Bowman L. Clarke is Professor of Philosophy and former Head of the department at the University of Georgia, Athens, Georgia.
The following article appeared in Process Studies, pp. 116-124, Vol. 9, Numbers 3-4, Fall, Winter, 1979. Process Studies is published quarterly by the Center for Process Studies, 1325 N. College Ave., Claremont, CA 91711. Used by permission. This material was prepared for Religion Online by Ted and Winnie Brock.
Dr. Clarke presents a critique of Lee F. Werth’s critique of Whitehead’s Theory of Extensive Connection. Clarke says, "If Werth has succeeded in demonstrating anything, it is the need for someone to cast Whitehead’s theory of extensive connection and abstraction in a systematic form and in logical notation."
In a recent article, "The Untenability of Whitehead’s Theory of Extensive Connection" (PS 8:37-44), Lee F. Werth argues that Whitehead’s definition of a straight line "contradicts a derivable theorem that no geometrical element a is ever incident in a geometrical element b" (PS 8:37). Werth goes on to maintain that "from this theorem we can demonstrate that, according to the explicit theory of extensive connection Whitehead proposes, all geometrical elements are points" (PS 8:37). He is quite right in pointing out that if he is correct, then this has very serious consequences for some of Whitehead’s other theories: "it follows that Whitehead’s geometrical account of presentational immediacy and of the strain-locus is untenable in its present form" (PS 8:37). Werth’s argument needs seriously to be examined, not just for its consequences for Process and Reality, but also for the sake of the theory of extensive connection and abstraction. It is important in its own right. Or, at least Whitehead thought so. He began working on his theory at least as early as 1914 and continued to reformulate it and revise it until the Gifford Lectures of 1928 (see AE 214, footnote).
The nervus probandi of Werth’s argument is his Theorem 22, which he states as follows:
Theorem 22.The relation of covering, if it is to hold between any two abstractive sets always holds symmetrically.
In the concluding paragraph of his article, for example, he makes this claim:
To refute my argument requires the satisfaction of two criteria. Asymmetrical coverage between abstractive sets must be possible, i.e., the refutation of theorem 22. It must also be established that it is not always possible to construct a covered set a such that a’s associated geometrical element can be said to be incident in ’s associated geometrical element (for if a covered set can always be constructed where a does not cover its coverer, then all geometrical elements are not points). (PS 8:44)
This claim is far too strong. All that is needed to refute Werth’s argument is to show that he has not made his case -- that is, to show that his argument is invalid, or if valid, unsound. Although I shall consider his criteria later, I would like first to attempt to demonstrate (I) that Werth’s Theorem 22 is ambiguous as stated; (II) that what he says follows from it does not in fact follow; (III) that it does not follow from what he says it does; and (IV) that under either interpretation it is false.
In examining Werth’s theorems and proofs it will useful (if not necessary) to introduce some logical tools. Let us assume the classical two-functional senential logic, classical quantificational theory with identity, and some form of set theory. We will let the individual variables ‘. . . , x,y,z,’ range over Whiteheadian regions, the variables, ‘a, b, c range over sets of regions, and the variables, ‘a', b', c' , . . .,‘ range over sets of sets of regions. The English ‘if . . . then . . .,’ ‘and,’ ‘or,’ and ‘not’ will be used in the truth-functional sense of the horseshoe, dot, wedge, and tilde. The quantifiers, parentheses, and brackets will be used in their conventional way. With this logical machinery we can symbolize the following relevant Whiteheadian definitions, where ‘Cx,y’ is taken as primitive and is a rendering of ‘x is extensionally connected with y.’ I shall follow Whitehead’s own numbering in Part IV of Process and Reality and give in parentheses his own formulation.
DEF 2. ‘Ix,y’ = def ‘ (z) (if Cz, y then Cz,x)’
(Region A is said to ‘include’ region B when every region connected with B is also connected with A.)
DEF 3. ‘Ox,y’ = def ‘ (Ez) (Ix,z and Iy,z)’
(Two regions are said to ‘overlap,’ when there is a third region which they both include)
DEF 7. ‘ECx,y’ = def ‘Cx,y and not Ox,y’
(Two regions are ‘externally’ connected when (i) they are connected, and (ii) they do not overlap.)
DEF 8. ‘TIx,y’ = def ‘Ix,y and (Ez) (ECz,x and ECz,y)’
(A region B is ‘tangentially’ included in a region A, when (i) B is included in A, and (ii) there are regions which are externally connected with both A and B.)
