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AN ACCOUNT OF PEIRCE’S PROOF OF PRAGMATISM
Ver. 2.0 (08-15-07)

JEREMIAH MCCARTHY
jeremiahmcc@hotmail.com

NOTICE TO THE READER: This paper originally appeared in The Transactions of the Charles S. Peirce Society XXVI(1), 1990. It is being made available here at the ARISBE website for purposes of critical feedback prior to a possible revision.

I

INTRODUCTION

  In this paper I will give an account of the proof of pragmatism that is presented in the “Lectures on Pragmatism” (5.14-5.212).[1] The proof has four major parts or phases. The first three establish the truth of pragmatism ostensibly by securing a conclusion about the nature of belief (see 5.27). The fourth adds refinements to show that pragmatism need not rule out hypotheses involving continuity. My discussion will be confined to the first three parts. But before I discuss the “Lectures” I want to consider the subject of the proof of pragmatism in general.

  The question of the proof has aroused some interest lately, but the commentators tend to give the “Lectures” the cold shoulder. Roberts hardly mentions the “Lectures” and concentrates on material for a series of papers in The Monist begun in 1904.[2] The series was to contain a proof of pragmatism and, since the series was never completed, Roberts concludes that Peirce was never able to assemble his proof.[3] In his discussion of the proof Esposito concentrates on the material for The Monist and ignores the “Lectures.”[4] He too leaves the impression that Peirce never managed to give a satisfactory proof in the manuscripts for The Monist or elsewhere.

  From my point of view Fisch’s opinions on the subject of the proof are more satisfactory.[5] He at least recognizes an attempt at a proof in the “Lectures” as well as in several other places. However, as the title of his paper suggests (“The ‘Proof’ of Pragmatism”), Fisch maintains a skeptical attitude about how successful these attempts were.

  The opinions of the commentators raise two questions for me. (1) What evidence is there that the “Lectures on Pragmatism” contains a proof? (2) Concerning any such proof, what was Peirce’s opinion of it?

  As to the first question, the “Lectures” purports to be an investigation into the truth of pragmatism (5.34, 5.35) which Peirce, in 1903, regarded as having originally been supplied with an inadequate proof (5.28). This together with the identification of the problem as one about the nature of belief and Peirce’s statement of the thesis he needs to prove strongly suggest the presence of a proof. Of course, the “Lectures” may be just a lot of philosophical rambling that adds up to nothing, but that was at least not Peirce’s judgment, as we can see by considering the second question.

  In one of the manuscripts from “The Basis of Pragmaticism” group[6] Peirce tells us that “the line of thought that made the truth of pragmaticism quite evident” to him was that presented in “seven carefully written lecturers delivered in Harvard University”[7] – the “Lectures on Pragmatism.” The argument, though “far from being a simple one . . . yet still left too may difficulties,”[8] but the difficulties cannot have been so great as seriously to impair the argument. Peirce says that he has spent “four solid months in criticism” of the argument preparatory to restating it – not to giving a different argument.[9]

  Fisch brings up some unfavorable evidence on this matter that we need to look at. In a manuscript later than the “Basis” material but part of the same effort for The Monist, “Peirce says he began the series ‘under the clear conviction that no valid argument had ever been put forth for the truth of Pragmatism.’”[10] This looks devastating. Did Peirce change his mind between writing the earlier manuscript and this one? There is little context for the remark that Fisch quotes, but comparison of the fragment containing the statement with others in the same group puts it beyond a doubt that Peirce is talking about other pragmatists here. He can scarcely be speaking of himself when he says that he is “struck by the apparent unconcern of the strenuous preachers of pragmatism at their dogma’s so long floating in the air like a cloud without the support of any solid argument.”[11] Shortly after this criticism Peirce concludes, “So I resolved that, Dr. Carus consenting, I would set forth in the Monist, in as intelligible a form as I could, the argument that had finally put to rest those doubts of the truth of Pragmaticism that through a number of years had troubled my own mind.”[12]

  We have good evidence, then, that (1) the “Lectures on Pragmatism” contains a proof of pragmatism and that (2) Peirce thought the proof was substantially correct. On the basis of the evidence we can also make a conjecture about the later work on the proof of pragmatism. It is that from the beginning to the end of the abortive Monist project Peirce was only trying to give an improved version of the “Lectures” proof. The evidence for the conjecture is that Peirce was substantially satisfied with this proof when he wrote the “Basis” material dated 1904[13] from which I have quoted. Then in material written in the spring of 1908,[14] from which I have quoted, Peirce seems to be satisfied with the same argument. The material nearly spans the whole Monist project.

  A remaining problem about the “Lectures” proof is whether we have all of it. Fisch thinks not[15] and with some reason. The “Lectures on Pragmatism” consists of seven lectures. On the day after the seventh lecture, Peirce delivered an eighth on multitude and continuity which is related to the “Lectures”:

    . . . the substance of all sound argumentation about pragmatism has, as I conceive it, been already given in the previous lectures . . . . I must, however, except from this statement the logical principles which I intend to state in tomorrow evening’s lecture on multitude and continuity. (5.201)

The manuscript of this lecture has been lost.[16] Perhaps we may take heart because Peirce says that the argument that convinced him of the truth of pragmatism was given in seven lectures, not eight.[17] Anyhow, these reservations seem to apply to pragmatism and continuity, which I will have nothing to say about.

  One last word about the “Lectures” manuscripts. As the “Lectures” stands in the Collected Papers the argument goes nowhere because the sixth lecture is truncated. A successful analysis cannot be given on the basis of the published material. An unpublished notebook (MS 316) contains the end of the sixth lecture.

II

FIRST PHASE

  In this part, after some preliminaries, I will treat the first phase of the proof. I will be primarily concerned to reconstruct Peirce’s argument to show that logic is a branch or species of ethics and to give the backing for its premisses. The first phase advances the proof by providing the non-psychological basis for a theory of belief that Peirce thinks he needs but did not have in 1878. The theory that begins to emerge suggests that if we want to know what the nature of belief or assent is, we ought to find out what the purpose of belief or assent is.

  One suggestion. I have quoted as much material as I dare, but the best way to read this paper is with the text of the “Lectures” in front of you.

*   *   *

  Peirce’s original formulation of pragmatism, which he never gave up (see 5.415), took the form of a maxim.

Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then our conception of these effects is the whole of our conception of the object. (5.402; also 5.2, 5.18, 5.422, 5.438)

The doctrine put into the indicative mood states that

the entire intellectual purport of any symbol consists in the total of all the general modes of rational conduct which, conditionally upon all the possible different circumstances and desires, would ensue upon the acceptance of the symbol. (5.438; see also 5.9, 5.18, 5.412)

  In the “Lectures” Peirce asks what the proof is that “the possible practical consequences of a concept constitute the sum total of the concept” (5.27). The original exposition of pragmatism in “How To Make Our Ideas Clear” (5.338-410) justified the doctrine by arguing that “belief consists mainly in being deliberately prepared to adopt the formula believed in as the [sic] guide to action. If this be in truth the nature of belief, then undoubtedly the proposition believed in can itself be nothing but a maxim of conduct. That I believe is quite evident” (5.27). But the paper of 1878 gave this proposition about the nature of belief a psychological basis which Peirce no longer finds acceptable (5.28). The problem for the “Lectures on Pragmatism” is to provide the proposition with an adequate proof. A point to note here is that, as it turns out, Peirce’s proof gives results about the content of beliefs from which we could infer, if we wanted to, what the nature of belief is. The two results are so intimately related that the reversal seems to make no difference to the proof and is easy to overlook.

  The first lecture sets out the problem, shows that it is a problem – i.e. that there is a real question about the truth of pragmatism – and gives a plan for the subsequent inquiry. The next three lectures are taken up by the first part of the plan, the exposition of the categories – phenomenological in the second and third as well as metaphysical in the fourth. The fourth lecture has been severely cut in the Collected Papers. At the beginning of it Peirce announces that he will discuss metaphysics – even though logic can receive no support from metaphysics – in order to familiarize his audience with the categories before he proceeds further.[18] This confirms what one suspected already, viz., that the second, third, and fourth lectures were intended more to familiarize the audience with Peirce’s modes of thought than to contribute anything directly to the proof. In any case that is my opinion and I will have little more to say about them.[19]

  In the fifth lecture we finally arrive at the first part of the proof. This comes at 5.130 immediately following the statement, “Just at this point we begin to get upon the trail of the secret of pragmatism after a long and apparently aimless beating about the bush.” The subsequent argument is intended to show that logic depends on esthetics and ethics. The argument allows Peirce to apply the category of purpose to belief in a non-psychological way and indicates where to look for the purpose that interests him.

  The first impression of the argument at 5.130 is that it is a mess and that impression is quite correct. One of the problems is that as one reads through the passage, one comes upon the word “inference” used equivocally for illations that have been critically accepted and for illations that have not been, but might be. “Argument” seems to be reserved for approved inferences, but then “inference” is used a couple of lines later where one would expect “argument.”

  I will quote as much of the important passage at 5.130-131 as I need to for the reader to understand what I am doing. I will reserve the word “inference” for illations in general, whether critically accepted or not. I will use “argument” and “reasoning” for inferences subjected to the check of logical criticism. This will require me to replace Peirce’s words a couple of times. Numbers in brackets key parts of the quoted material to premisses and conclusions of the reconstructed argument that follows.

