It has been observed? that a distinction may be drawn between at least two ways in which descriptive phrases may be employed.  (I) A group of men is discussing the situation arising from the death of a business acquaintance, of whose private life they know nothing, except that (as they think) he lived extravagantly, with a household staff which included a butler. One of them says "Well, Jones' butler will be seeking a new position".  (2) Earlier, another group has just attended a party at Jones' house, at which their hats and coats were looked after by a dignified individual in dark clothes and a wing-collar, a portly man with protruding ears, whom they heard Jones addressing as "Old Boy", and who at one point was discussing with an old lady the cultivation of vegetable marrows.  One of the group says "Jones' butler got the hats and coats mixed up". i The speaker in example (1) could, without impropriety, have inserted after the descriptive phrase "Jones' butler" the clause "whoever he may be". It would require special circumstances to make a corresponding insertion appropriate in the case of example (2). On the other hand we may say, with respect to example (2), that some particular individual has been 'described as', 'referred to as', or 'called' Jones' butler by the speaker; furthermore, any one who was in a position to point out that Jones has no butler, and that the man with the protruding ears was Jones gardener, or someone hired for the occasion, would also be in a position to claim that the speaker had misdescribed that individual as Jones' butler. No such comments are in place with respect to example (I). (ii) A schematic generalized account of the difference of type between examples (I) and (2) might proceed along the following lines. Let us say that X has a dossier for a definite description & if there is a set of definite descriptions which includes &, all the members of which X supposes (in one or other of the possible sense of 'suppose") to be satisfied by one and the same item. In a type (2) case, unlike a type (I) case, the speaker intends the hearer to think (via the recognition that he is so intended) (a) that the speaker has a dossier for the definite description & which he has used, and (b) that the speaker has selected from this dossier at least partlyin the hope that the hearer has a dossier for & which 'overlaps' the speaker's dossier for & (that is, shares a substantial, or in some way specially favoured, subset with the speaker's dossier). In so far as the speaker expects the hearer to recognize this intention, he must expect the hearer to think that in certain circumstances the speaker will be prepared to replace the remark which he has made (which contains 8) by a further remark in which some element in the speaker's dossier for & is substituted for d. The standard circumstances in which it is to be supposed that the speaker would make such a replacement will be (a) if the speaker comes to think that the hearer either has no dossier for &, or has one which does not overlap the speaker's dossier for & (i.e., if the hearer appears not to have identified the item which the speaker means or is talking about), (b) if the speaker comes to think that & is a misfit in the speaker's dossier for , i.e., that & is not, after all, satisfied by the same item as that which satisfies the majority of, or each member of a specially favoured subset of, the descriptions in the dossier. In example (2) the speaker might come to think that Jones has no butler, or that though he has, it is not the butler who is the portly man with the protruding ears, etc., and whom the speaker thinks to have mixed up the hats and coats. (iii) If in a type (2) case the speaker has used a descriptive phrase (e.g.,  "Jones' butler") which in fact has no application, then what the speaker has said will, strictly speaking, be false; the truth-conditions for a type (2) statement, no less than for a type (I) statement, can be thought of as being given by a Russellian account of definite descriptions (with suitable provision for unexpressed restrictions, to cover cases in which, example, someone uses the phrase "the table" meaning thereby "the table in this room"). But though what, in such a case, a speaker has said may be false, what he meant may be true (for example, that a certain particular individual [who is in fact Jones' gardener] mixed up the hats and coats).  Let us introduce two auxiliary devices, italics and small capital let-ters, to indicate to which of the two specified modes of employment a reported use of a descriptive phrase is to be assigned. If I write "S said 'The Fis G'," I shall indicate that S was using "the F" in a type  (1), non-identificatory way, whereas if I write "S said "THE F is G",' I shall indicate that S was using "the F" in a type (2), identificatory way.  It is important to bear in mind that I am not suggesting that the differencebetween these devices represents a difference in the meaning or sense which a descriptive phrase may have on different occasions; on the con-trary, I am suggesting that descriptive phrases have no relevant systematic duplicity of meaning; their meaning is given by a Russellian account.  We may now turn to names. In my type (1) example, it might be that in view of the prospect of repeated conversational occurrences of the expression "Jones' butler," one of the group would find it convenient to say "Let us call Jones' butler 'Bill'." Using the proposed supplementa-tion, I can represent him as having remarked "Let us call Jones' butler  'Bill'." Any subsequent remark containing "Bill" will have the same truth-value as would have a corresponding remark in which "Jones' butler" replaces "Bill". If Jones has no butler, and if in consequence it is false that Jones' butler will be seeking a new position, then it will be false that Bill will be seeking a new position.  