This is a preprint version of the paper which
appeared in definitive form in the10th Anniversary Issue of
*Axiomathes*,
Centro Studi per la Filsofia Mitteleuropa, Trento, Italy, 1997,
53-82 and
in* Husserl or Frege, Meaning, Objectivity
and Mathematics*, by Claire Ortiz Hill and
Guillermo Rosado Haddock, Chicago: Open Court, 2000. The published
versions should be consulted for all citations.

In 1894 Gottlob Frege published a scathing attack (Frege 1894) on the Cantorian theory of psychological abstraction Edmund Husserl espoused in the Philosophy of Arithmetic (Husserl 1891). Frege denounced that aspect of Husserl's attempt to provide sound foundations for arithmetic as "naive". Any view according to which a statement of number is not a statement about a concept or about the extension of a concept, Frege said, is naive for "when one first reflects on number, one is led by a certain necessity to such a conception" (p. 197). Had Husserl used the term 'extension of a concept' in the way, he, Frege, thought fit, he wrote, they "should hardly differ in opinion about the sense of a number statement" (pp. 201-02).

Yet despite that show of confidence, Frege's apparent self-assurance quickly evaporated less ten years later upon learning of the contradiction of the class of all classes that are not members of themselves from Bertrand Russell. Frege immediately responded to Russell (Frege 1980a, 130-32) that it was the law about extensions that was at fault, and that its collapse seemed to undermine the foundations of arithmetic proposed in the Basic Laws of Arithmetic.

So it was a considerably less confident man who wrote in the appendix to the 1903 volume of The Basic Laws of Arithmetic that hardly anything more unfortunate could befall a scientific writer than to find the foundations of his work shaken after it was completed. He would have gladly dispensed with his law about extensions, Frege confessed, had he known of any substitute for it. At least, he consoled himself, he could draw comfort in the fact that misery loves company for he believed that everybody who had made use of extensions of concepts, classes, sets in proofs was in the same position he was (Frege 1980b, 214).

Frege never believed that his law about extensions recovered from the shock it had sustained from Russell's finding. And, as it happens, his review of Husserl's book actually contains some of the most forceful statements Frege ever made in favor of them. For Frege actually expressed profound reservations about using extensions throughout his career, and many statements he made during his lifetime belie the aplomb with which he carried out his caustic attack upon Husserl's theories. Frege confessed several times that he had never actually become completely reconciled to introducing extensions into his reasoning. He alluded to having been "led by a certain necessity" (Frege 1894, 197) or to having been "constrained to overcome" his "resistance" to them (Frege 1980a, 191). By the end of his life he was warning that talk of extensions "easily... can get one into a morass" and that it leads "into a thicket of contradictions" (Frege 1979a, 55).

Here I propose to cast a cold eye on the logical moves which led Frege into that "morass" in an attempt to lay bare just what lie behind his conviction that everybody who made use of extensions of concepts, classes, sets in proofs was in the same position he was. I then study Russell's battle with the "thicket of contradictions". Finally, I call for lucidity regarding the differences between mere equality and full identity which are lost in the abstraction process. I must, of course, begin by defining logical abstraction.

1. What Logical Abstraction Is

Logical abstraction is a technique by which one singles out what is common among the members of a given set of objects. Through it a property is isolated and the particular equivalence obtaining between objects possessing that property comes to be regarded as an identity (on paper). A common predicate is interpreted as a common relation to a new term, the class of all those terms that are equal in terms of the property indicated by the predicate. The class of terms having that particular relation then replaces the common property inferred from the equivalence relation chosen, and all the other properties which might have normally served to distinguish those objects from each other, or from other objects equal to them in the same respect, are "abstracted" out of the picture. Once deleted on paper, the properties which originally might have marked 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.

Willard Van Orman Quine has provided the following description of the procedure:

given a condition '---' upon x, we form the class x^--- whose members are just those objects x which satisfy the condition. The operator 'x^' may be read 'the class of all objects x such that'. The class x^--- is definable, by description, as the class y to which any object x will belong if and only if --- (Quine 1961a, 87).

And Quine has provided this example of how that procedure might applied in actual practice:

It may happen that a theory dealing with nothing but concrete individuals can conveniently be reconstrued as treating of universals, by the method of identifying indiscernibles. Thus consider a theory of bodies compared in point of length. The values of the bound variables are physical objects, and the only predicate is 'L', where 'Lxy' means 'x is longer than y'. Now where '~Lxy Ù ~Lyx, anything that can be truly said of x within this theory holds equally for y and vice versa. Hence it is convenient to treat '~Lxy Ù ~Lyx' as 'x = y'. Such identification amounts to reconstruing the values of our variables as universals, namely lengths, instead of physical objects (Quine 1961b, 117).

The talk of universals engaged in, Quine explains, may be regarded "merely as a manner of speaking - through the metaphorical use of the identity sign for what is really not identity but sameness of length.... In abstracting universals by identification of indiscernibles, we do no more than rephrase the same old system of particulars" (Quine 1961b, 118).

Many have found logical abstraction to be an attracive and convenient technique for translating various familiar expressions, and those having to do with numbers especially, into the notation of the extensional logic favored by so many logicians. Through logical abstraction one might introduce new objects into reasoning by translating many expressions into extensional language which would not on the surface seem to lend themselves to extensional treatment. Talk of properties is transformed into talk of classes as every monadic predicate comes to have a class as an extension, --the class of all things of which the predicate is true. The notation Fy comes to mean y is a member of the class F, the class of all objects fulfilling a given condition, - the class of those things that are equal in that particular respect (Quine 1961b, 120-21). Many things people want to accomplish can, in fact, be accomplished by leaving unwanted properties out of the picture (and, one might add, might not be accomplishable were they properly taken into account).

2. Frege Begins Abstracting Away Properties in Foundations of Arithmetic

Arithmetic was the point of departure for the ideas that led Frege to develop his logical theories. And the symbolic language he hoped might one day "become a useful tool for the philosopher" was modelled after the language of arithmetic (Frege 1879, Preface).

