Peirce’s 1870 “Logic of Relatives” • Comment 10.11

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.

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.

One application of the comma operator yields the following dyadic relatives.

Another application of the comma operator generates the following triadic relatives.

Assuming the associativity of multiplication among dyadic relatives, the product may be computed by a brute force method to yield the following result.

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 the product is multiplied out along the following lines, where the trinomials of the form are the only ones producing a non‑null result, namely,

So we have that

In closing, observe how the teridentity relation has turned up again in this context, as the second comma‑ing of the universal term itself.

Resources

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

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