Aristotle Quine

Aristotle's Response to Quine's Objections to Modal LogicAuthor(s): Alan CodeSource: Journal of Philosophical Logic, Vol. 5, No. 2 (May, 1976), pp. 159-186Published by: SpringerStable URL: http://www.jstor.org/stable/30226139 .Accessed: 27/10/2014 20:30

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ARISTOTLE'S RESPONSE TO QUINE'S OBJECTIONS TO MODAL LOGIC

ALAN CODE

I. INTRODUCTION

In Section II it is shown that some of the Quinean objections to modal logic found in [15] can be transferred to the notions used to describe and account for temporal change.1 The remainder of the paper is devoted to a demonstration of the thesis that Aristotle's predecessors had already formulated problems similar to the temporal puzzles so generated, and that the three most prominent reactions to Quine's puzzles were also anticipated by certain ancient Greek philosophers. Furthermore, Aristotle's own reaction - as manifested in his analysis of the elements of change in Physics A7 - can be seen to involve concepts which easily lend themselves to the kind of semantical analysis which has recently enhanced our understanding of modality. Let us begin, then, by getting clear on just what problems I have in mind.

II. STATEMENT OF THE PROBLEM

It is now common knowledge in the philosophical community that the extensionality principle known as the law of the substitutivity of identicals (SI) fails to hold within the scope of modal operators (e.g., "It is necessary that ...") and psychological verbs (e.g., "... believes that "). It is because of this fact that such contexts have come to be called non- extensional. Closely related to the failure of SI are several well known problems involving quantification into such contexts.

Less widely publicized (but long recognized by logicians concerned with the analysis of tensed discourse) is the fact that the same problems beset us with any verb whatsoever once we venture beyond the confines of the present tense. That is, the tense of a verb is by itself sufficient to render the scope of that verb non-extensional.

This is most easily demonstrated when the statement contains an explicit, definite temporal adverb. This condition is met in the following:

(1) The President resigned last August.2 Journal of Philosophical Logic 5 (1976) 159-186. All Rights Reserved Copyright A 1976 by D. Reidel Publishing Company, Dordrecht-Holland


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"resigned" is a simple past tense verb, and "last August" is an adverbial modifier which serves to make explicit the span of time to which the verb refers us. We also note the fact that (1) contains a singular term (i.e., "the President") as the subject expression - which term we find on the left of an identity sign in:

(2) the President = Ford Given that the identity expressed in (2) holds, we may assert that the object referred to by the phrase "the President" is the same object as that referred to by "Ford." Aside from certain semantical facts (such as that "O" denotes 0), what is true of an object is quite independent of the language in which we describe it and talk about it. So in the light of (2) we should expect that anything than one may truly assert about the President may equally well be truly asserted about Ford. This is simply to endorse the principle SI.3 This idea has been encapsulated by Quine in the following way: ... given a true statement of identity, one of its two terms may be substituted for the other in any true statement and the result will be true ([15], p. 139).

Instances of this principle are easy to find. The President is skiing in Colorado; therefore Ford is skiing in Colorado. Ford will veto the bill next week; so the President will veto the bill next week. All seems well and good. Nevertheless, this principle of substitutivity seems to break down once from (1) and (2) we attempt to infer:

(3) Ford resigned last August. (3) is clearly false in spite of the fact that (1) and (2) are (at the time of this writing) true, so we apparently have a case in which SI fails.

In addition to SI, the quantificational principle of existential generaliz- ation (EG) seems to flounder when brought up against (1) and (2). For since the President resigned last August, there is some object such that it resigned last August. In other words, we are entitled by EG to infer from (1) that:

(4) (3x)(x resigned last August). But if (4) is true, we may legitimately ask what object it is that resigned last Fall. Is it the President; i.e., Ford? This cannot be correct, though, since it conflicts with the fact that (3) is false. Hence we are presented with a


ARISTOTLE'S RESPONSE TO QUINE'S OBJECTIONS 16i

problem as to how one interprets quantification into temporal contexts, and lacking a solution it appears that the inference from (1) to (4) cannot be made.4

It should not be thought that this type of problem depends upon the use of temporal adverbs, since one can find equally persuasive examples which contain no adverbs. Using the letter "P" as a past tense operator (so that where A is our translation of S, "'P"A is our translation of the past tense of S), consider the attempt to infer:

(5) P(Ford = Nixon) from

(6) P(the President = Nixon) and

(2) the President = Fords The verbs that we use to describe change are susceptible to like problems.

To help make this clear we should first do a little stage setting. Just prior to becoming the President, Ford was the Vice-President. Let us suppose for the sake of example that Rockefeller was the Speaker of the House just prior to his becoming the Vice-President. Furthermore, suppose that the law stipu- lates that the moment that the Vice-President ascends to the Presidency that the Speaker of the House becomes the Vice-President. Keeping this story in mind, we find that there is some interval bf time at which the following are both true:

(7) The Vice-President becomes the President (8) The Speaker of the House becomes the Vice-President.

Since (7) and (8) are both true at the same time, it seems to follow by EG that:

(9) (3x)(x becomes the President & the Speaker of the House becomes x)

Now what object is this that (a) becomes the President and (b) has the Speaker of the House become it? It is not Ford, since he was never Speaker of the House. But neither is it Rockefeller, since he is not now the President. Furthermore, the Speaker of the House does not become Rockefeller. Thus


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we are once again brought face to face with the problem of interpreting existential quantification.

The alert reader will have noticed that there is no interval of time at which both (2) and (7) are true, and thus SI would not sanction an inference from these premises to:

(10) The Vice-President becomes Ford. However, using the present perfect of "becomes" we can create problems for SI. Still keeping our story in mind, imagine that we are present at the historic moment at which Ford became President. At this moment the following four statements are (according to our fabrication) true:

(11) The Vice-President has just become the President. (12) The Speaker of the House has just become the Vice-President. (13) the President = Ford (14) the Vice-President = Rockefeller

Now, using the principle SI we attempt to infer the following:

(15) The Vice-President has just become Ford. (from 11 and 13) (16) Rockefeller has just become Ford. (from 14 and 15) (17) Rockefeller has just become the President. (from 11 and 14;

or from 13 and 16)

(15)-(17) are all false on our story. SI again seems to fail.6 Since we have the same problems showing their heads with modal state-

ments, and with tensed statements, and with statements using the verb "becomes," we have prima facie reason to suppose that in each of the three cases the difficulties should be handled in the same way. Consequently, in searching for a solution to the anomalies presented above we would do well to explore the responses that philosophers have given when confronted with the same puzzling facts with regards to modality. With an eye to just this end, we will consider three predominant views about modality. In the course of this discussion, it will be seen that in an attempt to deal with the problems about becoming which were just presented, certain Greek philosophers anticipated all three of these views.