According to Frege's theory of arithmetic, as set out in the Foundations of Arithmetic (Frege 1884), numbers are always independent objects which as such are qualified to figure in identity statements, and any appearance to the contrary could "always be got around" because expressions which in everyday discourse do not seem to name independent objects can be rewritten so that they do (§§ 55-57).

Moreover, Frege held that with numbers it was "a matter of fixing the sense of an identity" (p. x, §§ 62, 106). And it became his aim "to construct the content of a judgement which can be taken as an identity such that each side of it is a number" (§ 63). So finding "a means for arriving at that which is to be regarded as being identical" (§ 63) became an integral part of his project to provide a deeper foundation for the theorems of arithmetic.

Not wanting to introduce a special definition of identity for this, but wishing rather "to use the concept of identity, taken as already known as a means for arriving at that which is to be regarded as being identical" (§ 63), Frege explicitly adopted Leibniz's principle that "things are the same as each other, of which one can be substituted for the other without loss of truth" (§ 65). Of his choice he wrote:

This I propose to adopt as my own definition of equality. 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).

As Frege was writing Leibniz's formula right into the foundations of his logic, however, he modified Leibniz's dictum in an important way. Not one ever to adhere slavishly to usual linguistic practice, Frege advocated rephrasing statements in ways which eliminate distinctions obtaining in ordinary language. So, in the passage cited above he has recommended rephrasing the statement 'The segments are equal in length' as 'the length of the segments is equal or the same', and 'the surfaces are identical in color' as 'the color of the surfaces is identical'.

In so doing he adjusted Leibniz's principle to meet his own ends by deciding to translate sentences of natural languages into his symbolic language in a way he thought might do away with the differences between being identical (complete agreement in all respects) and equal (only agreement in this respect or that). For although Leibniz's law defines identity, complete coincidence, Frege, here as elsewhere, explicitly maintained that for him "whether we use 'the same' as Leibniz does, or 'equal' is not of any importance. He would insist over and over again that for him there was no difference between equality and identity. For example in the preface to his 1893 Basic Laws of Arithmetic (p. ix), he explained that he had chosen to use the ordinary sign of equality in his symbolic language because he has convinced himself that it is used in arithmetic to mean the very thing that he wished to symbolize.

To achieve these goals Frege appealed to a process of logical abstraction (though he never used that term) by which statements in which certain objects are said to be equivalent in terms of a certain property predicated of them are transformed into statements affirming the identity-equality of abstract objects formed out of those properties. In the Foundations of Arithmetic (§ 65) he used the following examples to illustrate his move:

5) 'The segments are equal in length'

and

6) 'The surfaces are equal in color'

which he wished to see reformulated as

7) 'The lengths of the segments are equal'

and

8) 'The color of the surfaces is equal'.

Here, Frege has changed the statements of sameness of concrete properties predicated of concrete objects in 5) and 6) into statements 7) and 8) which affirm the equality-identity of abstract objects, in this case surfaces and lengths. He believes he has thus transformed statements about objects which are equal under a certain description into statements expressing a complete identity. By erasing the difference between identity and equality, he is in fact arguing that being the same in any one way is equivalent to being the same in all ways. However, he realized that many of the inferences that could be made by appealing to such a principle would lead to evidently false and absurd conclusions.

3. The Problems Frege Addressed from the Beginning

Frege himself acknowledged that left unmodified the procedure just described was liable to produce nonsensical conclusions, or be sterile and unproductive (Frege 1884, § 66-67). For example, he realized that his definition of identity only afforded logicians a means of recognizing an object as the same again if determined in a different way, but it did not account for all the ways in which it could be determined.

To illustrate some nonsensical consequences of defining identity in this way, Frege carried the reasoning involved in his example of the identity of two lines one step further. However, the points he wishes to make can be made more graphically by leaving the world of abstract geometrical figures for denizens of a more material one.

So, parroting Frege's reasoning in Foundations of Arithmetic § 66, I propose to illustrate his point about possible nonsensical consequences of his definition by appealing to a more concrete case. Suppose it has finally been determined to be true that:

the man who fired the shots from behind the grassy knoll is identical with the man who killed John Kennedy.

Frege's definition of identity would then afford us a means of identifying the man who fired the shots from behind the grassy knoll again in those cases in which he is referred to as the man who killed John Kennedy. But, as Frege recognized, this means does not provide for all the cases. For instance, it could not decide for us whether Lee Harvey Oswald was the man who fired the shots from behind the grassy knoll. While any informed person would consider it perfectly nonsensical to confuse Lee Harvey Oswald with the man who fired the shots from behind grassy knoll, this would not, Frege would acknowledge, be owing to his definition. For it says nothing as to whether the statement:

'the man who fired the shots from behind the grassy knoll is identical with Lee Harvey Oswald'

should be affirmed or denied, except for the one case where Lee Harvey Oswald is given in the form of 'the man who killed John Kennedy'. Only if we could lay it down that if Lee Harvey Oswald was not the man who killed John Kennedy, (still following Frege's reasoning), could our statement be denied, while if he was that man, our original definition would decide whether it is to be affirmed or denied. But then we have obviously come around in a circle, Frege acknowledged. For in order to make use of this definition we should have to know already in every case whether the statement 'Lee Harvey Oswald is identical with the man who killed John Kennedy' was to be affirmed or denied.

Left as it was his definition was unproductive, Frege further judged, because in adopting this way out, we would be presupposing that an object can only be given in one single way. For otherwise, (still parroting Frege's reasoning, but using our more material example), it would not follow from the fact that Lee Harvey Oswald was not introduced by our definition that he could not have been by means of it. "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).

