Peirce and Philo*

Jay Zeman

Charles Peirce, logician and philosopher, contributed notably to the theory of the conditional. Actually, from his perspective and in his terminology it is better, as we shall see, to link his work on the conditional with his discussions of the hypothetical proposition. Peirce spoke often of the consequentia de inesse,1 the concept of which is intimately linked with the material, or "Philonian" conditional; indeed, we shall see him calling himself a Philonian. And it is not uncommon to hear Peirce—at least prior to the last decade of his life—declared a Philonian, whose fundamental analysis of the conditional was essentially the same as that of Philo (and of more modern types like Russell and like Quine).

In this paper, I intend first to examine Peirce’s understanding of "Philonian"; I will then look at the Philonian or "de inesse" conditional in the context of his overall logical thought. It is commonly held that Peirce in his early years held to the "nominalistic" Philonian conditional, and only later "surrendered" it in favor of a more "realistic" view; the study we are here undertaking will indicate that this does not adequately reflect his position.

As we shall see, Peirce at one time or another called himself a Philonian. He was, of course, also quite aware that Philo is historically paired with his teacher Diodorus. That pairing, in fact, is so integral to the meaning of "Philonian" that without consideration of it, "Philonian conditional" becomes a somewhat pretentious term for "truth-functional implication,"

When he discusses the historical pairing I have mentioned, Peirce employs "Diodoran" as the adjective derived from the name of Diodorus. Our considerations here are complicated by contemporary discussions; an index of the complication is the fact that the usual contemporary form of the adjective is "Diodorian." Arthur Prior, one of the rediscoverers of the possible worlds approach to modal logic,2 used Diodorus as a take-off point in his work (see Prior 1955, 1957); Prior, and modal logicians following him, employed this latter form of the relevant adjective. Prior at first thought that what he called the "Diodorian modal system" supplied semantics for S4; this misapprehension was rather quickly dispelled, however, and ‘Diodorian modality’ gravitated to its contemporary usage: generically, it means the class of modal logics with reflexive, transitive, and linear (total-ordered) relational frames (that is, the systems containing S4.3); specifically, it refers to one of these systems, "D," the system in whose frames "possible worlds" are "discrete" (see Zeman 1973).

Peirce’s discussions of the Philo-Diodorus debate are among those in which he anticipates the discovery of possible-worlds modal semantics; we must not, however, conclude that his term "Diodoran" means anything like the contemporary "Diodorian." We find him saying that

Cicero informs us that in his time there was a famous controversy between two logicians, Philo and Diodorus, as to the signification of conditional propositions. Philo held that the proposition "if it is lightening it will thunder" was true if it is not lightening or if it will thunder and was only false if it is lightening but will not thunder. Diodorus objected to this. Either the ancient reporters or he himself failed to make out precisely what was in his mind, and though there have been many virtual Diodorans since, none of them have been able to state their position clearly without making it too foolish. Most of the strong logicians have been Philonians, and most of the weak ones have been Diodorans. For my part, I am a Philonian; but I do not think that justice has ever been done to the Diodoran side of the question. The Diodoran vaguely feels that there is something wrong about the statement that the proposition, "If it is lightening it will thunder," can be made true merely by its not lightening (NEM 4:169; this is MS 441 (1898)).

Although Peirce comments elsewhere that "the Diodoran view seems to be the one which is natural to the minds of those, at least, who speak the European languages" (CP 3.441), he takes (as is suggested above) a rather dim view of those who have advocated this position. He contrasts the Philonian and Diodoran positions thus:

According to the Philonians, "If it is now lightening it will thunder," understood as a consequence de inesse, means "It is either not now lightening or it will soon thunder." According to Diodorus, and most of his followers (who seem here to fall into a logical trap), it means "It is now lightening and it will soon thunder" (CP 3.442).

