Abstract: Evidence is drawn together to connect sources of inconsistency that Frege discerned in his foundations for arithmetic with the origins of the paradox derived by Russell in Basic Laws I and then with antinomies, paradoxes, contradictions, riddles associated with modal and intensional logics. Examined are: Frege's efforts to grasp logical objects; the philosophical arguments that compelled Russell to adopt a description theory of names and a eliminative theory of descriptions; the resurfacing of issues surrounding reference, descriptions, identity, substitutivity, paradox in the debates concerning modal and intensional logics; the development of the New Theory of Reference. I consider this to be the philosophical ground upon which the debates regarding that theory should take place.
It is one of the great merits of Frege's work that he created a system so clear that, to borrow a phrase from Matthias Schirn "all expressions wear their logical form on their sleeves" (Schirn 1996, 122). It was the first purpose of his formal language, Frege explained in the Preface Begriffsschrift, "to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed, so that its origin might be investigated (Frege 1879, 6). In the preface to his Basic Laws of Arithmetic I (Frege 1893), he explained that his ideal had been to set out his reasoning in such a way that one could see upon what the whole construction rested (p. vi). He said that he had "drawn together everything that can facilitate a judgment as to whether the chains of inference are cohesive and the buttresses firm" and added that if "any one perchance finds anything faulty, he must be able to indicate exactly where, to his thinking, the mistake lies" (p. vii). He believed that he had essentially attained his goal (p. vi).
That being so, we should not be obliged, as Georg Boolos suggested in "Whence the Contradiction?", to "guess at Frege's trains of thought", only to conclude that "we cannot explain how the serpent entered Eden", that Frege had been "hoodwinked"(Boolos 1996, 249-50). We should be able to start where Frege started, test with him the validity of his chain of inferences and discern the source of the problems whose origin we wish to investigate.
I propose to do that here. I investigate the origins of the contradiction derived by Russell in Frege's Basic Laws I in order to show that the source of antinomies, paradoxes, contradictions associated with modal and intensional logics is found there too. I first investigate the theory of reference of Frege's Foundations of Arithmetic (Frege 1884), then turn to study the philosophical arguments that compelled Russell to adopt a description theory of names and a eliminative theory of descriptions. I finally contend that such investigations shed needed light on issues surrounding the hows and whys of the New Theory of Reference that grew up in connection with modal and intensional logics. This is part of a broader quest to draw the implications of Kurt Gödel's observation that close examination shows that the set theoretical paradoxes "are a very serious problem, not for mathematics, however, but rather for logic and epistemology" (Gödel 1964, 258).
2. Frege on Picking Out Self-subsistent Objects and Recognizing Them as the Same Again
"The prime problem of arithmetic," Frege affirmed at the end of the appendix to Basic Laws II on Russell's paradox, "may be taken to be the problem: How do we apprehend logical objects, in particular numbers?" (Frege 1903, 224).
Now this problem is indeed acute in Frege's case because for him the reference of words was "at every point…the essential thing for science" (Frege 1979, 123). Frege had to have objects because that was the way he had designed his logic. "Only in the case of objects can there be any question of identity (equality)", he held (Frege 1979, 182; 120). And he had put identity statements at the very heart of his project to provide foundations for the theorems of arithmetic (Frege 1884, x, §§55-67, 106). He insisted that "in science and wherever we are concerned about truth, we are not prepared to rest content with sense, we also attach a reference to proper names and concept-words; and if through some oversight, say, we fail to do this, then we are making a mistake that can easily vitiate our thinking" (Frege 1979, 118, 122). It was Frege's need for objects that had compelled him to introduce the extensions and Basic Law V, his law about extensions, that he believed led to Russell's contradiction.
Frege's problem arises in Foundations of Arithmetic in this form. Having determined to his satisfaction that "the content of a statement of number is an assertion about a concept", Frege starts to define 0 as the number that "belongs to a concept, if the proposition that a does not fall under that concept is true universally, whatever a may be". He starts to define 1 as the number that "belongs to a concept F, if the proposition that a does not fall under F is not true universally, whatever a may be, and if from the propositions 'a falls under F' and 'b falls under F' it follows universally that a and b are the same". In that case, he concludes, the number (n + 1) would have the number that "belongs to a concept F, if there is an object a falling under F and such that the number n belongs to the concept 'falling under F, but not a'" (§ 55).
However, Frege promptly confesses: "It was only an illusion that we have defined 0 and 1; in reality we have only fixed the sense of the phrases 'the number 0 belongs to' 'the number 1 belongs to'; but we have no authority to pick out the 0 and 1 here as self-subsistent objects that can be recognized as the same again" (§ 56). He explains that:
strictly speaking we do not know the sense of the expression 'the number n belongs to the concept G' any more than we do that of the expression 'the number (n + 1) belongs to the concept F'… we can never --to take a crude example-- decide by means of our definitions whether any concept has the number JULIUS CAESAR belonging to it, or whether that same familiar conqueror of Gaul is a number or is not. Moreover we cannot by the aid of our suggested definitions prove that, if the number a belongs to the concept F and the number b belongs to the same concept, then necessarily a = b. Thus we should be unable to justify the expression 'the number which belongs to the concept F', and therefore should find it impossible in general to prove a numerical identity, since we should be quite unable to achieve a determinate number (§ 56).
That attempt had failed, he explained in one of the concluding sections of the book, "because we had defined only the predicate which we said was asserted of the concept, but had not given separate definitions of 0 or 1, which are only elements in such predicates. This resulted in our being unable to prove the identity of numbers" (§106).
Now, it does not take very much imagination or logical acuity to see what great problems with inference would inevitably immediately arise were Frege not to rise to the occasion and find a definition of number not saddled with such a blatant basic problem of denotation and identity. However, Frege did not dream of leaving things there and so the challenge represented by the Julius Caesar problem itself could never have lived on to haunt philosophers' consciences were not shades of it to be found also haunting Frege's subsequent attempts to define numbers. They are. So Richard Heck was not exaggerating when he wrote that we shall "not fully understand Frege's philosophy until we understand the enormous significance the question how we apprehend logical objects, and the Caesar problem, had for him" (Heck 1993, 287).