4. Frege Espouses Extensions

So seeing that he could not by these methods alone obtain concepts with sharp limits to their application, nor therefore, for the same reasons, any satisfactory concept of number either, Frege was led to introduce extensions to guarantee that an identity holding between two concepts could be transformed into an identity of extensions, and conversely (Frege, 1884, § 67; also § 107). He hoped thereby to eliminate the undesirable consequences (§§ 66-67, 107) he saw accumulating around his theory of number and he devoted several sections of Foundations of Arithmetic to discussing the pros and cons of introducing them (§§ 68-73). However, in one of its concluding sections he wrote of extensions that: "This way of getting over the difficulty cannot be expected to meet universal approval, and many will prefer other methods of removing the doubt in question. I attach no decisive importance even to bringing in the extensions of concepts at all" (§ 107).

Although Frege ended the Foundations of Arithmetic claiming that he was not attaching any decisive importance to bringing in extensions, he did not propose any alternative and by the time it came to actually proving his theory of number in the Basic Laws of Arithmetic, he had managed to quiet the reservations he would later confess having had about introducing them. He explained in the preface to the book that extensions had taken on "great fundamental importance." Their introduction, he now maintained was "an important advance which makes for far greater flexibility" (pp. ix-x). "In fact," he wrote there, "I even define number itself as the extension of a concept, and extensions of concepts are, according to my definitions, graphs. So we just cannot do without graphs" (p. x). As far as he could see, he wrote, his basic law about extensions of concepts was the only place in which a dispute could arise. This, he believed, would be the place where the decision would have to be made. (p. vii). Then, a year later he came out fighting for extensions in the 1894 review of Husserl.

In Basic Laws, 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). "This possibility," he wrote then, "must be regarded as a law of logic, a law that is invariably employed, even if tacitly, whenever discourse is carried on about extension of concepts. The whole Leibniz-Boole calculus of logic rests upon it. One might perhaps regard this transformation as unimportant or even as dispensable. As against this, I recall the fact that in my Grundlagen der Arithmetik I defined a Number as the extension of a concept..." (Frege 1893, § 9).

In §§ 146-47 of the 1903 Basic Laws II, he characterized extensionality writing:

If a (first-level) function (of one argument) and another function are such as always to have the same value for the same argument, then we may say instead that the graph of the first is the same as that of the second. We are then recognizing something common to the two functions... We must regard it as a fundamental law of logic that we are justified in thus recognizing something common to both, and that accordingly we may transform an equality holding generally into an equation (identity) (Frege 1980b, 159-60).

Frege never believed that any proof could be supplied that would sanction such a transformation. So he devised Basic Law V, or Principle V, to mandate the view of identity, equality and substitutivity his system required. By transforming "a sentence in which mutual subordination is asserted of concepts into a sentence expressing an equality," Basic Law V would permit logicians to pass from a concept to its extension, a transformation which, Frege held, 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).

5. Strong Extensionality

Now that we have looked at Frege's characterization of extensionality, it is very important to look at how extensions could help Frege solve the particular problems with identity and substitutivity in connection with which they were introduced in the Foundations of Arithmetic.

First of all, Frege was conscious of the fact that the logician's job was initially made simpler because she initially only had to recognize the object as given in the particular way stipulated by her identity statement. But Frege realized that his definition afforded no means of recognizing that object as the same again when given in a different way. So returning to our Kennedy example, suppose that the wife of a man who fired at Kennedy from behind the grassy knoll has come forward with incontrovertible evidence that her husband, one Mr. Knoll, fired shots at Kennedy and killed him. She was actually with her husband behind the grassy knoll that day, but was too blinded by love for him to tell the police what she knew. She also wanted to protect her family. But Mr. Knoll has since died and her conscience is tormenting her. In addition to her eyewitness account she has produced authentic diaries in which her husband gave details of his plans. So, (changing the descriptions to expressions that name directly so as to avoid any problems deriving solely from the fact that descriptions appear in the putative identity statement):

'Mr. Knoll is identical with Kennedy's assassin'

is a true statement. Initially, it looks as if the case is closed. We can now substitute Mr. Knoll's name every time we find a reference to Kennedy's assassin. However, our identity statement is completely blind as concerns other alternatives. And once we give the matter further thought we find that Kennedy's assassin has so often been identified with Lee Harvey Oswald that substitution in most contexts would yield nonsense. For example, most of the Warren Report would become complete nonsense.

Now as concerns Frege's second point regarding the sterility of the procedure. It is certain that the single fact given by our identity is a highly informative statement. But just because our x is both F and G, this does not mean that if something is F it is G. We have only learned that Mr. Knoll killed Kennedy. But only on paper has Basic Law V ruled out the possibility that Lee Harvey Oswald, or someone else, might also have fired at Kennedy, and so might also have been Kennedy's assassin. Kennedy did not die instantly, and numerous factors may have finally conspired to bring about his death.

However, with Frege's principle of extensionality, we put on logical blinders. It mandates that since Mr. Knoll is identical to Kennedy's assassin, then anyone who was Kennedy's assassin was Mr. Knoll. Our concepts have acquired the sharp limits our quest for knowledge and substitution requires (Frege 1884, § 67; also § 107). Which is all fine on paper, but extensionality will not of itself keep Lee Harvey Oswald from slipping back into the picture as, for example, someone who might also have been Kennedy's assassin, but was definitely not Mr. Knoll. It may seem at first attractively simple to obliterate distinctions between identity and equality, but 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 have a rule stipulating that equality is to function as identity. For informative identity statements are the breeding ground of contradictions in extensional systems. The seed of contradictions derivable in extensional systems lies buried in them, --something, of course, wholly unacceptable to Frege, a man who developed a symbolic language whose stated first purpose was to to provide "the most reliable test of validity for a chain of inferences and to point out every presupposition that tries to sneak in unnoticed" (Frege 1879, 6).

In his most confident moments, however, Frege believed that Basic Law V could bring logicians out of what he called the "queer twilight" of identity in which he saw mathematicians performing their logical conjuring tricks (Frege 1980b, 151-52). But the procedure he had hoped would bring identity out of the penumbra into the clear light of day had actually 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, with the result that, for example, a class might seem to be a member of itself.