Far from being connected with the contemporary "Diodorian" modal logic (which, from an important perspective, fits perfectly within Peirce’s own theory of possibility), the Diodoran position as understood by Peirce seems to require a sort of "existential import" for the antecedent condition (in the above case, "It is now lightening"). A function with the meaning ascribed above by Peirce to the conditional of "Diodorus, and most of his followers," is, of course, not really a conditional, but is much closer in meaning to a conjunction. This won’t do, but Peirce is unwilling simply to reject the "Diodoran" strategy. Presumably, this approach expresses something about the conditional that can be taken as common-sense wisdom about it; perhaps Peirce himself can come to the aid of the inept Diodoran, and can

fit him out with a better defence than he has ever been able to construct for himself, namely, that in our ordinary use of language we always understand the range of possibility in such a sense that in some possible case the antecedent shall be true (NEM 4:169).

Taken thus, the "Diodoran" position is not really in competition with the Philonian, but might be considered complementary to it, reflecting something to do with ordinary uses of language.

This sense of "Diodoran" stands in contrast to the technical modal "Diodorian" of the last couple of decades: Peirce’s reading of Diodorus is quite different from Prior’s. And Peirce’s understanding of Philo/Diodorus also differs from another way in which these Ancients may be compared. It is possible to see Philo vs. Diodorus as the preliminary bout of a card of fights which has more recently matched C. I. Lewis with the Russellians and W. v. O. Quine with Ruth Marcus. The most visible issue in the disputes I mention here is that of the nature of modal propositions, and in particular, of the relationship of "implication" to modality. In the terminology of Lewis, this is the question of "material" vs. "strict" implication. And in terminology often used in discussing broad movements in western philosophy, it is the matter of "nominalism" vs. "realism." Peirce’s understanding of Philo vs. Diodorus does not seem, as our quotes indicate, to reflect this contrast. I emphasize this so that we won’t unconsciously read current approaches back into Peirce.

Now, it is generally accepted that a development can be traced over the years in Peirce’s thought. As one commentator has put it,

Peirce’s philosophy is like a house which is being continually rebuilt from within. Peirce works now in one wing, now in another, yet the house stands throughout, and in fact the order of the work depends upon the house itself since modification of one part necessitates the modification of another. And although entire roofs are altered, walls moved, doors cut or blocked, yet from the outside the appearance is ever the same (Murphey 4).

Well, perhaps; I think that it is well to examine carefully the extent to which "modifications" actually occur, however. The richness of a later position may not at all mean that its earlier forebears have been abandoned. It may be that these earlier positions are, in fact, so well established as to be unproblematic with respect to the later position; three decades after the Illustrations of the Logic of Science series, for example, Peirce was writing expansive footnotes on those papers—but hardly rejecting Pragmatism as laid out there.

Now, coming near matters which will be near to our concern here, in tracing "Peirce’s Progress from Nominalism to Realism," another author tells us that

Since the proof of pragmaticism, and thereby of realism, can be most cogently stated in existential graphs, the series [of Monist articles beginning in 1905] proceeds to a fresh exposition of the graphs (CP 4.530-572). But the Monist printer’s ink is scarcely dry on that when Peirce hits upon an improvement to enable them to represent different kinds of possibilities; and, confronted by this improvement, his other self, his nominalist self, surrenders his last stronghold, that of Philonian or material implication (Fisch 1986).3

CP 4.546 is even more difficult to see as the "surrender of [a] stronghold," although its emphasis is on the reality of possibles (see my discussion of the EG rule given in CP 4.569; the "suiciding wife" example is connected to this, and it is further discussed in CP 4.580).

Actually, a fairly careful study of material available from the period in which this "surrender" is supposed to have taken place doesn’t give much evidence of it; in fact, there is at this period very little testimony of concern with the conditional as such. Now I certainly don’t believe that a dearth of discussions specifically regarding the conditional indicates that Peirce had lost interest in it. I believe, rather, that Peirce’s theory of the conditional had taken its basic form long before, and that while there would be elaborations and amplifications of that theory, that basic form—which was integral to his thought—would maintain itself intact.