It is peculiar to the nature of argument that no argument can exist without being referred to some special class of arguments. [1] [Reasoning] consists in the thought that the inferred conclusion is true because in any analogous case an analogous conclusion would be true. This classification is not a mere qualification of the [inference]. It essentially involves an approval of it – a qualitative approval.[20] Now such self-approval supposes self-control. [2,3] Not that we regard our approval as itself a voluntary act, but that [4] we hold the act of inference, which we approve, to be voluntary . . . . [5] Now, the approval of a voluntary act is a moral approval . . . . That is right action which is in conformity with ends which we are deliberately prepared to adopt . . . . [6] The righteous man is the man who controls his passions, and makes them conform to such ends as he is deliberately prepared to adopt as ultimate . . . . [7] A logical reasoner is a reasoner who exercises great self-control in his intellectual operations; and therefore [10] the logically good is simply a particular species of the morally good . . . . An ultimate end of action deliberately adopted . . . must be a state of things that reasonably recommends itself in itself aside from any ulterior consideration. [11] It must be an admirable ideal, having the only kind of goodness such an ideal can have; namely, esthetic goodness. . . .

    If this line of thought be sound, [12] the morally good will be the esthetically good specially determined by a peculiar superadded element; and [16] the logically good will be the morally good specially determined by a special superadded element. (5.130-131)

  The following is a reconstruction of the argument. The first premiss would be

(1) All inferences are subject to approval.

Using three or four letters in parentheses to express the appropriate predicates, we may symbolize the premiss as

(1*) (x)[(INF)x ⊃ (SUB)x].

Peirce slips in another premiss which shows how he thinks of inferences,

(2) All inferences are acts.

In symbols we have

(2*) (x)[(INF)x ⊃ (ACT)x].

He maintains as well that

(3) If an act is subject to approval, then it is voluntary;

(3*) (x){[(ACT)x · (SUB)x] ⊃ (VOL)x}.

From (1), (2), and (3) we may conclude that

(4) All inferences are voluntary acts;

(4*) (x){(INF)x ⊃ [(ACT)x · (VOL)x]}.

Peirce wants to say that

(5) If a voluntary act is good, then it is morally good;

(5*) (x){[(ACT)x · (VOL)x · (GOOD)x] ⊃ (MOR)x}.

Also

(6) Moral goodness consists in the conformity of a voluntary act to an ultimate end;

(6*) (x){(MOR)x ≡ [(ACT)x · (VOL)x · (CON)x]}.

Next we need to make explicit a somewhat unobvious assumption that Peirce makes about the subjects of logical value:

(7) Anything is logically good if and only if it is a good inference;

(7*) (x){(LOG)x ≡ [(INF)x · (GOOD)x]}.

It follows from (4) that

(8) A good inference is a good voluntary act;

(8*) (x){[(INF)x · (GOOD)x] ⊃ [(ACT)x · (VOL)x · (GOOD)x]}.

From (7) and (8) we have

(9) Whatever is logically good is a good voluntary act;

(9*) (x){(LOG)x ⊃ [(ACT)x · (VOL)x · (GOOD)x]}.

On the basis of (5) and (9) we may conclude that

(10) Whatever is logically good is morally good;

(10*) (x)[(LOG)x ⊃ (MOR)x].

Next we have a premiss about esthetic goodness:

(11) Anything is esthetically good just in case it conforms to an ultimate end;

(11*) (x)[(EST)x ≡ (CON)x].

Peirce’s conclusion about the morally good being the esthetically good may be put as follows, from (6) and (11):

(12) Anything is morally good just in case it is an esthetically good voluntary act;

(12*) (x){(MOR)x ≡ [(ACT)x · (VOL)x · (EST)x]}.

This is Peirce’s conclusion with the “superadded element” made explicit. To get the second equivalence at the end of the quoted material is a little more difficult. It follows from (7) and (10) that

(13) Whatever is logically good is a morally good inference;

(13*) (x){(LOG)x ⊃ [(INF) · (MOR)x]}.

To obtain the converse we need the innocuous premiss that

(14) Whatever is morally good is good;

(14*) (x)[(MOR)x ⊃ (GOOD)x].

This premiss together with (7) will allow us to prove that

(15) Whatever is a morally good inference is logically good;

(15*) (x){[(INF)x · (MOR)x] ⊃ (LOG)x}.

Then from (13) and (15) we obtain the second equivalence at the end of the quoted material with its “superadded element” made explicit:

(16) Anything is logically good just in case it is a morally good inference.

(16*) (x){(LOG)x ≡ [(INF)x · (MOR)x]}.

  The conclusions (12) and (16) are unfortunately not quite to the point in the context of the proof, so we need to deduce some conclusions that are more to the point and which Peirce in fact seems to rely on. The real object the argument is to show that logical goodness, being a form of moral goodness, is conformity to an ultimate end. So from (6) and (13) we have:

(17) Whatever is logically good is an inference that conforms to an ultimate end;

(17*) (x){(LOG)x ⊃ [(INF)x · (CON)x]}.

A conditional proof relying on (4), (6), (7), and (14) will give us

(18) Any inference that conforms to an ultimate end is logically good;

(18*) (x){[(INF)x · (CON)x] ⊃ (LOG)x}.

Then from (16) and (18) we have :

(19) Logical goodness consists in the conformity of an inference to an ultimate end;

(19*) (x){(LOG)x ≡ [(INF)x · (CON)x]}.

This takes belief as something formed by means of or subject to the check of logical criticism out of the realm of psychology and places it firmly in the context of logic conceived unpsychologically as a normative science.

  Now that we have the argument from the first phase reconstructed in a way that is reasonably perspicuous if a trifle crude, it is time to defend the premisses.

  Premiss 1. (line (1), keyed to [1]): The support for this in the material following [1] requires no comment except to say that Peirce evidently does not have unconscious inferences in mind.

  I assume that everybody will agree to premiss 1. Anybody who does not incapacitates himself from criticizing those who hold that 1. is true. He can register his disagreement, but he cannot say that he has good reasons for disagreeing or that others have bad reasons for holding 1. By his own admission he has one preference and we have another. What more need be said?

  Premiss 2. (line (2), keyed to [2,3]): This premiss at least agrees with common usage in representing inferences as acts: inferring is something that people do. Consistently with Peirce’s usage we can regard an inference as the act of accepting one proposition on the basis of others, whether critically or not. But there is a whiff of something objectionable here. People will say that an inference, or at least a valid inference, has a certain objectivity that somebody’s act of assent cannot have. Accordingly, it will be said, inferences cannot really be acts even though we habitually speak of them in that way.

  In what sense, then, are inferences acts? The act of inference is the thought that one proposition is true and, so, another is.[21] Here what is done is to accept an illation. If I do not, then I do not make an inference. So in this sense inference or argument is an activity. If the premiss really does assure the truth of the conclusion in the way I think, then the relation holds objectively. That is, it is independent of what anybody thinks about it or about anything else.[22] But there is another sense in which inference is an activity. There is no deductive inference if I do not associate replicas of propositions or of their subjects and predicates according to rules of association for deductive inference. This association is something done and it can be done right or wrong. If I associate a replica of “~Q” with a replica of “(P ⊃ Q) · ~P” I get it wrong and make a bad inference. It may seem that textbook inferences that nobody makes are still inferences, but what sense can we make of these except by means of a fiction that they are associations of replicas that somebody has actually made? The nature of non-deductive inference as purposive activity is even more evident, as far as Peirce is concerned. These inferences depend for their validity on the human activities in which they are embedded. Whether these inferences are valid depends on how the premisses were obtained and not merely on formal rules for associating replicas, as in deduction. This will be evident enough when we look at induction and abduction. This more radical dependency on conduct does not, however, prevent the relation of assurance from being objective in these cases too.

  Premiss 3. (line (3), keyed to [2,3]): The problem with premiss 3. is to show that it does not drag in metaphysics. If the premiss is metaphysical, the argument will be contrary to Peirce’s reiterated assertions that metaphysics depends on logic and not vice versa (1.625, 2.36, 2.121, 8.158). In view of Peirce’s customary opinions, we ought to say that the premiss is not metaphysical. There are observations at 5.130 that back this up. According to Peirce there are some mental and physical operations that it would be idle to criticize. By “idle” he means “inefficacious.” Some mental and physical operations are unaffected by criticism. That some acts are voluntary and some are not in this sense is a fact which we take with us to the study of metaphysics, not a result of that study. Finally, when Peirce speaks of acts being subject to approval or disapproval he of course does not have idle criticism in mind.

  Premisses 4. and 5. (lines (5) and (6), keyed to [5] and [6]): These premisses are backed by a theory of moral ideals and moral criticism adumbrated in 5.130. A fuller account is given at 5.133-136. Recovering the theory behind premisses 4. and 5. is anything but easy. In order to get some control over the material I will divide it into two parts. The first, at 5.133, expresses Peirce’s partial agreement with and reservations about Kant’s notion of the categorical imperative. The second part, 5.134-136, is mostly an attempt to explicate what is involved in the consistent pursuability of an aim, Peirce’s version of what it is to act in conformity with reason.