6. The Shaking of Frege's Foundations for Arithmetic

When informed of Russell's paradox of the class of all classes which are not members of themselves in 1902 (Frege 1980a, 130-31), Frege immediately designated the logical transformations legitimized by Basic Law V as being the source of the problem. Basic Law V was false, he recounted in the appendix to Basic Laws of Arithmetic II (Frege 1980b, 214-24). The way he had introduced extensions was not legitimate (p. 219), and the interpretation he had so far put on the words 'extension of a concept' needed to be corrected.

In the several texts in which he pinpoints what he believed was the source of the difficulties, he consistently cites Basic Law V's transformation of concepts into objects for extensional treatment as being at fault (Frege1980a ,54-56, 191; 1979, 181-82, 269-70). There was nothing, he decided, to stop him 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 we 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 and so the second part of his law fails (Frege 1980b, 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. This, however, really abolishes the extension of the concept" he concluded (Frege 1980b, 221).

An example will make Frege's problem clearer. To illustrate the point he is making we might turn to the modern analogue of his 'the number of Jupiter's moons is 4', i.e., the putative identity statement Quine made famous:

1) '9 is the number of the planets'.

According to Basic Law V, supposing this to be a true identity, the number which belongs to the concept 9 is the same as that which belongs to the concept the number of the planets. But the converse does not necessarily hold. For though 9 may be the number of the planets, we cannot infer that whatever falls under one falls under the other. There is nothing other than Basic Law V to guarantee that all objects that fall under '9' fall under 'the number of the planets' as well. For instance, according to materialistic astronomy:

2) The ninth planet, Pluto, was discovered in 1930.

3) The planet Pluto may be an anomaly and not a planet at all, but a giant asteroid flung into its present position when it had a close gravitational encounter with one of the outer planets.

4) A Planet X may exist far beyond Pluto, which would explain apparent irregularities in Neptune's orbit.

5) According to astronomers the dark matter accounting for more than 90% of the total mass of the universe could be made of giant planets.

So, while Frege's Basic Law V would secure for us that all objects that fall under '9' fall under 'the number of the planets' as well, according to materialistic astronomy, by substitution into (1) 9 may equal 8, or 10, or an as yet undetermined number of planets. And this indeterminacy causes Frege's system to go haywire.

Frege finally decided that all efforts to repair his logical edifice were destined to failure. By 1912 he had laid down his extensions and conceded defeat. As he wrote for an article by Philip Jourdain:

And now we know that when classes are introduced, a difficulty, (Russell's contradiction) arises.... Only with difficulty did I resolve to introduce classes (or extents of concepts) because the matter did not appear to me to be quite secure --and rightly so as it turned out. The laws of numbers are to be developed in a purely logical manner. But numbers are objects…. Our first aim was to obtain objects out of concepts, namely extents of concepts or classes. By this I was constrained to overcome my resistance and to admit the passage from concepts to their extents... I confess... I fell into the error of letting go too easily my initial doubts (Frege 1980a, 191).

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. The difficulties which this idiosyncrasy of language entangles us in are incalculable" (Frege 1979, 270).

"From this," Frege wrote, "has arisen the paradoxes of set theory which have dealt the death blow to set theory itself" (Frege 1979, 269).

7. Intellectual Sorrow Descends Upon Bertrand Russell

Intellectual sorrow, Bertrand Russell has said, descended upon him in full measure when the contradiction about classes which are not members of themselves put an end to the logical honeymoon he was having when he began writing the Principles of Mathematics (Russell 1959, 56). One of the things he found once he set out to find out exactly how and why Frege's theories could have given rise to the contradiction was that one could not generally suppose that objects which all have a certain property form a class which is in some sense a new entity distinct from the objects making it up (Russell 1973, 171).

Unbridled use of the principle of logical abstraction was producing fake objects, which in turn were causing the worrisome contradictions.

In Mathematical Logic, Quine showed how easily class abstraction leads to Russell's contradiction:

The usual way of specifying a class is by citing a necessary and sufficient condition for membership in it. Such is the method when one speaks of "the class of all entities x such that...," appending one or another matrix. The class of all entities x such that x writes poems, e.g., is the class of poets... Despite its sanction from the side of usage and common sense, however, this method of specifying classes leads to trouble. Applied to certain matrices, the prefix 'the class of all entities x such that' produces expressions which cannot consistently be regarded as designating any class whatever. One matrix of this kind, discovered by Russell, is 'i(x Î x)'; there is no such thing as the class of all entities x such that ~(x Î x). For suppose w were such a class. For every entity x, then,

x Î w = ~(x Î x).

Taking x in particular as w itself, we are led to the contradiction:

(w Î w) = ~(w Î w). (Quine 1940, § 24)

9. The Distinction of Logical Types

Early in his search for a solution to the problem of the paradoxes, Russell believed that "the key to the whole mystery" would be found in the distinguishing of logical types (Russell 1903, § 104). So he established a hierarchy of classes according to which the first type of classes would be composed of classes made up entirely of particulars, the second type composed of classes whose members are classes of the first type, the third type composed of classes whose members are classes of the second type, and so on. The types obtained would be mutually exclusive, making the notion of a class being a member of itself meaningless. No totality of any kind could be a member of itself. "In this way," Russell believed, "we obtain a series of types, such that, in all cases where formerly a paradox might have emerged, we now have a difference of type rendering the paradoxical statement meaningless" (Russell 1973, 201).

Russell believed that the theory of types he developed led "both to the avoidance of contradictions, and to the detection of the precise fallacy which has given rise to them" (Russell 1927, 1). And he believed that no solution to the contradictions was technically possible without it. However, he ultimately became aware that it was not "the key to the whole mystery" (see Russell 1919, 135; 1956, 333; Quine, 1961a, 91-92).