DEF 9. ‘NTIx,y’= def ‘Ix,y and not (Ez) (ECz,x and ECz,y)’
(A region B is ‘non-tangentially "included in Region A when (i) B is included in A, and (ii) there is no third region whil is externally connected with both A and B.)
All the above relations, both primitive and defined, hold between regions. We shall now introduce a series of Whiteheadian definitions which characterized certain sets of regions and sets of sets of regions.
DEF 10. ‘ABSa’ = def ‘ (x) (y)[if (x a and y c a) then (NTIx,y or NTIy,x)] and not (Ex)[x a and (y) (ify eathen NTIy,x)]’
(A set of regions is called an ‘abstractive set, when (i)any two members of the set are such that one or them includes the other non-tangentially, (ii) there is no region included in every member of the set.)1
DEF 11. ‘COVa,b’ def ‘ABSa and ABSb and (x)[if x a then (Ey) (y b and NTIx,y)]’
(An abstractive set a is said to ‘cover’ an abstractive set , when ever)’ member of the set a includes some members of the set .)
DEF. 12. ‘EQVa,b’ = def ‘CO Va,b and COVb,a’ ERC (T
(Two abstractive sets are said to be ‘equivalent’ when each set covers the other.)
DEF. 13. ‘GEOEa'’= def ‘ (b) (c)[if (b a' and c a') then EQVb,c] and not (Eb) (Ec) (b a' and not c a' and EOVb,c)’
(A geometrical element is a complete group of abstractive sets equivalent to each other, and no one equivalent to any abstractive set outside the group.)
The ease with which these definitions can be rendered into the logical notation demonstrates the careful, precise, and systematic way in which Whitehead has presented his theory in Part IV of Process and Reality. (Alas, had only the rest of Process and Reality been written as precisely, carefully, and systematically.)
With the logical tools and these Whiteheadian definitions, let us now examine Werth’s Theorem 22 as stated above. It is difficult to know precisely what Werth is asserting here. The difficulty lies in knowing the force of the phrase ‘if it [covering] is to hold between any two abstractive sets’ and the force of the term ‘always.’ The usual definition of symmetry is as follows:
1.0 ‘R SYM’ = def ‘ (x) (y) (if Rx,y then Ry,x),’
where ‘SYM’ designates the class of symmetrical relations. There is no such thing as a relation’s being sometimes symmetrical and sometimes not (i.e., nonsymmetrical). It either is symmetrical or it is nonsymmetrical. Therefore, the force of the conditional (‘if it holds’) and the ‘always’ could be taken to assert something like this:
THM 22'. If (Ea) (Eb) COVa,b then (c) (d) (if COVc,d then COVd,c).
It is mysterious to me, however, why Werth would wish to put this condition on the symmetry of "covering," for THM 22 follows from the symmetry of the relation itself, but not vice versa. Therefore, it could be that the force of the conditional and the ‘always’ of Theorem 22 is intended to assert something like this: It is always the case that if ‘x covers y’ holds then the reverse also holds. If so, then Theorem 22 would appear as follows:
THM 22’. (a) (b) (if COVa,b then COVb,a).
In discussing his Theorem 26, Werth tells us that Theorem 26 "follows from the symmetry of coverage (22)" (PS 8:40). Also, he later speaks of Theorem 22 as "the theorem of the symmetry of coverage"(PS 8:42);and again, "the theorem of symmetrical coverage (22)" (PS 8:44). This leads me to think that it is THM 22’, rather than THM 22, which he has in mind. At any rate Werth’s statement of Theorem 22 is ambiguous, since the two formulations (THM 22 and THM 22’) are not logically equivalent.
Let us now examine two of Werth’s theorems which he claims follow from Theorem 22. Theorem 26, which we mentioned above, he tells us "follows from the symmetry of coverage (22)." This theorem he states as follows:
When every member of an arbitrary geometrical element a covers every member of an arbitrary geo metrical element b, every member of geometrical element b will cover every member of geometrical element a.
This theorem appears to be quite unambiguous and would be rendered in the logical notation as,
(a') (b')[if (GEOEa' and GEOEb' and (c) (d)[if (c a' and d C b) then COVc,d]) then (c) (d)[if (d b'and c a') then COVd,c]].