3. Numbers as Self-subsistent Objects for Frege
"Precisely because it forms only an element in what is asserted," Frege argued as he developed his philosophy of arithmetic in Foundations, "the individual number shows itself for what it is, a self-subsistent object". For him, this was indicated by the presence of the definite article 'the' in expressions like 'the number 1', which served "to class it as an object" and in arithmetic "this self-subsistence comes out at every turn, as for example in the identity 1 + 1 = 2" (§57; §106).
Any appearance that numbers were not to be construed substantivally, Frege averred, could "always be got around". To illustrate this point, he showed how the statement 'Jupiter has four moons' could be rewritten to read 'the number of Jupiter's moons is four', where the word 'is' is to mean 'is identical with' or 'is the same as'. This, he reasoned, yields an identity statement affirming that both of those expressions referred to the same object that was being called 'Four' or 'the Number of Jupiter's moons'. The second example he proposed was "it is the same man that we call Columbus and the discoverer of America" (§§57-58).
However, this self-subsistence of numbers, Frege insisted, "is not to be taken to mean that a number word signifies something when removed from the context of a proposition…." (§60). So the problem became one of defining "the sense of a proposition in which a number word occurs". And, for Frege, the answer lie in the very fact that number words stand for self-subsistent objects, which he considered, "is enough to give us a class of propositions which must have a sense, namely those which express our recognition of a number as the same again. If we are to use the symbol a to signify an object, we must have a criterion for deciding in all cases whether b is the same as a…" (§62; §106).
In the case at hand, Frege continues, "we have to define the sense of the proposition 'the number which belongs to the concept F is the same as that which belongs to the concept G'; that is to say, we must reproduce the content of this proposition in other terms, avoiding the use of the expression 'the Number which belongs to the concept F". This, he concludes, will provide the needed general criterion for the identity of numbers; when "we have thus acquired a means of arriving at a determinate number and of recognizing it again as the same, we can assign it a number word as its proper name" (§62).
4. Frege's Criterion for Identity
By defining numerical identity or equality in terms of one-to-one correlation, David Hume had provided such a means of arriving at a determinate number and of recognizing it again. However, Frege found that Hume's principle too raised "certain logical doubts and difficulties, which ought not to be passed over without examination". It "is not only among numbers that the relationship of identity is found", Frege reasoned. So it seemed to him "to follow that we ought not to define it specially for the case of numbers", but we "should expect the concept of identity to have been fixed first" and the concept of number "to be determined in the light of our definition of numerical identity". The aim was "to construct the content of a judgement which can be taken as an identity such that each side of it is a number" (§63).
He began doing this by showing how one could rewrite 'line a is parallel to line b' as an identity statement reading 'the direction of line a is identical with the direction of line b', "through removing what is specific in the content of the former and dividing it between a and b." The "only trouble with this is", he admits, "that this is to reverse the true order of things.…Our convenient proof is only made possible by surreptitiously assuming, in our use of the word 'direction', what was to be proved" (§64).
Frege realized that his definition of the proposition 'line a is parallel to line b' is to mean the same as 'the direction of line a is identical with the direction of line b', "departs to some extent from normal practice, in that it serves ostensibly to adapt the relation of identity, taken as already known, to a special case, whereas in reality it is designed to introduce 'the direction of line a', which only comes into it incidentally." So, he tested his procedure against the well-known laws of identity. For this he chose Leibniz's principle that "things are the same as each other, of which one can be substituted for the other without loss of truth", which he then adopted as his own definition of identity. However, while writing that well-known principle of substitutivity of identicals right into the very foundations of his logic, Frege decided to adjust Leibniz's principle to meet his own ends and to translate sentences of natural languages in such a way as to do away with the differences between being identical (complete agreement in all respects) and equal (only agreement in this respect or that). Although Leibniz's law defines identity, complete coincidence, Frege insisted that for him there was no between equality and identity:
Whether we use 'the same', as Leibniz does, or 'equal', is not of any importance. 'The same', may indeed be thought to refer to complete agreement in all respects, 'equal', only to agreement in this respect or that; but we can adopt a form of expression such that this distinction vanishes. For example, instead of 'the segments are equal in length', we can say 'the length of the segments is equal', or 'the same', and instead of 'the surfaces are equal in color', 'the color of the surfaces is equal'... in universal substitutability all the laws of identity are contained (§65).
Here, Frege has changed statements of sameness of concrete properties predicated of concrete objects into statements that affirm the equality-identity of abstract objects, in this case surfaces and lengths. He believes he has thus transformed statements about objects that are equal under a certain description into statements expressing a complete identity. By erasing the difference between identity and equality, he is in fact asserting that being the same in any one way is equivalent to being the same in all ways. All the other properties serving to distinguish those objects from one another, or from other objects equal to them in the same respect, are "abstracted" out of the picture; deleted on paper, those properties marking any difference between the mere equality and the full identity of the objects are presumably expected to simplify matters by vanishing entirely from the reasoning.
5. Reference, Identity and Substitution in Dickens' A Tale of Two Cities
At this point, I wish to introduce an example from A Tale of Two Cities, by Charles Dickens that can be used to judge the cohesiveness of the chains of inferences and the firmness of the buttresses involved as we proceed from Frege's very old theory of reference to the new theory of reference. The story goes somewhat like this. The Marquis de St. Evrémond voluntarily relinquishes a title and a rank bound to a system that is frightful to him in order to live otherwise and elsewhere earning his living by giving instruction in the French language and literature under the name of Charles Darnay.
