## Six Ways of Looking at a Triadic Relation ⌬ 1

A triadic relation and its converses form a set of $3! = 6$ triadic relations all together, six grammatically and rhetorically different ways of representing what is logically the same information.  Peirce illustrates the situation as follows, with six variations on the theme of giving.

So in a triadic fact, say, for example

$A ~\mathrm{gives}~ B ~\mathrm{to}~ C$

we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object.  We say that the proposition has three logical subjects.  We regard it as a mere affair of English grammar that there are six ways of expressing this:

$\begin{array}{lll} A ~\mathrm{gives}~ B ~\mathrm{to}~ C & \qquad & A ~\mathrm{benefits}~ C ~\mathrm{with}~ B \\ B ~\mathrm{enriches}~ C ~\mathrm{at~expense~of}~ A & \qquad & C ~\mathrm{receives}~ B ~\mathrm{from}~ A \\ C ~\mathrm{thanks}~ A ~\mathrm{for}~ B & \qquad & B ~\mathrm{leaves}~ A ~\mathrm{for}~ C \end{array}$

These six sentences express one and the same indivisible phenomenon.

### References

• Peirce, C.S., “The Categories Defended”, Harvard Lectures on Pragmatism : Lecture 3 (MS 308, delivered on 9 April 1903).  Published in Collected Papers (CP 5.66–81, 88–92, in part), Harvard Lectures (HL 167–188), Essential Peirce : Volume 2 (EP 2, 160–178).
• 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.  Volume 5 : Pragmatism and Pragmaticism, 1934.  (Cited as CP).
• Peirce, C.S., Pragmatism as a Principle and Method of Right Thinking : The 1903 Harvard Lectures on Pragmatism, Patricia Ann Turrisi (ed.), State University of New York Press, Albany, NY, 1997.  (Cited as HL).
• Peirce, C.S., The Essential Peirce : Selected Philosophical Writings, Volume 2 (1893–1913), Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1998.  (Cited as EP 2).