Now THM 26 does not follow from the conditional form of Theorem 22 (THM 22), but it does follow from the unconditional form (THM 22’). The proof is rather simple and is worth detailing. From THM 22', by quantification theory, we get
2.0 If COVc,d then COVd,c,
and by sentential logic we can get
2.1 If (c a and d b) then (if COVc,d then COVd,c).
This by sentential logic is equivalent to
2.2 If [if (c a and d b) then COVc,d] then [if (d b and c a) then COVd,c].
By quantification theory we can obtain
2.3 If (c) (d)[if (c a and d b) then COVc,d] then (c) (d)[if (d b and c a) then COVd,c].
From this, by sentential logic, we can get
2.4 If (GEOEa' and GEOEb') then ( (c) (d)[if (c a' and d b') then COVc,d] then (c) (d)[if (d b' and c a') then COVd,c]).
And this, by sentential logic, is logically equivalent to
2.5 If (GEOEa' and GEOEb' and (c) (d)[if (c a' and d c b') then COVc,d]) then (c) (d)[if (d b' and c a') then COVd,c].
From 2.5 by quantification theory we can obtain THM 26. Here Werth is quite correct.
This proof leads me to think that Theorem 22 should be rendered as THM 22’. But be that as it may, Theorem 27 is, in turn, said to follow from Theorem 26 and DEF 12. Let us examine this proposal. This theorem is stated as follows:
The members of geometrical element a and the members of the geometrical element b are equivalent to each other.
(p. 120 ff.) What is intended here appears to be unambiguous enough and would be rendered as,
THM 27. (a) (b) (if [GEOEa' and GEOEb''] then (c) (d)[if (c a' and d b') then EQVc,d]).
Unfortunately THM 27 does not follow from TEM 26 and DEE 12. The difficulties can be seen easily if we use DEE 12 to rewrite THM 26 in a definitionally equivalent form; that is,
3.0 (a) (b) (if [GEOEa' and GEOEb'] then (c) (d)[if (c a' and d b') then (COVc,d and COVd,c)]).
Comparing 3.0 with an equivalent form of THM 26 might give us a clue as to why Werth has been led astray. By sentential logic THM 26 is logically equivalent to
3.1 (a) (b) (if [GEOEa' and GEOEb'] then (c) (d)[if (c a’ and d b’) then (if COVc,d then COVd,c)]).
An examination will reveal that 3.0 and 3.1 are identical except for the final ‘and’ of 3.0 and the final ‘if... then...’ of 3.1. Perhaps Werth has confused the form ‘p and q’ with the form ‘if p then q.’ Be that as it may, this much is clear: 3.0 does not follow from 3.1 and THM 27 does not follow from THM 26 and DEE 12.
Now since THM 27 is a major step in the proof of Werth’s main thesis that all geometrical elements are points and, consequently, that there are no straight lines, we must conclude that his argument is invalid.
Let us now examine Werth’s proof of Theorem 22. Werth tells us that Theorem 22 follows from Theorem 20 and Theorem 21. Let us look at Theorem 20 first. It is stated as follows:
Theorem 20. If every member of an abstractive set includes some members of another abstractive set, it will always be the case that every member of the latter abstractive set includes some member of the former abstractive set.
On the surface of things, this theorem appears to be unambiguous and should be rendered as
THM 20. (a) (b)[if (ABSa and ABSb and (x)[if x a then (Ey) (y b and NTIx,y)]) then (x)[if x b then (Ey) (y a and NTIx,y)]].
That this is a proper rendition of Theorem 20 into logical notation is supported by the fact that Theorem 22 under one interpretation (i.e., THM 22') follows from it. In fact, THM 20 by DEF 11 is logically equivalent to THM 22'. The proof is rather simple and probably worth stating. THM 20, by sentential logic is equivalent to
4.0 (a) (b)[if (ABSa and AESb and (x)[if x a then (Ey) (y e b and NTIx,y)]) then (ABSb and ABSa and (x)[if x b then (Ey) (y a and NTIx,y)])], which, by the substition of DEF 11, is equivalent to
4.1 (a) (b) (if COVa,b then COVb,a),
or THM 22'. THM 20 is not equivalent, however, to THM 22, but it does entail THM 22, since THM 22' entails THM 22.