My own investigations lead me to believe that Peirce’s relationship to the Philonian conditional is quite complex; although there was an increased emphasis on the reality of thirds and firsts in the later Peirce, his view of the conditional de inesse sees it as second to a larger third from about the days of the Illustrations of the Logic of Science, anyway.

I note that in many locations Peirce speaks not of conditional propositions, but of hypotheticals. The term "hypothetical" suggests a strong link between mathematical logic and philosophy for Peirce; we note its role in the thought of Kant, for example, who was a major influence on Peirce, while on the other hand, seeing that Peirce’s discussions of "hypotheticals" are almost always located in the context of his symbolic logic. Let’s look at what he says in the well-known Philosophy of Notation article of 1885:

To make the matter clear, it will be well to begin by defining the meaning of a hypothetical proposition, in general. What the usages of language may be does not concern us; language has its meaning modified in technical logical formulae as in other special kinds of discourse. The question is what is the sense which is most usefully attached to the hypothetical proposition in logic? Now the peculiarity of the hypothetical proposition is that it goes out beyond the actual state of things and declares what would happen were things other than they are or may be. The utility of this is that it puts us in possession of a rule, say that "if A is true, B is true," such that should we hereafter learn something of which we are now ignorant, namely that A is true, then, by virtue of this rule, we shall find that we know something else, namely, that B is true. There can be no doubt that the Possible, in its primary meaning, is that which may be true for aught we know, that whose falsity we do not know.4 The purpose is subserved, then, if throughout the whole range of possibility, in every state of things in which A is true, B is true too. The hypothetical proposition may therefore be falsified by a single state of things, but only by one in which A is true while B is false. States of things in which A is false, as well as those in which B is true, cannot falsify it. If, then, B is a proposition true in every case throughout the whole range of possibility, the hypothetical proposition, taken in its logical sense, ought to be regarded as true, whatever may be the usage of ordinary speech. If, on the other hand, A is in no case true, throughout the whole range of possibility, it is a matter of indifference whether the hypothetical be understood to be true or not, since it is useless. But it will be more simple to class it among true propositions, because the case in which the antecedent is false do not, in any other case, falsify a hypothetical. This, at any rate, is the meaning which I shall attach to the hypothetical proposition in general, in this paper (CP 3.374).

I note especially in the above Peirce’s speaking of "a proposition true in every case throughout the whole range of possibility"; the notion of quantification over a range of possibility is central here. And we see him in 1902 saying that

the quantified subject of a hypothetical proposition is a possibility, or possible case, or possible state of things. In its primitive state, that which is possible is a hypothesis which in a given state of information is not known, and cannot certainly be inferred, to be false. The assumed state of information may be the actual state of the speaker, or it may be a state of greater or less information. Thus arise various kinds of possibility (CP 2.347).

The notion of quantification over a range of possibilities is a basic theme in Peirce’s work; in 1902, he himself sees his understanding of hypotheticals in these terms as going back even earlier than the Philosophy of Notation paper:

In a paper which I published in 1880, I gave an imperfect account of the algebra of the copula. I there expressly mentioned the necessity of quantifying the possible case to which a conditional or independential proposition refers. But having at that time no familiarity with the signs of quantification which I developed later, the bulk of the chapter treated of simple consequences de inesse. Professor Schröder accepts this first essay as a satisfactory treatment of hypotheticals; and assumes, quite contrary to my doctrine, that the possible cases considered in hypotheticals have no multitudinous universe. This takes away from hypotheticals their most characteristic feature (CP 2.349).

Peirce probably refers here to the well-known paper of CP 3.154 ff.; he notes there that

De Morgan, in the remarkable memoir with which he opened his discussion of the syllogism . . . has pointed out that we often carry on reasoning under an implied restriction as to what we shall consider as possible, which restriction, applying to the whole of what is said, need not be expressed. The total of all that we consider possible is called the universe of discourse, and may be very limited. One mode of limiting our universes by considering only what actually occurs, so that everything which does not occur is regarded as impossible (CP 3.174).