ARISTOTLE'S RESPONSE TO QUINE'S OBJECTIONS 163

III. QUINE AND PARMENIDES

Since it was Quine who first brought our attention to the SI and EG problems for modal contexts, we may start by considering his own reaction. Actually, throughout his career he has endorsed several different views, but the main thrust of his remarks is that in the light of such difficulties one should do away with the use of modal operators on the grounds that the modal notions themselves are inherently confused. Transferring this position to our present concern, we see that one possible response to the problems about tense is to altogether abandon the practice of tensing verbs. Historical- ly, this option was advocated by Aristotle's illustrious forerunner, Parmen- ides. It was he who stated:

S.. the only account that still remains is [of this] route: that it is... it was not ever [in the past], nor will it be [in the future] ... ([6], Fr. 8; 1, 2, 5).

Parmenides is here urging us to do without tense operators in much the same way as Quine urges us to do away with modal operators.

Furthermore, still keeping Quine's strictures against modal logic in mind, we might react to our other paradoxes (the ones about becoming) by simply denying that change ever takes place. We find Parmenides maintaining this position when he says:

... coming to be and perishing have been driven afar... all these are just names which mortals have laid down believing them to be true: to come to be and to perish; to both be and not be; to alter in place; and to change with regard to bright color ([6], Fr. 8; 27-8, 38-41).

An examination of Fragment 8 of Parmenides' poem is not in itself sufficient to show that the considerations which led to these conclusions were at all like the problems presented in Section II. However, in Physics A8 Aristotle tells us what the arguments were that purported to show the impossibility of change (along with his own diagnosis of what goes wrong in them).' The argument starts out with the premise that that which comes to be either comes to be from that which is or from that which is not. In order to prevent the argument from being hopelessly bad, we should take this as elliptical for:

(18) That which is c either comes to be that which is 4 from that which is 4 or from that which is not 4.

Aristotle does not challenge (18), and it certainly has the ring of truth. That


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which is red came to be a red thing from either a red thing or a non-red thing. The next step in the argument is to point out that the first alternative is ruled out because that which is @ could not come to be 4 from that which is already 4 - e.g., if my table comes to be red, I could not have started out with a red table. For in such a case no change takes place. However, this would mean that (given (18)) if something has come to be 4, then the following is true:

(19) That which is not k has become that which is 5. Aristotle also accepts (19). In general, whenever a change takes place, a statement of the form given in (19) must be true. For instance, if my black table comes to be a red table, then that which is not red has come to be that which is red.

The problem that we now encounter is that, given the truth of (19), we are now at a loss when asked for the referent of the expression "that which is not 4)." It cannot be that which is 4, for then we would have:

(20) That which is not 4 = that which is 4. (20) is not true since nothing can both have and lack the same property. In addition to this, (19) together with (20) would entail by means of SI that:

(21) That which is 4 has become that which is 4. But (21) has already been ruled out.

So we cannot identify that which is not 4 with that which is 4. But neither can we identify it with anything else, since if (19) is true it must be the same thing which first is not 4 and is then 4. Thus we seem to been stuck with this:

(22) It is impossible to specify the referent of the phrase " that which is not 4" in (19).

If a statement of the form given in (19) must be true whenever a change takes place, and if (22) is correct, it follows that change is impossible. How- ever, Aristotle does not accept the Parmenidean conclusion, as we can tell from his statement:

... we may assume on the basis of induction that some or all natural things are chang-

ing. ([3d], A2, 185a12-14)



Thus Aristotle must reject (22) and provide us with an intelligible answer to the question, "What is the referent of the phrase 'that which is not k?'" In the follpowing sections we shall try to determine how he answered this question.

To help the reader to see that this problem is the same kind of problem that we discussed in the previous section, we should note that by EG (19) gives us:

(23) (3 x)(x has become that which is 4) But what is this object which has become #? Suppose that my table has just become that which is red. Then is the object we are looking for my table; i.e., that which is red? But this conflicts with the fact that that which is red did not become that which is red.

Quite briefly, we shall now look at how we can interpret Parmenides as having made this type of point. I take it that when Parmenides asks "For what birth will you look for for it? From whence could it possibly have grown?"' he is asking us to specify the object which became it. In other words, on the assumption that that which is 4 has come to be, he wants us to fill in the blank in:

(24) has become that which is 4. And when he goes on to say in the next sentence that "I will not allow you to say or to think that [it came from] what is not [it] - for it is neither sayable nor thinkable in what manner that which is not [it] exists,"'9 he is claiming that we cannot fill in the blank with "that which is not p," and that the reason is that we are unable to specify the referent of that expression. In short, then, he is arguing that if a change takes place, then a statement like (19) is true, but since (22) is true, (19) is not true, and hence no change can take place.

This would provide Parmenides with an argument to show that change is impossible. If we assume that if an object existed in the past (or will exist in the future) then it must have (or lack) at least one property which it lacked (or had) in the past (or will lack (or have) in the future), then it is a simple matter to use these same arguments to show that no object existed in the past (or will exist in the future). Thus it would follow that objects do not have past or future temporal parts. This would then provide him with an argument to show that tensed discourse should be abandoned.


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We have now seen that one possible response to our Quine-like problems about becoming is to claim that the Quinean questions cannot be answered, and hence there is no such thing as change. We shall now examine two different attempts to provide answers to the Quinean questions.

IV. SMULLYAN AND THE FOURTH CENTURY B.C. REPARSERS

A second response is suggested by A. Smullyan's use of Russell's theory of descriptions [17] to disarm Quine's modal puzzles (see Smullyan's [19]). What Smullyan did was to treat all descriptions as contextually defined and then to paraphrase them away before allowing applications of SI or EG. Dropping the temporal adverb "last August" (since it is not needed to make the relevant points) and still using "P" as a past tense operator, we may now employ this technique to rewrite (1)-(4) as follows:

(1') P(3x)((y)(y is President iffx = y) and x resigns) (2') (3x)((y)(yO is President iffx = y) and x = Ford) (3') P(Ford resigns) (4') (3x)P(x resigns) Once this has been done, it is plain that (3') does not follow from (1')

and (2') by SI, nor does (4') follow from (1') by EG. Of course, if we interpret the quantifiers as ranging over all past, present and future objects, we shall want (1') to entail (4') - but this entailment will not be an instance of EG. At any rate, the truth of (4') is not problematic in the way that (4) is, since when asked what object it is that satisfies the matrix we may unhesi- tantly answer: Nixon.