There is, however, a difficulty here. Werth tells us that Theorem 22 follows from Theorem 20 and 21. Since THM 20 and THM 22' are logically equivalent, it is a mystery as to how Theorem 21 fits into the picture. Werth states Theorem 21 as follows:
Theorem 21. The two abstractive sets of 20 cover each other.
The phrase ‘the two abstractive sets of 10’ is puzzling. Theorem 20, it appears, is about all pairs of, or any two, abstractive sets; if not, then THM 22' does not follow. We certainly cannot reason from two sets to all pairs of sets. Perhaps Werth only means to assert: Two abstractive sets of 20 cover each other, or
THM 21. COVa,b and COVb,a;
that is, an instantiation of THM 20. Theorem 21, however, is said to follow from Theorem 20 by DEF 11. This, however, is certainly not the case. By DEE 11, the most that we can get from THM 20 by instantiation, as we have seen, would be
5.0 If COVa,b then COVb,a.
Here again we have the same apparent fallacy which we saw in the proof of Theorem 27; that is, an attempt to go from ‘if p then q’ to ‘p and q.’ This strengthens the belief that this was Werth’s error in the proof of Theorem 27.
Since, however, THM 21 is not needed in the proof of THM 22' or THM 22, let us not tarry here and proceed to look at Werth’s proposed proof of Theorem 20. He tells us that this theorem follows from Theorem 11 and 13. Theorem 11 is stated in this way:
Theorem 11. Every member of an abstractive set a includes some members of the set ; and Theorem 13 in this way:
Theorem 13. Every member of an abstractive set includes some members of the set .
Now since is a particular kind of set -- namely, one "determined by a" (see Theorem 4,5, 6, and 7, from which Theorem 11 and 13 are said to follow, or to follow from theorems which follow from these) -- then cannot be universally quantified. On the other hand, since every abstractive set, according to the argument, "determines" some set , a can be universalized. Also, the a of both theorems concerns the same set and the of both theorems concerns the same set -- a is the "determinor" and the "determinee" in both cases. In order to keep the quantifiers straight and since Theorem 20 is said to follow from the conjunction of 11 and 13, let us conjoin them before we attach the quantifiers. In this wax’ we obtain the following.
THM 11 (13). (a)[if ABSa then (Eb) (ABSb and (x)[if x a then (Ey) (y b and NTIx,y)] and (x)[if x b then (Ey) (y a and NTIx,y)])].
Now, I have absolutely no quarrel with Werth about THM 11 (13). It does follow from Whitehead’s definitions and assumptions along with set theory. There is, however, only one difficulty with THM 11 (13); it does not entail THM 22 or THM 22'. 1 do not believe there is any question about the notation of THM 11 (13). It is a straightforward rendering of Werth’s often repeated thesis: Every abstractive set a determines some abstractive set , such that a covers , and , in turn, covers a. I think it will be easy to see how Werth was probably led astray in his proof if we recast THM 11 (13) in a logically equivalent form; that is,
6.0 (a) (Eb)[if ABSa then (ABSb and (x)[if x a then (Ey) (y b and NTIx,y)] and (x)[if x b then (Ey) (y a and NTIx,y)])].
Now from 6.0 by sentential logic and quantification theory, we can gets2
6.1 (a) (Eb)[if ABSa and (ABSb) then ((x)[if a then (Ey) (y b band NTIx,y)] and (x)[x b then (Ey)(y a and NTIx,y)])].
A comparison of 6.1 to a logically equivalent form of THM 20 will, I think, give us a clue. That form is this:
6.2 (a)(b)[if (ABSa and ABSb) then (if (x)[if x a then (Ey)(y band NTIx,y)] then (x)[x b then (Ey)(y a and NTIx,y)])].
Here, again, we have the ‘if p then q’ and ‘p and q’ difference, which we have encountered before. In addition, however, we have the problem of different quantifiers. This shows us why Theorem 11 and 13 together do not entail Theorem 20. Under no circumstances does ‘ (a) (Eb)p’ yield ‘ (a)(b)p’. Thus, THM 20 does not follow from TI-IM 11 (13).
It appears that throughout his argument Werth has confused two different statements:
7.0 (a) (b) (if COVa,b then COVb,a), or THM 22', which is logically equivalent to 6.2; and
7.1 (a)(Eb) (COVa,b and COVb,a),
which is equivalent to 6.1. This confusion would, I think, explain the invalid proposed proof of Theorem 27, the invalid proposed proof of Theorem 20, the mysterious appearance of Theorem 21 in the proof of 22, and the ambiguity of Theorem 22.