And a universe so limited would be, of course, the universe of the de inesse; it seems clear to me, even without Peirce’s 1902 testimony, that even in 1880 he considers this realm just a limiting case of a broader domain which cannot be ignored; in fact, in another 1880 paper, we find a harbinger of the thought which was to be made explicit with the development of notations for quantification:

To express the proposition: "If S then P," first write


for this proposition. But the proposition is that a certain conceivable state of things is absent from the universe of possibility. Hence instead of A we write

B B 5

Then B expresses the possibility of S being true and P false. Since, therefore, SS denies S, it follows that (SS,P) expresses B. Hence we write


(CP 4.14).

This is close to the time of the "Illustrations of the Logic of Science"; it might be argued that, with that series of essays and with the significant contributions to symbolic logic dating from this period, we have the beginnings of Peirce’s maturity as a philosopher-logician. And his theory of the conditional—or, more generally, of the hypothetical—is consistent from this point on, and is integral to his thought as a whole.

Examination of Peirce’s later work in logic strongly supports this position. A decade after the Philosophy of Notation paper, Peirce had changed notations. The algebraic notations we have been examining were successful vehicles for his deductive logic; for a logic, however, it is not enough that a notation be mathematically correct. The purpose of a logic is not efficient calculation. The purpose is to provide an appropriate representation of the process of necessary deductive reasoning; one of the measures of appropriateness, Peirce tells us, is iconicity (Zeman 1986, 13). And Peirce felt that his Existential Graphs were a more appropriate representation of the subject matter of logic. The Graphs provide an alternative and arguably more iconic (than the algebras) representation of a number of features of deductive logic. The feature we are interested in at this point is the relationship between the de inesse conditional and the "hypothetical."6

The most basic of the signs of the Existential Graphs is the "Sheet of Assertion" (SA); SA is itself a graph, which represents whatever is true about the universe of discourse; effectively, it represents that universe of discourse. And

A proposition de inesse relates to a certain single state of the universe, like the present instant. Such a proposition is altogether true or altogether false. But it is a question whether it is not better to suppose a general universe, and to allow an ordinary proposition to mean that is sometimes or possibly true. Writing down a proposition under certain circumstances asserts it. Let these circumstances be represented in our system of symbols by writing the proposition on a certain sheet (CP 4.376).

It seems natural to employ a given SA to represent what Peirce calls the "quasi-instantaneous" state of the general universe; Peirce sometimes employs SA in this manner, and sometimes suggests other representations (or at least that there are other representations; see ibid.; we shall eventually note at least two specific cases in which Peirce sets separate "quasi-instantaneous" states upon the same sheet). In 1903 Peirce tells us that

If a system of expression is to be adequate to the analysis of al necessary consequences,7 it is requisite that it should be able to express that an expressed consequent, C, follows necessarily from an expressed antecedent, A. The conventions hitherto adopted do not enable us to express this. in order to form a new and reasonable convention for this purpose we must get a perfectly distinct idea of what it means to say that a consequent follows from an antecedent. It means that in adding to an assertion of the antecedent an assertion of the consequent we shall be proceeding upon a general principle whose application will never convert a true assertion into a false one. . . . But before we can express any proposition referring to a general principle, or as we say, to a "range of possibility," we must first find means to express the simplest kind of conditional proposition, the conditional de inesse, in which "If A is true, C is true means only that, principle or no principle, the addition to an assertion of A of an assertion of C will not be the conversion of a true assertion into a false one. That is, it asserts that the graph of Fig 1, anywhere on the sheet of assertion, might be transformed into the graph of Fig 2 without passing from truth to falsity.