  At 5.133 Peirce interprets the categorical imperative as a test of an esthetic ideal, that is, as a test of something proposed as good in itself in the realm of action. “The instant that an esthetic ideal is proposed as an ultimate end of action . . . a categorical imperative pronounces for or against it.” This strikes one as a peculiar reading of Kant for the following reasons. A proposed esthetic ideal is something possibly good in itself. But Kant says again and again that there is nothing good in itself but a good will. This produces the impression that actions done by a good will might co-incidentally be good as means but that otherwise they have no value at all. An action is good in itself or good as a means. A good will is the only thing that is good in itself. An action is not a good will. Therefore, actions are either good as means or have no value. This is not what Kant holds.

Every practical law represents a possible action as good and therefore as necessary for a subject whose actions are determined by reason. . . . If the action would be good solely as a means to something else, the imperative is hypothetical; if the action is represented as good in itself and therefore as necessary, in virtue of its principle, for a will which of itself accords with reason, then the imperative is categorical.[23]

  An act endorsed by the categorical imperative, if done from the motive of duty, is intrinsically good. The principle corresponding to the maxim of the action in such cases (in Kant’s terms, a practical law[24]) must also be good in itself and although the passage quoted implies this, I do not find any explicit statements to that effect in Kant. But it would be a peculiar and untenable position to hold that the acts are good in themselves, but the principles on which they are done are not. Rather, it seems that the acts have value because of the principles they express and the principles have value because they express a good will and all these values are intrinsic. This line of thought brings us to Peirce’s position in 5.133. A categorical imperative pronounces for or against a principle of conduct – an esthetic ideal – proposed as an intrinsic value.

  Peirce thinks that he parts company with Kant over the status of the verdict given by the categorical imperative. We are not compelled to act according to it. Hence, its deliverances can be subjected to effective criticism. This is as much as to say that the categorical imperative is not infallible. So the judgment that the principle in question gives a basis for consistent action might be correct or incorrect. In admitting this Peirce does not deny that an adequate ethical theory would be Kantian in a certain sense.

Any aim whatever which can be consistently pursued becomes, as soon as it is unfalteringly adopted, beyond all possible criticism . . . . An aim which cannot be adopted and consistently pursued is a bad aim. It cannot properly be called an ultimate aim at all. The only moral evil is not to have an ultimate aim. (5.133)

An action that is not (or that would not be) endorsed by a categorical imperative is presumably an action without an ultimate aim and hence an evil act.[25]

  Now we are ready to take a look at the second part of the material. There is a problem here that we need to address at the outset. It is that the terms “ultimate aim,” “ultimate end,” “esthetic ideal [of conduct]” and the like are systematically ambiguous. We know from sources other than the “Lectures” that reason is the ultimate end for Peirce (e.g. 1.615). As we have seen in the discussion of Kant, when reason is what is ultimately good, we will have two different types of ends or ideals that are ends in themselves: on the one hand reason, on the other the principles or aims that express reason or to which reason gives rise. Rationality is “‘conformity to a widely general principle’”(8.152). But it is utterly impossible to take conformity to general principles as your principle without getting yourself some general principles besides that one to conform to. These lower-level principles are expressions of reason, an ultimate end, and so merit the name of “ultimate end” themselves. They are not the means to some intrinsic good. They are expressions or manifestations of it. So when Peirce refers to ultimate ends and the like it is sometimes hard to tell which type he has in mind.

  As I said, I believe that 5.134-136 is mostly devoted to trying to explicate what is involved in the consistent pursuability of an aim, Peirce’s version of what it is to act in conformity with reason. Unfortunately reason or Reason keeps an uncharacteristically low profile in this material. Yet there can be little doubt that reason as an ultimate end is what Peirce is occupied with here. The only problems in this direction arise from the ambiguity just noted.

  An aim that is capable of consistent pursuit Peirce characterizes as one that “would be pursued under all possible circumstances – that is, even though the contingent facts ascertained by the special sciences were entirely different from what they are” (5.134). I take it that Peirce does not mean to include the case in which the contingent facts are so very different from the present ones that there are no moral agents. Notice that in his characterization of an ultimate aim here, Peirce cannot be referring to a lower-level principle.

  Let us examine the two requirements for an ultimate aim that Peirce states and then consider how well reason fits them. The first is that the aim “should accord with a free development of the agent’s own esthetic quality” (5.136). The second is that “it should not ultimately tend to be disturbed by the reactions upon the agent of that outward world which is supposed in the very idea of action” (5.136).

  The esthetic quality of something is approximately what it is like in itself (5.132). So for an end to be consistently pursuable it will need to remain in accord with what the agent as a whole is like even when he undergoes development. The other requirement appears to be that the end should not tend to be frustrated or eliminated by the experience of acting on it. For an end to be ultimate, both conditions must be fulfilled at the same time (5.136).

  The requirements for a particular esthethic ideal to be capable of consistent pursuit seem to be about the same. First, it should appeal to the agent.[26] Second, following the ideal should not lead to such conflicts and troubles that the agent will be led to change it. So by the acceptance of particular ideals and their trial and modification in the face of experience, and the trial of the modification and the modification of the modification, and the trial of that, and so on, men and women find out what they ought to be and life attains such rationality as it has.

It is by the indefinite replication of self-control upon self-control that the vir is begotten, and by action, through thought, [a man] grows an esthetic ideal, not for the behoof of his own poor noddle merely, but as the share which God permits him to have in the work of creation. (5.402n.3)

  Recall that Peirce says that the two requirements need to be satisfied together. In that case “the esthetic quality toward which the agent’s free development tends and the ultimate action of experience on him are parts of one esthetic total” (5.136). When the agent’s esthetic preferences in the realm of conduct and his experience are working together, they will produce satisfactory ideals of conduct, and the sum total of these is a kind of self-ideal for the agent that I would identify with the esthetic ideal referred to in the last quotation but one. This is the esthetic quality toward which the agent’s free development tends – remember, this is not an ethicist’s hot air but an effective ideal, one that the agent tries to keep embodied, and so it is what he is like – or at least approximates to being like – in himself. The effects of esthetic preference on the one hand and experience on the other go to produce a single thing, the agent’s ideal. When the conditions are not satisfied together they produce two things: an ideal, which attracts, and experience of it, which does not attract. This is not one thing, an esthetic total, because it cannot as a whole be an object of desire or will. This is the kind of situation which we endeavor to rid ourselves of in the criticism of ideals. When reason breaks down, if it does, such an impasse is reached and there the matter stands. There will be an irremovable dose of irrationality in the agent’s conduct.

  We can see now how reason fulfills the requirements for an ultimate end. It is capable of pursuit through any changes in the esthetic quality of the agent and has no tendency to be disturbed by the action of experience on him, even in the possible cases in which experience frustrates the pursuit of reason. This ethics, perhaps, deserves the name “Kantian” because it is reason as consistent willing that is at the foundation of it. But it is somewhat un-Kantian in appearance, which Peirce seems to have been happy enough about. He says that the definition of an ultimate end must not be “reduced to a mere formalism” (5.134) which sounds like a criticism of Kant. Peirce’s conception of reason is not a mere formalism because it essentially involves the agent’s peculiarities. What attracts or repels him is what makes an end consistently pursuable, not some formal property of the end. This gives Peirce, unlike Kant, a nice naturalistic explanation of how reason becomes effective in people’s lives. Second, the fact that there is no guarantee against having the end break down on occasion shows that we are dealing with a contingent fact about the way the world is, and not with metaphysical smoke.

  Such is Peirce’s ethics, as far as I can make it out from the “Lectures” and collateral information.[27]

  One more related puzzle and I will leave this. Reason, as the summum bonum, is not just an aim of conduct but a cosmic end. All things, not just human conduct, contribute to it. To apply this just to conduct, if all things contribute to reason, how do we have any basis for the distinction between good and bad? Is Peirce committed to the idea that nothing is morally bad? I think not. Even what we categorize as a divagation from reason must somehow contribute to it. We have to admit that. And so it is. What diverges from reason thereby provides an opportunity for reason to go to work to make things more rational. We have seen this in the pattern of moral criticism. And if an act does not diverge from reason? Then it is an expression of reason. And that too contributes to its embodiment in the world, but in a different way. Most ends are not like this. If my aim is to preserve human life and I act on it, I may succeed in expressing it by preserving a human life. (Or by trying to. That too is an expression of it. But I must not get caught in the details here.) If I violate the end by taking a human life, I do not thereby provide an opportunity for preserving life.

  Now at last we can get back to premisses 4. and 5. which for lo these many pages I have been aiming to justify. According to premiss 5. the moral goodness of an act consists in its conformity to an ultimate end. In Peirce’s case this is conformity to reason. Conformity to reason entails that the agent will not have occasion to find fault with the act through experience indefinitely extended. I hear a voice with a British accent ask, “But is it good?” Well, if that isn’t moral goodness then I, like Peirce, don’t know what would be. This may be called the “Closed Question Argument.” Once this is settled we can see that premiss 4. is secure also. The judgment that an act is unqualifiedly good commits one to the judgment that it is morally good.

  Premiss 6. (line (7), keyed to [7]): I take it that premiss 6. is a reasonable construction of Pierce’s remarks about a logical reasoner. The subsequent discussion makes it clear that 6. is what he has in mind. The premiss is a trouble-spot for Peirce because there appear to be at least two kinds of logical value, truth and validity, and truth seems to be the more fundamental since we use it to define validity. Premiss 6. says that there is only one kind of logical value and it is validity. Defending 6. is essential to the proof. It makes logical value a value that qualifies something that people do. If this turns out to be false or only true in part – say if truth and validity are both fundamental – then Peirce will be unable to show that logical value is a kind of moral value and the proof collapses.