This technique also allows us to answer Parmenides' question. When my table turns from black to red, it is the table which was first black (and so, non-red) and then red. However, in actually analysing statements containing the verb "becomes" we get something rather complicated. For instance, (11) would get analyzed something like this:

(11') (3x)(9 t)(t < now & (Vy)(y is Vice-President iffx =y) at t & (Vt')((t < t' & t' < now) - ((x is not President) at t')) & (Vy)(y is President iffx = y)).'o



In this analysis of (11) we find that not only have the phrases "the Vice- President" and "the President" been paraphrased away, but so has the verb "become." What this shows is that if we adopt the Russellian approach, we will not be able to treat becoming as a two-place relation between individuals. No statement of the form "X becomes Y" can be analyzed in terms of (i) the individual assigned to "X", (ii) the individual assigned to "Y", and (iii) a two-place relation (expressed by "becomes") which holds between these two individuals. However attractive we may find the present theory, the fact just noted makes it impossible for us to claim that Aristotle advocated the Russellian approach, since it may be clearly seen in Physics A7 that he regarded becoming as just such a two-place relation. Before setting out a semantical treatment of becoming which does have this feature, we shall first look at two independent reasons for supposing that Aristotle did not endorse a Russellian analysis.

First of all, Aristotle analyzes time as the number of change with respect to the before and after ([3d] Al 1, esp. 220a24-5). Without dwelling on this point, it is sufficient to note that (as evidenced by (11')) the Smullyan- esque approach requires one to analyze change in terms of time, and not the other way around as Aristotle would have it.

More important for present purposes, however, is Aristotle's expressed attitude towards reparsing singular terms as general terms. It will be remem- bered that we paraphrased (2) t la Russell as:

(2') (3x)((y)(y is President iffx = y) and x = Ford) One way to think about what is going on here is to consider "(3 x)((y)(y is President iffx = y) and x = z)" as a complex predicate (i.e., a complex general term) meaning "z Presidentizes." Hence we may think of (2') as meaning "Ford Presidentizes." What we have done, then, is to analyze what is apparently an identity statement like:

(2) the President = Ford as a subject/predicate statement like:

(2") Ford Presidentizes. We have taken a statement involving the word "the" and replaced it by a statement containing only general terms and quantifiers (our complex predicate "Presidentizes") and the proper name "Ford."


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Now Aristotle stressed the equivalence of statements like (2) and (2"), and in so doing apparently advocated this policy of reparsing singular terms as general terms. He states that: there is no difference between

(a) [the] man is [the] healthy [thing] and (a') [the] man is healthy; nor is there a difference between

(b) [the] man is [the] walking [thing] or [the] cutting [thing] and (b') [the] man walks or cuts. ([3c]J7, 1017a27-30)." The general principle which Aristotle accepts may be put in the following way if we let "F" be the general term obtained from the singular term "the

(25) The following statements are equivalent: (a) a = the ( (b) Fa

This would certainly suggest (contrary to what I have said) that Aristotle would have adopted a Russellian solution to the temporal problems. How- ever, if this were actually the case he would not have allowed that (25a) expressed a binary relation which obtains between some object signified by the name "a" and some object signified by the phrase "the p." What we find, though, is that even though certain ancient thinkers who preceded Aristotle (but who were nonetheless more recent than Parmenides, Melissus and Heraclitus) held such an opinion, Aristotle himself did not.

We learn in the early pages of the Physics ([3d], A2, 185b25-186a3) that these other ancient thinkers were worried that statements like (25a) might make "the same [thing] prove to be both one and many at the same time," and consequently they "changed their mode of speech, [and said] that the man is not [the] white [thing], but rather has-been-whitened, and [he] is not [the] wallking [thing], but walks."12 The ancient thinkers to whom he refers us endorsed the practice of reparsing because they thought that the word "is" (uTL) expressed a binary relation between two objects, and apparently were construing that relation as one which could only hold between two objects if they were the same object. Consequently they were concerned with the fact that two different objects (namely, a and the k) are being called the same object in (25a).

In view of the fact that there will probably be more than just one # thing (for instance, more than just one white thing) in the world, we might be tempted to suppose that it was this fact that caused them to worry about



statements like (25a). If both Socrates and Coriscus are white, we should be able to say that Socrates is the white thing if we want to say that Coriscus is the white thing. But would not this commit us to saying that Socrates is Coriscus?13

Nonetheless, this is not the problem that they had in mind, since Aristotle continues the passage now under scrutiny by claiming that the reparsers were right in thinking that statements like (25a) relate two different objects, but wrong in thinking that there is anything troublesome about this. If the alleged problem with (25a) rested upon the type of considerations we gave in the last paragraph, it is hardly possible for Aristotle to maintain that in that sense one object really is the same as some different object.

It sounds as though Aristotle is trying to draw a distinction between the "is" of identity and the "is" of predication, but this too is mistaken. He makes his point by saying that being and one are not spoken of in only one way, so we must try to find some distinction which would be applicable to both notions and not merely the former. In Metaphysics A (the so-called philosophical lexicon) we find the material necessary to make the required distinction. Roughly, what Aristotle is claiming is that there are two notions of 'same object,' and that two objects with different persistence conditions (although not the same object in the strong sense) are in a weak sense of "same" the same objects if they are currently sharing all spatial parts. Corresponding to this two-fold notion of sameness, is a two-fold notion of being and also a two-fold notion of unity. A little later we will go into this in more detail.

The upshot of this is that although Aristotle considered the process of reparsing perfectly legitimate, we may go on treating statements like:

(2) the President = Ford as expressing a binary relation between two objects which are in one sense the same and in another sense not the same. Thus he considered, but did not adopt, a Smullyan-esque response to a problem about sameness. And if the grounds for asserting that Socrates and this white thing are not strictly speaking the same object are that their persistence conditions are different, we are lead to suspect that he would adopt the same avenue of response for the problems about change. After all, if Socrates and this white thing have different persistence conditions, there will be some changes which the one


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can survive that the other cannot. For instance, Socrates goes on existing even when he stops being white and pale, but this same change would drive the white thing out of existence. All of this suggests that he would have advocated a semantical solution to the becoming paradoxes which treated the logical form of statements about change in a manner which does not involve a radical departure from surface form. In other words, he will want to treat statements like "The Vice-President has become the President" as expressing a binary relation between the individual specified by "the Vice- President" and the individual specified by "the President." We may hope that the theory about to be presented coincides with Aristotle's own point of view.