Figure_01.GIF (500 bytes) Figure_02.GIF (766 bytes)
Figure 1 Figure 2

This conditional de inesse has to be expressed as a graph in such a way as distinctly to express in our system both a and c and to exhibit their relation to one another. To assert the graph thus expressing the conditional de inesse, it must be drawn upon the sheet of assertion, and in this graph the expressions of a and c must appear; and yet neither a nor c must be drawn upon the sheet of assertion. How is this to be managed? (CP 4.435).

Peirce goes on to develop the typical alpha-graph representation of the conditional. Significantly, he stresses iconicity:

In order to make the representation of the relation between [antecedent and consequent] iconic, we must ask ourselves what spatial relation is analogous to their relation (ibid).

His solution is the representation in Figure 3:

Figure_03.GIF (1062 bytes)
Figure 3

I have remarked that Peirce will give us ways of dealing with more than one "quasi-instantaneous state" on a given sheet of assertion; he also, however, describes sheets themselves as representing such states:

. . . in the gamma part of the subject all the old kinds of signs take new forms. . . . Thus in place of a sheet of assertion, we have a book of separate sheets, tacked together at points, if not otherwise connected. For our alpha sheet, as a whole, represents simply a universe of actual existent individuals, and the different parts of the sheet represent facts or true assertions concerning that universe. At the cuts we pass into other areas, areas of conceived propositions which are not realized. In these areas there may be cuts where we pass into worlds which, in the imaginary worlds of the outer cuts, are themselves represented to be imaginary and false, but which may, for all that, be true, and therefore continuous with the sheet of assertion itself, although this is uncertain. You may regard the ordinary blank sheet of assertion as a film upon which there is, as it were, an undeveloped photograph of the facts in the universe. I do not mean a literal picture, because its elements are propositions, and the meaning of a proposition is abstract and altogether of a different nature from a picture. But I ask you to imagine all the true propositions to have been formulated; and since facts blend into one another, it can only be in a continuum that we can conceive this to be done. This continuum must clearly have more dimensions than a surface or even than a solid; and we will suppose it to be plastic, so that it can be deformed in all sorts of ways without the continuity and connection of parts being ever ruptured. Of this continuum the blank sheet of assertion may be imagined to be a photograph. When we find out that a proposition is true, we can place it wherever we please on the sheet, because we can imagine the original continuum, which is plastic, to be so deformed as to bring any number of propositions to any places on the sheet we may choose (CP 4.512).

So the "alpha sheet . . . represents simply a universe of actual existent individuals, and the different parts of the sheet represent facts or true assertions concerning that universe." We seem to have "gates"8 into other universes: "At the cuts we pass into other areas, areas of conceived propositions which are not realized."9 The model for the relationship between the universe of all possibles and the actual universe is given geometrically (actually, topologically). Further, Peirce is not interested only in "the" universe of actual existent fact, but as well in certain other subsets of the realm of all possibles—he is interested in those subsets which could themselves constitute existential universes.

. . . Now, qualities are not, properly speaking, individuals. . . . Nevertheless, within limitations, which include most ordinary purposes, qualities may be treated as individuals. At any rate, however, they form an entirely different universe of existence. it is a universe of logical possibility. As we have seen, although the universe of existential fact can only be conceived as mapped upon a surface by each point of the surface representing a vast expanse of fact, yet we can conceive the facts [as] sufficiently separated upon the map for all our purposes; and in the same sense the entire universe of logical possibilities might be conceived to be mapped upon a surface. Nevertheless, if we are going to represent to our minds the relation between the universe of possibilities and the universe of actual existent facts, if we are going to think of the latter as a surface, we must think of the former as three-dimensional space in which any surface would represent all the facts that might exist in one existential universe (CP 4.514).