  Peirce takes up the problems raised by premiss 6. (5.142) and wastes no time in adding another by apparently blurring the distinction between soundness and validity in his discussion. I will just remark that we will see, when his idea of validity is explicated, that there is really no difference, ultimately, between sound and valid inferences. Unsoundness reduces to invalidity.

  The treatment of the problem begins with a discussion of perceptual judgments preparatory to an attempt to define truth in terms of validity.

  Perceptual judgments are unrepeatable and, consequently, neither true nor false but merely veracious (5.142). Veracity is a kind of moral rather than logical goodness of representations (5.137, 5.141). It is “the conformity of an assertion to the speaker’s or writer’s belief” (5.570). A judgment is essentially an assertion to oneself, a self-notification of belief (5.29), so the concept of veracity applies to judgments.

  The backing for the alleged unrepeatability of perceptual judgments is not entirely clear, but it must have to do with the nature of percepts as momentary presentations which, once gone, cannot be recovered and reinspected (5.115). Hence, a judgment I make about a percept’s nature is unrepeatable.[28]

  There are only two ways in which a perceptual judgment might be verified or refuted. Either we might repeat the observation and compare the results – which we have already ruled out – or else try to verify or refute the perceptual judgment through its logical relations to non-perceptual judgments. But “at most we can say of a perceptual judgment that its relation to other perceptual judgments is such as to permit a simple theory of the facts” (5.142). The only evidence I have about the character of the precept is the perceptual judgment. If I throw it out, then whatever gets substituted for it as correctly representing the percept will be dictated by my theory of the facts. In other words, the facts will be altered to fit the theory (see 8.148). To use Peirce’s example, if I judge that I see a clean white surface and then look again more carefully and judge that I see a soiled surface, I have no right to say that the first percept was really of a soiled surface (5.142). The proposition that the surface – which is not a momentary presentation – is really soiled is a theory of the facts. If I decide on the basis of the theory to deny the first perceptual judgment, then I alter the facts to make them fit the theory.

  In view of all this, the argument that Peirce had in mind for the mere veracity of perceptual judgments must have been something like the following. There is no possible sound argument that permits the attribution of a truth value to a perceptual judgment. The premisses of an argument represent a condition under which the conclusion is true (2.93). But no condition can be given under which a perceptual judgment either corresponds or fails to correspond to its object. Therefore, no attribution of a truth value is permissible. This does not prevent the judgment from representing a belief about a percept, so it will have some degree of veracity.

  Once the premiss about perceptual judgments has been presented, Peirce gives a pair of arguments designed to show that inferences are the primary subjects of logical value and that the logical goodness of propositions can be defined in terms of the logical goodness of inferences. I will concentrate on the first and easier to understand of the arguments.

[Every perceptual judgment is merely veracious.] Now consider any other judgment I may make. That is a conclusion of inferences ultimately based on perceptual judgments, and since these are indisputable, all the truth which my judgment can have must consist in the logical correctness of those inferences. (5.142)

  The word “indisputable” must mean that perceptual judgments are indisputable as to truth value because they have none, not because they are indubitably true. The premiss that every judgment not a perceptual judgment is the conclusion of an inference is not argued for here but reflects Peirce’s customary idea about mental activity. He argues for this premiss in the seventh lecture at 5.192. Since this idea is already familiar I am going to assume it and not try to investigate or justify it. So the argument for premiss 6. is that if a non-perceptual judgment can be obtained only by inferences from perceptual judgments not themselves true or false, then the only factor that determines the truth or falsity of a non-perceptual judgment is the validity of the inferences by which it was obtained. Therefore, the truth of a proposition consists in the fact that it can be the conclusion of a valid inference from veracious perceptual judgments.

The only difference between material truth and the logical correctness of argument is that the latter refers to a single line of argument and the former to all the arguments which could have a given proposition or its denial for their conclusion. (5.142)

In other words, a proposition is false just in case every inference having it as conclusion is invalid. A proposition is true just in case there is some valid inference having it as conclusion.

  Two implications of this argument that are worth looking at are that the logical operators of propositional and predicate calculus can be defined in terms of valid inference and that valid inference itself can be explicated without using the concept of truth.

  To take the first of the implications, if we examine the arguments at 5.142 we see that Peirce is trying to work in terms of assertibility conditions for propositions in order to define truth in terms of validity, so he would very likely want to give definitions for logical operators in terms of assertibility conditions.

  In intuitionist logic the logical operators are defined in terms of constructive provability and the definitions for them are given in terms of assertion conditions.[29] The concept of constructive provability is obviously inappropriate for Peirce’s needs so we should substitute valid inferability by any means. We can say that p is assertible if and only if “It has been found that p” is assertible. This is virtually what Peirce does at 5.142. Unfortunately the broadened intuitionist definition of disjunction runs the cart into the ditch. We have to say that “p or q” is assertible if and only if “It has been found that p” is assertible or “It has been found that q” is assertible. This would not be consistent with Peirce’s ideas about mathematical proof and it also gives too strong a reading for contingent propositions. Can I really assert “A Democrat or a Republican will win the next presidential election” if and only if I can assert “A Democrat will win the next presidential election” or “A Republican will win the next presidential election”? Cases like this abound so we need to try something different.

  There is another way to give assertibility conditions for logical operators which has been discussed by Michael Dummett[30] and Hilary Putnam.[31] The versions of the redefinition discussed by Dummett and Putnam are not the same and I think that Peirce’s idea is closer to what Dummett suggests in respect of Peirce’s generalized notion of proof. But Putnam’s technique of definition is essential for us because it preserves the theorems of classical logic and saves us from the unacceptably strong reading of disjunction. As before, to say that p has been validly derived from veracious perceptual judgments is to say, for short, that it has been found that p. Here is how the redefinition goes.

(1) To assert that p is to assert that it has been found that p.

(2) To assert that ~p is to assert that there is no valid inference to p and that it has been found that there is no valid inference to p. That will be assertible if and only if it has been found that ~p (see 5.142, second argument).

(3) To assert that p · q is to assert that p and to assert that q.

(4) To assert that p v q is to assert that it has been found that

~(~p · ~q).

(5) To assert that p ⊃ q is to assert that it has been found that

~(p · ~q).[32]

(6) To assert that (∃x)Φx is to assert that it has been found that, for a suitably chosen α, Φα (see 5.154).

(7) To assert that (x)Φx is to assert that it has been found that for any α you care to choose, Φα (see 5.154).

  This seems to give a sufficiently verificationist conception of the logical operators to be congenial to pragmatism and settle the question of whether a definition in terms of valid inference is available.

  Now let us consider the truth definition that Peirce gives near the end of 5.142. This, essentially, gives a definition of “p is true” in terms of the assertibility condition “p has been validly derived from veracious perceptual judgments.” I want to judge this according to the material criterion of adequacy for an account of truth laid down by Tarski,[33] viz., that every equivalence of the form

(T)     X is true if, and only if, p

should be assertible, where “X” is the name of the sentence to which “p” refers. Thus,

“Snow is white” is true if, and only if, snow is white

is an instance of such an equivalence.

  For Peirce T-equivalences will be interpreted as stating assertibility conditions. Peirce is committed, for every proposition, to an equivalence of the following sort:

(I)     X is true if, and only if, it has been found that p.

If Peirce’s account is adequate according to his own lights, the T-type and I-type equivalences ought themselves to be equivalent. We can see that they are. “p” is assertible if and only if “It has been found that p” is assertible, so every T-equivalence can be transformed into an I-equivalence and conversely.

  The material condition of adequacy will be satisfied if we can show that every T-type equivalence is assertible. This will be the case just in case their constituents are assertible so, to make it more explicit what T-type equivalences involve, we can explicate them as

(T₁)     If X is true, then p and if p then X is true.

And this may be further explicated as

(T₂)     ~[(X is true) · ~p] · ~[p · ~(X is true)].

We can assert T-type equivalences just in case we can assert corresponding T₂-type equivalences. Can we always find out that it is not the case that both X is true and ~p ? Since “X” is the name of the sentence referred to by “p,” from “X is true” we can infer “p.” Consequently, the assumption that there is a valid inference to [(X is true) · ~p] permits us to deduce a contradiction, which permits us to infer that there is no such valid inference. The same goes, mutatis mutandis, for the other conjunct of T₂, so there is a method of showing, for each T-type equivalence, that it is assertible. Therefore, all such equivalences are assertible, so Peirce’s account is materially adequate.

  Since in each case to assert that a proposition p is true is to assert that it has been found that p, we might generalize in an un-Tarskian way and say that truth is valid inferability. If this is not satisfactory we might consider the following lemma. Having found that p is short for having found a valid inference to p. So if “It has been found that p” is assertible, then “p is validly inferable” is assertible. The converse obviously holds, so for each occurrence of “It has been found that p” we may substitute the expression “p is validly inferable.” This allows us to say that to assert of any sentence X that it is true, is the same as to assert that p is validly inferable, where “X” is the name of the sentence referred to by “p.”

