Peirce’s 1870 “Logic of Relatives” • Comment 11.5

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.

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 be an arbitrary dyadic relation.  The following properties of can then be defined.

If is tubular at then is known as a partial function or a pre-function from to frequently signalized by renaming with an alternate lower case name, say and writing

Just by way of formalizing the definition:

To illustrate these properties, let us fashion a generic enough example of a dyadic relation, where and where the bigraph picture of is shown in Figure 30.

If we scan along the dimension from to we see that the incidence degrees of the nodes with the domain are in that order.

If we scan along the dimension from to we see that the incidence degrees of the nodes with the domain are in that order.

Thus, is not total at either or since there are nodes in both and having incidence degrees less than

Also, is not tubular at either or since there are nodes in both and having incidence degrees greater than

Clearly then the relation cannot qualify as a pre-function, much less as a function, on either of its relational domains.

Resources

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

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