So moving from the topological icon of the basic Sheet of Assertion (considered to be a surface) as representing the universe of actual existent fact, we take an appropriate space (that "surface" plus another dimension) as representing the overall realm of possibles; other surfaces within that space would then represent other possible existential universes. The "book of separate sheets, tacked together at points, if not otherwise connected" of CP 4.512 is an approximation of this "possibility space"; somehow the cuts (which are intimately associated with the conditional) are means of passing from one possible existential universe to another. It seems clear to me that the model here proposed for "the relation between the universe of possibilities and the universe of actual existent facts" is also the model for the relation between the hypothetical (which is concerned with the universe of possibles in general) and the conditional de inesse (which focuses on conditions at just one "quasi-instantaneous" state.

Peirce does not at this point see it possible to exploit this model adequately:

In endeavoring to begin the construction of the gamma part of the system of existential graphs, what I had to do was to select, from the enormous mass of ideas thus suggested, a small number convenient to work with. It did not seem to be convenient to use more than one actual sheet at one time; but it seemed that various different kinds of cuts would be wanted (ibid.).

He suggests the "broken cut" as a way of entering the broader universe of possibility; thus

Figure_04.GIF (715 bytes)
Figure 4

does not assert that it does not rain. it only asserts that the alpha and beta rules do not compel me to admit that it rains, or what comes to the same thing, a person altogether ignorant, except that he was well versed in logic so far as it [is] embodied in the alpha and beta parts of existential graphs, would not know that it rained (CP 4.515).

He sketches out rules for this cut; our purpose at this moment is not to explore these rules in detail except insofar as that exploration helps us to investigate the model for the relationship of the possible to the realm of the de inesse as Peirce tries to lay it out. The broken cut is a "possibly not" operator, which Peirce uses in conjunction with the standard cut (simple negation) to define the usual modal operators. Apropos of the aim of this paper, we note that he employs this notation along with some different new signs:

CP 4.517 It must be remembered that possibility and necessity are relative to the state of information.

Figure_05.GIF (991 bytes)
Figure 5

Of a certain graph g let us suppose that I am in such a state of information that it may be true and may be false; that is I can scribe on the sheet of assertion Figs. 5 and 6.

Figure_06.GIF (576 bytes)
Figure 6

Now I learn that it is true. This gives me a right to scribe on the sheet Figs 5, 6, and 7.

Figure_07.GIF (317 bytes)
Figure 7

But now relative to this new state of information Fig 6 ceases to be true; and therefore relatively to the new state of information we can scribe Fig 8.


Figure_08.GIF (952 bytes)
Figure 8

Presumably, Fig 6 was "scribable" at one state of our information, and ceased being so at a later state; Peirce will now suggest within the graphs ways of representing such states (as we noted earlier, he had been talking about such states and about quantification over them since the 1880's); some kind of organized representation is needed, since

CP 4.518. You thus perceive that we should fall into inextricable confusion in dealing with the broken cut if we did not attach to it a sign to distinguish the particular state of information to which it refers. And a similar sign has then to be attached to the simple g, which refers to the state of information at the time of learning that graph to be true. I use for this purpose cross marks below, thus:

Figure_09.GIF (1715 bytes)
Figure 9

These selectives are very peculiar in that they refer to states of information as if they were individual objects. They have, besides, the additional peculiarity of having a definite order of succession, and we have the rule that from Fig. 10 we can infer Fig. 11.10

Figure_10.GIF (525 bytes) Figure_11.GIF (1010 bytes)
Figure 10 Figure 11


These signs are of great use in cleaning up the confused doctrine of modal propositions as well as the subject of logical breadth and depth.