  From Tarski’s point of view the trouble with this as an account of truth – whatever he might think of the substance of it – is that it is not given in relation to an exactly specified language. That is essential to ensure that the use of the concept of truth does not produce paradoxes.[34] I can do nothing to remedy this, but I should point out that Peirce’s broad notion of proof is not an obstacle to correcting the defect, at least in Tarski’s opinion. He remarks that the only exactly specified languages that have been developed are formal deductive systems, but that a language could have an exactly specified structure without being such a system. He seems to be referring to languages in which non-deductive inferences could be used.[35]

  The second implication of premiss 6. that I mentioned was that validity can be defined without recourse to the concept of truth. Peirce does mention truth when he describes the thought involved in the critical acceptance of an inference: “the thought that the inferred conclusion is true because in any analogous case an analogous conclusion would be true” (5.130). This may do as a description of the judgment of validity, but it can be of no use as a general description of validity. But following out the matter really takes us into the second phase of the argument, which is largely devoted to this issue, and I am not quite prepared to begin that yet. I will, however, point out that the validity of an argument, to put it vaguely, lies in performing what it promises (see 2.779-781). If Peirce can describe the promise of argument and show that the different forms of inference perform what they promise without recourse to truth, the problem will be solved.

  Premiss 7. (line (11), keyed to [11]): Esthetic goodness is the goodness of what is good in itself. “Any kind of goodness consists in the adaptation of its subject to its end” (5.158). Put these two propositions together and you get premiss 7. Whether this premiss is quite right or not is not very important as long as it is true that moral goodness is conformity to an ultimate end.

  Premiss 8. (line (14), my addition): To say that what is morally good is good seems to be a mere tautology. It instantiates the principle that any individual having a first-level property that has a second-level property, has the first-level property.

*   *   *

  The upshot of the first phase of the proof is that logical goodness is goodness of inference and goodness of inference, which is validity, is to be understood as conformity to an ultimate end. This advances the proof because Peirce is interested in the nature of belief or assent insofar as it is voluntary. Voluntary belief is belief that is subject to the check of logical criticism through the criticism of inference. Since inference is a purposive activity which controls belief, belief in general must have a purpose. A reasonable conjecture, which Peirce makes, is that the nature of belief can be uncovered by finding out its purpose[36] and that its purpose can be understood by investigating the aim of reasoning in general.

III

SECOND PHASE

  In the second phase of the proof, Peirce is concerned with determining what the purpose of reasoning is. To this end Peirce launches into a long discussion about the nature of deduction. This aims to give insight into what the purpose of inference is, that is, what makes inference valid. Then brief discussions of deduction, induction, and abduction are presented (1) to show what the property of each inference is, under Peirce’s analysis, that makes it valid, and (2) to show that the inference, as analyzed, has the validity-making property. The discussion of abduction is a possible exception to (2). The briefer discussion of deduction brings in a connection between deduction, meaning, and purpose which is relevant to the third phase of the proof. As I reconstruct the argument, all of this is primarily in aid of defending premiss 6. and setting the stage for results about abduction and meaning in the third phase.

  The part of the discussion of deduction that is most important is at 5.147-157, divided between the end of the fifth and the beginning of the sixth lecture. I will try to describe what is going on here in a way that will make evident the bearing of the material on both the method of deduction and the purpose of reasoning.

  Part of the material, 5.151-155, is concerned with generality and quantification. Peirce argues that perceptual judgments contain general elements in their predicates, but that they are never general in the sense of being universal quantifications. The subject or subjects of all perceptual judgments are singulars, viz., percepts or parts of percepts. He asserts that nevertheless a universal proposition will be deducible from every perceptual judgment in virtue of the generality of its predicate and he gives an example to show how. I will change only the format so that it will be less confusing. Suppose we symbolize the relation “c is apparently subsequent to b” as

(SEQ)cb.

This may be taken to represent a perceptual judgment. The definition of apparent subsequence is

(x)(y)(z)((SEQ)zy ≡ {[(SEQ)yx ⊃ (SEQ)zx] · ~(SEQ)xx}).

In words, the relation of apparent subsequence is irreflexive and has a property allied to transitivity which I have never seen outside Peirce’s writings and whose name, if it has one, I do not know. It entails transitivity but transitivity does not entail it, so we might call it “strong transitivity.” From the singular proposition expressing the perceptual judgment and the definition we can deduce a universal proposition:

(x)[(SEQ)bx ⊃ (SEQ)cx]

whatever b is apparently subsequent to, c is apparently subsequent to.

  Not very impressive, is it? Of course we can deduce a universally quantified proposition from a perceptual judgment if we can always drag in some appropriate universally quantified premiss. Peirce would defend using the non-perceptual premiss because it is only a definition of the predicate of the perceptual judgment. If his procedure here is trivial or illegitimate, that should show itself in the discussion that follows.

  The example was in aid of indicating a solution to the problem about geometrical proof mentioned at the end of the preceding lecture. When a student does a proof in geometry, he observes a diagram, a single thing. Peirce claims that at a certain point in the proof he uses as an example, the student must be able to perceive that something will hold, not just in the case of the diagram he is looking at, but in any such case. Otherwise the proof will not have the generality it needs. Let us turn to this example to see what we can learn from it and its implications about (a) the perception of generality, (b) the way deductive proof works, and (c) the aim or promise of reasoning in general.

  The proof that Peirce takes as an example uses a diagram like the following.

A line BD abuts another AC at an ordinary point B, to form two angles, ABD and DBC. It is taken for granted that these are not right angles. The proposition to be proved is that the sum of ABD and DBC is equal to two right angles. To do the proof we need to erect a perpendicular EB to AC at B. We know that ABE is a right angle. The sum of EBD and DBC is a right angle. Therefore, the sum of all three angles is equal to two right angles. The angle ABD is equal to the sum of ABE and EBD, so the sum of the two angles ABD and DBC is equal to two right angles. Q.E.D.[37]

  The problem here is that the student sees only in the particular case that the perpendicular lies in one of the adjacent angles. For the proof to have the requisite generality the student must be able to assert that in any case the perpendicular will be in one of the angles. It seems that the student is able to perceive something general. Moreover, the general in question seems to be a universally quantified proposition. But how can the conditions for the assertibility of a universal proposition hold when the student has nothing to go on but a perceptual judgment or two about a single object?

  In aid of clarifying the issue, let me say what the issue is not. Peirce is not merely worried about the conditions for the valid application of the rule of universal generalization. That is, if we point out that a generalization is permissible when it does not depend on any special assumptions about the individuals reasoned about and that some rule of universal generalization which we propose to use ensures this, Peirce will be unimpressed. He would, perhaps, point to some step in our proof and remark that we have observed that the inferential step holds in this case. But for the proof to be general we must be able to say that in every such case an analogous inference will hold. How can we say that on the basis of observing one particular group of symbols? That is the question. If we say that any inference of the observed form is valid, that only asserts the proposition the assertibility of which is at issue. This is the type of problem that concerns Peirce here.

  When we proceed from premiss to conclusion of an inference we are supposed to see that in any analogous case an analogous conclusion would be true (5.130). That is the justification for drawing the inference or, at least, it is a principle that we need to be able to assert if we are to make the inference. Similarly, in the example from geometry, we are supposed to see that in any case the perpendicular constructed must lie in one or the other of the adjacent angles. Neither of these principles can be perceptual judgments. But perceptual judgments must give us the right to assert them. That is, each principle will be assertible if and only if a certain perceptual judgment can be made. This is where the analogy comes in with the case of apparent subsequence: a perceptual judgment confers the right to assert a universally quantified proposition.

  In the geometrical example the perceptual judgment is supposed to be that the constructed perpendicular must lie in one or the other of the pair of adjacent angles. To assert that the perpendicular, EB, must lie in angle ABD or in angle DBC is to assert that, for any x, any y, and any z, if x is a perpendicular analogous to EB and y and z are angles analogous to ABD and DBC, then x will lie in y or in z and not in both y and z.

  In both the apparent-subsequence example and the geometrical example a perceptual judgment having singular subjects gets explicated by a universally quantified proposition. Perceived generality is the generality of the predicate of a perceptual judgment. An explication of the predicate plus the original perceptual judgment permits the inference and, hence, the assertion of a universally quantified proposition.

  This takes care of topic (a) from p.23, the perception of generality. The discussion was in aid of investigating topic (b), how deductive proof works, since the perception of generality is supposed to be involved in all deductive inference. The geometrical example seems to clear up the question. For if we could not say, on the basis of our perception of a diagram or the like, that in every analogous case, if the diagrammed premiss were true, the diagrammed conclusion would be true, then we could not make the deductive inference. Absent the assertibility of the general principle, the inference would be open to criticism as not doing what deductive inference purports to do, not giving a conclusion which cannot be false if the premisses are true.

  As for topic (c), the aim or promise of reasoning in general, when Peirce brings up the subject at 5.146 he talks about the “rationale” of deductive reasoning. None of the three types of reasoning that Peirce recognizes is reducible to any of the others, “yet the only rationale of these methods is essentially Deductive or Necessary. If then we can state wherein the validity of Deductive reasoning lies, we shall have defined the foundation of logical goodness of whatever kind.” Peirce’s usual idea about logical goodness, a.k.a. validity, is that an inference is valid when it performs what it promises (2.779-781). Every inference professes to get at the truth in some way and, if it lives up to its profession, then it is valid. This is just for the inference to conform to its professed purpose. The rationale of something is its controlling principle and a purpose can be a controlling principle. So it is a fair conjecture that the slippery word “rationale” should be interpreted as “purpose.” Hence, whatever the purpose of deduction turns out to be, that – perhaps generalized – will be the purpose of inference. The material at 5.158-159 further confirms this.