Now, as I suggested above, 7.1 does follow in Whitehead’s system. But this does not make "covering" a symmetrical relation. It merely makes more than one abstractive set converge to the same geometric lement. This is harmless enough. In fact, Whitehead recognizes this fact. This is why he defines a geometric element in terms of an equivalent set of converging abstractive sets (DEE 13), rather than in terms of one converging abstractive set. A geometric element is a set of converging abstractive sets, all converging to the same geometric element.
I would now’ like to argue that not only has Werth not proven Theorem 22 nor proven his main thesis which he claims follows from it, but that Theorem 22, in either of its formulations, is false. Consider, for example, Diagram 1. Let X be the set, X = [1, 2, 3,...], and Y be the set, Y [a, b, c,...], and Z the set, Z [A, B, C, D,...], where the exemplified pattern is repeated ad infinitum. Now all three sets, X, Y, and Z, will fulfill the definition of an abstractive set. In each set, choose any two regions: one will nontangentially include the other -- the first half of DEF 10. Also, since the same pattern of interlocking regions is repeated over and over, there will be no smallest member of any of the three sets -- the second half of DEE 10. Now set Z, according to DEF 11, will cover both set X and set Y, but neither X nor Y will cover Z. These three sets serve as a counter-example to both TEM 22 and THM 22’. Consequently, both forms of Theorem 22 are false. In fact, the associated geometric element of set Z will be a "segment" between the two "end points" associated with set X and Y (see Definitions 18 and 19, PR 457).
I would like now’ to say something about Werth’s two criteria (quoted above) which he feels must be fulfilled for a refutation of his argument. Let us first consider the second criterion: "It must also be established that it is not always possible to construct a covered set such that ’s associated geometrical element can be said to be incident in a’s associate geometrical element" (PS 8:44). This criterion rests upon Werth’s apparent assumption, which is false, that 7.0 entails 7.1. The fact that given any abstractive set a, we can always construct an abstractive set /3, which it covers and which in turn covers it, is perfectly harmless and does not entail that covering is symmetrical.
In his first criterion Werth tells us that in order to refute his argument one must refute his proposed Theorem 22. To refute a proposed theorem usually means to prove within the given system the negation of that proposed theorem. Now if Werth’s contention, which he has not proven, that Whitehead’s system is contradictory (or inconsistent) were true (PS 8:37), then, as a matter of fact, Theorem 22 in both its forms would be provable. In fact, any theorem whatsoever would be provable. The problem of contradictory, or inconsistent, systems is a tricky business. As Whitehead once said, one of the "prevalent habits of thought" which must be repudiated is the "belief that logical inconsistencies can indicate anything else than some antecedent errors" (PR viii). Inconsistencies can usually be patched up. When Russell proved his now famous paradox, the story is that he sent a copy of it to Frege and that Frege responded with a postal card saying, "Alas, arithmetic totters." Well, arithmetic still stands. Alas, it was only Frege who tottered. If Werth has succeeded in demonstrating anything, it is the need for someone to cast Whitehead’s theory of extensive connection and abstraction in a systematic form and in logical notation. The theory is too important, too subtle, and too complex to be handled in any other way.
1In formulating this definition, I am following Werth (PS 8:42) rather than the text of Process and Reality. In the second half Whitehead uses the term included’ rather than nontangentially included.’ I rather think this is a slip on the p art of Whitehead. This difference, however, is quite irrelevant to our argument. As Werth says, "actually ‘includes’ as opposed to ‘nontangentially includes’ is a pseudo-issue" (PS 8:42).
2The proof here is rather clumsy, so I have not included it. It is based on the quantification theorem, ‘If (x) (if p then q) then [if (Ex)p then (Ex)q],’ with ‘if ABSb then (ABSb and (x)[if x a then (Ey)y b and NTIx,y)] and (x) [if x b then (Ey) (y b and NTIx,y[) and (x) [I x b then (Ey)(y a and NTIx,y)]) ‘p’’ and ‘If (ABSa and ABSb) then ((x)[if x a then (Ey)(y b and NTIx,,y)] and (x)[if x b then (Ey)(y a and NTIx,y([)’ replacing ‘q’. The antecedent of the quantification theorem will be tautology, and 6.0 with the ‘a’ instantiated will be the antecedent of the consequent of the theorem.