Cf: Peirce’s 1870 “Logic of Relatives” • Selection 2 https://inquiryintoinquiry.com/2014/01/29/peirces-1870-logic-of-relatives-s… All, We continue with §3. Application of the Algebraic Signs to Logic. Peirce’s 1870 “Logic of Relatives” • Selection 2 ================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#S… <QUOTE CSP> Numbers Corresponding to Letters ================================ I propose to use the term “universe” to denote that class of individuals about which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan’s, is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains. I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men (men), the number of “tooth of” would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus, [t]. (Peirce, CP 3.65) </QUOTE> Peirce’s remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse as usually to escape explicit mention, others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly slipped into the stream of thought that it’s all too easy to overlook their significance, that all I can do to highlight their impact is to dress them up in different words, whose main advantage is being more jarring to the mind’s sensibilities. • This mapping of letters to numbers, or logical terms to mathematical quantities, is the very core of what quantification theory is all about, definitely more to the point than the mere “innovation” of using distinctive symbols for the so-called quantifiers. • The mapping of logical terms to numerical measures, to express it in current language, would probably be recognizable as some kind of morphism or functor from a logical domain to a quantitative co-domain. • Notice that Peirce follows the mathematician’s usual practice, then and now, of making the status of being an individual or a universal relative to a discourse in progress. • It is worth noting that Peirce takes the plural denotation of terms for granted — or what’s the number of a term for, if it could not vary apart from being one or nil? • I also observe that Peirce takes the individual objects of a particular universe of discourse in a generative way, as opposed to a totalizing way, and thus these contingent individuals afford us with a basis for talking freely about collections, constructions, properties, qualities, subsets, and higher types built up thereon. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 4 https://inquiryintoinquiry.com/2014/01/31/peirces-1870-logic-of-relatives-s… All, Here is the next part of §3. Application of the Algebraic Signs to Logic. Peirce’s 1870 “Logic of Relatives” • Selection 4 ================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#S… <QUOTE CSP> The Signs for Addition ====================== The sign of addition is taken by Boole so that x + y denotes everything denoted by x, and, besides, everything denoted by y. Thus m + w denotes all men, and, besides, all women. This signification for this sign is needed for connecting the notation of logic with that of the theory of probabilities. But if there is anything which is denoted by both terms of the sum, the latter no longer stands for any logical term on account of its implying that the objects denoted by one term are to be taken besides the objects denoted by the other. For example, f + u means all Frenchmen besides all violinists, and, therefore, considered as a logical term, implies that all French violinists are besides themselves. For this reason alone, in a paper which is published in the Proceedings of the Academy for March 17, 1867, I preferred to take as the regular addition of logic a non-invertible process, such that m +, b stands for all men and black things, without any implication that the black things are to be taken besides the men; and the study of the logic of relatives has supplied me with other weighty reasons for the same determination. Since the publication of that paper, I have found that Mr. W. Stanley Jevons, in a tract called “Pure Logic, or the Logic of Quality” [1864], had anticipated me in substituting the same operation for Boole’s addition, although he rejects Boole’s operation entirely and writes the new one with a “+” sign while withholding from it the name of addition. It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations. The conception of taking together involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive. Addition being taken in this sense, nothing is to be denoted by zero, for then x +, 0 = x whatever is denoted by x; and this is the definition of zero. This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of zero and that of nothing, and because we shall thus have [0] = 0. (Peirce, CP 3.67) </QUOTE> A wealth of issues arises here that I hope to take up in depth at a later point, but for the moment I shall be able to mention only the barest sample of them in passing. The two papers precedinge this one in CP 3 are Peirce’s papers of March and September 1867 in the Proceedings of the American Academy of Arts and Sciences, titled “On an Improvement in Boole’s Calculus of Logic” and “Upon the Logic of Mathematics”, respectively. Among other things, these two papers provide us with further clues about the motivating considerations that brought Peirce to introduce the “number of a term” function, signified here by square brackets. In setting up a correspondence between “letters” and “numbers”, Peirce constructs a structure-preserving map from a logical domain to a numerical domain. That he does this deliberately is evidenced by the care that he takes with the conditions under which the chosen aspects of structure are preserved, along with his recognition of the critical fact that zeroes are preserved by the mapping. Incidentally, Peirce appears to have an inkling of the problems that would later be caused by using the plus sign for inclusive disjunction, but his advice was overridden by the dialects of applied logic that developed in various communities, retarding the exchange of information among engineering, mathematical, and philosophical specialties all throughout the subsequent century. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 6 https://inquiryintoinquiry.com/2014/02/05/peirces-1870-logic-of-relatives-s… Note. Please follow the link above for better formatting of Peirce's text, as some of his typographical distinctions are lost in the transcript below. All, We continue with §3. Application of the Algebraic Signs to Logic. Peirce’s 1870 “Logic of Relatives” • Selection 6 ================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#S… The application of a relation is one of the most basic operations in Peirce’s logic. Because relation applications are so pervasive and because Peirce treats them on the pattern of algebraic multiplication, the part of §3 concerned with “The Signs for Multiplication” will occupy our attention for many days to come. <QUOTE CSP> The Signs for Multiplication (cont.) A conjugative term like “giver” naturally requires two correlates, one denoting the thing given, the other the recipient of the gift. We must be able to distinguish, in our notation, the giver of A to B from the giver to A of B, and, therefore, I suppose the signification of the letter equivalent to such a relative to distinguish the correlates as first, second, third, etc., so that “giver of ── to ──” and “giver to ── of ──” will be expressed by different letters. Let “g” denote the latter of these conjugative terms. Then, the correlates or multiplicands of this multiplier cannot all stand directly after it, as is usual in multiplication, but may be ranged after it in regular order, so that: gxy will denote a giver to x of y. But according to the notation, x here multiplies y, so that if we put for x “owner” (o), and for y “horse” (h), goh appears to denote the giver of a horse to an owner of a horse. But let the individual horses be H, H′, H″, etc. Then: h = H +, H′+, H″ +, etc. goh = go(H +, H′+, H″ +, etc.) = goH +, goH′+, goH″ +, etc. Now this last member must be interpreted as a giver of a horse to the owner of “that” horse, and this, therefore must be the interpretation of goh. This is always very important. “A term multiplied by two relatives shows that the same individual is in the two relations.” If we attempt to express the giver of a horse to a lover of a woman, and for that purpose write: gℓwh, we have written giver of a woman to a lover of her, and if we add brackets, thus, g(ℓw)h, we abandon the associative principle of multiplication. A little reflection will show that the associative principle must in some form or other be abandoned at this point. But while this principle is sometimes falsified, it oftener holds, and a notation must be adopted which will show of itself when it holds. We already see that we cannot express multiplication by writing the multiplicand directly after the multiplier; let us then affix subjacent numbers after letters to show where their correlates are to be found. The first number shall denote how many factors must be counted from left to right to reach the first correlate, the second how many more must be counted to reach the second, and so on. Then, the giver of a horse to a lover of a woman may be written: g₁₂ℓ₁wh = g₁₁ℓ₂hw = g₂₍₋₁₎hℓ₁w, Of course a negative number indicates that the former correlate follows the latter by the corresponding positive number. A subjacent zero makes the term itself the correlate. Thus, ℓ₀ denotes the lover of “that” lover or the lover of himself, just as goh denotes that the horse is given to the owner of itself, for to make a term doubly a correlate is, by the distributive principle, to make each individual doubly a correlate, so that: ℓ₀ = L₀ +, L′₀ +, L″₀ +, etc. A subjacent sign of infinity may indicate that the correlate is indeterminate, so that: ℓ_∞ will denote a lover of something. We shall have some confirmation of this presently. If the last subjacent number is a one it may be omitted. Thus we shall have: ℓ₁ = ℓ, g₁₁ = g₁ = g. This enables us to retain our former expressions ℓw, goh, etc. (Peirce, CP 3.69–70) </QUOTE> Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Sets as Sums https://inquiryintoinquiry.com/2014/02/06/peirces-1870-logic-of-relatives-s… Comment on Peirce’s 1870 “Logic of Relatives” • Sets as Sums ============================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, Peirce’s way of representing sets as logical sums may seem archaic, but it’s quite often used in mathematics and remains the tool of choice in many branches of algebra, combinatorics, computing, and statistics to this day. Peirce applied this genre of representation to logic in fairly novel ways and the degree to which he elaborated its use in the logic of relative terms is certainly original with him, but this particular device, going