The Reign of Terror finds Darnay him France. He is arrested and imprisoned in the prison of La Force and tried by the people's Tribunal who want to know whether he is the Marquis de St. Evrémond, at heart and by descent an aristocrat. The Tribunal finds him guilty of having been born into an obnoxious family of aristocrats, a notorious oppressor of the People, enemy of the Republic (and of equality). Along with 51 other prisoners, he is condemned to die within 24 hours and imprisoned at the Conciergerie. No one has ever escaped from the Conciergerie and it is necessary that 52 heads fall corresponding to those of 52 prisoners condemned to death. Darnay is prisoner no. 22.
For personal reasons, Sydney Carton, who refers to himself as a self-flung away, wasted, drunken, poor creature of misuse, who should never be better than he was, should but sink lower, and be worse, and whom Dickens describes as "the idlest and most unpromising of men", "the jackal", "the fellow of no decency", of careless and slovenly, if not debauched appearance, decides to replace de St. Evrémond and be guillotined in his place.
The day on which the 52 heads of the 52 condemned prisoners were to fall, Carton enters the vermin haunted cell of the condemned man and exchanges cravats, boots, and coats with him. Carton takes the ribbon from the prisoner's hair and tells him to shake out his hair like his own, drugs the prisoner, and calls a spy to have the man he has replaced taken to a coach. Carton then dresses himself in the clothes the prisoner had laid aside, combs back his hair, and ties it with the ribbon the prisoner had worn. A gaoler eventually comes to find Citizen Evrémond. It is a dark winter day, and what with the shadows within, and what with the shadows without, one could but dimly discern the prisoners. As prisoner 22, Carton takes Evrémond's place in the tumbril and dies in his place.
We have our identity:
Charles Darnay = Charles de St. Evrémond
The Marquis de St. Evrémond; the occupant of cell 22 of the Conciergerie; the guillotined person no. 22, etc.
The Marquis de St. Evremond has hair on his head.
Charles Darnay is the author of certain crimes against the people
The man presumed to be Charles de St. Evremond = Sydney Carton
52 is the number of prisoners found guilty that day by the Tribunal.
52 is the number of prisoners that day in the tumbrils.
52 is the number de these prisoners actually guillotined that day.
Charles Darnay = prisoner 22
Citizen Evrémond = prisoner 22
Sydney Carton = prisoner 22
Several one to one correspondences, among them: The number of condemned people from the prison of La Force (F) and the number of people guillotined (G).
And antinomies resulting from substitution, for example:
Citizen Evrémond is prisoner no. 22
Prisoner 22 was guillotined.
Citizen Evrémond was guillotined.
6. Applying Frege's Reasoning to Our Life and Death Situation
So, taking Frege's reasoning into the cells of the Conciergerie, we have:
(1) The Marquis de St. Evrémond has the same destiny as prisoner 22.
Which translated in Frege's way yields:
(2) The destiny of the Marquis de St. Evrémond is identical to the destiny of prisoner 22.
Whether we use 'the same', complete agreement in all respects, or 'equal', only agreement in this respect or that, Frege stresses, is not of any importance because "we can adopt a form of expression such that this distinction vanishes" (Frege 1884, §65). Holding that "it is actually the case that in universal substituability all the laws of identity are contained", Frege maintains that to justify our proposed definition of the destiny of a man, we should have to show that it is possible, if
(1) The Marquis de St. Evrémond has the same destiny as prisoner 22.
"the destiny of the Marquis de St. Evrémond"
"the destiny du prisoner 22".
The task is simplified, he admits, by the fact that we are being taken initially to know nothing that can be asserted about the destiny of a man except one thing, that it coincides with the destiny of prisoner 22. "We should thus have to show only that substitution was possible in an identity of this one type, or in judgement-contents containing such identities of this kind as constituent elements" (Frege 1884, §65).
7. Recognizing Objects When Given in a Different Way
Frege was perfectly conscious that left unmodified the procedure just described was liable to lead to false or nonsensical conclusions, or be sterile and unproductive (Frege 1884, §§66-67). To illustrate this, he carried the reasoning involved in his example of the identity of two lines one step further. Using our life or death example, according to Frege's reasoning, in proposition (2) "The destiny of Marquis de St. Evrémond is identical to the destiny of prisoner 22", the destiny of the Marquis de St. Evrémond plays the part of an object, and our definition affords us a means of recognizing this object in case it could happen to crop up in another guise (in einer andern Verkleidung), say as the destiny du prisoner 22, but this means does not provide for all the cases. It does not, still imitating Frege, for example, decide for us whether England is the same as the destiny of a man condemned to death. The definition does not say whether the statement:
(3) "The destiny of the Marquis de St. Evrémond est identical with q"
is to be affirmed or denied except in the one case where q is given in the form of "the destiny of prisoner 22".
Left as it was his definition was unproductive, Frege further judged, because in adopting this way out, we would be presupposing that an object could only be given in a single way.
"All identities would then amount simply to this," Frege then wrote, "that whatever is given to us in the same way is to be reckoned as the same. This is, however, a principle so obvious and sterile as not to be worth stating. We could not, in fact, draw from it any conclusion which was not the same as one of our premisses." Surely though, he concluded, identities play such an important role in so many fields "because we are able to recognize something as the same again even although it is given in a different way" (Frege 1884, §67; §107).
8. Apprehending Logical Objects as Extensions
Frege introduced extensions in order to overcome the undesirable consequences that he saw gathering about his theory of number. He hoped that they would guarantee that an identity holding between two concepts could be transformed into an identity of extensions, and conversely (Frege, 1884, §§66-67; also §107). In that case, parroting Frege's reasoning once again, if, in accordance with (1), The Marquis de St. Evrémond has the same destiny as prisoner 22, then the extension of the concept "man with the same destiny as that of the Marquis de St. Evrémond" is identical to the "the extension of the concept person having destiny of prisoner 22". And conversely, if the extensions of the two concepts just named are identical, then the Marquis de St. Evrémond has the same destiny as prisoner 22. To apply this means of definition to the case at hand, we must substitute concepts for men, and for sharing the same destiny, the possibility of correlating one to one the objects that fall under the one concept with those that fall under the other (Frege 1884, §68).