For one thing, he realized that the theory only solved some of the paradoxes for the sake of which he had invented it. Deeper problems caused the old contradiction to break out afresh and he realized that "further subtleties" would be needed to solve them. For, though Russell's theory of types restores some of the logical structure Frege had eclipsed when talk of classes and their extensions opened the door to the illegitimate inferences in the first place, the theory just treats the symptoms of the malady. It does not come to terms with the problem as to how and why the logical types had become confused to begin with, --as to what had made Frege try to obtain objects out of concepts in the first place. It does not adequately address the deeper logical problems concerning identity and equality which had tempted Frege to introduce classes and a law saying that a class could be predicated of its own extension. The quest to uproot the paradoxes would require further investigations into what classes were.

10. Whither Classes?

Having evaded some of the contradictions by distinguishing between various types of objects, and having proposed a hierarchy of types, Russell was obliged to come to some conclusions regarding the ontological status of classes. He decided that classes could not be independent entities. Regarding them as such leads inescapably to the contradiction about the class of classes which are not members of themselves. He had originally believed that:

When we say that a number of objects all have a certain property, we naturally suppose that the property is a definite object, which can be considered apart from any of all of the objects, which have, or may be supposed to have, the property in question. We also naturally suppose that the objects which have the property form a class, and that the class is in some sense a new single entity, distinct, in general, from each member of the class (Russell 1973, 163-64).

According to Frege's theory, Russell explained,

Whatever a class may be, it seems obvious that any propositional function fx determines a class, namely the class of objects satisfying fx. Thus 'x is human' defines the class of human beings, 'x is an even prime' defines the class whose only member is 2, and so on. We can then (so it would seem) define what we mean by 'x is a member of the class u', or 'x is a u' as we may say more shortly. This will mean: 'There is some function f which defines the class u and is satisfied by x'. We then need an assumption to the effect that two functions define the same class when they are equivalent, i.e. such that for any value of x both are true or both false. Thus 'x is human' and 'x is featherless and two-legged' will define the same class. From this basis the whole theory of classes can be developed (Russell 1973, 171).

Any such object that might be proposed, he believed, presupposed the notion of class, i.e. an object uniquely determined by a propositional function, and equally determined by any equivalent propositional function (Russell, 1903, § 489). However, he had become convinced that this was precisely the kind of reasoning that had gotten himself, Frege and others involved in the contradictions. For we cannot, then, escape the contradiction, Russell explained:

For it is essential to an entity that it is a possible determination of x in any propositional function fx; that is, if fx is any propositional function, and a any entity, fa must be a significant expression. Now if a class is an entity, 'x is a u' will be a propositional function of u; hence, 'x is an x' must be significant. But if 'x is an x' is significant, the best hope of avoiding the contradiction is extinguished (Russell 1973, 171).

The idea that classes were not entities shed some light on the ontological nature of classes by saying what they were not, but Russell had to do more than that. For the contradictions were, however, a distressing after-effect of introducing the extensions Frege had reluctantly appealed to because he saw that without them the theory of abstraction he was prescribing could produce nonsensical conclusions, or be sterile and unproductive (Frege 1884, §§ 66-67). Frege had resorted to extensions out of necessity and they served a definite purpose in his system.

And Russell was well aware of the need for classes. "The reason," he wrote in the closing pages of the Principles of Mathematics,

which led me, against my inclination to adopt an extensional view of classes, was 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. The only single entity I have been able to discover is the class as one... (Russell 1903, § 486).

And he believed 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).

So Russell could not very well just demolish classes. The trick was to find a way of making them disappear from the reasoning in which they were present without really completely letting go of them. If, he reasoned, "we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming that there are classes without being compelled to make the opposite assumption that there are no classes" (Russell 1919, 184).

11. Having Your Classes and Deleting Them Too

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 he would have to find a way of correlating objects and concepts in a way which correlated mutually subordinate concepts with the same object. He suggested 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 meanings (Frege 1979, 181-82).

But it was precisely that sort of recipe for making what was incomplete behave as if it were complete in identity statements that Russell's paradox had cast doubt upon, and the problems that procedure caused were precisely the ones Russell hoped to circumvent through a new theory of classes based on his theory of definite descriptions by which 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 1980b, 214).

While struggling to get to the bottom of the problem of the fake objects created by logical abstraction, Russell had discovered some parallels existing between the problems which arise when classes are treated as objects and problems which come up when descriptions are treated as names. These analogies plus the success he had with his 1905 theory of definite descriptions gave him an idea as to how classes might be analyzed away much as descriptions had been, and thus gave him a concrete idea as to how he might sweep his problems away.

Statements containing descriptions had proven amenable to further analysis. So once Russell was satisfied that classes and descriptions both fell into the same logical category of non-entities represented by incomplete symbols, he decided 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 'Scott is the author of Waverley' might be rewritten 'Scott wrote Waverley; and it is always true of c that if c wrote Waverley, c is identical with Scott' (Russell 1956, 55). In this construal of the sentence, what was not identical has been made to be equivalent. A symbol deemed equatable with 'Scott' has gone proxy for the description and 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. Unadulterated class abstraction was generating contradictions, but the theory of definite descriptions was a different way of making an object fit to go proxy for what was said about it. By adapting the theory to class symbols, he thought he might acquire a "method of obtaining an extensional function from any given function of a function," which was precisely what the theory of classes needed (Russell 1927, 187).

12. Problems With Equivalent Propositional Functions

Russell, however, also realized that resolving the paradoxes would mean coming to terms with problems with classes caused by the fact that it was "quite self-evident that equivalent propositional functions are often not identical" (Russell 1903, § 500). And these problems display certain formal similarities with the puzzles about descriptions which had impressed upon Russell the need to remove descriptions from statements in which they were present in the first place.