under the handle of “generating functions”, goes way back, well before anyone thought of sticking a flag in set theory as a separate territory or of trying to fence off our native possessions of classes and collections with explicit decrees of axioms. And back in the days when a “computer” was simply a person who computed, well before the advent of electronic computers we take for granted today, mathematicians commonly used generating functions as a rough and ready sort of addressable memory to organize, store, and keep track of their accounts on a wide variety of formal objects. Let’s look at a few simple examples of generating functions, much as I encountered them during my own first adventures in the Realm of Combinatorics. Suppose we are given a set of three elements, say, {a, b, c}, and we are asked to find all the ways of choosing a subset from this collection. We can represent this problem setup as the problem of computing the following product: (1 + a)(1 + b)(1 + c). The factor (1 + a) represents the option e have, in choosing a subset of {a, b, c}, to exclude the element “a” (signified by the “1”), or else to include it (signified by the “a”), proceeding in a similar fashion with the other elements in their turn. Probably on account of all those years I flippered away playing the oldtime pinball machines, I tend to imagine a product like this being displayed in a vertical array: (1 + a) (1 + b) (1 + c) I picture this as a playboard with six bumpers, the ball chuting down the board in such a way that it strikes exactly one of the two bumpers on each of the three levels. So a trajectory of the ball where it hits the “a” bumper on the 1st level, hits the “1” bumper on the 2nd level, hits the “c” bumper on the 3rd level, and then exits the board, represents a single term in the desired product and corresponds to the subset {a, c}. Multiplying out the product (1 + a)(1 + b)(1 + c), one obtains: 1 + a + b + c + ab + ac + bc + abc. This informs us that the subsets of choice are: ∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}. And so they are. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Proto-Graphical Syntax https://inquiryintoinquiry.com/2014/02/12/peirces-1870-logic-of-relatives-p… Comment on Peirce’s 1870 “Logic of Relatives” • Proto-Graphical Syntax ====================================================================== https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, It is clear from our last Selection that Peirce is already on the verge of a graphical syntax for the logic of relative terms. Indeed, it is likely he had already reached that point in his own thinking some time before. For instance, it seems quite impossible for a person with any graphical sensitivity whatever to scan that last variation on “giver of a horse to a lover of a woman” without drawing or at least imagining lines of identity to connect the corresponding marks of reference, as shown in the following Figure. Figure 3. Giver of a Horse to a Lover of a Woman https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-glwh.png Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 8 https://inquiryintoinquiry.com/2014/02/17/peirces-1870-logic-of-relatives-s… Note. Please follow the link above for the proper formatting of Peirce's text, as many of his typographical distinctions are lost in the following transcript. All, We continue with §3. Application of the Algebraic Signs to Logic. Peirce’s 1870 “Logic of Relatives” • Selection 8 ================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#S… <QUOTE CSP> The Signs for Multiplication (cont.) Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives. Now the absolute term “man” is really exactly equivalent to the relative term “man that is ──”, and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term. Then “man that is black” will be written: m,b. But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one. Then: ℓ,sw will denote a lover of a woman that is a servant of that woman. The comma here after ℓ should not be considered as altering at all the meaning of ℓ , but as only a subjacent sign, serving to alter the arrangement of the correlates. In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates. So: m,,b,r interpreted like goh means a man that is a rich individual and is a black that is that rich individual. But this has no other meaning than: m,b,r or a man that is a black that is rich. Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number. If, therefore, ℓ,,sw is not the same as ℓ,sw (as it plainly is not, because the latter means a lover and servant of a woman, and the former a lover of and servant of and same as a woman), this is simply because the writing of the comma alters the arrangement of the correlates. And if we are to suppose that absolute terms are multipliers at all (as mathematical generality demands that we should), we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series “that is ── and is ── and is ── etc.” Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the question is, What are to be the implied subjacent numbers for these implied correlates? Any term may be regarded as having an infinite number of factors, those at the end being ones, thus: ℓ,sw = ℓ,sw,1,1,1,1,1,1,1, etc. A subjacent number may therefore be as great as we please. But all these “ones” denote the same identical individual denoted by w ; what then can be the subjacent numbers to be applied to s , for instance, on account of its infinite “that is”'s? What numbers can separate it from being identical with w ? There are only two. The first is zero, which plainly neutralizes a comma completely, since s,₀w = sw and the other is infinity; for as 1^∞ is indeterminate in ordinary algebra, so it will be shown hereafter to be here, so that to remove the correlate by the product of an infinite series of ones is to leave it indeterminate. Accordingly, m,_∞ should be regarded as expressing “some” man. Any term, then, is properly to be regarded as having an infinite number of commas, all or some of which are neutralized by zeros. “Something” may then be expressed by: 1_∞. I shall for brevity frequently express this by an antique figure one 1. “Anything” by: 1₀. I shall often also write a straight 1 for anything. (Peirce, CP 3.73) </QUOTE> Regards. Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.2 https://inquiryintoinquiry.com/2014/02/18/peirces-1870-logic-of-relatives-c… All, I continue with my commentary on CP 3.73, developing the “Othello” example as a way of illustrating Peirce’s formalism. Peirce’s 1870 “Logic of Relatives” • Comment 8.2 ================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… In the development of the story so far, we have a universe of discourse characterized by the following equations. 1 = B +, C +, D +, E +, I +, J +, O b = O m = C +, I +, J +, O w = B +, D +, E This much forms a basis for the collection of absolute terms to be used in the example. Let us now consider how we might represent an exemplary collection of relative terms. Consider the genesis of relative terms, for example: “lover of____” “betrayer to____of____” “winner over of____to____from____” We may regard these fill-in-the-blank forms as being derived by a kind of rhematic abstraction from the corresponding instances of absolute terms. The following examples illustrate the relationships existing among absolute terms, relative terms, relations, and elementary relations. • The relative term “lover of____” can be derived from the absolute term “lover of____Emilia” by removing the absolute term “Emilia”. Iago is a lover of Emilia, so the relate-correlate pair I:E is an element of the dyadic relation associated with the relative term “lover of____”. • The relative term “betrayer to____of____” can be derived from the absolute term “betrayer to Othello of Desdemona” by removing the absolute terms “Othello” and “Desdemona”. Iago is a betrayer to Othello of Desdemona, so the elementary relative term I:O:D is an element of the triadic relation associated with the relative term “betrayer to____of____”. • The relative term “winner over of____to____from____” can be derived from the absolute term “winner over of Othello to Iago from Cassio” by removing the absolute terms “Othello”, “Iago”, and “Cassio”. Iago is a winner over of Othello to Iago from Cassio, so the elementary relative term I:O:I:C is an element of the tetradic relation associated with the relative term “winner over of____to____from____”. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.4 https://inquiryintoinquiry.com/2014/02/19/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 8.4 ================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, To familiarize ourselves with the forms of calculation available in Peirce’s notation, let us compute a few of the simplest products we find in the “Othello” universe. Here are the absolute terms: 1 = B +, C +, D +, E +, I +, J +, O b = O m = C +, I +, J +, O w = B +, D +, E Here are the dyadic relative terms: ℓ = B:C +, C:B +, D:O +, E:I +, I:E +, O:D s = C:O +, E:D +, I:O +, J:D +, J:O Here are a few of the simplest products among these terms: ℓ1 = lover of anything = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D) × (B +, C +, D +, E +, I +, J +, O) = B +, C +, D +, E +, I +, O = anything except J ℓb = lover of a black = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D) × O = D ℓm = lover of a man = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D) × (C +, I +, J +, O) = B +, D +, E ℓw = lover of a woman = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D) × (B +, D +, E) = C +, I +, O s1 = servant of anything = (C:O +, E:D +, I:O +, J:D +, J:O) × (B +, C +, D +, E +, I +, J +, O) = C +, E +, I +, J sb = servant of a black = (C:O +, E:D +, I:O +, J:D +, J:O) × O = C +, I +, J sm = servant of a man = (C:O +, E:D +, I:O +, J:D +, J:O) × (C +, I +, J +, O) = C +, I +, J sw = servant of a woman = (C:O +, E:D +, I:O +, J:D +, J:O) × (B +, D +, E) = E +, J ℓs = lover of a servant of____ = (B:C +, C:B +, D:O +, E:I +, I:E +, O:D) × (C:O +, E:D +, I:O +, J:D +, J:O) = B:O +, E:O +, I:D sℓ = servant of a lover of____ = (C:O +, E:D +, I:O +, J:D +, J:O) × (B:C +, C:B +, D:O +, E:I +, I:E +, O:D) = C:D +, E:O +, I:D +, J:D +, J:O Among other things, one sees the relative terms ℓ and s do not commute, in other words, ℓs is not equal to sℓ. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.5 https://inquiryintoinquiry.com/2014/02/20/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 8.5 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, I'm breaking the email version of this Comment into sections on account of the abundance of Figures and attachments in it. [Section 1] Since multiplication by a dyadic relative term is a logical analogue of matrix multiplication in linear algebra, all the products computed above can be represented by “logical matrices”, that is, by arrays of boolean {0, 1} coordinate values. Absolute terms and dyadic relatives are represented as 1-dimensional and 2-dimensional arrays, respectively. The equations defining the absolute terms are given again below, first as logical sums of individual terms and then as n-tuples of boolean coordinates. Display 1. Othello Universe https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-uni… Since we are going to be regarding these tuples as column arrays, it is convenient to arrange them into a table of the following form. Display 2. Othello Column Array https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-col… Here are the dyadic relative terms again, followed by their representation as coefficient matrices, in this case bordered by row and column labels to remind us what the coefficient values are meant to signify. Display 3. ℓ = B:C +, C:B +, D:O +, E:I +, I:E +, O:D https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-log… Display 4. s = C:O +, E:D +, I:O +, J:D +, J:O https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-log… To be continued ... Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 8.5 https://inquiryintoinquiry.com/2014/02/20/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 8.5 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, Last but not least, here are the matrix representations of the two products of dyadic relatives we computed earlier. ℓs = lover of a servant of ——— https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-log… sℓ = servant of a lover of ——— https://inquiryintoinquiry.files.wordpress.com/2021/12/lor-1870-othello-log… Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Selection 9 https://inquiryintoinquiry.com/2014/02/23/peirces-1870-logic-of-relatives-s… All, We continue with §3. Application of the Algebraic Signs to Logic. Peirce’s 1870 “Logic of Relatives” • Selection 9 ================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#S… <QUOTE CSP> The Signs for Multiplication (cont.) It is obvious that multiplication into a multiplicand indicated by a comma is commutative,¹ that is, s,ℓ = ℓ,s This multiplication is effectively the same as that of Boole in his logical calculus. Boole’s unity is my 1, that is, it denotes whatever is. 1. It will often be convenient to speak of the whole operation of affixing a comma and then multiplying as a commutative multiplication, the sign for which is the comma. But though this is allowable, we shall fall into confusion at once if we ever forget that in point of fact it is not a different multiplication, only it is multiplication by a relative whose meaning — or rather whose syntax — has been slightly altered; and that the comma is really the sign of this modification of the foregoing term. (Peirce, CP 3.74) </QUOTE> Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.2 https://inquiryintoinquiry.com/2014/02/26/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 9.2 ================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, In setting up his discussion of selective operations and their corresponding selective symbols, Boole writes the following. <QUOTE Boole> The operation which we really perform is one of “selection according to a prescribed principle or idea”. To what faculties of the mind such an operation would be referred, according to the received classification of its powers, it is not important to inquire, but I suppose that it would be considered as dependent upon the two faculties of Conception or Imagination, and Attention. To the one of these faculties might be referred the formation of the general conception; to the other the fixing of the mental regard upon those individuals within the prescribed universe of discourse which answer to the conception. If, however, as seems not improbable, the power of Attention is nothing more than the power of continuing the exercise of any other faculty of the mind, we might properly regard the whole of the mental process above described as referrible to the mental faculty of Imagination or Conception, the first step of the process being the conception of the Universe itself, and each succeeding step limiting in a definite manner the conception thus formed. Adopting this view, I shall describe each such step, or any definite combination of such steps, as a “definite act of conception”. (Boole, Laws of Thought, 43) </QUOTE> Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.4 https://inquiryintoinquiry.com/2014/02/27/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 9.4 ================================================ https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, Boole rationalizes the properties of what we now call “boolean multiplication”, roughly equivalent to logical conjunction, in terms of the laws which govern selective operations. Peirce, in his turn, taking a radical step of analysis which has seldom been recognized for what it would lead to, does not consider this multiplication to be a fundamental operation, but derives it as a by-product of relative multiplication by a comma relative. In this way Peirce makes logical conjunction a special case of relative composition. This opens up a wide field of inquiry, “the operational significance of logical terms”, but it will be best to advance bit by bit and to lean on simple examples. Back to Venice and the close-knit party of absolutes and relatives we entertained when last stopping there. Here is the list of absolute terms we had been considering before: Figure 1. Absolute Terms 1 M N W https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-abs… Here is the list of “comma inflexions” or “diagonal extensions” of those terms: 1, = anything that is____ = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O m, = man that is____ = C:C +, I:I +, J:J +, O:O n, = noble that is____ = C:C +, D:D +, O:O w, = woman that is____ = B:B +, D:D +, E:E One observes the diagonal extension of *1* is the same thing as the identity relation _1_. Earlier we computed the following products, obtained by applying the diagonal extensions of absolute terms to the same set of absolute terms. m,n = man that is a noble = C +, O n,m = noble that is a man = C +, O w,n = woman that is a noble = D n,m = noble that is a woman = D From that we take our first clue as to why the commutative law holds for logical conjunction. More in the way of practical insight could be had by working systematically through the collection of products generated by the operational means at hand, namely, the products obtained by appending a comma to each of the terms 1, m, n, w then applying the resulting relatives to those selfsame terms again. Before we venture into that territory, however, let us equip our intuitions with the forms of graphical and matrical representation which served us so well in our previous adventures. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Discussion 5 http://inquiryintoinquiry.com/2022/01/08/peirces-1870-logic-of-relatives-di… Re: Conceptual Graphs https://lists.cs.uni-kassel.de/hyperkitty/list/cg@lists.iccs-conference.org… ::: Peiyuan Zhu https://lists.cs.uni-kassel.de/hyperkitty/list/cg@lists.iccs-conference.org… <QUOTE PZ:> I’m studying imprecise probabilities which initially works as an extension in Boole's Laws of Thoughts. It seems like Boole was solving a set of algebraic equations for probabilities where some of the probabilities do not have precise values therefore need to be bounded. Has anyone studied Boole’s algebraic system of probabilities? Is Peirce extending Boole's algebraic system in his Logic of Relatives? </QUOTE> Dear Peiyuan, Issues related to the ones you mention will come up in the Selections and Commentary I'm posting on Peirce's 1870 Logic of Relatives, the full title of which, “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”, is sufficient hint of the author's intent, namely, to extend the correspondence Boole discovered between the calculus of propositions and the statistics of simple events to a correspondence between the calculus of relations and the statistics of complex events, contingency matrices, higher order correlations, and ultimately the full range of information theory. But it will take a while to develop all that … Resource ======== Peirce’s 1870 “Logic of Relatives” • Overview https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-o… Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.5 (complete) https://inquiryintoinquiry.com/2014/03/04/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 9.5 https://inquiryintoinquiry.com/2014/03/04/peirces-1870-logic-of-relatives-c… All, Thinking it would be useful to have the various forms of absolute and relative terms before us all on one page, I've revamped the graphics to the point where they have a good chance of fitting. So let's see ... ⁂ ⁂ ⁂ Peirce’s comma operation, in its application to an absolute term, is tantamount to the representation of that term’s denotation as an idempotent transformation, which is commonly represented as a diagonal matrix. Hence the alternate name, “diagonal extension”. An idempotent element x is given by the abstract condition that xx = x but elements like those are commonly encountered in more concrete circumstances, acting as operators or transformations on other sets or spaces, and in that action they will often be represented as matrices of coefficients. Let’s see how this looks in the matrix and graph pictures of absolute and relative terms. Absolute Terms ============== Display 1. Absolute Terms 1 M N W https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-abs… Previously, we represented absolute terms as column arrays. The above four terms are given by the columns of the following Table. Column Arrays ============= Display 2. Column Arrays https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-col… The types of graphs known as “bigraphs” or “bipartite graphs” can be used to picture simple relative terms, dyadic relations, and their corresponding logical matrices. One way to bring absolute terms and their corresponding sets of individuals into the bigraph picture is to mark the nodes in some way, for example, hollow nodes for non‑members and filled nodes for members of the indicated set, as shown below. Dichromatic Nodes ================= Display 3. Dichromatic Nodes 1 M N W https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-dic… Diagonal Extensions =================== Display 4. Diagonal Extensions 1 M N W https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-dia… Naturally enough, the diagonal extensions are represented by diagonal matrices. Diagonal Matrices ================= Display 5. Diagonal Matrices 1 M N W https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-dia… Cast into the bigraph picture of dyadic relations, the diagonal extension of an absolute term takes on a very distinctive sort of “straight-laced” character, as shown below. Idempotent Bigraphs =================== Display 6. Idempotent Bigraphs 1 M N W https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-othello-ide… Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 9.7 https://inquiryintoinquiry.com/2014/03/08/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 9.7 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, From this point forward we may think of idempotents, selectives, and zero-one diagonal matrices as being roughly equivalent notions. The only reason I say “roughly” is that we are comparing ideas at different levels of abstraction in proposing those connections. We have covered the way Peirce uses his invention of the comma modifier to assimilate boolean multiplication, logical conjunction, and what we may think of as “serial selection” under his more general account of relative multiplication. But the comma functor has its application to relative terms of any arity, not just the zeroth arity of absolute terms, and so there will be a lot more to explore on this point. But now I must return to the anchorage of Peirce’s text and hopefully get a chance to revisit this topic later. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.1 https://inquiryintoinquiry.com/2014/03/10/peirces-1870-logic-of-relatives-c… Re: Peirce’s 1870 “Logic of Relatives” • Selection 10 (CP 3.75) https://inquiryintoinquiry.com/2014/03/09/peirces-1870-logic-of-relatives-s… Peirce’s 1870 “Logic of Relatives” • Comment 10.1 ================================================= https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, What Peirce is attempting to do at CP 3.75 is absolutely amazing. I did not run across anything on a par with it again until the mid 1980s when I began studying the application of mathematical category theory to computation and logic. Gauging the success of Peirce’s attempt would take a return to his earlier paper “Upon the Logic of Mathematics” (1867) to pick up the ideas about arithmetic he sets out there. Another branch of the investigation would require us to examine the syntactic mechanics of “subjacent signs” Peirce uses to establish linkages among relational domains. The indices employed for this purpose amount to a category of diacritical and interpretive signs which includes, among other things, the comma functor we have just been discussing. Combining the two branches of this investigation opens a wider context for the study of relational compositions, distilling the essence of what it takes to relate relations, possibly complex, to other relations, possibly simple. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.3 https://inquiryintoinquiry.com/2014/03/25/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 10.3 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, We have been using several styles of picture to illustrate relative terms and the relations they denote. Let’s now examine the relationships which exist among the variety of visual schemes. Two examples of relative multiplication we considered before are diagrammed again in Figures 11 and 12. Figure 11. Lover of a Servant of a Woman https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-lsw-2.0.png Figure 12. Giver of a Horse to a Lover of a Woman https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-glwh-2.0.png Figures 11 and 12 employ one of the styles of syntax Peirce used for relative multiplication, to which I added lines of identity to connect the corresponding marks of reference. Forms like these show the anatomy of the relative terms themselves, while the forms in Table 13 and Figure 14 are adapted to show the structures of the objective relations they denote. Table 13. Relational Composition L ◦ S https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-relational-… Figure 14. Relational Composition L ◦ S https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-relational-… There are many ways Peirce might have gotten from his 1870 Notation for the Logic of Relatives to his more evolved systems of Logical Graphs. It is interesting to speculate on how the metamorphosis might have been accomplished by way of transformations acting on these nascent forms of syntax and taking place not too far from the pale of its means, that is, as nearly as possible according to the rules and permissions of the initial system itself. In Existential Graphs, a relation is represented by a node whose degree is the adicity of that relation, and which is adjacent via lines of identity to the nodes that represent its correlative relations, including as a special case any of its terminal individual arguments. In the 1870 Logic of Relatives, implicit lines of identity are invoked by the subjacent numbers and marks of reference only when a correlate of some relation is the rèlate of some relation. Thus, the principal rèlate, which is not a correlate of any explicit relation, is not singled out in this way. Remarkably enough, the comma modifier itself provides us with a mechanism to abstract the logic of relations from the logic of relatives, and thus to forge a possible link between the syntax of relative terms and the more graphical depiction of the objective relations themselves. Figure 15 demonstrates this possibility, posing a transitional case between the style of syntax in Figure 11 and the picture of composition in Figure 14. Figure 15. Anything that is a Lover of a Servant of Anything https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-universal-b… In this composite sketch the diagonal extension 1 of the universe *1* is invoked up front to anchor an explicit line of identity for the leading rèlate of the composition, while the terminal argument w is generalized to the whole universe *1*. Doing this amounts to an act of abstraction from the particular application to w. This form of universal bracketing isolates the serial composition of the relations L and S to form the composite L ◦ S. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.5 https://inquiryintoinquiry.com/2014/04/06/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 10.5 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, We have sufficiently covered the application of the comma functor to absolute terms, so let us return to where we were in working our way through CP 3.73 and see whether we can validate Peirce’s statements about the commafications of dyadic relative terms and the corresponding diagonal extensions to triadic relations. <QUOTE CSP> But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one. Then ℓ,sw will denote a lover of a woman that is a servant of that woman. The comma here after ℓ should not be considered as altering at all the meaning of ℓ , but as only a subjacent sign, serving to alter the arrangement of the correlates. (Peirce, CP 3.73) </QUOTE> Just to plant our feet on a more solid stage, let us apply this idea to the Othello example. For this performance only, just to make the example more interesting, let us assume that Jeste (J) is secretly in love with Desdemona (D). Then we begin with the modified data set: Display 1 https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-comment-10.… And next we derive the following results: Display 2 https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-comment-10.… Now what are we to make of that? If we operate in accordance with Peirce’s example of goh as the “giver of a horse to an owner of that horse” then we may assume the associative law and the distributive law are in force, allowing us to derive this equation: Display 3 https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-comment-10.… Evidently what Peirce means by the associative principle, as it applies to this type of product, is that a product of elementary relatives having the form (R:S:T)(S:T)(T) is equal to R but that no other form of product yields a non-null result. Scanning the implied terms of the triple product tells us that only the case (J:J:D)(J:D)(D) = Jis non‑null. It follows that: Display 4 https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-comment-10.… And so what Peirce says makes sense in this case. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.7 https://inquiryintoinquiry.com/2014/04/18/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 10.7 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, Here is what I get when I analyze Peirce’s “giver of a horse to a lover of a woman” example along the same lines as the dyadic compositions. We may begin with the mark-up shown in Figure 19. Figure 19. Giver of a Horse to a Lover of a Woman https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-glwh-2.0.png If we analyze this in accord with the spreadsheet model of relational composition then the core of it is a particular way of composing a triadic “giving” relation G ⊆ T × U × V with a dyadic “loving” relation L ⊆ U × W so as to obtain a specialized type of triadic relation (G ◦ L) ⊆ T × V × W. The applicable constraints on tuples are shown in Table 20. Table 20. Relational Composition G ◦ L https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-relational-… The hypergraph picture of the abstract composition is given in Figure 21. Figure 21. Anything that is a Giver of Anything to a Lover of Anything https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-composite-o… Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.8 https://inquiryintoinquiry.com/2014/04/22/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 10.8 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, Our progress through the 1870 Logic of Relatives brings us in sight of a critical transition point, one which turns on the “teridentity” relation. The markup for Peirce’s “giver of a horse to an owner of it” is shown again in Figure 22. Figure 22. Giver of a Horse to an Owner of It https://inquiryintoinquiry.files.wordpress.com/2022/01/lor-1870-goh.png The hypergraph picture of the abstract composition is given in Figure 23. Figure 23. Anything that is a Giver of Anything to an Owner of It https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-e280a2goe28… If we analyze this in accord with the spreadsheet model of relational composition then the core of it is a particular way of composing a triadic “giving” relation G ⊆ X × Y × Z with a dyadic “owning” relation O ⊆ Y × Z in such a way as to determine a specialized dyadic relation (G ◦ O) ⊆ X × Z. Table 24 schematizes the associated constraints on tuples. Table 24. Relational Composition G ◦ O https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-relational-… So we see the notorious teridentity relation, which I left equivocally denoted by the same symbol as the identity relation 1, is already implicit in Peirce’s discussion at this point. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.9 https://inquiryintoinquiry.com/2014/04/23/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 10.9 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… <QUOTE AMSB> Ergo in numero quo numeramus repetitio unitatum facit pluralitatem; in rerum vero numero non facit pluralitatem unitatum repetitio, vel si de eodem dicam “gladius unus mucro unus ensis unus”. Therefore in the case of that number by which we number, the repetition of ones makes a plurality; but in the number consisting in things the repetition of ones does not make a plurality, as, for example, if I say of one and the same thing, “one sword, one brand, one blade”. Boethius (Anicius Manlius Severinus Boethius, c. 480–524 A.D.), De Trinitate (The Trinity Is One God Not Three Gods), The Theological Tractates, H.F. Stewart, E.K. Rand, S.J. Tester (trans.), New Edition, Loeb Classical Library, Harvard/Heinemann, 1973. </QUOTE> All, The use of the concepts of identity and teridentity is not to identify a thing-in-itself with itself, much less twice or thrice over — there is no need and thus no utility in that. I can imagine Peirce asking, on Kantian principles if not entirely on Kantian premisses, “Where is the manifold to be unified?” The manifold requiring unification does not reside in the object but in the phenomena — in the appearances which might have been appearances of different objects but are bound by the indicated identities to be just so many aspects, facets, parts, roles, or signs of one and the same object. Notice how the various identity concepts actually functioned in the last example, where they had the opportunity to show their behavior in something like their natural habitat. Figure 23. Anything that is a Giver of Anything to an Owner of It https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-e280a2goe28… The use of the teridentity concept in the “giver of a horse to an owner of it” is to say the thing appearing with respect to its quality under an absolute term, “a horse”, the thing appearing with respect to its existence as the correlate of a dyadic relative, “a potential possession”, and the thing appearing with respect to its synthesis as the correlate of a triadic relative, “a gift”, are one and the same thing. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.11 Peirce’s 1870 “Logic of Relatives” • Comment 10.11 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, Let us return to the point where we left off unpacking the contents of CP 3.73. Here Peirce remarks that the comma operator can be iterated at will. <QUOTE CSP:> In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates. So m,,b,r interpreted like goh means a man that is a rich individual and is a black [person] that is that rich individual. But this has no other meaning than m,b,r or a man that is a black [person] that is rich. Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number. (Peirce, CP 3.73) https://inquiryintoinquiry.com/2014/02/17/peirces-1870-logic-of-relatives-s… </QUOTE> Again, let’s check whether this makes sense on the stage of our small but dramatic model. Let’s say Desdemona and Othello are rich and, among the persons of the play, only they. On this premiss we obtain a sample of absolute terms sufficiently ample to work through Peirce’s example. Display 1. Absolute Terms https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.… One application of the comma operator yields the following dyadic relatives. Display 2. Dyadic Relatives https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.… Another application of the comma operator generates the following triadic relatives. Display 3. Triadic Relatives https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.… Assuming the associativity of multiplication among dyadic relatives, the product m,b,r may be computed by a brute force method to yield the following result. Display 4. m,b,r = O https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.… This says that a man that is black that is rich is Othello, which is true on the premisses of our present universe of discourse. Following the standard associative combinations of goh, the product m,,b,r is multiplied out along the following lines, where the trinomials of the form (X:Y:Z)(Y:Z)(Z) are the only ones producing a non‑null result, namely, (X:Y:Z)(Y:Z)(Z) = X. Display 5. m,,b,r = O https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.… So we have that m,,b,r = m,b,r. In closing, observe how the teridentity relation has turned up again in this context, as the second comma‑ing of the universal term itself. Display 6. Teridentity 1,, https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-10.… Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 10.12 https://inquiryintoinquiry.com/2014/04/26/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 10.12 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_1#C… All, Potential ambiguities in Peirce’s two versions of the “rich black man” example can be resolved by providing them with explicit graphical markups, as shown in Figures 28 and 29. Figure 28. Man that is Black that is Rich https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-mbr.png Figure 29. Man that is a Rich Individual and is a Black Person that is that Rich Individual https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-mbr-1.png On the other hand, as the forms of relational composition become more complex, the corresponding algebraic products of elementary relatives, for example, (x:y:z)(y:z)(z), will not always determine unique results without the addition of more information about the intended linkings of terms. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.1 https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.1 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… Dear Reader, We have reached a suitable place to pause in our reading of Peirce’s text — actually, it’s more like a place to run as fast as we can along a parallel track — where I can pay off a few of the expository IOUs I’ve been using to pave the way to this point. The more pressing debts that come to mind are concerned with the matter of Peirce’s “number of” function that maps a term t into a number [t], and with my justification for calling a certain style of illustration the “hypergraph picture” of relational composition. As it happens, there is a thematic relation between these topics, and so I can make my way forward by addressing them together. At this point we have two good pictures of how to compute the relational compositions of dyadic relations, namely, the bigraph representation and the matrix representation, each of which has its differential advantages in different types of situations. But we lack a comparable picture of how to compute the richer variety of relational compositions involving triadic or higher adicity relations. As a matter of fact, we run into a non-trivial classification problem simply to enumerate the different types of compositions arising in those cases. Therefore let us inaugurate a systematic study of relational composition, general enough to articulate the “generative potency” of Peirce’s 1870 Logic of Relatives. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.3 https://inquiryintoinquiry.com/2014/04/30/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.3 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, Before I can discuss Peirce’s “number of” function in greater detail I will need to deal with an expositional difficulty I have been carefully dancing around all this time, but one which will no longer abide its assigned place under the rug. Functions have long been understood, from well before Peirce’s time to ours, as special cases of dyadic relations, so the “number of” function is already to be numbered among the class of dyadic relatives we’ve been dealing with all this time. But Peirce’s manner of representing a dyadic relative term mentions the “rèlate” first and the “correlate” second, a convention going over into functional terms as making the functional value first and the functional argument second. The problem is, almost anyone brought up in our present time frame is accustomed to thinking of a function as a set of ordered pairs where the order in each pair lists the functional argument first and the functional value second. Syntactic wrinkles of this sort can be ironed out smoothly enough in a framework of flexible interpretive conventions, but not without introducing an order of anachronism into Peirce’s text I want to avoid as much as possible. This will require me to experiment with various styles of compromise. Among other things, the interpretation of Peirce’s 1870 “Logic of Relatives” can be facilitated by introducing a few items of background material on relations in general, as regarded from a combinatorial point of view. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.5 https://inquiryintoinquiry.com/2014/05/02/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.5 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, Everyone knows the right sort of diagram can be a great aid in rendering complex matters comprehensible. With that in mind, let’s extract what we need from the Relation Theory article to illuminate Peirce’s 1870 Logic of Relatives and use it to fashion what icons we can within the current frame of discussion. Relation Theory https://oeis.org/wiki/Relation_theory For the immediate present, we may begin with dyadic relations and describe the most frequently encountered species of relations and functions in terms of their local and numerical incidence properties. Let P ⊆ X × Y be an arbitrary dyadic relation. The following properties of P can then be defined. Display 1 https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.… If P ⊆ X × Y is tubular at X, then P is known as a “partial function” or a “pre-function” from X to Y, frequently signalized by renaming P with an alternate lower case name, say “p”, and writing p : X ⇀ Y. Just by way of formalizing the definition: Display 2 https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.