In his next book, Basic Laws I, Frege argued that the generality of an identity could always be transformed into an identity of courses of values and conversely, an identity of courses of values may always be transformed into the generality of an identity. By this he meant that if it is true that (x) F(x) = G(x), then those two functions have the same extension and that functions having the same extension are identical (Frege 1893, §§9, 21; Frege 1903).
Frege never believed that any proof could be devised to sanction such a transformation. So he created Basic Law V to mandate the view of identity, equality, and substitutivity that he required. By transforming "a sentence in which mutual subordination is asserted of concepts into a sentence expressing an equality," his law would permit logicians to pass from a concept to its extension, a transformation which, Frege considered, could "only occur by concepts being correlated with objects in such a way that concepts which are mutually subordinate are correlated with the same object" (Frege 1979, 182; Frege 1979, 118). His law, however, authorized various illicit logical moves which yielded contradictory results by letting logicians to put their symbols to wrong uses and allowing type ambiguities to creep into reasoning unnoticed. The most famous of these was, of course, the contradiction derived by Russell: both (wÎw) Ù ~ (wÎw).
9. Russell's Paradox, Extensions, and Grasping Logical Objects
Upon learning of Russell's paradox, Frege tested the validity of the chain of inferences leading to the contradiction and determined that it was his Basic Law V that was at the origin of the contradiction. The contradiction indicated to him that the transformation of the generality of an identity into an identity of ranges of values was not always permissible, that the law was false, that his explanations did not suffice to secure a reference for his combinations of signs in all cases (Frege 1980a, 131-32). He confessed that he had been long reluctant to use classes, but that that was the only answer that he had found to the question as to how we apprehend logical objects (Frege 1980a, 140-41).
Frege studied the problem in an appendix to Basic Laws II (Frege 1903). There was nothing, he found, to stop one from transforming an equality holding between two concepts into an equality of extensions in conformity with the first part of his law, but from the fact that concepts are equal in extension one cannot infer that whatever falls under one falls under the other. The extension may fall under only one of the two concepts whose extension it is. This can in no way be avoided….(Frege 1903, 214n. f, 218-23). "If in general, for any first-level concept, we may speak of its extension, then the case arises of concepts having the same extension, although not all objects that fall under one fall under the other as well" (Frege1903, 221). This is, of course, the drama of substitution, reference and identity of the prison of the Conciergerie, for:
If F is the set of 52 prisoners of La Force condemned to die within 24 hours
G, the set of these 52 prisoners actually guillotined
F = G and (x) F(x) implies that G(x)
There was a one to one correspondence between the extension of concept F and that of concept G, but that does not mean that (x) if F(x) then G(x). No matter how attractive it may seem at first to obliterate distinctions between identity and equality, the differences between x and y when they are joined together by the equals sign to make an informative statement do not just go away because we make a law that stipulates that equality is to function as identity. Extensionality alone can not keep someone from slipping into the company of the prisoners of La Force and being guillotined in the place of one of them. Frege, as he himself realized, still did not have a firm hold on the reference.
10. The Definite Article and Pseudo-objects
When specifically asked about the causes of the paradoxes of set theory, Frege explained that the "essence of the procedure which leads us into a thicket of contradictions" consisted in regarding the objects falling under F as a whole, as an object designated by the name 'set of Fs', 'extension of 'F', or 'class of Fs' etc. (Frege 1980a, 55). He wrote that the paradoxes of set theory
arise because a concept e.g. fixed star, is connected with something that is called the set of fixed stars, which appears to be determined by the concept --and determined as an object. I thus think of the objects falling under the concept fixed star combined into a whole, which I construe as an object and designate by an proper name, 'the set of fixed stars'. This transformation of a concept into an object is inadmissible, for the set of fixed stars only seems to be an object, in truth there is no such object at all. (Frege 1980a, 54; 55)
"The definite article," he explained, "creates the impression that this phrase is meant to designate an object, or, what amounts to the same thing, that 'the concept star' is a proper name, whereas 'concept star' is surely a designation of a concept and thus could not be more different from a proper name" (Frege 1979, 270).
Russell also came to see the problems raised by the logical behavior of expressions having the form 'the So-and-so', which seem to denote objects and often figure in identity statements as if they do. He also saw a link between problems with descriptions and the paradoxes he was trying to avoid. Clear access to reference was needed and the paradoxes showed both Frege and Russell that they were still in need of an effective technique for guaranteeing it.
In his famous article "On Denoting" Russell reasoned that: "If a is identical to b, whatever is true of the one is true of the other, and either may be substituted for the other in any proposition without altering the truth or falsehood of that proposition". Substituting our Reign of Terror example into Russell's well known train of thought, we have: Now the Tribunal wished to know whether Darnay was the Marquis de St. Evrémond; and in fact Darnay was the Marquis de St. Evrémond. Hence we may substitute Darnay for the Marquis de St. Evrémond, and thereby prove that the Tribunal wished to know whether Darnay was Darnay. However, an interest in the law of identity can hardly be attributed to that revolutionary Tribunal (Russell 1905, 47-48; Russell 1959, 63-64). This and other puzzles about denoting led Russell to develop his theory of definite descriptions, which in the context of this essay may be called "the old theory of reference".
11. Drawing Objects Out of Descriptions and Laying Hold of the Extension of a Class
In his preface to the 1980 edition of Naming and Necessity, Saul Kripke asks how Russell came to propose a theory plainly incompatible with our direct intuitions of rigidity of reference. In response to his own question, Kripke first suggests that one reason is that Russell did not himself consider modal questions and the question of the rigidity of names in natural language was rarely explicitly considered after him. Kripke secondly answers that it seemed to Russell that various philosophical arguments made a description theory of names and a eliminative theory of descriptions necessary, seemed to him to compel adoption of his theory" (Kripke 1980, 14). I examine Kripke's second answer next and shall take up his first one later.