One of the reasons Russell had had to find a way to get rid of descriptions was that, while an informative statement of the form 'the author of Waverley is the author of Marmion' could be true and seem to be a genuine identity statement in virtue of the fact that both descriptions were true of the same individual, the two descriptions were obviously not themselves identical, nor were they always intersubstitutable salva veritate. If the only thing that mattered were that both descriptions are true of the same person, Russell observed, then any other phrase true of Scott would yield the same statement. Then 'Scott is the author of Marmion' would be the same as 'Scott is the author of Waverley', which is obviously not so, since from the one we learn that Scott wrote Marmion and from the other that he wrote Waverley, but the former statement tells us nothing about Waverley and the latter nothing about Marmion. Properly reconstrued, the descriptions would disappear and the statement transformed into: 'Someone wrote Waverley and no one else did, and that someone else wrote Marmion and no one else did' (Russell 1917, 217-18).

This way of correlating mutually subordinate descriptions with the same object could help solve a parallel problem in the paradox plagued theory of classes, e.g. the problem of two propositional functions having the same graph without everything that is true of one being true of the other. For just as descriptions are tied to a particular characterization of an object, so classes are formed by specifying the definite property giving the class. Moreover, just as two different descriptions might be true of the very same object, so a single class of objects might be defined in different ways, each one corresponding to a different sense of the class name. Yet as Russell observed "if a and b be different class-concepts such that x Î a and x Î b are equivalent for all values of x, the class-concept under which a falls and nothing else will not be identical with that under which falls b and nothing else" (Russell 1903, § 488).

Russell was pleased to point out, however, that the theory of classes inspired by his theory of descriptions would leave intact all the fundamental properties desired of classes, the principal one of these being that "two classes are identical when, and only when, their defining functions are formally equivalent" (Russell 1927, 189). "The incomplete symbols which take the place of classes serve the purpose of technically providing something identical in the case of two functions having the same extension," he believed (Russell 1927, 187).

Presto chango! By an act of logical prestidigitation, the theory of classes has been rendered "symbolically satisfactory." We can have our classes and delete them too. Statements verbally concerned with classes have been reduced to statements that are concerned with propositions and propositional functions. Logicians can effectively pass from a class to its extension on paper without incurring contradictory results. We have avoided contradictions arising from supposing that classes are entities and acquired a technique for laying hold of the extension of a class. In addition, functions having the same extension would be identical. And all our efforts have brought us ever nearer to the deep reasons why Frege's introducing extensions into his logic caused the paradoxes in the first place.

13. The Ultimate Source of the Contradictions

Russell considered the technique he devised for making incomplete symbols obey the same formal rules of identity as symbols which directly represent objects to be a breakthrough in solving the paradoxes and a host of other problems (Russell 1959, 49, 60; Grattan- Guinness 1972, 106-07; 1977, 70, 79-80, 94 and note; 1975, 475-88; Kilmister 1984, 102, 108, 123, 138; Hill 1997). However, he realized that incomplete symbols could only obey the same formal rules of identity as symbols referring to objects in so far as "we only consider the equivalence of the resulting variable (or constant) values of propositional functions and not their identity" (Russell 1927, 83), an observation which brings us right back to the problems concerning logical abstraction, identity and substitutivity which first obliged Frege to introduce extensions.

So to assess the ultimate effectiveness of the various techniques Russell invented for evading the paradoxes, it is important to take a close look at what Russell once called the "ultimate source" of the contradictions. In a statement which brings us all the way back to Frege's original idea that "the existence of different names for the same content is the very heart of the matter if each is associated with a different way of determining the content" (Frege 1879, § 8) and to his original reasons for introducing extensions, Russell affirmed that the cause of his and Burali-Forti's contradiction was to be found in that:

if x and y are identical, fx implies fy. This holds in each particular case, but we cannot say it holds always, because the various particular cases have not enough in common. This distinction is difficult and subtle... the neglect of it is the ultimate source of all the contradictions which have hitherto beset the theory of the transfinite (Russell 1973, 188).

These contradictions most intimately concerned with identity and the formal equivalence of functions proved especially hard to stamp out. For them Russell had to develop a new tactic more directly aimed at the theory of identity which had brought about the confusion of types and the reification of incomplete symbols in the first place.

In particular, Russell still faced the following predicament: The contradictions had taught him that there was a hierarchy of logical types. If contradiction producing vicious circle fallacies were to be avoided, functions would have to be divided into types, and all talk of functions would then necessarily be limited to some one type, which would effectively make statements about all functions true with a given argument, or all properties of a some given object, meaningless.

However, Russell was perfectly aware that "it is not difficult to show that the various functions which can take a given object a as argument are not all of one type" (Russell 1919, 189), and, even "that the functions which can take a given argument are of an infinite series of types" (Russell 1919, 190). By various technical devices we could, he once reminded readers, "construct a variable which would run through the first n of these types, where n is finite, but we cannot construct a variable which will run through them all, and, if we could, that mere fact would at once generate a new type of function with the same arguments, and would set the whole process going again" (Russell 1919, 190). So whatever selection of functions one makes there will always be other functions which will not be included in the selection.

Russell nonetheless believed that "it must be possible to make propositions about all the classes that are composed of individuals, or about all the classes that are composed of objects of any one logical 'type.' If this were not the case, many uses of classes would go astray...." (Russell 1919, 185). "If mathematics is to be possible," he believed, "it is absolutely necessary... that we should have some statements which will usually be equivalent to what we have in mind when we (inaccurately) speak of 'all properties of x.' ... Hence we must find, if possible, some method of reducing the order of a propositional function without affecting the truth or falsehood of its values" (Russell 1927, 166; 1956, 80).

13. The Axiom of Reducibility

So to cope with contradictions arising from necessary talk of 'all properties,' or 'all functions,' Russell introduced the axiom of reducibility. This specially designed axiom would be "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 provide a way of dealing with any function of a particular argument by means of some formally equivalent function of a particular type. It would thus yield most of the results which would otherwise require recourse to the problematical notions of all functions or all properties, and so legitimize a great mass of reasoning apparently dependent on such notions (Russell 1927, 56).

For Russell, 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 expunged and 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 which a less grudging admission of classes is apt to entail" (Russell 1927,166-67; 1956, 82; 1919, 191). Russell leaned on the axiom of reducibility at every crucial point in his definition of classes in Principia Mathematica (pp. 75-81).

