Inquiry Into Inquiry

Peirce’s 1870 “Logic of Relatives” • Comment 12.2

Let us make a few preliminary observations about the operation of logical involution which Peirce introduces in the following words.

In ordinary arithmetic the involution or the exponentiation of to the power is the repeated application of the multiplier for as many times as there are ones making up the exponent

In analogous fashion, the logical involution is the repeated application of the term for as many times as there are individuals under the term   On Peirce’s interpretive rules, the repeated applications of the base term are distributed across the individuals of the exponent term   In particular, the base term is not applied successively in the manner that would give something on the order of “a lover of a lover of … a lover of a woman”.

By way of example, suppose a universe of discourse numbers among its elements just three women,   In Peirce’s notation the fact may be written as follows.

In that case the following equation would hold.

The equation says a lover of every woman in the aggregate is a lover of that is a lover of that is a lover of   In other words, a lover of every woman in the universe at hand is a lover of and a lover of and a lover of

The denotation of the term is a subset of which may be obtained by the following procedure.  For each flag of the form with collect the subset of elements which appear as the first components of the pairs in and then take the intersection of all those subsets.  Putting it all together, we have the following equation.

It is instructive to examine the matrix representation of at this point, not the least because it effectively dispels the mystery of the name involution.  First, we make the following observation.  To say is a lover of every woman is to say loves if is a woman.  That can be rendered in symbols as follows.

Reading the formula as “ loves if is a woman” highlights the operation of converse implication inherent in it, and this in turn reveals the analogy between implication and involution which accounts for the aptness of the latter name.

The operations defined by the formulas in the boolean domain are tabulated as follows.

It is clear the two operations are isomorphic, being effectively the same operation of type   All that remains is to see how operations like these on values in induce the corresponding operations on sets and terms.

The term determines a selection of individuals from the universe of discourse which may be computed via the corresponding operation on coefficient matrices.  If the terms and are represented by the matrices and respectively, then the operation on terms which produces the term must be represented by a corresponding operation on matrices, which produces the matrix   In short, the involution operation on matrices must be defined in such a way that the following equation holds.

The fact that denotes individuals in a subset of tells us its matrix representation is a 1‑dimensional array of coefficients in indexed by the elements of   The value of the matrix at the index in is written and computed as follows.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

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Peirce’s 1870 “Logic of Relatives” • Comment 12.1

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.

A few concepts from the article on Relation Theory, touched on again in Comment 11.7, will also be useful.

The local flags of the relation are defined as follows.

The applications of the relation are defined as follows.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

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On to the next part of §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 12

References

• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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Peirce’s 1870 “Logic of Relatives” • Comment 11.24

We come to the last of Peirce’s observations about the “number of” function in CP 3.76.

NOF 4.4

There are problems with the printing of text at this point.  To recall the conventions used in this transcription, the italic figure denotes the dyadic identity relation while the antique figure denotes what Peirce otherwise defines as

Collected Papers CP 3 gives which does not make sense.  Chronological Edition CE 2 gives the 1’s in different styles of italics but reading the equation as 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 is defined as the average number of things related by the identity relation to one individual, and so it makes sense that where is the set of non-negative integers

With respect to the relative term in the syntactic domain and the number in the non-negative integers we have the following.

At long last, then, the “number of” mapping has another one of the properties required of an arrow from logical terms in to real numbers in

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

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Peirce’s 1870 “Logic of Relatives” • Comment 11.23

Peirce’s description of logical conjunction and conditional probability via the logic of relatives and the mathematics of relations is critical to understanding the relationship between logic and measurement, in effect, the qualitative and quantitative aspects of inquiry.  To root that connection firmly in mind, I will try to sum up as succinctly as possible, in more current notation, the lesson we ought to take away from Peirce’s last “number of” example, since I know the account I have given so far may appear to have wandered widely.

NOF 4.3

Viewed in different lights the formula presents itself as an aimed arrow, fair sampling, or statistical independence condition.  The concept of independence was illustrated in the previous installment by means of a case where independence fails.  The details of that counterexample are summarized below.

The condition that “men are just as apt to be black as things in general” is expressed in terms of conditional probabilities as which means that the probability of the event given the event is equal to the unconditional probability of the event

In the Othello example it is enough to observe that while in order to recognize the bias or dependency of the sampling map.

The reduction of a conditional probability to an absolute probability, as is one of the ways we come to recognize the condition of independence, via the definition of conditional probability,

By way of recalling the derivation, the definition of conditional probability plus the independence condition yields the following sequence of equations.

As Hamlet discovered, there’s a lot to be learned from turning a crank.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

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Peirce’s 1870 “Logic of Relatives” • Comment 11.22

Let’s look at that last example from a different angle.

NOF 4.3

Viewed in various lights the formula 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 and   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 empirical if fictional universe of discourse.

We have the following data.

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.

Assuming is not zero, the next equation follows.

As before, it is convenient to represent the absolute term by means of the corresponding idempotent term  

Let is next consider the bigraph for the following relational product.