So what philosophical arguments might have compelled Russell to adopt his famous theory? To understand this, let us begin with the obvious fact that Russell inherited Frege's problems. In the preface to Principia Mathematica, Russell explains that a "very large part of the labour involved in writing the present work has been expended on the contradictions and paradoxes which have infected logic and theory of aggregates" (Russell 1927, vii), in the introduction that his "system is specially framed to solve the paradoxes which, in recent years, have troubled students of symbolic logic and the theory of aggregates" (Russell 1927, 1).
Russell said that once he had sat down to remove the source of that infection, he devoted almost all his time to problems with denotation that he believed tied to them. "The whole theory of definition, of identity, of classes, of symbolism, and of the variable is wrapped up in the theory of denoting" he had written in Principles of Mathematics (Russell 1903, §56). And Russell always said that his theory of definite descriptions represented his first breakthrough in his efforts to find a solution to the paradoxes (Russell 1944, 13-14; Russell 1959, 60-61; Grattan-Guinness 1972, 106-07; Grattan-Guinness 1975, 475-88; Grattan-Guinness 1977, 70, 79-80, 94 and note; Kilmister 1984, 102, 108, 123, 138).
As we saw, to "lay hold upon the extension of a concept," Frege had proposed transforming "a sentence in which mutual subordination is asserted of concepts into a sentence expressing an identity." Since only objects could figure in identity statements, he realized that he would have to find a way of correlating objects and concepts which correlated mutually subordinate concepts with the same object. He suggested that this might be achieved by translating language asserting mutual subordination into statements of the form 'the extension of the concept X is the same as the extension of the concept Y' in which the descriptions would then be regarded as proper names as indicated by the presence of the definite article. But by permitting such a transformation, Frege realized, one is conceding that such proper names have references (Frege 1979, 181-82). However, it was precisely that sort of recipe that Russell's paradox had cast doubt upon. Russell would try to circumvent problems that that procedure caused through a new theory of classes based on his theory of definite descriptions. In so doing, he hoped to realize Frege's goal of correlating classes with extensions in such a way that concepts which are mutually subordinate would be correlated with the same objects (re. Frege 1903, 214).
12. The Need for a Single Object to Represent an Extension
Russell recognized: "the necessity of discovering some entity determinate for a given propositional function, and the same for any equivalent propositional function. Thus 'x is a man' is equivalent (we will suppose) to 'x is a featherless biped', and we wish to discover some one entity which is determined in the same way by both these propositional functions" (Russell 1903, § 486). He was convinced that "without a single object to represent an extension, Mathematics crumbles. Two propositional functions which are equivalent for all values of the variable may not be identical, but it is necessary there should be some object determined by both. Any object that may be proposed, however, presupposes the notion of class... an object uniquely determined by a propositional function, and determined equally by any equivalent propositional function" (Russell 1903, §489).
On the other hand, the contradiction of the classes that do not belong to themselves had persuaded Russell that one could not in general suppose that objects having a certain property constitute a class that is an entity distinct from the objects making it up. So Russell set out in search of a way of dealing with classes as symbolic fictions by which one could avoid having to assume that there are classes without being compelled to assume that there are no classes (Russell 1919, 184).
Parallels that he spotted existing between the problems arising when classes are treated as objects and those arising when descriptions are treated as names suggested to him how classes might be analyzed away much as descriptions had been by his 1905 theory of definite descriptions. Just as descriptions are bound to the particular way of characterizing the object, so classes are formed by specifying the definite property giving the class. And just as two different descriptions could be true of the same object, so a single class of objects could be defined in diverse ways, each one corresponding to another sense of the class name.
Classes were false abstractions, Russell decided, in the same sense as 'the present King of England' or 'the present King of France'. So he sought to extend his ideas about analyzing away descriptions to include class symbols. He reasoned that since:
we cannot accept "class" as a primitive idea. We must seek a definition on the same lines as the definition of descriptions, i.e. a definition which will assign a meaning to propositions in whose verbal or symbolic expression words or symbols apparently representing classes occur, but which will assign a meaning that altogether eliminates all mention of classes from a right analysis of such propositions. We shall then be able to say that the symbols for classes are mere conveniences, not representing objects called "classes," and that classes are in fact, like descriptions, logical fictions, or (as we say) "incomplete symbols" (Russell 1919, 181-82).
According to Russell's theory of definite descriptions: "There is a term c such that fx is always equivalent to 'x is c'" (Russell 1919, 178). That being so the putative identity statement 'Darnay is the Marquis de St. Evrémond' might be rewritten 'Darnay inherited the title the Marquis de St. Evrémond, and is always true of c that if c inherited the title Marquis de St. Evrémond, then c est identical to Darnay (Russell 1905, 55). In this rendering of the sentence, what was not identical has been made to be equivalent. A symbol deemed equatable with 'Darnay' has gone proxy for the description and, note well, this symbol will generally be "obedient to the same formal rules of identity as symbols which directly represent objects" (Russell 1927, 83). This means of drawing objects out of descriptions provided Russell with a practical model of how to make non-entities function as entities without incurring contradictory results. The theory of definite descriptions was his way of making an object fit to go proxy for what was said about it (Russell 1927, 187).
So a procedure explicitly analogous to the theory of definite descriptions was duly integrated into Principia Mathematica and Introduction to Mathematical Philosophy. Russell said that he was satisfied that although the real meaning of the notation that he had ultimately adopted was very complicated, it had "an apparently simple meaning which, except at certain crucial points, can without danger be substituted in thought for the real meaning" (Russell 1927, 1). He had avoided the contradictions arising from the supposition that classes are entities and acquired a means of laying hold of the extension of a class. Henceforth, incomplete symbols, like descriptions and classes, would obey the same formal rules of identity as symbols directly representing objects.
13. Further Guaranteeing Substitutivity
Russell was aware, however, that classes or descriptions could only obey the same formal rules of identity to the extent that we consider only the equivalence of the values of variables (or of constants) resulting from it and not their identity (Russell 1927, 83) and that brought him right back to the problems with reference, identity, and substitutivity that had compelled Frege to introduce extensions. In addition, the contradictions most closely connected to identity and the formal equivalence of functions proved particularly difficult to uproot.