Russell considered that many of the proofs of Principia "become fallacious when the axiom of reducibility is not assumed, and in some cases new proofs can only be obtained with with considerable labour" (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). 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).

"But in virtue of the axiom of reducibility," Russell writes in Principia Mathematica *13, "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). For if one assumes the axiom of reducibility, then

every property belongs to the same collection of objects as is defined by some predicate. Hence there is some predicate common and peculiar to the objects which are identical with x. This predicate belongs to x, since x is identical with itself; hence it belongs to y, since y has all the predicates of x; hence y is identical with x. It follows that we may define x and y as identical when all the predicates of x belong to y... (Russell 1927, 57; 1973, 243).

So by virtue of the axiom of reducibility, one might have the properties of identity and equality upon which the logic of Principia Mathematica is built. Russell thought of it as a generalized form of Leibniz's principle of the identity of indiscernibles (Russell 1919, 192; 1927, 57; 1973, 242). So he was finally back to square one, i.e. to the reasons why Frege's theory of identity had made him appeal to extensions in the first place.

14. Why All the Logical Acrobatics?

What had made such a radical measure necessary?

The axiom of reducibility was another attempt to rub out the differences between equality and full identity. In his search for a criterion for deciding whether in all cases x is the same as y, Frege had turned to Leibniz's principle of substitutivity of identicals (Frege 1884, § 65). Then he adopted a form of expression by which being the same in one way would be the same as being the same in all ways, making the differences between equality and identity seem to go away. After that he reluctantly introduced extensions as a way of artificially rectifying the problems with substitutivity which that attempt to equate equality and identity had caused (Frege 1884, §§ 66-67). Basic Law V would guarantee that an identity holding between two concepts could be transformed into an identity of extensions and conversely, that functions having the same extensions were identical --and it leads to Russell's paradox.

Russell struggled long and hard with the problems buried in Frege's theory of identity, and he devised some sharp logical instruments to erase the differences between equality and identity which had not vanished from reasoning as obligingly as hoped. Wielding his theory of types and his technique for analyzing away classes, Russell began sweeping away the tangled web Frege began to weave when he adopted the inference wrecking practice of equating identity with lesser forms of equivalence. By adroitly wiping out a wealth of intensions and casting them into logical oblivion, the axiom of reducibility would sweep Principia Mathematica nearly clean of intensions and so win new territory for extensional ontology.

Yet in spite of all that might be achieved by means of the axiom, Russell expressed reservations about it reminiscent of those Frege had expressed regarding Basic Law V. Russell deemed the axiom "only convenient, not necessary" (Russell 1919, 192), and even called it "a dubious assumption" and a "defect" (p. 193). "This axiom," he admitted, "has a purely pragmatic justification: it leads to the desired results, and to no others. But clearly it is not the sort of axiom with which we can rest content" (Russell 1927, xiv).

Russell, who had once written that "Mathematics, rightly viewed, possesses not only truth, but supreme beauty", concluded that the "solution of the contradictions... seemed to be only possible by adopting theories which might be true but were not beautiful" and that the "splendid certainty" he had "always hope to find in mathematics had become lost in a bewildering maze" (Russell 1959, 155-57). What had appeared so convenient, simple, and austerely beautiful had spawned error, contradiction, ugliness and messiness.

15. Frege's and Russell's Problems Live On

But Russell's struggle with the paradoxes was not the end of the story. No matter how convenient and attractive abstraction may seem to be as a technique for translating expressions into the popular notation of extensional logic, the properties marking the difference between equality and identity do not docilely submit to logical measures designed to wipe them out. They easily slip back into reasoning unawares to sow surprise, antinomy, nonsense, confusion and contradiction, frustrating the aims of the brave new logic which it was hoped might wipe them out with the stroke of a pen. So they have survived the campaign to extirpate them which has finally served to demonstrate the reality and ireradicability of what it was trying to remove from logical reasoning.

In particular, evidence of the logical violence done began to surface again when philosophers working on modal and intensional logics dared to try to bring light to some of dark areas of logic where philosophers had been warned not to tread. Taking a bolder attitude toward limning the true and ultimate structure of reality than many of their contemporaries, philosophers like Jaakko Hintikka and Ruth Barcan Marcus ventured beyond the narrow confines strong extensional calculi impose on philosophical reasoning and began working to increase the depth and utility of the standard languages and to develop intensional languages capable of investigating epistemic and deontic contexts and of analyzing the many non-extensional statements which figure significantly in the empirical sciences, law, medicine, ethics, engineering, politics, and ordinary philosophy but which have been deemed unfit for study because they complicate matters by not conforming to the rigid standards for admission into the stark, sterile logical world Quine and so many others have found so beautiful. And their work has been instrumental in pulling the deep issues underlying the puzzles, contradictions, and paradoxes haunting logical arguments out of the shadowy netherworld to which they were being consigned and into the clear light of day. So even as opponents of intensional and modal logics battled to contain logical reasoning within the narrow confines of strong extensional calculi, more and more reasons for not shoving reasoning into an extensional mold began gathering right in the "beautiful" world they were so intent upon preserving.

16. Marcus on Distinguishing Between Identity and Lesser Forms of Equivalence

Founder of quantified modal logic, Ruth Barcan Marcus has been one of the staunchest and most eloquent advocates of lucidity regarding the differences between identity and weaker forms of equivalence which explicit or implicit extensionalizing principles would extinguish. As part of her ongoing campaign to expand classical logic to deal with larger areas of discourse, she has drawn attention to ambiguities regarding equality and identity that have slipped into logical reasoning and are present there now. In particular, she has drawn attention to the extent to which extensional logical systems are dependent on (1) directly, or indirectly imposing restrictions prohibiting some intensional functions, and (2) equating identity with a weaker form of equivalence (Marcus 1960).