  It is anything but clear what the insight into the aim of reasoning might be that we are supposed take away from the discussion of deduction.[38] Peirce seems to be plugging away at the proposition that we can perceive that certain facts are without exception associated with certain others. Deduction seems not to aim at informing us of anything more than such connections between facts. The contribution made by the material on the perception of generality is to show that the aim of reasoning involves the cognition of general laws. This emphasis is evident in Peirce’s characterization of the “use of thinking,” “the end of argumentation . . ., what it ultimately leads to” (5.159).

    But the saving truth is that there is a Thirdness in experience, an element of Reasonableness to which we can train our own reason to conform more and more. If this were not the case, there could be no such thing as logical goodness or badness; and therefore we need not wait until it is proved that there is a reason operative in experience to which our own can approximate. We should at once hope that it is so, since in that hope lies the only possibility of any knowledge. (5.160)

  Now I will try to say what I think this difficult passage means.

  The reason in things is the element of generality in things. To train our reason to conform to the reason in things is to train it to be governed by the element of generality in things. This is opposed to being governed by those merely subjective generals that are peculiar to the thinking of this or that person or group. For example, if I think that all revolutionaries are bloodthirsty and I also think that George Washington was a revolutionary, I know I can conclude by means of a valid deduction that George Washington was bloodthirsty. The conclusion is false and it is false because it is inferred from a false premiss, “All revolutionaries are bloodthirsty.” The inscription in quotes is the icon of a replica of a symbol. The symbol that the icon-in-quotes device refers to is a law of association. It leads me to associate replicas of the symbol “is bloodthirsty” with whatever replicas of the symbol “is a revolutionary” are associated with. So, for any replicated name with which a replica of the second symbol is associated, I will be able to associate, with a replica of that name, a replica of the first symbol, by means of a valid deduction. Thus, the proposition “All revolutionaries are bloodthirsty” may be understood as a law for associating replicas of symbols.

  The law is a normative one. It operates through a representation of itself. Without a replica of the symbol to represent the law, the association via valid inferences will not occur. And because the law works like this, the associations can go right or wrong. They can conform to the law or fail to.

  Now the proposition “All revolutionaries are bloodthirsty” – supposing I or somebody else really believed it – would be a subjective general. For it would merely be a law that governed the thinking of certain persons. If these persons stop making the association governed by the symbol, that will be the end of the law.

  What would it take for the law to be objective, to be part of the reason in things? Evidently what would be needed is an absolute association between being a revolutionary and being bloodthirsty, so that every revolutionary there is or ever would be, would turn out to be bloodthirsty. This law would be, no matter what anybody thought about it, or about revolutionaries. In other words, it would be an objective general, according to Peirce’s definition of “objective.”[39]

  Supposing that there were such an objective law and that it were known, would that very law be the law that produced or governed the associations of replicas in our thinking? That is the question about 5.160. If the answer is yes, then we have an example of the growth of reason. The objective law will bring associations of replicas into conformity with itself. And if it does so, the very law itself must be a symbol.

  A negative answer to the question might be supported in the following way. The objective law ranges over revolutionaries. The law of mental association ranges over replicas. If two laws are the same their domains must be the same. In this case the domains are different. Therefore the laws are different. This answer won’t do because the principle it uses simply begs the question about the growth of reason. It assumes that a law cannot come to govern something which it had not governed previously.

  Our question comes to this. Is there a possible case in which the association of replicated symbols happens, but in which the law which supposedly governs this activity and the association of properties represented by the replicated symbols, fails to operate? We have assumed that the proposition in question, “All revolutionaries are bloodthirsty,” is true. Consequently, every association of replicas that is logically possible will be paired with associated properties represented by the replicas. There is no possible case in which the law of mental association is operative, but not the law about revolutionaries. To show the logical impossibility that one law should be operative in the absence of another is to show that the phenomena in question are under the control of the same law. Therefore, a true proposition is an objective symbol or objective law. And in cognition, the mind of the knowing subject comes to be governed by the laws or symbols which are the objects of knowledge.[40]

  Now we know what the discussion of deduction and the perception of generality was aiming at and what the promise (or purpose) of reasoning is.

  For Peirce to make good on his idea that validity is the fundamental logical value, he must be able to say what the validity-making property of each type of inference is without referring to truth. I take it that the discussions of deduction, induction, and abduction at 5.161, 5.170, and 5.171, respectively, are attempts to do that. In this material there is a single reference to truth, which can be eliminated.

  First, deduction.

In deduction, or necessary reasoning, we set out from a hypothetical state of things which we define in certain abstracted respects . . . . We consider this hypothetical state of things and are led to conclude that . . . wherever and whenever the hypothesis may be realized, something else not explicitly supposed in that hypothesis will be true invariably. Our inference is valid if and only if there really is such a relation between the state of things supposed in the premisses and the state of things stated in the conclusion. (5.161)

  The reference to truth may be removed by restating the second sentence as “Wherever and whenever the hypothesis may be realized, something else not explicitly supposed in that hypothesis will be realized invariably.” “Realized,” however, is uncomfortably close to “true.” We might avoid this by saying that a deduction is valid if and only if the facts represented in its premisses are invariably associated with the facts represented in its conclusion. Facts are usually thought of as ontological counterparts of propositions. But we can see from the account of the purpose of reasoning that facts and propositions are really the same thing. Facts are represented in premisses and conclusions via replicas and the reasoning is itself a replica whose parts are associated by a law of invariable association of the facts or propositions replicated, if the reasoning is deductively valid.

  As Peirce has already argued at length, the way we assure ourselves that the connection between facts is invariable is by examining a diagram of the facts and seeing that the connection is invariable. He comes back to this at 5.164 to show that, as he analyzes deductive inference, the method used in drawing deductive conclusions will give valid inferences. This hardly amounts to more than reiterating what he has said already.

  At 5.165-166 Peirce takes the opportunity of discussing deduction to connect meaning and purpose with deductive inference, which is relevant to the third phase of the argument. In the same place Peirce also seems to want to defend the propriety of bringing in the definition in the apparent subsequence example. The defense appears to be that in bringing in the definition we are doing no more than explicating a purpose. When we resort to the definition we assert, in effect, that we will employ the predicate “(SEQ)xy” to apply to just those pairs of objects to which the predicate “{[(SEQ)yz ⊃ (SEQ)xz] · ~(SEQ)zz}” applies. I suppose that this is legitimate on the principle (which Peirce does not state here) that we must be competent to say what our own purposes encompass. The sort of purpose at issue is a final cause that operates through a representation. Unknowable parts of a purpose are parts that we cannot represent and, hence, are not parts of the purpose.

  To apply this to the apparent subsequence case, if we intend to use the concept “(SEQ)xy” to apply to objects which do not have the relation to themselves – i.e., if that is part of our purpose – then we must be able to represent the irreflexivity of apparent subsequence to ourselves in order to check whether the use of the concept in cases where the relation would be reflexive conforms to our intentions about how the concept should apply. If we cannot represent the irreflexivity of the concept for some reason, then we cannot exercise a check on its application in cases where irreflexivity would be violated. We cannot say or think “No, the concept doesn’t apply here because it’s irreflexive and this relation isn’t” because we cannot, for some reason, replicate the appropriate proposition. So it could not be part of our purpose to use the concept “(SEQ)xy” to apply only to cases in which the relation is irreflexive. An essential representation for making the purpose operative would be unavailable.

  No special powers or principles are brought in at this point that are not already available from ethics. In the ethical context obligations to make conduct conform to purposes extend as far as conduct that can be controlled. Conduct that can be controlled can be represented as conforming or failing to conform to a purpose. If a purpose is unknowable, it cannot be represented. If it cannot be represented, then criticism and control with respect to it are impossible. So there can be no obligation to conform to an unknowable purpose.

  Turning to induction (5.170), we find that its validity-making property is that it will allow us to discover how well a hypothesis will be sustained by experimental tests in the future.

The reason that [it] must do so is that is that our theory, if it be admissible even as a theory, simply consists in supposing that [a series of] experiments will in the long run have results of a certain character. But I must not be understood as meaning that experience can be exhausted, or that any approach to exhaustion can be made. What I mean is that if there be a series of objects, say crosses and circles, this series having a beginning but no end, then whatever may be the arrangement or want of arrangement of these crosses and circles in the entire endless series must be discoverable to an indefinite degree of approximation by examining a sufficient finite number of successive ones beginning at the beginning of the series. (5.170)

  The principle behind the method of induction is that every term in the relevant series of experiments or observations occupies a finite ordinal place from the beginning of the series. Consequently, any regularity characteristic of the whole series must be exhibited by the time some finite ordinal place in the series is reached. If not, then, since each element of the series occupies a finite ordinal place, the character is not exhibited in the series at all, contrary to the hypothesis.

  This is Peirce’s idea, essentially, but it is not quite what he says at 5.170. The natural way to think of the series of experiments or observations is like this. Suppose that the series is a series of throws with a die and suppose that 1/6 of the throws in the series are sixes. Then we would suppose that, at some ordinal place in the series of observations of throws, the ratio of sixes to total throws will be close to 1/6 and will remain at least that close for the remainder of the series. So the series seems to acquire a trait at a certain ordinal place, which it always retains. But Peirce says, “Every such term has a finite ordinal place from the beginning and therefore, if [the series] presents any regularity for all finite successions from the beginning, it presents the same regularity throughout” (5.170). Now how is it that the series I used as an example could exhibit the same regularity for all finite successions from the beginning?