… To illustrate these properties, let us fashion a generic enough example of a dyadic relation, E ⊆ X × Y, where X = Y = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, and where the bigraph picture of E is shown in Figure 30. Figure 30. Dyadic Relation E https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-30.j… If we scan along the X dimension from 0 to 9 we see that the incidence degrees of the X nodes with the Y domain are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0 in that order. If we scan along the Y dimension from 0 to 9 we see that the incidence degrees of the Y nodes with the X domain are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0 in that order. Thus, E is not total at either X or Y since there are nodes in both X and Y having incidence degrees less than 1. Also, E is not tubular at either X or Y since there are nodes in both X and Y having incidence degrees greater than 1. Clearly then the relation E cannot qualify as a pre-function, much less as a function on either of its relational domains. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.7 https://inquiryintoinquiry.com/2014/05/05/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.7 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, We come now to the special cases of dyadic relations known as functions. It will serve a dual purpose in the present exposition to take the class of functions as a source of object examples for clarifying the more abstruse concepts of Relation Theory. To begin, let us recall the definition of a “local flag” L_{a @ j} of a k-adic relation L. Display 1 https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.… For a dyadic relation L ⊆ X × Y the notation for local flags can be simplified in two ways. First, the local flags L_{u @ 1} and L_{v @ 2} are often more conveniently notated as L_{u @ X} and L_{v @ Y}, respectively. Second, the notation may be streamlined even further by making the following definitions. Display 2 https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.… In light of these conventions, the local flags of a dyadic relation L ⊆ X × Y may be comprehended under the following descriptions. Display 3 https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.… The following definitions are also useful. Display 4 https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.… A sufficient illustration is supplied by the earlier example E. Figure 35. Dyadic Relation E https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-30.j… Figure 36 shows the local flag E_{3 @ X} of E. Figure 36. Local Flag E_{3 @ X} https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-36.j… Figure 37 shows the local flag E_{2 @ Y} of E. Figure 37. Local Flag E_{2 @ Y} https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-37.j… Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.9 https://inquiryintoinquiry.com/2014/05/07/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.9 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, Among the variety of regularities affecting dyadic relations we pay special attention to the c-regularity conditions where c is equal to 1. Let P ⊆ X × Y be an arbitrary dyadic relation. The following properties can be defined. Definitions 1 https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.… We previously examined dyadic relations exemplifying each of these regularity conditions. Then we introduced a few bits of terminology and special-purpose notations for working with tubular relations. Definitions 2 https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.… We arrive by way of this winding stair at the special cases of dyadic relations P ⊆ X × Y variously described as “1-regular”, “total and tubular”, or “total prefunctions” on specified domains, X or Y or both, and which are more often celebrated as “functions” on those domains. If P is a pre-function P : X ⇀ Y that happens to be total at X then P is known as a “function” from X to Y, typically indicated as P : X → Y. To say a relation P ⊆ X × Y is “total and tubular” at X is to say P is 1-regular at X. Thus, we may formalize the following definitions. Definitions 3 https://inquiryintoinquiry.files.wordpress.com/2022/02/lor-1870-comment-11.… For example, let X = Y = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and let F ⊆ X × Y be the dyadic relation depicted in the bigraph below. Figure 39. Dyadic Relation F https://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-39.j… We observe that F is a function at Y and we record this fact in either of the manners F : X ← Y or F : Y → X. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.11 https://inquiryintoinquiry.com/2014/05/09/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.11 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, The preceding exercises were intended to beef-up our “functional literacy” skills to the point where we can read our functional alphabets backwards and forwards and recognize the local functionalities immanent in relative terms no matter where they reside among the domains of relations. These skills will serve us in good stead as we work to build a catwalk from Peirce’s platform of 1870 to contemporary scenes on the logic of relatives, and back again. By way of extending a few very tentative planks, let us experiment with the following definitions. • A relative term “p” and the corresponding relation P ⊆ X × Y are both called “functional on rèlates” if and only if P is a function at X. We write this in symbols as P : X → Y. • A relative term “p” and the corresponding relation P ⊆ X × Y are both called “functional on correlates” if and only if P is a function at Y. We write this in symbols as P : X ← Y. When a relation happens to be a function, it may be excusable to use the same name for it in both applications, writing out explicit type markers like P : X × Y, P : X → Y, and P : X ← Y, as the case may be, when and if it serves to clarify matters. From this current, perhaps transient, perspective, it appears our next task is to examine how the known properties of relations are modified when aspects of functionality are spied in the mix. Let us then return to our various ways of looking at relational composition and see what changes and what stays the same when the relations in question happen to be functions of various kinds at some of their domains. Here is one generic picture of relational composition, cast in a style that hews pretty close to the line of potentials inherent in Peirce’s syntax of this period. Figure 44. Anything that is a P of a Q of Anything https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-universal-b… From this we extract the “hypergraph picture” of relational composition. Figure 45. Relational Composition P ◦ Q https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-p-e297a6-q-… All the information contained in these Figures can be expressed in the form of a constraint satisfaction table, or “spreadsheet picture” of relational composition. Table 46. Relational Composition P ◦ Q https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-relational-… The following plan of study then presents itself: to see what easy mileage we can get in our exploration of functions by adopting the above templates as the primers of a paradigm. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.13 https://inquiryintoinquiry.com/2014/05/14/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.13 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, As we make our way toward the foothills of Peirce’s 1870 Logic of Relatives there are several pieces of equipment we must not leave the plains without, namely, the utilities variously known as arrows, morphisms, homomorphisms, structure-preserving maps, among other names, depending on the altitude of abstraction we happen to be traversing at the moment in question. As a moderate to middling but not too beaten track, let’s examine a few ways of defining morphisms that will serve us in the present discussion. Suppose we are given three functions J, K, L satisfying the following conditions. • J : X ← Y • K : X ← X × X • L : Y ← Y × Y • J(L(u, v)) = K(Ju, Jv) Our sagittarian leitmotif can be rubricized in the following slogan. • “The J-image of the L-product is the K-product of the J-images.” Figure 47 presents us with a picture of the situation in question. Figure 47. Structure Preserving Transformation J : K ← L https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-structure-p… Table 48 gives the constraint matrix version of the same thing. Table 48. Structure Preserving Transformation J : K ← L https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-j-makes-k-f… One way to read the Table is in terms of the informational redundancies it summarizes. For example, one way to read it says that satisfying the constraint in the L row along with all the constraints in the J columns automatically satisfies the constraint in the K row. Quite by design, that is one way to understand the equation J(L(u, v)) = K(Ju, Jv). Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.15 https://inquiryintoinquiry.com/2014/05/15/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.15 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, I’m going to elaborate a little further on the subject of arrows, morphisms, or structure-preserving mappings, as a modest amount of extra work at this point will repay ample dividends when it comes time to revisit Peirce’s “number of” function on logical terms. The “structure” preserved by a structure-preserving map is just the structure we all know and love as a triadic relation. Very typically, it will be the type of triadic relation that defines the type of binary operation that obeys the rules of a mathematical structure known as a “group”, that is, a structure satisfying the axioms for closure, associativity, identities, and inverses. For example, in the case of the logarithm map J we have the following data. • J : Reals ← Reals (properly restricted) • K : Reals ← Reals × Reals where K(r, s) = r + s • L : Reals ← Reals × Reals where L(u, v) = u ⋅ v Real number addition and real number multiplication (suitably restricted) are examples of group operations. If we write the sign of each operation in brackets as a name for the triadic relation that defines the corresponding group, we have the following set-up. • J : [+] ← [⋅] • [+] ⊆ Reals × Reals × Reals • [⋅] ⊆ Reals × Reals × Reals It often happens that both group operations are indicated by the same sign, usually one from the set { ⋅ , ∗ , + } or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used. In such a setting, our chiasmatic theme may run a bit like one of the following two variants. • “The image of the sum is the sum of the images.” • “The image of the product is the sum of the images.” Figure 50 presents a generic picture for groups G and H. Figure 50. Group Homomorphism J : G ← H https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-group-homom… In a setting where both groups are written with a plus sign, perhaps even constituting the same group, the defining formula of a morphism, J(L(u, v)) = K(Ju, Jv), takes on the shape J(u + v) = Ju + Jv, which looks analogous to the distributive multiplication of a factor J over a sum (u + v). That is why morphisms are regarded as generalizations of linear functions and are frequently referred to in those terms. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.17 https://inquiryintoinquiry.com/2014/05/27/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.17 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, I think the reader is beginning to get an inkling of the crucial importance of the “number of” function in Peirce’s way of looking at logic. It is one plank in the bridge from logic to the theories of probability, statistics, and information, in which setting logic forms but a limiting case at one scenic turnout on the expanding vista. It is one of the ways Peirce forges a link between the “eternal”, logical, or rational realm and the “secular”, empirical, or real domain. With that note of encouragement and exhortation, let us return to the details of the text. NOF 2 ===== <QUOTE CSP:> But not only do the significations of = and < here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write 5 < 7 is to say that 5 is part of 7, just as to write f < m is to say that Frenchmen are part of men. Indeed, if f < m , then the number of Frenchmen is less than the number of men, and if v = p , then the number of Vice‑Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities. (Peirce, CP 3.66) https://inquiryintoinquiry.com/2014/01/30/peirces-1870-logic-of-relatives-s… </QUOTE? Peirce is here remarking on the principle that the measure v on logical terms preserves or respects the prevailing relations of implication, inclusion, or subsumption which impose an ordering on those terms. In these passages, Peirce is using a single symbol “<” to denote the usual linear ordering on numbers, but also what amounts to the implication ordering on logical terms and the inclusion ordering on classes. Later he will introduce distinctive symbols for logical orders. The links among terms, sets, and numbers can be pursued in all directions and Peirce has already indicated in an earlier paper how he would construct the integers from sets, that is, from the aggregate denotations of terms. I will try to get back to that another time. We have a statement of the following form. • If f < m then the number of Frenchmen is less than the number of men. This goes into symbolic form as follows. • f < m ⇒ [f] < [m]. In this setting the “<” on the left is a logical ordering on syntactic terms while the “<” on the right is an arithmetic ordering on real numbers. The question that arises in this case is whether a map between two ordered sets is “order-preserving”. In order to formulate the question in more general terms, we may begin with the following set-up. • Let X₁ be a set with the ordering <₁. • Let X₂ be a set with the ordering <₂. An order relation is typically defined by a set of axioms that determines its properties. Since we have frequent occasion to view the same set in the light of several different order relations, we often resort to explicit specifications like (X, <₁), (X, <₂), and so on to indicate a set with a given ordering. A map F : (X₁, <₁) → (X₂, <₂) is order-preserving if and only if a statement of a particular form holds for all x and y in (X₁, <₁), namely, the following. • x <₁ y ⇒ F(x) <₂ F(y). The “number of” map v : (S, <₁) → (R, <₂) has just this character, as exemplified in the case at hand. • f < m ⇒ [f] < [m] • f < m ⇒ v(f) < v(m) The “<” on the left is read as proper inclusion, in other words, “subset of but not equal to”, while the “<” on the right is read as the usual less than relation. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.18 https://inquiryintoinquiry.com/2014/05/28/peirces-1870-logic-of-relatives-c… [forgot the main link] Peirce’s 1870 “Logic of Relatives” • Comment 11.18 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, An “order-preserving map” is a special case of a structure-preserving map and the idea of preserving structure, as used in mathematics, means preserving some but not necessarily all the structure of the source domain in the transition to the target domain. In that vein, we may speak of “structure preservation in measure”, the suggestion being that a property able to be qualified in manner is potentially able to be quantified in degree, admitting answers to questions like, “How structure-preserving is it?” Let’s see how this applies to Peirce’s “number of” function v : S → R. Let “—<” denote the implication relation on logical terms, let “≤” denote the less than or equal to relation on real numbers, and let x, y be any pair of absolute terms in the syntactic domain S. Then we observe the following relationships. • x —< y ⇒ v(x) ≤ v(y) Equivalently: • x —< y ⇒ [x] ≤ [y] Nowhere near the number of logical distinctions on the left sides of the implication arrows are typically preserved as one passes to the linear orderings of real numbers on their right sides but that is not required in order to call the map v : S → R order-preserving, or what is known as an “order morphism”. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.20 https://inquiryintoinquiry.com/2014/06/01/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.20 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, We come to the last of Peirce’s statements about the “number of” function, first quoted in Selection 11 and again with the whole set in Comment 11.2. https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-s… https://inquiryintoinquiry.com/2014/04/30/peirces-1870-logic-of-relatives-c… NOF 4.1 ======= <QUOTE CSP:> The conception of multiplication we have adopted is that of the application of one relation to another. […] Even ordinary numerical multiplication involves the same idea, for 2 × 3 is a pair of triplets, and 3 × 2 is a triplet of pairs, where “triplet of” and “pair of” are evidently relatives. If we have an equation of the form • xy = z and there are just as many x’s per y as there are, per things, things of the universe, then we have also the arithmetical equation, • [x][y] = [z]. (Peirce, CP 3.76) </QUOTE> Peirce here observes what may be called a “contingent morphism”. On a condition he gives, the mapping of logical terms to their corresponding numbers preserves the multiplication of relative terms after the fashion of the following formula. • v(xy) = v(x) v(y). Equivalently: • [xy] = [x][y]. The condition for this to hold is expressed by Peirce in the following manner. “There are just as many x’s per y as there are, per things, things of the universe.” Peirce’s phrasing on this point is admittedly hard to parse but I think if we stick with his story to the end we can see what he is driving at. NOF 4.2 ======= <QUOTE CSP:> For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then • [t][f] = [tf] holds arithmetically. (Peirce, CP 3.76). </QUOTE> Now that is something we can sink our teeth into and trace the bigraph representation of the situation. It will help to recall our first examination of the “tooth of” relation and to adjust the picture we sketched of it on that occasion. Transcribing Peirce’s example: Let m = man and t = tooth of___. Then v(t) = [t] = [tm]/[m]. That is to say, the number of the relative term “tooth of” is equal to the number of teeth of humans divided by the number of humans. In a universe of perfect human dentition this gives a quotient of 32. The dyadic relative term t determines a dyadic relation T ⊆ X × Y, where X contains all the teeth and Y contains all the people under discussion. To make the case as simple as possible and still cover the point, suppose there are just four people in our universe of discourse and just two of them are French. The bigraph product below shows the pertinent facts of the case. Figure 52. Bigraph Product T ◦ F https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-bigraph-pro… In this picture the order of relational composition flows down the page. For convenience in composing relations, the absolute term f = Frenchman is inflected by the comma functor to form the dyadic relative term f, = Frenchman that is___, which in turn determines the idempotent representation of Frenchmen as a subset of mankind, F ⊆ Y × Y. Now let’s see if we can use this picture to make sense of the following statement. NOF 4.2 ======= <QUOTE CSP:> For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then • [t][f] = [tf] holds arithmetically. (Peirce, CP 3.76). </QUOTE> In statistical terms, Peirce is saying this: If the population of Frenchmen is a *fair sample* of the general population with regard to the factor of dentition, then the morphic equation, • [tf] = [t][f], whose transpose gives the equation, • [t] = [tf]/[f], is every bit as true as the defining equation in this circumstance, namely, • [t] = [tm]/[m]. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.22 https://inquiryintoinquiry.com/2014/06/04/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.22 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, Let’s look at that last example from a different angle. NOF 4.3 ======= <QUOTE CSP:> So if men are just as apt to be black as things in general, • [m,][b] = [m,b], where the difference between [m] and [m,] must not be overlooked. (Peirce, CP 3.76) https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-s… </QUOTE> Viewed in various lights the formula [m,b] = [m,][b] presents itself as an aimed arrow, fair sampling, or stochastic independence condition. Peirce’s example assumes a universe of things in general encompassing the denotations of the absolute terms m = man and b = black. That allows us to illustrate the case in relief, by returning to our earlier staging of Othello and examining the premiss that “men are just as apt to be black as things in general” within the frame of that that empirical if fictional universe of discourse. We have the following data. • b = O • m = C +, I +, J +, O • *1* = B +, C +, D +, E +, I +, J +, O • b, = O:O • m, = C:C +, I:I +, J:J +, O:O • *1*, = B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O The “fair sampling condition” amounts to saying men are just as likely to be black as things in general are likely to be black. In other words, men are a fair sample of things in general with respect to the predicate of being black. On that condition the following equation holds. • [m,b] = [m,][b]. Assuming [b] is not zero, the next equation follows. • [m,] = [m,b]/[b]. As before, it is convenient to represent the absolute term b = black by means of the corresponding idempotent term b, = black that is___. Let is next consider the bigraph for the following relational product. • m,b = man that is black. We may represent that in the following equivalent form. • m,b, = man that is black that is___. Figure 53. Bigraph Product M,B, https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-othello-pro… The facts of the matter in the Othello case are such that the following formula holds. • m,b = b. And that in turn is equivalent to each of the following statements. • m ∧ b = b • b ⇒ m • b —< m Those last implications puncture any notion of statistical independence for b and m in the universe of discourse at hand but it will repay us to explore the details of the case a little further. Putting all the general formulas and particular facts together, we arrive at the following summation of the situation in the Othello case. If the fair sampling condition were true, it would have the following consequence. • [m,] = [m,b]/[b] = [b]/[b] = 1. On the contrary, we have the following fact. • [m,] = [m,*1*]/[*1*] = [m]/[*1*] = 4/7. In sum, it is not the case in the Othello example that “men are just as apt to be black as things in general”. Expressed in terms of probabilities: • P(m) = 4/7 and P(b) = 1/7. If these were independent terms, we would have: • P(mb) = 4/49. In point of fact, however, we have: • P(mb) = P(b) = 1/7. Another way to see it is to observe that: • P(b|m) = 1/4 while P(b) 1/7. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 11.24 https://inquiryintoinquiry.com/2014/06/08/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 11.24 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… Note. Please see the blog post linked above for the proper formatting, as some of what's discussed below depends on the typography in various editions and transcriptions of Peirce's text. All, We come to the last of Peirce's observations about the “number of” function from CP 3.76. NOF 4.4 ======= <QUOTE CSP:> It is to be observed that • [\mathit{1}] = 1. Boole was the first to show this connection between logic and probabilities. He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of “expectation”. Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it. (Peirce, CP 3.76 and CE 2, 376) https://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-s… </QUOTE> There are problems with the printing of the text at this point. To recall the conventions we are using in this transcription, \mathit{1} is the italic 1 denoting the dyadic identity relation \mathfrak{1} is the “antique figure one” which Peirce defines as 1_∞ = “something”. Collected Papers CP 3 gives [\mathit{1}] = \mathfrak{1}, which does not make sense. Chronological Edition CE 2 gives the 1's in different styles of italics but reading the equation as [\mathit{1}] = 1 makes better sense if the latter “1” is the numeral denoting the natural number 1 and not the absolute term “1” denoting the universe of discourse. The quantity [\mathit{1}] is defined as the average number of things related by the identity relation \mathit{1} to one individual, and so it makes sense that [\mathit{1}] = 1 in N, where N is the set of non-negative integers {0, 1, 2, …}. With respect to the relative term \mathit{1} in the syntactic domain S and the number 1 in the non-negative integers N we have the following. • v(\mathit{1}) = [\mathit{1}] = 1. At long last, then, the “number of” mapping v : S → R has another one of the properties required of an arrow from logical terms in S to real numbers in R. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 12.1 https://inquiryintoinquiry.com/2014/06/10/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 12.1 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, To get a better sense of why Peirce’s formulas in Selection 12 mean what they do, and to prepare the ground for understanding more complex relational expressions, it will help to assemble the following materials and definitions. • X is a set singled out in a particular discussion as the “universe of discourse”. • W ⊆ X is the monadic relation, or set, whose elements fall under the absolute term w = woman. The elements of W are referred to as the “denotation” or “extension” of the term w. • L ⊆ X × X is the dyadic relation associated with the relative term ℓ = lover of___. • S ⊆ X × X is the dyadic relation associated with the relative term s = servant of___. • \mathsf{W} = Mat(W) = Mat(w) is the 1-dimensional matrix representation of the set W and the term w. • \mathsf{L} = Mat(L) = Mat(ℓ) is the 2-dimensional matrix representation of the relation L and the relative term ℓ. • \mathsf{S} = Mat(S) = Mat(s) is the 2-dimensional matrix representation of the relation S and the relative term s. The “local flags” of the relation L are defined as follows. • u ∗ L = L_u@1 = {(u, x) ∈ L} = ordered pairs in L with u in the 1st place. • L ∗ v = L_v@2 = {(x, v) ∈ L} = ordered pairs in L with v in the 2nd place. The “applications” of the relation L are defined as follows. • u ∙ L = proj₂(u ∗ L) = {x ∈ X : (u, x) ∈ L} = loved by u. • L ∙ v = proj₁(L ∗ v) = {x ∈ X : (x, v) ∈ L} = lover of v. Regards, Jon

Cf: Peirce's 1870 “Logic of Relatives” • Comment 12.2 (part 2 of 3) https://inquiryintoinquiry.com/2014/06/11/peirces-1870-logic-of-relatives-c… Peirce's 1870 “Logic of Relatives” • Comment 12.2 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, It is instructive to examine the matrix representation of ℓ^w at this point, not the least because it effectively dispels the mystery of the name “involution”. First, we make the following observation. To say j is a lover of every woman is to say j loves k if k is a woman. This can be rendered in symbols as follows. • j loves k ⇐ k is a woman. Reading the formula ℓ^w as “j loves k if k is a woman” highlights the operation of converse implication inherent in it, and this in turn reveals the analogy between implication and involution that accounts for the aptness of the latter name. The operations defined by the formulas x^y = z and (x ⇐ y) = z for x, y, z in the boolean domain B = {0, 1} are tabulated as follows. Table. Involution ≅ Implication https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-involution-… It is clear the two operations are isomorphic, being effectively the same operation of type B × B → B. All that remains is to see how operations like these on values in B induce the corresponding operations on sets and terms. To be continued ... Regards, Jon

Peirce’s 1870 “Logic of Relatives” • Comment 12.3 (part 1 of 2) https://inquiryintoinquiry.com/2014/06/12/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 12.3 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, We now have two ways of computing a logical involution raising a dyadic relative term to the power of a monadic absolute term, for example, ℓ^w for “lover of every woman”. The first method applies set-theoretic operations to the extensions of absolute and relative terms, expressing the denotation of the term ℓ^w as the intersection of a set of relational applications. Equation 1. Denotation Equation L^W https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-denotation-… The second method operates in the matrix representation, expressing the value of the matrix L^W = Math(ℓ^w) at an argument u as a product of coefficient powers. Equation 2. Matrix Computation L^W https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-matrix-comp… Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved. Next time we'll take up one such example. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Comment 12.4 https://inquiryintoinquiry.com/2014/06/14/peirces-1870-logic-of-relatives-c… Peirce’s 1870 “Logic of Relatives” • Comment 12.4 https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Part_2#C… All, Peirce next considers a pair of compound involutions, stating an equation between them analogous to a law of exponents from ordinary arithmetic, namely, (a^b)^c = a^{bc}. Law of Exponents (a^b)^c = a^(bc) https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-a5eb5ec-a5e… <QUOTE CSP:> Then (s^ℓ)^w will denote whatever stands to every woman in the relation of servant of every lover of hers; and s^(ℓw) will denote whatever is a servant of everything that is lover of a woman. So that (s^ℓ)^w = s^(ℓw). https://inquiryintoinquiry.files.wordpress.com/2022/03/lor-1870-s5el5ew-s5e… (Peirce, CP 3.77) https://inquiryintoinquiry.com/2014/06/09/peirces-1870-logic-of-relatives-s… </QUOTE> Articulating the compound relative term s^(ℓw) in set-theoretic terms is fairly immediate. Denotation Equation s^(ℓw) https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-denotation-… On the other hand, translating the compound relative term (s^ℓ)^w into its set-theoretic equivalent is less immediate, the hang-up being we have yet to define the case of logical involution raising one dyadic relative term to the power of another. As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example. Involution Example 2 ==================== Consider a universe of discourse X subject to the following data. • X = {a, b, c, d, e, f, g, h, i} • L = {b:a, b:c, c:b, c:d, e:d, e:e, e:f, g:f, g:h, h:g, h:i} • S = {b:a, b:c, d:c, d:d, d:e, f:e, f:f, f:g, h:g, h:i} Figure 56. Bigraph for the Involution S^L https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-bigraph-inv… There is a “servant of every lover of” link between u and v if and only if u∙S ⊇ L∙v. But the vacuous inclusions, that is, the cases where L∙v = ∅, have the effect of adding non‑intuitive links to the mix. The computational requirements are evidently met by the following formula. Matrix Computation for S^L https://inquiryintoinquiry.files.wordpress.com/2022/04/lor-1870-matrix-comp… In other words, (S^L)_xy = 0 if and only if there exists a p in X such that S_xp = 0 and L_py = 1. Regards, Jon

Cf: Peirce’s 1870 “Logic of Relatives” • Intermezzo https://inquiryintoinquiry.com/2014/06/18/peirces-1870-logic-of-relatives-i… Peirce’s 1870 “Logic of Relatives” https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview Update • 10 April 2022 All, This brings me to the end of the notes on Peirce’s 1870 Logic of Relatives I began posting to the web in various online discussion groups a dozen (now a score) years ago. There are apart from that only scattered notes and bits of discussion with other people I archived on the InterSciWiki talk page of the OEIS Wiki article linked above. I rushed through my last few comments a little too hastily, giving no more than sketches of proofs for Peirce’s logical formulas, and I won’t be reasonably well convinced of them until I examine a few more concrete examples and develop one or two independent lines of proof. So I have that much unfinished business to do before moving on to the rest of Peirce’s paper. But I’ll take a few days to catch my breath, rummage through those old notes of mine to see if they hide any hints worth salvaging, and then start fresh, raveling out the rest of Peirce’s clues to the maze of logical relatives. Regards, Jon