V. POSSIBLE WORLD SEMANTICS AND ARISTOTELIAN INDIVIDUALS

This brings us to a consideration of our third (and last) solution to our temporal puzzles. Many logicians have met with a considerable degree of success with a mode of semantical analysis which involves making use of a domain of reference points (often thought of as simply 'possible worlds'). Making use of this idea, we can start out with both a domain of possible worlds and a domain of objects associated with each possible world, and define individual concepts as the members of some set of functions from the former domain into the latter.

One of the advantages of this procedure is that if we assign individual concepts instead of simply individuals to the singular terms of our language, we are able to allow in a very straightforward manner for a distinction between identities which hold necessarily and those which hold only contin- gently. The sentence "a = b" is true at some possible world w if the individual concept assigned to "a" has the same value at w as does the individual concept assigned to "b." If the sentence is true in all (of a designated set of) possible worlds, the identity expressed is necessary; if it holds at some, but not all, such points it is contingent.

Once individual concepts have been introduced we are able to (following Bressan [4]) define an intensional style of predication by assigning to each n-place predicate a function from possible worlds to sets of n-tuples of individual concepts.14 Once this has been done Quine's questions about modality can be given clear answers, thus meeting his objections.



If we interpret our points of reference as moments of time instead of as possible worlds, we immediately have a solution to the problems outlined in Section II. Though this is not the only course open to us, I prefer to think of the domains of individuals associated with each point in time as space- time slices (from that time) of individuals. In this way the spatio-temporal coincidence of two space-time worms can be reflected by the coincidence of the individual concepts with which they are associated.

To help make this clear, let us look at a concrete example. To the name "Ford" we assign (on the intended interpretation) a function which picks out at each time the space-time slice of Ford which exists at that point (leaving it undefined otherwise). Notice that we may also allow for individual concepts which take their values from different men at different times. Thus "the President" gets assigned an individual concept which picks out slices of George Washington between 1789 and 1797, and then slices of John Adams from 1797-1801, and so on in this way picking out at each time the slice of the man who is President at that time. Since this function will take a slice of Gerald Ford as its current value, the statement:

(2) the President = Ford is at the present time true.

We will not delve into the mechanics of the matter, but it turns out that the present theory allows us to block the unwanted inferences in Section II and also enables us to answer the Quinean questions. To understand why this is so, we will first try to describe the theory in a way in which Aristotle could have conceived it, and then look at some of the textual evidence which supports attributing this theory to him.

Logicians often identify the proposition expressed by a sentence with the set of worlds in which that sentence is true. In a somewhat analogous manner we may, with reasonable results, identify an individual concept (as defined above) with the composite object which is the mereological sum of the values it takes for each time. Thus we identify the individual concept assigned "Ford" with that spatio-temporal continuant made up of the space- time slices of Ford - i.e., with Ford himself. Furthermore, the individual concept associated with the words "the President" is simply a discontinuous space-time worm which is the mereological sum of all the space-time slices taken from men when they were President. In this way we have identified each individual concept with a concrete physical object. If one objects to


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this identification, then we can say that each individual concept represents some such spatio-temporal worm.is

The President and Ford are different worms which are taking up the same space (i.e., sitting in the same chair) right now. Let us say that the President and Ford coincide at the present time. As they currently occupy the same spatial location, if right now I were to hit Ford with a snow ball, I would also be hitting the President with a snow ball. Likewise, when Ford skis down the slopes in Colorado, this also counts as a case in which the President is skiing down the slopes. Yet if ten years ago I had hit Ford, I would have missed the President entirely; and ten years ago Ford could go skiing and leave the President far behind. Thus even though Ford is now the same thing as the President, the two have different past and future histories.

To evaluate the truth of a past tense sentence rP",A about some

object a, we simply follow a's career back through time and see if eventually we come to a time at which it has the property designated by "&". Since we have allowed for the fusion and fission of individuals, there is no guarantee that the past careers of two individuals must be the same just because they coincide at the present moment. This explains why the inference from the statement that the President resigned last August ((1)) and the statement that the President is Ford ((2)) to the conclusion that Ford resigned last August ((3)) is invalid - as it should be. The fact that Ford and the Presi- dent are now the same (as expressed by (2)) does not require that Ford was ever in the extension of "resigns" just because the President was. As a matter of fact, (1) and (2) tell us absolutely nothing about Ford's past his- tory.

Without going into the technical details, we can also see how this kind of theory handles a predicate like "becomes." We are able to treat this verb as expressing a genuine relation between different individual objects (or, more technically, individual concepts). In general, the extension of "becomes" at a given time interval with initial boundary to and final boundary t1 will be the set of all ordered pairs of individual concepts [a, b] such that (i) the individual concept a picks out a space-time slice at to (call this slice so); (ii) the individual concept b picks out a space-time slice at t1 (call this slice s1); (iii) there is no t' such that to < t' and t' < t, and such that the individual concepts a and b have the same value for argument t'; and (iv) there is some ordering of space-time slices according to spatio-temporal conti- nuity which contains both so and s1.16



The Smullyan-esque response had us giving up the claim that "the Presi- dent = Ford" expresses an identity. Have we in fact done this once again? There are two ways of looking at what we have just done. We might say that we have continued construing this statement as an identity, but have now abandoned the principle SI. On the other hand, we might claim that SI remains intact, but that the statement in question does not express identity, but rather the weaker notion of temporal coincidence. In either case, however, we have accomplished something that was not allowed by our second solution - we have treated definite descriptions as denoting certain physical objects and also taken "=" and "becomes" as expressing relations between such objects.