  Here is the solution, which is perfectly consistent with what I have said. Any admissible hypothesis must be capable of being paraphrased to say that a series of inductive tests has a certain character in the long run. Hence, the hypothesis is equivalent to the proposition that, for any stated degree of approximation to the character sought, there is a finite ordinal place, α, sufficiently large such that all initial segments of the series terminating at it and at higher finite ordinal places exhibit the character to the desired degree of approximation, whereas this is false for every initial segment terminating at a finite ordinal place smaller than α. So every testable hypothesis will state a law of succession of terms in a series which holds for every finite ordinal place, beginning at the beginning of the series.

  Induction contributes to the purpose of reasoning by leading us to make associations governed by laws that will not lead us into expectations about the nature of experience that will be disappointed. That is, it produces conformity of our own reason to the reason in things.

  This takes care of induction without any fatal references to truth or some uncomfortably similar concept.

  Unfortunately, Peirce’s treatment of abduction is hard to understand. Most of the pieces make sense, but it is not clear how they are supposed to fit together.

  The validity-making property of abduction seems to be that it yields a testable hypothesis which gives us a chance of extending our knowledge (5.171).

  Peirce does not proceed to show that abduction has this property, but gives a justification by desperation. If abduction does not work, then learning or understanding anything is hopeless. This is a kind of justification, but it is not like the preceding two cases. In those Peirce gave reasons for holding that the methods of drawing inferences will do what he says they need to do in order to be valid. The present sort of justification merely defends the validity of abduction on the grounds that, if the inference is not valid, then our cognitive purposes are in vain. This is indeed a reason for holding that the inference has the validity-making property, but a reason of an odd sort. It is like the reason a poor swimmer has for holding that he can make it to shore when his boat has sunk and he has no life jacket and he has never swum so far before: a reason for holding a proposition that can be consistent with excellent evidence that the proposition is false.

  Some further perplexities.

    No reason whatsoever can be given for [abduction], as far as I can discover; and it needs no reason since it merely offers suggestions. (5.171)

If “reason” here means “justification” in the sense in which deduction and induction can be justified, then we can make sense of the first clause of the statement. Peirce may mean to refer to something else, viz., to the fact that we cannot give a reason for drawing one abductive conclusion rather than another from a given set of data to be explained. All we can say is that one hypothesis (or maybe more than one) occurs to us as a result of trying to explain the data and others do not. That is a statement of fact, not a rational defense. Whichever way we interpret “reason” in the first clause, it is consistent with what Peirce has already said.

  The second clause is a puzzle. The fact that something is a suggestion does not always exempt it from the need to be justified. However I turn and twist this second clause, I cannot see what Peirce could be getting at.

  But that’s not all. At 5.172-173 Peirce does seem to launch into a justification of abduction like the justifications of the other inferences. He points out that the history of scientific investigation, where the latter is understood very broadly, shows that humans have a faculty for guessing right about “the Thirdnesses, the general elements, of Nature” (5.173). If abduction gets a justification by desperation and no reason can be given for it and it needs no reason, then what is this? A pound of salami? I throw up my hands in despair. Fortunately, the confusion here – merely my own, no doubt – does not affect anything else.[41]

*   *   *

  The second phase of the proof gives us the result that the aim of inference, conformity to which makes an inference valid, is the conformity of our own reason to the reason in things. Each fundamental type of inference does this in a different way and each type of inference can be justified as valid without unremovable references to truth. Consequently, premiss 6. of the argument set out earlier can be defended successfully. There is only one kind of logical goodness and that is goodness of inference. Now the stage is set for the third phase.

IV

THIRD PHASE

  The third phase of the “Lectures” presents a problem in reconstruction. It is diffuse and although it is clear how material from the second phase is relevant to it, it is not so clear how material from the first phase is pertinent.

  We should consider that the third phase begins with the material omitted at the end of 5.174. The editors of the Collected Papers have left out the beginning of a new section introduced with the words “Let us now come to the question of the maxim of Pragmatism.”[42]

  Here Peirce begins to discuss meaning and here the reader is likely to be puzzled about how the treatment of meaning connects with the first phase. The connection is made through the relation of meaning and purpose which allows Peirce to explicate the concept of meaning by reference to the concept of ultimate purpose. Admittedly, the phrase “ultimate purpose” never makes an appearance here. Nevertheless it is not too hard to see that the crucial concept in the discussion is “ultimate meaning.” If Peirce is to defend the idea that this is not an empty concept, he will need to be able to show that there is such a thing as the ultimate purpose of an argument. This is what the results from the first phase allow him to do.

  Accordingly, Peirce connects meaning and purpose at 5.175. The purpose of an argument is “to determine an acceptance of its conclusion, and it quite accords with general usage to call the conclusion of an argument its meaning.” On the basis of this suggestion from common usage, Peirce adopts “meaning” as a technical term. “It seems natural to use the word meaning to denote the intended interpretant of a symbol” (5.175). Unfortunately we are never told what an intended interpretant is. It seems to be merely whatever interpretant it is the purpose of the user of the symbol to determine.

  Next Peirce extends his technical sense of “meaning” to cover terms and propositions. “The meaning of a proposition or term is all that that proposition or term could contribute to the conclusion of a demonstrative argument” (5.179). The basis for this is that deductive reasoning explicates the meaning of its premisses (5.176). So any deduction that a certain term or proposition permits will give a part of the meaning of the term or proposition in its conclusion.

  If we apply Peirce’s idea of meaning consistently we cannot quite say what Peirce does about the meaning of an argument. The conclusion of an argument is the part of its meaning that is made explicit, but not the whole of its meaning, as Peirce seems to imply. The meaning of an argument will include whatever its conclusion can contribute to the conclusion of a deduction that can be carried out with its aid. Peirce is committed to this because of the transitivity of “means.” The meaning of an argument that is explicit is its conclusion. The rest of its meaning is whatever its conclusion means.

  Peirce thinks that the technical sense of meaning is not sufficient for evaluating the pragmatic maxim (5.179). He does not say why, but the reason is pretty evident. An assertion may contain any blather whatever. So long as it is syntactically correct, it will permit the deduction of more blather which will count as part of the meaning of the argument. But the conclusion will be a meaning that is pragmatically meaningless. For example:

 (1) The warranted genuine Snark has a taste that is meagre and hollow, but crisp.

 (2) Therefore, the warranted genuine Snark has a taste that is hollow.

  This is why Peirce introduces the concept of an ultimate meaning (5.179). This concept is never very clearly connected with pragmatism, so the first order of business is to make up for that. According to pragmatism there are certain common components of meaning possessed by whatever qualifies as pragmatically meaningful. These components will relate actions in certain circumstances to perceptual results to be expected (5.18). I believe that it is these components of meaning that Peirce is trying to get at when he introduces the concept of an ultimate meaning. Pragmatism says that whatever else a pragmatically meaningful term or proposition contributes to the conclusion of a deduction, it always contributes (at least implicitly) information relating actions in certain circumstances to results to be expected. What a term or proposition always contributes to a deductive conclusion would be its ultimate meaning, analogous to an ultimate purpose, a purpose to which actions contribute, no matter what other purposes they contribute to.

  In an attempt to clarify what I think is going on, I will give an argument that connects together pragmatic meaning and ultimate meaning, showing what an ultimate meaning is, if pragmatism is true. I propose to read the predicate “(MEAN)x · (CON)x” as “x possesses an ultimate meaning” since that seems to be what “x is meaningful and x conforms to an ultimate end” comes to. So the first premiss is

(A) Pragmatic meaning is ultimate meaning;

(A*) (x){(PRAG)x ≡ [(MEAN)x · (CON)x]}.

We will be concerned with the pragmatic meaning of terms and proposition, so from (A) we deduce

(B) A term or proposition is pragmatically meaningful if and only if it possesses an ultimate meaning;

(B*) (x)({[(TERM)x v (PPN)x] · (PRAG)x} ≡ {[(TERM)x · (MEAN)x · (CON)x] v [(PPN)x · (MEAN)x · (CON)x]}).

As far as pragmatism is concerned, the possession by a term or proposition of pragmatic meaning consists in its capacity to contribute certain common meaning elements to deductive conclusions.

(C) A term or proposition has pragmatic meaning if and only if it involves the common meaning element peculiar to pragmatic meaning;

(C*) (x)({[(TERM)x v (PPN)x] · (PRAG)x} ≡ {[(TERM)x v (PPN)x] · (COMM)x}).

Further, pragmatism says that the common meaning element in question consists in information relating actions and circumstances of action to percepts to be expected. This information seems to be what Peirce had in mind by “practical consequences” so I will abbreviate the idea of involving such information as “(PCTL)x.”

(D) Anything involves the common meaning element peculiar to pragmatic meaning if and only if it involves practical consequences;

(D*) (x)[(COMM)x ≡ (PCTL)x].

Then from (B), (C), and (D) we can deduce

(E) The ultimate meaning of a term or proposition is the practical consequences it involves;

(E*) (x)({[(TERM)x · (MEAN)x · (CON)x] v [(PPN)x · (MEAN)x · (CON)x]} ≡ {[(TERM)x · (PCTL)x] v [(PPN)x · (PCTL)x]}).