In the type (2) example, also, one of the group might have found it convenient to say "Let us call Jones' butler 'Bill'," and his intentions might have been such as to make it a correct representation of his remark for me to write that he said "Let us call JONES' BUTLER 'Bill'." If his remark is correctly thus represented, then it will nor be true that, in all conceivable circumstances, a subsequent remark containing "Bill" will have the same truth-value as would have a corresponding remark in which "Bill" is replaced by "Jones's butler". For the person whom the speaker proposes to call "Bill" will be the person whom he meant when he said "Let us call JONES'S BUTLER 'Bill'," viz., the person who looked after the hats and coats, who was addressed by Jones as "Old Boy", and so on; and if this person turns out to have been Jones's gardener and not Jones's butler, then it may be true that Bill mixed up the hats and coats and false that Jones's butler mixed up the hats and coats. Remarks of the form "Bill is such-and-such" will be inflexibly tied, as regards truth-value, not to possible remarks of the form "Jones's butler is such-and-such", but to possible remarks of the form "The person whom X meant when he said  'Let us call Jones's butler "Bill"' is such-and-such".  It is important to note that, for a definite description used in the explanation of a name to be employed in an identificatory way, it is not required that the item which the explainer means (is referring to) when he uses the description should actually exist. A person may establish or explain a use for a name & by saying "Let us call THE F &" or "THE F iscalled &" even though every definite description in his dossier for "the F" is vacuous; he may mistakenly think, or merely deceitfully intend his hearer to think, that the elements in the dossier are non-vacuous and are satisfied by a single item; and in secondary or 'parasitic' types of case, as in the narration of or commentary upon fiction, that this is so may be something which the speaker non-deceitfully pretends or feigns.  So names introduced or explained in this way may be vacuous.  I may now propound the following argument in answer to the objection that any distinction in Q between individual constants and descriptive terms will be arbitrary.  (1) For a given definite description 6, the difference between a type  and type (2) employment is not to be construed as the employment of o in one rather than another of two systematically different senses of . A name a may be introduced either so as to be inflexibly tied, as regards the truth-value of utterances containing it, to a given definite description ô, or so as to be not so tied (6 being univocally employed); so the difference between the two ways of introducing a may reasonably be regarded as involving a difference of sense or meaning for a; a sense in which a may be said to be equivalent to a definite description and a sense in which it may not. It is, then, not arbitrary so to design Q that its individual constants are to be regarded as representing, among other linguistic items, names used with one of their possible kinds of meaning, namely that in which a name is not equivalent to a definite description. X. CONCLUDING REMARKS  I do not propose to attempt the important task of extending Q so as to include the representation of psychological verb-phrases, but I should like to point out a notational advantage which any such extension could be counted on to possess. There are clearly at least two possible readings of such a sentence as "John wants someone to marry him", one in which it might be paraphrased by "John wants someone or other to marry him" and another in which it might be paraphrased by "John wants a particular person to marry him" or by "There is someone whom John wants to marry him". Symbolizing "a wants that p" by Wap, and using the apparatus of classical predicate logic, we might hope to represent reading (1)by W°(3x) (Fxa) and reading (2) by (x) (WªFxa). But suppose that John wants Martha to marry him, having been deceived into thinking that his friend William has a highly delectable sister called Martha, though in fact William is an only child. In these circumstances one is inclined to say that "John wants someone to marry him" is true on reading (2), but we cannot now represent reading (2) by (3x) (WªFxa), since Martha does not exist.  The apparatus of Q should provide us with distinct representations for two familiar readings of "John wants Martha to marry him"  , VIZ., (a)  Wy F,ba, and (b) W9*F,b,a,. Given that Martha does not exist only  (b) can be true. We should have available to us also three distinct 3-quantificational forms (together with their isomorphs):  (i)  W93x,F,xzas;  (ii)  (iii)  Since in (iii) "x" does not dominate the segment following the 3-quantifier, (iii) does not have existential force, and is suitable therefore for representing "John wants a particular person to marry him" if we have to allow for the possibility that the particular person does not actually exist. [ and (ili) will be derivable from each of (a) and (b): (ii) will be derivable only from (a).]  I have in this paper developed as strong a case as I can in support of the method of treatment of vacuous names which I have been expounding.  Whether in the end I should wish to espouse it would depend on the outcome of further work on the notion of reference.  REFERENCES  1 Iam particularly indebted to Charles Parsons and George Boolos for some extremely helpful correspondence, to George Myro for countless illuminating suggestions and criticisms, and to Benson Mates for assistance provided both by word of mouth and via his book Elementary Logic, on which I have drawn a good deal.  • I owe the idea of this type of variant to George Myro, whose invaluable help was essential to the writing of this section.  9 c.g. by K. S. Donnellan, 'Reference and Definite Descriptions', Philosophical Review  75 (1966) 281-304; as may perhaps be seen from what follows, I am not sure that L am wholly sympathetic towards the conclusions which he draws from the existence of the distinction. h. P. Grice