From the point of view of this solution, we can give an explanation of the connection between the temporal problems and the becoming problems in Section II. Both types of perplexities find their source in the fact that descriptions can pick out different objects at different times. We only ran into difficulty because the phrase "the President" describes different men at different times. It is the phenomenon of change (broadly construed) which makes it the case that the descriptions applicable to objects can vary in this way. We started running into problems with the verb "becomes" because whenever a change takes place there will be some description which applies to the changing object after the change in question, but which did not apply to it just prior to (or during) the change. But we are not entitled to replace that description by just any expression which happens to have the same extension (if we want the result to be true). Furthermore, there will be a description which applied prior to the change, but no longer applies. And here we must be careful to replace that description with only another description which was co-extensive with the former at the start of the change (if we want the result to be true).

If we treat possible worlds as maximal chains on a domain of times, the difference between temporal and modal puzzles vanishes. Both temporal and modal operators will function as quantifiers over (actual and/or possible) times."17 In this case, it would be no mere coincidence that Quine's modal puzzles can be generated using tense operators or the verb "becomes." In all three cases we have a manifestation of the same semantic fact: descriptions can and do describe different things at different points of reference.

Since our third solution was suggested by Aristotle's comments on


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reparsing, we should now determine if the present theory is like the one Aristotle himself held.

VI. ARISTOTLE AND TEMPORAL COINCIDENCE

In two separate passages ([3c], E2, 1026b18-20; [3e], A11, 104b25-28) Aristotle mentions an argument which some sophists had used in an attempt to refute the principle that 'everything which is, but is not eternally, has come to be.' Although the argument is not spelled out in detail, it seems that we can reconstruct it like this:

(26) The musical thing has become the literate thing. (27) So the musical thing is now the literate thing. (28) So (by SI) the literate thing has become the musical thing.

This argument is simply a variant of (11)-(17) in Section II, and so the paradox about becoming which we presented earlier was actually thought up some 2,400 years ago and was familiar to Aristotle.

As we would expect, Aristotle reiected the above inference from (26) and (27) to (28). He held that it was invalid because (27) is an example of 'the coincidental' (r6 ov/3er6lrdk). One natural way to understand Aristotle here is to think of the musical and the literate as being spatio-temporal continuants which coincide in one another in the way that Ford and the President coincide. What we shall do in the remainder of this section is put forward some arguments which support the claim that Aristotle should be taken in this way. Then, in Section VII, we will show how Aristotle can actually be found using the concept of coincidental entities in his analysis of change in Physics A7.

In Metaphysics A9 Aristotle state the conditions under which things are called 'the same [things] in accordance with coincidence' (rabrd Ka'rd oavj4e3raKd). The word just translated as "coincidence" is from the perfect of the verb ovjpaco. One of the primary meanings of this verb is "to come together" or "to coincide." This certainly lends some credibility to the idea that Aristotle is about to discuss the type of notion that we have explicated in the previous section. I leave this point to the etymologists.

It should be pointed out that the traditional rendering of this verb by the translators of Aristotle's works is as "accidental," and most of the



passages in which he is translated as discussing 'the accidental' or 'accidental attributes' contain - in the Greek - the perfect of this verb. However, he thought that although it is necessary that a tringle possess two right angles, it is nonetheless coincidental (accidental).18 Thus we must be careful not to suppose that what is coincidental (or accidental) in his sense is also accidental in our sense. It would seem that he used the notion of coincidence to cover both (i) non-necessary (i.e., contingent) states of affairs and (ii) a subset of the set of necessary states of affairs. In light of the example just given, perhaps category (ii) contains descriptions of states of affairs which are non-definitionally necessary.

We will now go to the text to see what more we can learn about this notion.

We are told in Metaphysics A9 that

(29) The pale [thing] and the musical [thing] are the same according to coincidence

because

(30) they-are-coincidental-to the same thing.19 Once again we find phrases such as "the pale [thing]" which are construc- ted by placing the neuter singular definite article before a neuter singular adjective. Since such phrases were used to signify both the thing which is pale (an individual) and the attribute paleness, it is sometimes tacitly assumed that what is meant by (29) and (30) is that

(29') The pale individual and the musical individual are the same coincidentally,

because

(30') pallor and musicality are coincidentally instantiated in the same individual.

In other words, readers of Aristotle sometimes bank on the ambiguity of such phrases in their interpretation of A9.20

However, this cannot be the correct interpretation since in the Greek the phrases "the pale [thing]" and "the musical [thing]" occur only in (29) and are merely implied by the number of the verb in (30). Thus there is no room for an ambiguity interpretation - the subjects of the verb in (30) just are the subjects of the earlier clause (29).


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It is worth mentioning that the pair (29')-(30') is (if we take (30') as the analysis of (29')) an example of reparsing. Thus the fact that Aristotle could have used this type of analysis, but did not, helps strengthen the belief that he did not endorse the Smullyan-esque theory.

Two recent authors,21 seeing that the above interpretation will not do, have maintained that actually Aristotle was confused about identity, and that (29) and (30) are symptomatic of that confusion. They attempt to account for this confusion by claiming to find a mistake in the passage on coincidental unity at Metaphysics A6. Aristotle, so they say, failed to distinguish the question:

(31) When are a and b constituents of some third thing c (which is itself one thing)?

from the question:

(32) When are a and b one (and the same) thing? Since Aristotle confused these two questions, he mistakenly thought that the fact that the pale and the musical are constituents of some third thing (e.g., Coriscus) meant that the pale and the musical are one and the same thing.

In this way the supposedly incoherent A9 is diagnosed via a detour through A6. All that is needed now in order to make this work is to show that he thought that to say that a and b are one is just to say that a is the same as b. Since his confusion of (31) with (32) explains how he could have thought that the pale and the musical are coincidentally one, it would then likewise explain how he came to think that they are also coincidentally the same thing.

There are quite a few passages which support the move from the claim that a and b are the same to the claim that a and b are one, and vice versa. For example, there is the passage in the Topics where he freely moves from the claim that Socrates and the sitting thing are the same to the assertion that they are one ([3e], A7, 103a30-31). Even stronger evidence is obtained, however, when we note that the very same facts which lead him in A9 to say that a and b are coincidentally the same lead him in A6 to say that a and b are coincidentally one. Thus any argument which he would give to support the claim that the pale and the musical are coincidentally one is (for Aristotle) equally an argument that they are coincidentally the same thing.



Consequently, if his views about oneness involve confusing (31) with (32), then his views about sameness are confused in the same way. Yet on their most obvious readings (31) and (32) are so clearly different that it is hard to see how anybody - let alone Aristotle - could confuse them. The fact that my head, torso and limbs go together to make up my body does not even vaguely suggest that my head and my leg are the same thing. It is therefore extremely unlikely that Aristotle made such a gross error.