So Russell took aim at Frege's theory of identity. To cope with contradictions arising from necessary talk of 'all properties,' or 'all functions,' Russell introduced the axiom of reducibility, which was "equivalent to the assumption that 'any combination or disjunction of predicates is equivalent to a single predicate"' (Russell 1973, 250; 1927, 58-59) and would yield most of the results otherwise requiring recourse to all functions or all properties, legitimize much reasoning apparently dependent on such notions (Russell 1927, 56).
Russell considered that the axiom embodied all that was really essential in his theory of classes (Russell 1919, 191; 1927, 58). "By the help of the axiom of reducibility," Russell affirmed, "we find that the usual properties of classes result. For example, two formally equivalent functions determine the same class, and conversely, two functions which determine the same class are formally equivalent" (Russell 1973, 248-49). He came to believe classes themselves to be mainly useful as a technical means of achieving what the axiom of reducibility would effect (Russell 1919, 191). It seemed to him "that the sole purpose which classes serve, and one main reason which makes them linguistically convenient, is that they provide a method of reducing the order of a propositional function" (Russell 1927, 166). Classes were producing contradictions. They should be replaced with this axiom which seemed to him "to be the essence of the usual assumption of classes" and to retain "as much of classes as we have any use for, and little enough to avoid the contradictions" (Russell 1927,166-67; 1956, 82; 1919, 191).
Examination shows that Russell leaned on the axiom of reducibility at every crucial point in his definition of classes in Principia Mathematica (Russell 1927, 75-81). He believed that many of his proofs "become fallacious when the axiom of reducibility is not assumed" (Russell 1927, xliii). He also believed that without the axiom, or its equivalent, one would be compelled to regard identity as indefinable and to admit that two objects might agree in all their predicates without being identical (Russell 1927, 58). "But in virtue of the axiom of reducibility it follows that, if x = y and x satisfies yx, where y is any function... then y also satisfies yy." And this effectively made his definition of identity as powerful as if he had been able to appeal to all functions of x (Russell 1927, 168; also 57; 1973, 243). In particular, by resorting to the axiom of reducibility one might avoid a difficulty with the definition of identity which Russell explained as follows:
We might attempt to define "x is identical with y" as meaning "whatever is true of x is true of y," i.e., fx always implies fy." But here, since we are concerned to assert all values of "fx implies fy" regarded as a function of f, we shall be compelled to impose upon f some limitation which will prevent us from including among values of f values in which "all possible values of f" are referred to. Thus for example "x" is identical with "a" is a function of x; hence, if it is a legitimate value of f in "fx always implies fy," we shall be able to infer, by means of the above definition, that if x is identical with a, and x is identical with y, then y is identical with a. Although the conclusion is sound, the reasoning embodies a vicious-circle fallacy, since we have taken "(f)(fx implies fa)" as a possible value of fx, which it cannot be. If, however, we impose any limitation upon f it may happen, so far as appears at present, that with other values of f we might have fx true and fy false, so that our proposed definition of identity would plainly be wrong (Russell 1927, 49).
So, if the axiom of reducibility could be true, we could have the properties of identity and equality that we need. Russell thought of it as a generalized form of Leibniz's principle of the identity of indiscernibles (Russell 1919, 192; 1927, 57; 1973, 242). He was finally back to square one, back to the reasons why Frege's theory of identity had made him appeal to extensions in the first place.
But if the axiom of reducibility could do the trick, the Leibniz' law or Basic Law V could have. The problems were not uprooted. They began surfacing again, in another guise, when Willard Quine engaged in his campaign against intensional and modal logics.
14. Reference, Modality, and Paradox
In 1946 and 1947, three articles by Ruth Barcan (Marcus) on quantified modal logic appeared in the Journal of Symbolic Logic (Barcan 1946a, 1946b, 1947). In them, she stipulated that: x = y , iff necessarily x = y, a form of rigid reference. She also distinguished between identity in the strong sense and in the weak sense. Those papers were formal. We can put some flesh on the bones, however, by looking at her reasoning as mirrored in the reactions of others.
Willard Van Orman Quine reacted immediately (Quine 1947 a, b, c). As is well known, he had a wealth of objections to modal and intensional logics. To begin with I want to look at one whose error he eventually acknowledged (Quine 1962). In his review of "The Identity of Individuals in a Strict Functional Calculus of Second Order," he observed that Barcan had defined two relations of "identity" between individuals, a weak one holding between x and y wherever (F) (Fx É Fy), and a strong one holding only where (F) (Fx strictly implies Fy). And he remarks that:
As to be expected, only the strong kind of identity is subject to a law of substitutivity valid for all the modal contexts.
It should be noted that only the strong identity is therefore interpretable as identity in the ordinary sense of the word. The system is accordingly best understood by reconstruing the so-called individuals as "individual concepts". For example, the physical planet which is both the evening star and the morning star should not be reckoned as a value of the individual variables, lest it turn out not to be identical (in the full sense) with itself. On the other hand, two distinct concepts of Evening Star and Morning Star are available as values of the variables without fear of paradox…. It should be noted further that the primitive idea of abstraction need not have been assumed in this system, for it could have been introduced by contextual definition … (Quine 1947c, 95-96).
In "The Problem of Interpreting Modal Logic", Quine had contended that Barcan's version of quantified modal logic was committed to an curiously idealistic ontology which repudiates material objects (Quine 1947a, 43, 47); he had sought to show that it had "queer ontological consequences. It leads us to hold that there are no concrete objects (men, planets, etc.), but rather that there are only, corresponding to each supposed concrete object, a multitude of distinguishable entities…. It leads us to hold, e.g. that there is no such ball of matter as the so-called planet Venus, but rather at least three distinct entities: Venus, Evening Star, and Morning Star" (Quine 1947a, 47).