  This argument tells us what we get when we equate pragmatic meaning with ultimate meaning. It tells us what ultimate meaning is as far as the doctrine of pragmatism is concerned. The question for Peirce is whether his logical theory makes ultimate meaning the same thing that pragmatism makes it. Specifically, the question for him is whether he can deduce (E) on the basis of his previous results and without any reference to pragmatism. If Peirce can do so, then he will have given a proof of pragmatism.

  It is pretty clear that Peirce is committed to the proposition that

(20) If logical goodness is the conformity of an inference to an ultimate end, then ultimate meanings of terms and propositions are common meaning elements which they involve;

(20*) (x){(LOG)x ≡ [(INF)x · (CON)x]} ⊃ (x)({[(TERM)x v (PPN)x] · [(MEAN)x · (CON)x]} ≡ {[(TERM)x v (PPN)x] · (COMM)x}).

This seems justifiable. Certainly the conclusion of any logically good argument will be meaningful in the non-ultimate sense and for the argument to conform an ultimate end would seem to amount to its conclusion advancing the growth of reason. Whatever information the conclusion conveys that contributes to the aim of argument – as long as it is information that is common to all conclusions that advance the aim – will be an ultimate meaning. So given Peirce’s conception of logical goodness, ultimate meanings are common meaning elements of terms and propositions, where it is understood that the terms and propositions of interest are those that figure in the conclusions of logically good inferences. So from (19) and (20) we can infer

(21) The ultimate meaning of a term or proposition is the same as the common meaning element that it involves;

(21*) (x)({[(TERM)x v (PPN)x] · [(MEAN)x · (CON)x]} ≡ {[(TERM)x v (PPN)x] · (COMM)x}).

  The question may arise here as to whether we have introduced a narrower concept of meaning than the one that figures in the proof of (E). For we did not say anything there about terms and propositions of interest being restricted to those that turn up in the conclusions of veracious arguments. However, I think that we ought to understand the restriction to apply in the proof of (E). Throughout the “Lectures” Peirce makes it clear that he is treating argument as a purposive activity. So whatever pragmatism has to say about the constituents of argument will be understood by Peirce to apply to veracious argument. Indeed, Peirce would say that an argument not veracious doesn’t conform to the purpose of argument. It is a willful deception or a textbook illustration and is not used to get at information that will contribute to the growth of reason. We only classify such arguments as good or bad, I suppose Peirce would say, by means of the fiction that they are veracious.

  The next item of business is whether the common meaning element is the practical consequences involved in whatever possesses such an element.

  The unprinted notebook containing the end of the sixth lecture (MS 316) gives us the material we need to finish the third phase of the proof. As we have seen, we should be on the look-out for a common meaning element, information that is conveyed by any term or proposition that can turn up in the conclusion of a logically good inference. To ascertain what this element is, supposing that there is one, Peirce asks what the purpose of a term is (MS 316, p.44). The question quickly changes to what the purpose of assent to a proposition is, since a term is of no use unless it occurs in a proposition that is assented to (MS 316, p.44, p.58). Whether or not all terms originate in perception, a term that is of any use must occur in a proposition that is assented to. Peirce assumes that whatever term is in question is not wholly perceptual, since he wants to defer the question of whether all terms originate in perception (MS 316, pp.56-58). So the terms Peirce is interested in are terms that occur in the conclusions of arguments, terms in propositions assent to which is not compulsory, so terms in propositions not perceptual. Concerning such terms he asks what reasons there can be for assenting to propositions containing them and what information such propositions convey (MS 316, pp.57-58).

  The discussion has gone slightly off the track at this point because Peirce has brought up his three cotary propositions that figure in the seventh lecture and has dismissed them until later. Since one of these propositions implies that all terms originate wholly in perception, he seems to think that his dismissal of the propositions leaves him in the position of assuming that no terms originate wholly in perception. Not only is this a non-sequitur. It also leaves the discussion about the end of a term disconnected from (or at least not obviously connected with) his previous results. Peirce gives no appropriate justification for the ensuing discussion of the question in terms of the theory of abduction, no indication of why this is a suitable strategy.

  Fortunately the omission is an oversight and not a flaw intrinsic to the argument. The justification is this. If a term is going to find its way into the conclusion of an inference, it has to get introduced into reasoning somehow. If there is any common cognitive purpose that all terms and propositions subserve, they should share a common meaning element and that should be revealed by looking at the type of argument that introduces new content into reasoning. Abduction is the only type of argument that introduces new content into reasoning, so we should see what content, if any, is always introduced by abduction if we want to find out what Peirce’s logical theory has to say about the common meaning element.

  So we come to the question of what makes a hypothesis admissible. “The true maxim of abduction is that which Auguste Comte endeavored to formulate when he said that any hypothesis might be admissible if and only if it was verifiable” (MS 316, p.59). Comte may have had verification by direct observation in mind, which is not how verification ought to be understood.

But what must and should be meant is, that the hypothesis must be capable of verification by induction. Now induction, or experimental inquiry, consists in comparing perceptual predictions deduced from a theory with the facts of perception predicted and in taking the measure of agreement observed as the provisional and approximative . . . measure of the general agreement of the theory with fact.

    It thus appears that a conception can only be admitted into a hypothesis in so far as its possible consequences would be of a perceptual nature; which agrees with my original maxim of pragmatism as far as it goes. (MS 316, pp.59-60)

Thus ends the sixth lecture.

  To unpack this a little, Peirce’s theory of induction requires that, for a hypothesis to be testable, it must be capable of yielding perceptual predictions. As for testing,

An experiment, says Stöckhardt, in his excellent School of Chemistry, is a question put to nature. Like any interrogatory, it is based on a supposition. If that supposition be correct, a certain sensible result is to be expected under certain circumstances which can be created, or are at any rate to be met with. (5.168)

So here we have the information that any validly abduced hypothesis must convey. If the hypothesis has nothing to say about sensible results to be encountered in consequence of actions that bring about certain conditions, or at least bring us into them, then the hypothesis is not admissible. Every term and proposition that can be of use in reasoning must convey this information, whatever else it conveys. So

(22) To possess the common meaning element is to involve practical consequences;

(22*) (x) [(COMM)x ≡ (PCTL)x].

From (21) and (22) we can deduce

(23) The ultimate meaning of a term or proposition is the practical consequences it involves;

(23*) (x)({[(TERM)x · (MEAN)x · (CON)x] v [(PPN)x · (MEAN)x · (CON)x]} ≡ {[(TERM)x · (PCTL)x] v [(PPN)x · (PCTL)x]}).

This is the proposition that we have been aiming to prove.

V

LOOSE ENDS

  The proof raises a number of issues that I cannot take up here so I will just mention some of them. Perhaps others will take them up.

  (1) Peirce raises the issue of the pragmatic meaning of mathematical propositions when he brings up the practical consequences, if any, of the proposition that the diagonal of a square is incommensurable with the side. This issue is never resolved in the “Lectures.” The answer must lie in practical consequences relating to the manipulation of diagrams and results to be expected. What would a pragmatic analysis of the proof of incommensurability look like? Would it be successful?

  (2) The theory of meaning that Peirce embraces looks uncomfortably like Ayer’s weak verificationism in its first formulation, albeit weak verificationism with cosmic significance.[43] Does Berlin’s counter-example or some version of it shoot down Peirce as it shot down Ayer?[44] If so, then Peirce’s pragmatism belongs on the ash-heap with 1930's verificationism and we would, perhaps, be well-advised to stop making a fuss over it.

  (3) The proof makes extensive use of an observation-theory distinction that many people would now find suspect. In the “Lectures” Peirce turns into a foundationalist. Is this essential to the proof, as it appears to be, or could the proof be articulated with perceptual judgments counted as those which have no justification, but which there is no reason to doubt? With this concept of perceptual judgments, the latter would not be immune from doubt and would presumably have truth values.

  (4) What, if anything, is the relation of Peirce’s pragmatism to reductionism? Is he committed to the idea that the meaning of “There is a Princeton Graphics Max-12 monitor setting on the computer” is reducible to statements describing percepts? If so, there is another possible reason for discarding pragmatism.

  (5) The role of abduction in the discussion of meaning seems much more suited to non-deductive than to mathematical reasoning. What is the role of abduction in mathematical reasoning? If there is a problem here, then the account of ultimate meaning will go wrong.

  (6) It is just false that abduction is the only inference that introduces new content into reasoning. In one sense of “content” logical addition introduces new content too. Does this undermine the proof?

  (7) The biggest loose end of all is the fourth phase of the proof. Why is Peirce worried about the admissibility of hypotheses involving continuity? Just as a conjecture, I would say that it is related to Peirce’s analysis of induction. Any admissible hypothesis must be translatable into a proposition about a denumerably infinite series. A hypothesis involving continuity will be satisfiable in a nondenumerably infinite domain. This being so, how do we translate it into a proposition about a denumerably infinite domain and still have a hypothesis involving continuity? If we cannot, then the hypothesis will be inadmissible. Induction will not be capable of testing it. If my conjecture is right, the solution to Peirce’s problem might lie in the direction of the Löwenheim-Skolem Theorem. It is very unfortunate that we do not know what Peirce said in the supplementary eighth lecture.

  All these problems need to be addressed and my reconstruction of the proof can no doubt be improved upon. But I hope that this paper can be a useful step in understanding the proof of pragmatism.

NOTES

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