Strangely enough, when White - the proponent of the view that Aristotle confused (31) with (32) - explains why Aristotle failed to distinguish the two questions, he provides us with the information needed to exonerate Aristotle from the charge of confusion. What White urges is that such entities as the pale and the musical are not parts of Coriscus in the way that his arms and legs are parts (i.e., spatial parts), but are rather spatio- temporal parts of the four-dimensional continuant, Coriscus.22 If this is the case, then (keeping in mind the relationship between oneness and sameness) (31) and (32) amount to:

(31') Under what conditions are a and b spatio-temporal parts of some third (possibly distinct) spatio-temporal continuant c?

and

(32') Under what conditions are a and b coincidentally the same (in the sense sketched in Section V)?

If we suppose that a and b are individuals which exist at the present time, then these questions both receive the same answer, namely:

(33) under the condition that either (i) a is a spatio-temporal part of b

or (ii) b is a spatio-temporal part of a or (iii) a and b are both spatio-temporal parts of some third

spatio-temporal continuant c.

Since the two questions have the same answer, we may agree with White that Aristotle failed to distinguish (31) from (32), and also admit the connection between oneness and sameness, without attributing any confusion to our author.

Not only does this move clear Aristotle from a charge of confusion, but it also provides the key for a coherent interpretation of (29) and (30). For if we take "x is a spatio-temporal part ofy" to mean "x coincides in y" or


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"x is coincidental toy," then (33) is transformed into both (a) a statement of the conditions under which (according to Aristotle) a and b are coincidentally one ([3c], A6, 1015b 16-34), and (b) a statement of the conditions under which a and b are coincidentally the same ([3c], A9, 1017b27-1018a4). Thus we may read the pair (29)-(30) as asserting that the pale thing and the musical thing are the same according to coincidence because there is something of which both the pale thing and the musical thing are spatio-temporal parts. Such a reading would then bring Aristotle's own doctrine in line with the theory explained in Section V. The pale thing and the musical thing are coincidentally the same in exactly the same way that Ford and the President are the same thing.

In the light of Metaphysics A7 we can further claim that there is, corresponding to coincidental oneness and coincidental sameness, coincidental being. The same facts (i.e., those given in (33)) which were relevant to the former two notions also make it true that a is b (in the coincidental sense). Hence we have now succeeded in finding a notion which encompasses all three of these concepts - which is what we had promised to do in Section IV.

We are now in a better position to understand Aristotle's comments on reparsing in the Physics A2 passage. Imagine somebody who felt that the statement that the President is Ford ((2)) cannot be taken as an identity between individuals since (as we saw in Section II) there are things true of the one that are not true of the other. Aristotle would be perfectly happy to concede that in some sense or another the two individuals were different, but since they are currently coinciding there is a weaker notion (temporal coincidence) which can be expressed by "is" and which makes the statement true.

Furthermore, we are not bloating our ontology with new and strange objects such as the President, the pale, the musical, etc., since each of these allegedly new entities coincides in some man at any time at w~hich one of them exists. The President is nothing over and above the men who were, are or will be President, for example. So talk of an individual such as the Presi- dent does not introduce any new matter into our ontology, but merely allows us to carve up the matter that we already have in a different way.

For reasons of space we cannot here properly develop the relationship between these ideas and Aristotle's well known theory of categories. None- theless, since there are a number of questions about coincidence which can



only be answered by looking at his categorial classifications, we must say a few words about this here.

First of all, one may wonder how artificial space-time worms may get before they stop counting as individuals. For instance, would Aristotle allow a spatio-temporal object made up of completely random segments to count as an individual?

To this question I can provide only a partial answer. There are some individuals which Aristotle treats as the most fundamental - and these he calls primary substances. Since all of the other individuals with which we are concerned (the pale, the musical, etc.) are specified in terms of a predicate from a non-substance category, and since they are individuals, we must suppose that they are the individuals in the categories other than substance. It is an Aristotelian dogma that individual substances are the primary furni- ture of the universe, and items in other categories have a derivative sort of existence which is dependent upon the primary substances which they are in. The fact noted above that the President is 'nothing over and above the men who were, are or will be President' suggests that perhaps we can expli- cate (at least in part) this Aristotelian dogma in terms of the notion of coincidence. Non-substantial individuals are dependent for their existence upon the substances which they are in for the simple reason that they are spatio-temporal chunks of such substances. Primary substances are the spatio-temporal objects in which all of the other spatio-temporal individuals coincide.

Until we can make sense of the distinction between substance and acci- dent, this remains at best a suggestion. But still, we have found some con- straints on the kinds of collections of spatio-temporal parts which Aristotle would count as individuals. An Aristotelian individual is a spatio-temporal object which (a) coincides in a substance whenever it exists, and (b) is specified in terms of a predicate from one of Aristotle's categories (and hence has that predicate true of it for its duration).

There is, however, a further question for which I have no answer, namely, "May an Aristotelian individual coincide in more than one substance?" There are two ways in which this may take place. One way is for the individual to behave as the President did, coinciding in many differert substances over a period of time, but coinciding in just one substance at any given time. Another way in which an individual could coincide in more than one substance is if we were allowed to take the mereological sum of two or


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more non-substantial individuals to give us a new non-substantial individual. For instance, is the mereological sum of all of the individual white things itself an individual white thing? These are questions which must eventually be answered, but I currently cannot answer them.23

VII. ARISTOTLE'S USE OF COINCIDENCE

In the first paragraph of the last section (in (26)-(28)) we saw how certain sophists had anticipated our argument about the verb "becomes," and that Aristotle thought that the fact that (27) was coincidental invalidated the argument. This is, I think, sufficient evidence to show that he was aware of the intensional nature of the verb. Having investigated Aristotle's concept of the coincidental at some length, and having considered alternative responses to the intensionality problem, we have helped to establish the plausibility of the claim that our third solution captures some of the important features of Aristotle's own thought about change. To help con- firm this view, we should now try to show how that theory works in helping us to understand his own discussion of this topic in Physics A7.

Physics A7 begins with the claim that when an unmusical man becomes a musical man, the following four statements are true:

(34) The unmusical [thing] becomes the musical [thing]. (35) The man becomes the musical [thing]. (36) The man and the unmusical [thing] are [prior to the change]

one in number but two in form and account.

(37) The unmusical [thing] perishes [as a result of the change] while the man remains.