We may represent that in the following equivalent form.

The facts of the matter in the Othello case are such that the following formula holds.

And that in turn is equivalent to each of the following statements.

Those last implications puncture any notion of statistical independence for and 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.

On the contrary, we have the following fact.

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:

  and  

If these were independent terms, we would have:

In point of fact, however, we have:

Another way to see it is to observe that:

  while  

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

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Peirce’s 1870 “Logic of Relatives” • Comment 11.21

One more example and one more general observation and we’ll be caught up with our homework on Peirce’s “number of” function.

NOF 4.3

The protasis, men are just as apt to be black as things in general, is elliptic in structure and presents us with a potential ambiguity.  If we had no further clue to its meaning, it might be read as either one of the following statements.

1. Men are just as apt to be black as things in general are apt to be black.
2. Men are just as apt to be black as men are apt to be things in general.

The second interpretation, if grammatical, is pointless to state, since it equates a proper contingency with an absolute certainty.  So I think it is safe to assume the following paraphrase of what Peirce intends.

• Men are just as likely to be black as things in general are likely to be black.

Stated in terms of conditional probability, we have the following equation.

From the definition of conditional probability:

Equivalently:

Taking everything together, we have the following result.

That, of course, is the definition of independent events, as applied to the event of being Black and the event of being a Man.  We may take that as the most likely reading of Peirce’s statement about frequencies:

The terms of that equation can be normalized to produce the corresponding statement about probabilities.

Let’s see if that reading checks out.

Let be the number of things in general.  Expressed in Peirce’s notation we have the equation   On the assumption that and are associated with independent events, we obtain the following sequence of equations.

As a result, we have to interpret = “the average number of men per things in general” as = “the probability of a thing in general being a man”.  That seems to make sense.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

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Peirce’s 1870 “Logic of Relatives” • Comment 11.20

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.

NOF 4.1

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.

Equivalently:

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 if we stick with his story to the end I think we can see what he’s driving at.

NOF 4.2

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:

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 determines a dyadic relation where contains all the teeth and 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.

In this picture the order of relational composition flows down the page.  For convenience in composing relations, the absolute term is inflected by the comma functor to form the dyadic relative term which in turn determines the idempotent representation of Frenchmen as a subset of mankind,

By way of a legend for the Figure, we have the following data.

We can use this picture to make sense of Peirce’s statement, repeated below.

NOF 4.2

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,

whose transpose gives the equation,

is every bit as true as the defining equation in this circumstance, namely,

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

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Peirce’s 1870 “Logic of Relatives” • Comment 11.19

Up to this point in the 1870 Logic of Relatives, Peirce has introduced the “number of” function on logical terms, such that and discussed the extent to which its use as a measure satisfies the relevant measure-theoretic principles, beginning with the following two.

1. The “number of” map exhibits a certain type of uniformity property, where the value of the measure on a uniformly qualified population is in fact actualized by each member of the population.
2. The “number of” map satisfies an order morphism principle, where the partial order of logical terms under implication or inclusion is reflected to a moderate degree by the linear order of their measures.

In Selection 4 Peirce takes up the action of the “number of” function on two types of more or less additive operations we normally consider in logic.

NOF 3.1

It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions.  (CP 3.67).

Peirce uses the sign “” to indicate what he calls the “regular non-invertible addition”, corresponding to the inclusive disjunction of logical terms or the union of their extensions as sets.

Peirce uses the sign “” to indicate what he calls the “invertible addition”, corresponding to the exclusive disjunction of logical terms or the symmetric difference of their extensions as sets.

NOF 3.2

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.  (CP 3.67).

A full interpretation of the above remark would require us to pick up the precise technical sense in which Peirce is using the word collection and that would take us back to his logical reconstruction of certain aspects of number theory, all of which I am putting off to another time, but it is still possible to get a rough sense of what he is saying relative to the present frame of discussion.

The “number of” map evidently induces some sort of morphism with respect to logical sums.  If this were true in the strictest sense, we could remove the question marks from the following dubious equations.

Equivalently:

Of course, things are not quite that simple when it comes to inclusive disjunctions and set‑theoretic unions, so it is usual to introduce the concept of a sub‑additive measure to describe the principle that does hold here, namely, the following.

Equivalently:

That is why Peirce trims his discussion of the point with the following hedge.

NOF 3.3

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.  (CP 3.67).

Finally, a morphism with respect to addition, even a contingently qualified one, must do the right thing on behalf of the additive identity element, as follows.

NOF 3.4

With respect to the nullity in and the number in we have the following equation.

In sum, therefore, it can be said:   A measure only serves which also preserves a due respect for the function of a vacuum in nature.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

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Peirce’s 1870 “Logic of Relatives” • Comment 11.18

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   Let denote the implication relation on logical terms, let denote the less than or equal to relation on real numbers, and let be any pair of absolute terms in the syntactic domain   Then we observe the following relationships.

Equivalently:

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 order-preserving, or what is known as an order morphism.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
• Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)