Frederic Fitch and Arthur Smullyan rose quickly to the defense of Barcan's position. In "The Problem of the Morning Star and the Evening Star", Fitch argued that there was really no reason to resort to individual concepts in order to make quantified modal logic consistent with the facts of astronomy, that modal logicians are free to deal directly with concrete individuals and to use the identity relation in relation to them. They might choose to regard the Morning Star and the Evening Star as proper names of individuals, or as descriptive phrases in the sense of Principia Mathematica *14. "Both of these ways are orthodox, natural, and available", he pointed out (Fitch 1949, 138). Smullyan showed Quine's argument fails and that no paradox arises if the Morning Star and the Evening Star are considered as being proper names of the same individual. (Smullyan 1947, 1948; Fitch 1949, 138-39).
For Fitch, the source of the apparent paradox seemed to lie in the ambiguity of the scope of the descriptions. As he explained, the "usage of ordinary language with respect to descriptive phrases is so vague that… that it is usually not possible to take at random various sentences of ordinary language and interpret them all correctly by some uniform use of Russell's theory of descriptions…." (Fitch 1949, 140). He also concluded that in Ruth Barcan's system "the axiom of extensionality need not be used at all, or only in such a way that (f = g) follows from ? (fx =xgx)" (Fitch 1949, 141).
The message conveyed did not fall on fertile ground. It was not fashionable, or welcome at all. As is well-known, Quine, the fashionable philosopher of the 1950s and 1960s, fought hard to contain logical reasoning within the confines of strong extensional calculi at all costs and made quashing modal and intensional logics one of the principal planks of his philosophical program. His attacks, of which Marcus was quite often the target, were very influential and appeared in his most influential works, for example "Reference and Modality" (Quine 1953), "Three Grades of Modal Development" (Quine 1953b), the chapter of Word and Object entitled "Flight from Intension" (Quine 1962, Chapter 6), Ontological Relativity (Quine 1969). Figuring on his list of the "varied sorrows of modal logic" were referential opacity, failure of substitutivity, of existential quantification, of inference, riddles about descriptions, bizarre ontology, confusions of use and mention, intensions, essentialism.
Meanwhile, defying Quine's opprobrium, Marcus continued to work to draw attention to ambiguities regarding equality and identity that had slipped into logical reasoning. She endeavored to point out the extent to which extensional systems depend on direct or indirect restrictions forbidding intensional functions and the extent to which they reduce identity to a weaker form of equivalence. She called for lucidity regarding the differences between identity and weaker forms of equivalence that explicit or implicit principles extensionality would ignore. She called a principle extensional if it either "(a) directly, or indirectly imposes restrictions on the possible values of the functional variables such that some intensional functions are prohibited or (b) it has the consequence of equating identity with a weaker form of equivalence" (Marcus 1960, 46). She urged "that the distinctions between stronger and weaker equivalences be made explicit before, for one avowed reason or another, they are obliterated" (Marcus 1960, 55), a request which would seem to be perfectly in keeping with the requirements of any logic priding itself on its clarity, and devised to keep ambiguity from slipping unawares into reasoning.
A meeting of the Boston Colloquium in 1962 marked a new step in the history of these ideas. In her paper on "Modalities and Intensional Languages", expounding on the themes discussed above, Marcus defended quantified modal logic (Marcus 1993, 3-35). In his reply to her, Quine explained his conviction that modern modal logic was conceived in sin (Quine 1962, 177-79). He condemned her conclusion that identity, substitutivity, and extensionality are things that come in grades as unacceptable (Quine 1962, 180). He saw trouble in the contrast that she developed between proper names and descriptions (Quine 1962, 181). He once again challenged the "champion" of quantified modal logic to explain away his puzzlement about the logical behavior of statements like "9 = the number of the planets" in modal contexts (Quine 1962, 183-84). He complained about the pernicious way in which Marcus distinguished between necessary proprieties and contingent proprieties, about "the invidious distinction between some traits of an object as essential to it… and other traits of it as accidental" that opens the door to essentialism (Quine 1962, 184).
Though Quine's perspective remained dominant and intensional and modal logics were despised and derided for decades, some philosophers began to take a more daring attitude toward limning the true and ultimate structure of reality and began venturing into proscribed territory beyond the narrow boundaries that strong extensional calculi would impose on philosophical reasoning.
Among those making pioneering contributions to the field were Jaakko Hintikka (Hintikka 1962, 1969, 1975; Hintikka and Davidson 1969), Stig Kanger, Saul Kripke and Dagfinn Føllesdal (Humphreys and Fetzer 1998). They worked to increase the depth and utility of the standard languages and to develop intensional languages capable of analyzing the many non-extensional statements deemed unfit for study because they complicated matters by not conforming to the rigid standards for admission into what even Frege had feared could be a sterile logical world.
The development of possible worlds logic eventually proved to be a particularly effective device for exposing logical form and viewing the inner workings of analytic philosophy's brave new logic. Explorers of possible worlds made discoveries that helped to confirm the results of Marcus's earlier ventures into the discreditable world of modality and intensionality. So, more and more reasons for not shoving reasoning into an extensional mold began to mount as those courageous enough not to flee at the sight of intensional and modal phenomena pulled deep issues underlying the puzzles, contradictions, and paradoxes to the surface and increasingly demonstrated the insufficiencies of strong extensional systems.
For one thing, possible worlds logic put a spotlight on problems associated with extensionality, identity, failures of substitutivity and existential generalization and so helped prepared the ground for a wider acceptance for ideas that went into the making of a "new" theory of reference as opposed to Russell's theory of definite descriptions. According to this "new" theory, there is a direct reference relation between names, proper names in particular, and their references; names refer to objects directly and not through descriptions.