In view of (37) it is hard to take the phrase "the unmusical," and hence also the phrase "the musical", as designating abstract entities - after all, unmusicality does not go out of existence when one of the many unmusical people becomes cultured and refined at the Academy. Furthermore, (34) could not be meant to express the claim that unmusicality changes into musicality. It means that an unmusical individual changes into a musical individual. Likewise, (35) is not saying that some man (or the species man) becomes musicality.

Thus a reasonable interpretation of this passage must both (a) treat the musical and the unmusical as individuals, and (b) treat the man who is



unmusical as distinct from the unmusical on the ground that according to (37) their persistence conditions are different. Parity of reason requires the man and the musical to also be distinct. Yet (36) requires us to treat the man and the unmusical as one in number. But then we find ourselves confronted with the problems about "becomes" which we presented in Section II. The verification of this claim is left for the reader.

In view of these problems, we have two alternatives. Either we claim that his discussion of the elements of change is incoherent, or we claim that the framework within which he is working provides an adequate solution to these problems. Since we have shown on the basis of other passages that Aristotle was aware of these problems, we can only suppose that he was aware of them when he wrote this passage. Hence the second alternative is by far the more natural.

Certainly he had some solution in mind, if we are right in supposing that he was aware of the problems. The theory of coincidence which we have sketched two sections back was developed in an attempt to account for Aristotle's response to these problems, and on the supposition that he was operating with that theory in Physics A7 we can make sense of this important passage. This gives us strong warrant for attributing this theory to him.

Since that theory is about individuals which persist through time, and which can coincide with one another, we can make sense of (34)-(37) as follows. We should see Aristotle as treating "becomes" as expressing an intensional binary relation between individuals in the various categories of being. In the case at hand, the unmusical and the musical are non-substantial individuals, and the man is an individual substance. The unmusical is the spatio-temporal hunk of the man prior to the change, and the musical is the hunk of the man after that change. Thus the unmusical, although it perishes, was coincidentally the same thing as the man prior to the change. Some of the textual exegesis relevant to the present interpretation can be found in [5], so we will not pursue the matter further here.

In Physics A8 Aristotle claims that the theory of the elements of change which he presented in Physics A7 provides us with a refutation of the argument against the possibility of change which we looked at in Section III. Briefly, that argument went wrong because it failed to distinguish (a) that which is not p (e.g., the unmusical), (b) that which is 4 (e.g., the musical) and (c) the underlying substratum which persists through the change, first being not 4 and then being k (e.g., the man). Once this third element is


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introduced, and its relationship to the other two elements is made clear, Aristotle thought that the alleged difficulties vanished. Hence one again we find him using our Section V analysis to handle problems with the verb "becomes."

A few words about methodology and anachronism are now in order. The aim of our enterprise is to discover what Aristotle's doctrine was on various points. Our purpose, therefore, is to talk about his theories and not in them. Of course we could have engaged in the latter, carefully restricting the discussion to an explicitly Aristotelian vocabulary. But this would only en- gender an illusion of understanding. It is important that we are clear about what is being done, lest the reader lets his fear that talk of space-time worms is anachronistic breed skepticism about the present interpretation. Such an objection would be as misplaced as the condemnation of the work of an historian who, in describing Herodotus' view of the geography of the world, talked in terms of miles instead of parasangs. Given that Aristotle believed that sensible individuals occupy spatial location, and given that he thought that in order for such objects to change they must endure through (at least some period of) time, it has to be the case that there exists a correct meta-theoretical account of his doctrine which utilizes the concept of a spatio-temporal worm. Furthermore, where we find him distinguishing such individuals by their persistence conditions, it likewise has to be the case that in our representation of his theory some worms 'coincide' in others.24

Thus Aristotle's own solution to the paradoxes about becoming involves dividing the individuals of our world into categories with the proviso that all of the non-substantial individuals coincide with primary substances. He is then enabled to treat statements of sameness between individual objects in different categories in a straightforward manner, and utilize these different kinds of objects in his analysis of becoming. To put the matter more pre- cisely, becoming will be treated as a relation between just such objects.

VIII. CONCLUSION

We have now seen that analogues to Quine's objections to modal logic had already been formulated by Aristotle's time. We have also seen that the problems have generated three major types of response. The same pattern of reaction to such puzzles is found in both eras. The initial reaction by some is to altogether abandon the locutions employed to describe temporal



(or modal) facts - a stage exemplified by both Parmenides and by Quine. Another stage is to regain confidence in the validity of the concepts in- volved, but to revise language to accommodate them. Finally, as the philosophical underpinnings of the concepts become better understood, we regain our original locutions and modes of expression, rendering them again unproblematic.

It would seem, then, that there was good reason to take seriously Quine's claim that quantified modal logic makes sense only if we 'revert' to Aris- totelian essentialism ([15], Section 3, pp. 155-6).

I have been helped in the writing of this paper by discussions with many people. In particular I would like to thank Terry Penner, Jonathan Bennett, Michael Byrd, Zane Parks, and Dennis Stampe.2s

The University ofBritish Columbia

NOTES

SWhen these problems are presented, the reader will very likely have a solution in mind already, and hence may find it hard to feel the pull of the problems. Towards this end, I suggest that the reader constantly remind himself of the analogy with the Quinean modal problems. It should also be stressed from the start that the three solu- tions that I discuss are not the only possible responses. 2 The phrase "the President" is short for "the President of the United States of America." An adequate treatment of such phrases should allow for a distinction between cases in which this phrase is used to talk about (a) the man who is in fact the President, and (b) the President qua President. In this paper, I consider only the latter uses. This is no weakness in what follows, however, since the distinction can easily be added to the theories actually discussed. The theory given in Section IV would rely on a Russellian scope distinction, while the theory in Section V would adopt the technique found in [21]. a But see fourth paragraph from end of Section V. 4 Notice that we will be ignoring the empty-name problem. s If the letter "P" is interpreted as a possibility operator, this is converted into a Quinean argument against modality. 6 The EG problems arise here also. 7 [3d], A8, 191a23-33. 8 [6], Fragment 8; 6-7. S[6], Fragment 8; 7-9. In case the reader objects to my interpreting "that which is

not" as "that which is not 4," I point out that "4" can be interpreted as "is an ex- istent." to This is not intended to be a full analysis of the verb "becomes," but at least a partial analysis of a particular statement using the verb. For a more detailed account, see [7].