"The usual reason given for reducing identity to equality," she has written, "is that it provides a simpler base for mathematics, mathematics being concerned with aggregates discussed in truth functional contexts, not with predicates in intensional contexts. Under such restrictive conditions, the substitution theorem can generally be proved for equal (formally equivalent) classes, with the result that equality functions as identity" (Marcus, 1960, p. 58).

Extensionality, Marcus explains, has acquired the undeserved reputation of being a clear, unambiguous concept, and as such well-suited to the needs of mathematics and the empirical sciences where, it is claimed, there is no need to traffic in fuzzy, troublesome non-extensional notions. However, strongly extensional functional calculi, she notes, are "inadequate for the dissection of most ordinary types of empirical statement" (Marcus 1993, 5). "Establishing the foundations of mathematics," she points out, "is not the only purpose of logic, particularly if the assumptions deemed convenient for mathematics do violence to both ordinary and philosophical usage" (Marcus 1960, 58). No single well-defined theory of extensionality exists, she argues, but only stronger and weaker principles of extensionality. She has 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 a logic which prides itself on its clarity, and was devised to keep ambiguity from creeping unawares into reasoning.

17. Abiding Differences

It is easy to find puzzles, contradictions, paradoxes, failures of basic logical principles, etc. which illustrate the need for lucidity regarding the differences between identity and weaker forms of equivalence. Now I want to draw attention to some undesirable consequences of failing to distinguish between identity and lesser forms of equivalence which I believe demonstrate the real need for consciously trafficking in intensional notions in order to control confusion and to draw the fine distinctions which are both germane and indispensable to many undertakings, scientific and otherwise.

Significant differences between equality and identity come to the fore in a much more tangible way and the issues at stake become clearer once we leave the realm of x's and y's, and a's and b's and turn to the reference of the signs. Since the logical ideas we are examining are supposed to be a useful tool for scientists and philosophers and since identity statements are only to be true or false on the basis of what they stand for, this is surely what we are supposed to do. There are, after all, few cases in which the difference between being called 'x' or being called 'y' really matters.

While strong extensional principles may prove appropriate in certain contexts, e.g., in criminal investigations or judicial proceedings where a person's guilt or innocence may be the sole determinant factor, relying upon them could have disastrous consequences in other contexts. For instance, in medical research and practice, extensionality could unnecessarily complicate situations and generate confusion, and even make the difference between life and death, sickness and health.

Consider this example. Doctors at Toronto's Hospital for Sick Children have discovered that the immune system of certain diabetics identifies a protein present on the surface of their insulin-producing cells as being the same as a protein present in cow's milk with which it is in many respects almost identical. Unable to distinguish between the two proteins, the immune system stimulates the body to attack and destroy its own insulin-producing cells in the pancreas causing juvenile onset diabetes which may lead to blindness, kidney failure and heart disease.

Marcus's point about identity and equivalence is most a propos here for the immune systems in question are identifying two proteins as being the same on the basis of compelling similarities; they are not picking out essential differences between the milk protein and the protein on the insulin-producing cells. In a case like this equating identity with lesser forms of equivalence may be having disastrous consequences for diabetics.

Intensional factors marking the differences between identity and lesser forms of equivalence also come into play in organ transplant operations where what is sought are organs which are alike in certain respects, but surely not identically the same as the diseased or defective organs they are meant to replace. A strong enough equivalence may obtain between different organs of the same type so that the host organism will not reject the transplanted organ as foreign because it is sufficiently like the organ it supplants. It is precisely intensional considerations that mark the difference between unwanted strict identity and the sought after equivalence powerful enough to keep the substitutivity of the organs from breaking down. To say of organ x that it has all the properties in common with organ y essential for a successful transplant is to make a weaker claim than to claim they are identical. The situation with bone-marrow transplants appears to be more complex than for organs because six different genetic identity markers are involved in matching tissue types.

And don't differences between equality and identity figure in many other dilemmas actually faced in medical practice today? Surely, many of the really challenging moral issues involved in abortion, euthanasia, medically assisted suicide, etc. turn on whether the person to be killed is in all essential respects the same as that person once was, or in the case of abortion the same as the person who will be if the pregnancy is not terminated. For instance, are patients in apparently irreversible comas identical to the people they were before they were in that unfortunate situation? Think of the innumerable things that could been have been predicated of them before which are no longer true, and the truly macabre propositions that could result from substitution rules which do not take sufficient account of the difference between equality and identity. Surely, such considerations are involved drawing the line between a mercy killing and a premeditated murder.

It is plain to see that in the above cases doctors, judges, medical researchers, and concerned individuals could not even seriously consider resorting to any conceptual tool which was so blunt, crude and blind as one which systematically disregards the differences between identity and lesser forms of equivalence. However inconvenient they may be on paper, and however uncongenial they may be to some philosophers, intensions can be the decisive factor in real life situations where failure to resort to them is sure to increase perplexity, engender confusion and complicate matters instead of bringing clarity and precision. Anyone insisting upon deintensionalizing the situations just cited would surely inspire contempt in the scientific community which fortunately would not rely on so crude a conceptual instrument in situations which cry out for the conscious intensional adjustment and progressive refinement of a conceptual tool which less finely tuned could have disastrous consequences. The blithe use of extensional notions could, in the case of the organ transplant operation, be dangerous enough to cause a patient to die of post-operative complications due to simplistic ideas about extensionality and scientific thinking.

I would now like to close with a warning Edmund Husserl made in Formal and Transcendental Logic in 1929, almost forty years after having condemned Frege's theory of equality and identity in Philosophy of Arithmetic. Mathematicians, Husserl observed, are not in the least interested in the different ways object may be given. For them objects are the same which have been correlated together in some self-evident manner. However, he contended, logicians who do not bewail the lack of clarity involved in this, or who claim that the differences do not matter are not philosophers since here it is a matter of insights into the fundamental nature of formal logic, and without a clear grasp of the fundamental nature of formal logic, one is obviously cut off from the great questions that must be asked about logic and its role in philosophy (Husserl 1929, 147-48).

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