In the 1980 preface to Naming and Neccesity, Saul Kripke, one of the philosophers whose name has been associated with this new theory of reference, told of the evolution of his own ideas on the matter. Most of his views, he recalled, had grown out of earlier formal work on model theory of modal logic and were formulated around 1963 to1964 (Kripke 1980, 3). He described the realization that inaugurated his work of 1963-64 in the following way: "Eventually I came to realize… that the received presuppositions against the necessity of identities between ordinary names were incorrect and that the natural intuition that the names of ordinary language are rigid designators can in fact be upheld. Part of the effort to make this clear involved the distinction between using a description to give a meaning and using it to fix a reference" (Kripke 1980, 5). He explained that his model theoretic work on modal logic had confirmed his conviction that the principle of the indiscernibility of identicals was self-evident and made it completely clear that the alleged counter-examples involving modal properties always turned out to turn on confusions. "It was clear that from (x) ? (x = x) and Leibnitz's law that identity is an 'internal' relation: (x) (y) (x = y) implies ? (x = y)…. If 'a' and 'b' are rigid designators, it follows that 'a = b', if true, is a necessary truth", he wrote (Kripke 1980, 3).
In this essay, I have drawn together evidence to connect sources of inconsistency that Frege himself discerned in his foundations for arithmetic to the origins of the paradox derived by Russell in Frege's Basic Laws I and to paradoxes, puzzles, riddles antinomies, contradictions that became major issues in later stages of analytic philosophy. Specifically, this has involved linking Frege's unsatisfied need specify referents in the way his system requires, the philosophical arguments that compelled Russell to adopt a description theory of names and a eliminative theory of descriptions, and the resurfacing of issues surrounding identity, substitutivity, paradox, descriptions, and rigid reference in the debates concerning modal and intensional logics.
In his introduction to Frege's Philosophy of Mathematics, William Demopoulos reflects that once the Julius Caesar problem is taken to show that Frege's contextual definition fails to specify the referents of numerical expressions, it is natural to want to know whether the problem does not appear in an analogous way in later stages of Frege's work. In particular, Demopoulos suggests, the same problem appears to iterate to extensions, so that it becomes reasonable to ask whether Frege's introduction of extensions really overcomes the difficulty that the Julius Caesar problem posed for numbers while not itself succumbing to a similar objection. Demopoulos further notes that this matter certainly needs to be addressed in connection with Basic Laws of Arithmetic where Frege introduces extensions in a way that is formally exactly analogous to his contextual definition of number (Demopoulos 1995, 8-12).
I have maintained that Frege's appeal to extensions did not overcome the difficulty that the Julius Caesar problem posed for him and that the problem reappeared in different guises both in later stages of Frege's philosophy and in later stages of analytic philosophy. I further suggest that, studied from the angle of Frege's unsatisfied need to specify referents in the way his system required and the connections that this has with the philosophical issues chronicled in this essay, Ruth Barcan Marcus' work on identity, necessity, and reference, right from the beginning in the 1940s, proves most lucid, far-reaching, and replete with indications regarding the implications of the set theoretical paradoxes for logic and epistemology.
Looking at the charges that Quine leveled against Marcus within the broader context that I have outlined here helps shed light on the issues raised and makes it improbable that, as Scott Soames once categorically stated, there "is no way that the formal system of Marcus' early papers could have significant consequences about ordinary names and descriptions in natural language" (Humphreys and Fetzer 1998, 71). Soames himself acknowledged that: "Although Quine's arguments were mistaken, they were enormously influential and they baffled large numbers of the profession for decades" (Humphreys and Fetzer 1998, 16; 14). As I see it the very nature of those very public and influential complaints established the significance of the formal system of Marcus' early papers to later theories about ordinary names and descriptions in natural language.
In her doctoral dissertation, Ruth Barcan Marcus already had her finger on the pulse of the problem of rigid reference. Postulating the necessity of identity is another response to the demands that Frege and Russell had tried to meet with the law of substitutivity of identicals, the introduction of extensions, Basic Law V, the theory of definite descriptions, the axiom of reducibility. Fitch's observation, cited above, that in Barcan's system there is lack of need for an axiom of extensionality merits study in this regard.
Likewise, Marcus' insistence that the apparent opacity of intensional contexts lies in the way that logicians like Quine use the terms 'identity', 'true identity', 'equality' in their attempts to blind themselves and others to differences between identity and weaker forms of equivalence has far-reaching implications. For the "paradoxes" associated with modal logic arise, not because someone has fiddled with the reference process in some improper way, thereby thwarting access to the objects we want to put in our theories and quantify over, but because, like the identity statements that Frege put identity at the heart of his logic, modal contexts force a produce a bifurcation in the reference of the statements that they govern and so display intensions eclipsed in other forms of discourse. Differences between identity and weaker forms of equivalence always obtain in any non-trivial case in which the difference between the signs in an identity statement corresponds to some intensional difference, some difference in the mode of presentation of the reference.
Frege and Russell both had to struggle with the fact that descriptions are opaque in perfectly extensional contexts. Like Frege's senses, descriptions serve to illuminate a single aspect of the thing that they would refer to. It is this partial illumination that is behind the impression of referential opacity that so vexed Quine. While a definite description of an object can alone suffice to fix the reference univocally, no description, however definite it may be, will suffice to define the reference to the point of blocking all the other descriptions identifiable with the same object which could come up to perturb the inference and change our truths into falsehoods or give truth to a lie.
In the 1970 lectures that went into the making of Naming and Necessity, Saul Kripke explained some of these same ideas in particularly accessible, very ordinary language. However, in my opinion the interest of the new theory of reference ultimately lies in its connection to the broader questions discussed above, which are more threatening than those dealt with in Naming and Necessity. For a full understanding of the nature and deepest implications of Quine's attacks on modal and intensional logics leads right into the heart of problems in the foundations that Frege laid for his logic. And as Edmund Husserl once warned the Göttingen Mathematical Society, the development of the sciences, has constantly shown that lack of clarity in the foundations ultimately wreaks its vengeance, that if certain levels of progress are reached, further progress is fettered by errors due to obscure methodological ideas (Husserl 1901, 431-32).
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