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' In translating this, it is possible to delete the bracketed words and to treat the passage as discussing not individuals, but universals not taken universally. It makes no difference to the point about repassing. The fact that musical Coriscus is (at On Sophistical Refutations 178b) treated as a such (as opposed to a this), and as the same type of entity as the walking, helps point towards my reading, though. (So does the treatment of such phrases in Section VI and Section VII below).

It is important to realize that in (a') the phrase "is healthy" is one word in the Greek. 12 The sense of the Greek is a little clearer if we depart from the grammatical structure and take the latter clause as:

... they did not say that: the man is the white thing,

but rather they said that: the man has-been-whitened, etc.

Again, as in the previous note, this could have been taken as a statement about universals not taken universally. 13 See note 23 for a further treatment of the problem of securing uniqueness for definite descriptions. 14 Following [4] as explicated in [12], and letting a

- P be the set of functions from

a into j, Xn the nth Cartesian product of X with itself, I a set of individual concepts, and T the set of times:

V(tn) eI (where t, is a term) V(F) e T-. 9 (In) (where F is an n-place predicate) for i e T,

V(Ft1,... tn)(i) = True iff # False iff


T(~ XeVKOV KOi T(~ CWV~flK&' TO aQt4~ 5174) abrc avifql iipe~B 20 In print, this view may be found in [10], where it is also apparently attributed to [16a]. 21 [8] and [22]. The view to be presented is certainly to be found in [22]; though I am not certain about [8], the tendency is clearly the same. 22 [22], esp. V-VII. White is cautious, however, and does not discuss this particular example, or even another Aristotelian example. 23 Two other points are worth mentioning. First, the much discussed problem of the status of things which are present in a subject but never predicable of a subject ([3a], 2, la20-b6, esp. la23-29; see [1], [2], [9], [11]) can be approached from a fresh direction. Since non-substantial individuals are each in some category or other, the four-fold classification in Categories 2 should have a slot for them. If so, it is they which are in a subject, but never predicated of a subject. Furthermore, our treatment of coincidental being suggests that paronyms ([3a], 1) are non-substantial individuals.

Secondly, descriptions such as "the pale thing" will usually fail to pick out anything unless we construe the points of reference more narrowly than we have. Let us add as many parameters to our specification of points of reference as is needed to secure unique reference (perhaps all that we will need is time and place). The singular terms of our language can now be assigned functions from these narrow points of reference to individual concepts as defined in the text of this paper. If I am not mistaken, this idea was first suggested by Dana Scott [18]. 24 Nonetheless, I think that [3d] shows that Aristotle did have the notion of a temporal part of an object. See especially All, Z4, Z5 and Z10. 2s I am indebted to Sandra Peterson's pioneering work on intensional contexts in Aristotle [14]. Also, for a fuller treatment of coincidental sameness see [13]. In this paper we have considered only numerical sameness, and have ignored specific, generic and analogical sameness (for which see [3c], A9, 1018a4-15; also [3e], 17).

BIBLIOGRAPHY

[ 1] John L. Ackrill, Aristotle's Categories and De Interpretatione, Clarendon Press (1963).

[2] R. E. Allen, "Individual Properties in Aristotle's Categories," Phronesis, 14 (1969).

[3] Aristotle (a) Categories (b) De Interpretatione (c) Metaphysics (d) Physics (e) Topics [all translations in this paper of (c) and (d) are based on text in [16a] and [16b]].

[4] Aldo Bressan, A General Interpreted Modal Calculus, Yale University Press (1972).

[5] Alan Code, "The Persistence of Aristotelian Matter," Philosophical Studies (forthcoming).

[6] H. Diels, Die Fragmente der Vorsokratiker (edited with additions by W. Kranz), 10th edition, Berlin (1961).


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[7] David R. Dowty, "Toward a Semantic Analysis of Verb Aspect and the English 'Imperfective' Progressive" (unpublished).

[8] Christopher Kirwan, Aristotle's Metaphysics: Books F, A, E, Clarendon Press (1971). [9] G. B. Matthews and S. M. Cohen, "The One and the Many," Review of Meta-

physics, 21(1968). [10] Fred D. Miller, Jr., "Did Aristotle Have the Concept of Identity?," Philosophical

Review, LXXXII (1973). [11] G. E. L. Owen, "Inherence," Phronesis, 10 (1965). [12] Zane Parks, "Investigations into Quantified Modal Logic" (unpublished). [13] F. J. Pelletier, "Sameness and Referential Opacity in Aristotle: A Look at Topics

A7, Metaphysics A6, a9 and Z6" (unpublished). [14] Sandra Peterson, TheMaskerParadox (unpublished PhD thesis, Princeton 1969). [15] W. V. O. Quine, "Reference and Modality," From a LogicalPoint of View,

Harper and Row, New York (1963). [16] W. D. Ross

(a) Aristotle's Metaphysics, Clarendon Press (1924) (b) Aristotle's Physics, Clarendon Press (1936).

[17] Bertrand Russell, "On Denoting," Mind (1905). [18] Dana Scott, "Advice on Modal Logic," in Philosophical Problems in Logic

(edited by K. Lambert), Reidel, 1970. [19] Arthur F. Smullyan, "Modality and Description," The Journal of Symbolic Logic

(1948). [20] Richmond H. Thomason, "Indeterminist Time and Truth-Value Gaps," Theoria

18:3 (1970). [21] Richmond H. Thomason and Robert C. Stalnaker, "Modality and Reference,"

Noi2s, II, #4 (1968). [22] Nicholas P. White, "Aristotle on Sameness and Oneness," Philosophical Review,

LXXX (1971).


Article Contentsp. [159]p. 160p. 161p. 162p. 163p. 164p. 165p. 166p. 167p. 168p. 169p. 170p. 171p. 172p. 173p. 174p. 175p. 176p. 177p. 178p. 179p. 180p. 181p. 182p. 183p. 184p. 185p. 186

Issue Table of ContentsJournal of Philosophical Logic, Vol. 5, No. 2 (May, 1976), pp. 159-298Aristotle's Response to Quine's Objections to Modal Logic [pp. 159-186]Epistemic Foundations of Logic [pp. 187-204]Knowledge and Necessity [pp. 205-208]An Empirical Hypothesis about Natural Semantics [pp. 209-236]Completeness of Quantum Logic [pp. 237-280]Kreisel, the Continuum Hypothesis and Second Order Set Theory [pp. 281-298]

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Aristotle Quine