In the first published paper in this series Peirce had suggested, in opposition to the Cartesian account, that all knowing involved an inferential sign process. In the second paper in the series he takes up the task of articulating in some detail his own theory of the structure of mental activity, that is, the structure of the internal sign process that is involved in knowing. Constructing this account, he is guided by his methodological strictures to the effect that any account of the internal (mental activity) must be in terms of the external (publicly accessible objects) and that, given the postulation of one structure, another is not to be introduced into the theory unless there are facts impossible to explain on the basis of the first.
Focusing on our public sign system, that is, language, as the paradigmatic external manifestation of mental activity, Peirce proceeds to construct an account of mental activity in terms of “inner speech.” Furthermore, he develops an holistic form of this tradition in which the basic mental unit is not the concept (the mental word) or even the judgment (the mental sentence) but rather the process of reasoning itself (the mental syllogism). Since it is then the structure (rather than the matter) of the sign process that is of primary importance, Peirce accordingly construes the process as one of drawing inferences, as syllogistic in nature. Next, drawing on his formal accounts of deduction, induction, and hypothesis, he proceeds to give an account not only of thinking but also of the other forms of mental activity (sensation, emotion, and attention) in terms of his syllogistic model. His final extrapolation of the model enables him to give a speculative account of the mind itself.
The third paper in the published series, “Grounds of Validity of the Laws of Logic: Further Consequences of Four Incapacities,” picks up some of the remaining questions outlined in MS 148 and finally comes to grips with Harris’s original challenge which had been the impetus for all three papers, namely, how can Peirce account for the objective validity of the laws of logic? The theories of cognition and reality were developed for the sake of providing just such an account, an account which begins with a justification of deduction and then broadens out to encompass a philosophical grounding of the general logic of science.
The point of continuity with the previous pieces is Peirce’s claim that every cognition results from an inference and that the structure of all mental activity is inferential. Can’t the question be raised—what reason do we have to believe that the principles of inference are true or correspond to anything in the real world? While not purporting to take seriously the stance of the absolute skeptic, Peirce does think it incumbent upon him to provide an account of the objective validity of the logical principles of inference. He proceeds to give an account of the validity of deduction, induction, and hypothesis; and his proffered “justifications” invoke the characteristic Peircean concepts of truth (as the ultimate agreement of investigators), reality (as that which is represented in that agreement), and community (as the ultimate ground of both logic and reality).
It would be difficult indeed to overstate the importance of these three papers in the Peircean corpus. That Peirce himself saw them as central is clear from his designation of them as Chapters 4, 5, and 6 of one of his major projected works, the 1893 “Search for a Method.” Most later commentators have seen them as the key to his overall philosophical orientation.
III
The 1870 Logic of Relatives Memoir
DANIEL D. MERRILL
Peirce’s “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of Boole’s Calculus of Logic” (DNLR) is one of the most important works in the history of modern logic, for it is the first attempt to expand Boole’s algebra of logic to include the logic of relations. The complex mathematical analogies which govern parts of this work make it obscure in spots; but the main thrust of its important innovations may be seen by placing it in the context of Peirce’s earlier logical studies, and by relating it to the work of Boole, De Morgan, and Benjamin Peirce.
The logical substructure of DNLR is a modified version of Boole’s algebra of classes, in which Peirce had shown an early interest.15 One modification is the use of the “inclusive” sense of logical addition, which Peirce had introduced by 1867.16 The other main modification is the replacement of Boole’s equality or identity sign (=) by the sign of illation or inclusion (
To this basically Boolean structure, Peirce adds a notation for relations and for operations upon relations, as well as laws governing those operations. Even then, though, the influence of Boole remains strong. While Peirce admits logical relations between relations, he most often considers logical relations that hold between class terms of which relation terms form a part.
Peirce’s interest in the logic of relations can be traced to several sources.17 Published and unpublished papers prepared around 1866 show a strong interest in the problems which relation terms present for the theory of categories.18 They are also concerned with different types of relations, such as the distinction between relations of equiparance and relations of disquiparance. His work at this time also shows an interest in arguments involving relations and multiple sub-sumptions. Such an argument is “Everyone loves him whom he treats kindly; James treats John kindly; hence, James loves John.” Peirce’s early treatment of these arguments is rather conservative, either reformulating them so as to apply the usual syllogistic forms, or using some principle of multiple subsumption which is construed as a natural generalization of the syllogism.
Unfortunately, the origins of the more powerful and, indeed, revolutionary techniques of DNLR are more obscure.19 Only two surviving documents provide a sustained insight into their origins. One is the so-called Logic Notebook (LN), which carries entries from 3 to 15 November 1868 in which several notations are devised and some basic identities are shown. Only the rudiments of DNLR may be found here. The same is true of the other source, a series of notes that Peirce wrote at about the same time to add to a projected republication of his American Academy papers of 1867. Note 4 in this set shows how an algebraic notation may be used to validate the following argument, which De Morgan had claimed could not be shown to be valid by syllogistic means:
Every man is an animal.
Therefore, any head of a man is a head of an animal.
Most unfortunately, the surviving parts of LN have no entries from 16 November 1868 through 5 October 1869, nor is there any other document which would allow us to trace the development of these techniques.
Peirce’s references to De Morgan in DNLR, as well as an undated comparison (in LN) between his notation and De Morgan’s, raise the question of De Morgan’s role in stimulating the work which led to DNLR.20 It must be noted, though, that there is little direct biographical