Note that Peirce refers to the new "cross marks" as selectives. This is a little unusual, since selectives are ordinarily letters of the alphabet. It is clear, however, that he wishes these signs to be (implicitly) quantified variables, which is precisely what selectives are. These selectives are in appearance much like lines of identity, which is Peirce’s preferred form of implicitly quantified variable in the graphs. I suggest that he thinks of them as selectives because they show the "order of succession" he refers to, and show it in a way that may not be quite as iconic as he would prefer; in the last two figures, the order of succession is indicated by the single or the double nature of the crossmark selectives. As we shall see, he also comes up with a representation of this situation employing lines of identity. Before we examine that however, we take note of another element that Peirce just mentions, the state of reflection:

There is not much utility in a double broken cut. Yet it may be worth notice that Fig. 7 and

Figure_12.GIF (912 bytes)
Figure 12

can neither of them be inferred from the other. The outer of two broken cuts is not only relative to a state of information but to a state of reflection. The graph asserts that it is possible that the truth of the graph g is necessary (CP 4.519).

Peirce is here exploring a theory of modality, and his exploration is continuous with the views on the conditional/hypothetical which had been part of his approach to logic since about 1880, anyway. He had then written about states of information (or of "things," or of "the speaker"), but had not had a formal mathematical mechanism especially for such states—he had held that they were capable of being represented in his quantified logic, as we have seen. With the Gamma Graphs, he has a wealth of signs to help him deal with these "states"; as he goes on:

CP 4.520. It becomes evident . . . that a modal proposition is a simple assertion, not about the universe of things, but about the universe of facts that one is in a state of information sufficient to know. [Fig 6] without any selective, merely asserts that there is a possible state of information in which the knower is not in a condition to know that the graph g is true. We should naturally express this by Fig. 13.

Figure_13.GIF (1025 bytes)
Figure 13

But this is tosay that there isaconceivable state of information in which the knower would know that g is true. [This is expressed by] Fig. 10.

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Figure 14

Now suppose we wish to assert that there is a conceivable state of information in which the knower would know g to be true and yet would not know another graph h to be true. We shall naturally express this by Fig. 14.

Here we have a new kind of ligature, which will follow all the rules of ligatures. We have here a most important addition to the system of graphs. There will be some peculiar and interesting little rules, owing to the fact that what one knows, one has the means of knowing what one knows—which is sometimes incorrectly stated in the form that whatever one knows, one knows that one knows, which is manifestly false (CP 4.521).

So Peirce applies ligatures—systems of lines of identity—to the states of information. He makes suggestions that go even further, introducing a sign specifically relating such states of information; this is effectively a special "spot" whose 2 "hooks" are marked by the A and the B below:

The truth is that it is necessary to have a graph to signify that one state of information follows after another. If we scribe

Figure_15.GIF (223 bytes)
Figure 15

to express that the state of information B follows after the state of information A, we shall have

Figure_16.GIF (1518 bytes)
Figure 16

This last is a version—employing lines of identity ("ligatures")—of the rule of necessitation done with "selectives" in Fig. 10-11.

So we see that Peirce from about 1880 on, anyway, was interested in what we recognize as "possible worlds" logic. He understood the conditional as appropriately interpretable within such a context; he never really "abandoned" the de inesse conditional, but through this entire period he understood this "material implication" as a second to the larger third of the "hypothetical," which cannot be understood outside this context of possible states of information, states which can be "quantified over"; he alludes to this quantification early on (in 1885, anyway). When he has, in the Existential Graphs, what he considers an adequate notation for dealing with such matters, he discusses and gives examples of such quantifications, as illustrated in the text and illustrations we have been presenting. I note that in this material, Peirce is very close to our notion of possible worlds semantics, and, indeed, may have something to teach us about it!



Fisch, Max.

1986, Peirce, Semeiotic, and Pragmatism, Bloomington: Indiana.

Murphey, Murray G.

1961, The Development of Peirce's Philosophy, Cambridge: Harvard.

Peirce, C. S.

1936-58, Collected Papers of C. S. Peirce, v. 1-6 ed. Charles Hartshorne and Paul Weiss, v. 7-8 ed. Arthur Burks, Cambridge: Harvard.

1976, The New Elements of Mathematics, v. 4, ed. Carolyn Eisele, The Hague: Mouton.

Prior, Arthur.

1955, "Diodorian Modalities," The Philosophical Quarterly 5, 205-13.

1957, Time and Modality, London.

Robin, Richard S.

1967, Annotated Catalogue of the Papers of Charles S. Peirce, Amherst: University of Massachusetts.

Zeman, J. Jay.

1973, Modal Logic: The Lewis Modal Systems, Oxford: Clarendon Press.

1986, "Peirce's Philosophy of Mathematics," Transactions of the Charles S. Peirce Society 22, 1-22.



*. Originally presented at the Peirce Sesquicentential Conference at Harvard University, September 1989; published in Studies in the Logic of Charles Sanders Peirce, Ed. Nathan Houser, Don D. Roberts, & James Van Evra, Bloomington: Indiana University Press, 1997, 402-17.

1. Which term Peirce credits to none other than:

Duns Scotus, who was a Philonian, [and who] as a matter of course, threw considerable light upon the matter by distinguishing between an ordinary consequentia, or conditional proposition, and a consequentia simplex de inesse. A consequentia simplex de inesse relates to no range of possibilities at all, but merely to what happens, or is true, hic et nunc. But the ordinary conditional proposition asserts not merely that here and now either the antecedent is false or the consequent is true, but that in each possible state of things throughout a certain well-understood range of possibility either the antecedent is false or the consequent true. So understood the proposition "If it lightens it will thunder" means that on each occasion which could arise consistently with the regular course of nature, either it would not lighten or thunder would shortly follow (NEM 4:169 [1898]).

2. The original discoverer was Peirce (see Zeman 1986, 9); the other rediscoverer was, of course, Kripke.

3. Fisch at this point refers to CP 4.580 (1906) and CP 4.546 (1906, "Prolegomenon"). Although Peirce in these passages does indeed advert to complications in the determination of the truth values of conditionals, I find it difficult to interpret the passages’ actual text as evidence of what we could call a "surrender of his last [nominalist] stronghold," especially in the light of what I will argue about the early Peirce’s view of the conditional. We might take as a summary of the arguments of these paragraphs the conclusion of CP 4.580:

Some years ago [1903 is suggested by the Editors of CP] when . . . I was led to revise [the] doctrine [that a mere possibility is an absolute nullity], in which I had already found difficulties, I soon discovered, upon a critical analysis, that it was absolutely necessary to insist upon and bring to the front, the truth that a mere possibility may be quite real. That admitted, it can no longer be granted that every conditional proposition whose antecedent does not happen to be realized is true, and the whole reasoning just given breaks down.

4. The Editors of CP at this point suggest looking at CP 3.527. This is an 1896 location which (especially in the light of Peirce’s own 1908 comments on it) must be regarded as transitional to the view of possibility developed by Peirce over the next decade in the Existential Graphs.

5. The propositional forms here are the result of this being Peirce’s presentation of "A Boolian Algebra with one constant," namely, "neither-nor."

6. We are about to examine some of Peirce’s work in the logic of "Possible Worlds." At this point I will note that his 1906 work in and about the Prolegomena to an Apology for Pragmaticism (CP 4.530 ff.) contains much potentially fruitful material in this direction, in the form of his "Tinctured Existential Graphs." This is the topic of a paper in itself; it takes somewhat different directions than those I am emphasizing here, and I will, in this paper, just note the existence of the "Tinctures," and promise an investigation of them at an early opportunity.

7. In the language of logic "consequence" does not mean that which follows, which is called the consequent, but means the fact that a consequent follows from an antecedent—csp.

8. As in contemporary science fiction!

9. This fits with his discussion of CP 4.435: "one should observe that the consequent of a conditional proposition asserts what is true, not throughout the whole universe of possibilities considered, but in a subordinate universe marked off by the antecedent. This is not a fanciful notion, but a truth."

10. As the editors of CP point out, this is effectively a rule of necessitation. It would appear that implicit in this rule is that the first figure appears on the Sheet of Assertion by logical rule rather than by contingency.