A idea of what Peirce means by an Interpretant and the part it plays in a triadic sign relation is given by the following passage.
It is clearly indispensable to start with an accurate and broad analysis of the nature of a Sign. I define a Sign as anything which is so determined by something else, called its Object, and so determines an effect upon a person, which effect I call its Interpretant, that the latter is thereby mediately determined by the former. My insertion of “upon a person” is a sop to Cerberus, because I despair of making my own broader conception understood. (Peirce 1908, Selected Writings, p. 404).
According to his custom of clarifying ideas in terms of their effects, Peirce tells us what a sign is in terms of what it does, the effect it brings to bear on a “person”. That effect he calls the interpretant of the sign. And what of that person? Peirce finesses that question for the moment, resorting to a “Sop to Cerberus”, in other words, a rhetorical gambit used to side‑step a persistent difficulty of exposition. In doing so, Peirce invokes the hypostatic abstraction of a “person” who conducts the movement of signs and embodies the ongoing process of semiosis.
Reference
- Peirce, C.S. (1908), “Letters to Lady Welby”, Chapter 24, pp. 380–432 in Charles S. Peirce : Selected Writings (Values in a Universe of Chance), Edited with Introduction and Notes by Philip P. Wiener, Dover Publications, New York, NY, 1966.
Resources
cc: Academia.edu • BlueSky • Laws of Form • Mathstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
are shown in the next two Figures. 
![[\mathcal{X}] \to [\mathcal{Y}]](https://s0.wp.com/latex.php?latex=%5B%5Cmathcal%7BX%7D%5D+%5Cto+%5B%5Cmathcal%7BY%7D%5D&bg=ffffff&fg=000000&s=0&c=20201002)
is implied any time we make use of one basis 
which happens to be included in another basis 
When discussing differential relations we usually have in mind the extended alphabet 
has a special construction or a specific lexical relation with respect to the initial alphabet 
one which is marked by characteristic types of accents, indices, or inflected forms.
‑gear curves, the indexing scheme results in the data of the next two Tables, showing one period for each orbit.
where 
may be read as a temporal parameter indicating the present time of the state and where 
is the decimal equivalent of the binary numeral 
with a subscript 
equal to the numerator of its rational index, taking for granted the constant denominator 
In that way the temporal succession of states can be reckoned by a parallel round‑up rule. Namely, if 
is any pair of adjacent digits in the state index 
then the value of 
in the next state is 
with the dyadic rationals (or the binary fractions) in the half-open interval 
Formally and canonically, a state 
is indexed by a fraction 
whose denominator is the power of two 
and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next.
are the values 
for 
where 
is defined as identical to 
To form the binary index 
of the state 
Expressed in algebraic terms, the rational index 
Those are the trajectories generated subject to the dynamic law 
where it’s understood all higher order differences are equal to 
and all higher differences 
are fixed, the state vectors vary only with respect to their projections as points of 
Thus there is just enough space in a planar venn diagram to plot the orbits and show how they partition the points of 
It turns out there are just two possible orbits, of eight points each, as shown in the following Figure.
The discussion to follow considers a class of trajectories having the property that 
for all 
greater than a fixed value 
and indulges in the use of a picturesque vocabulary to describe salient classes of those curves.
exhibited at each point of any trajectory one may consider. With respect to any point of the corresponding curve let us call that highest order differential feature 
‑gear curves.
and 
has a canonical expression 
in the set 
The tacit extension 
may then be expressed as a logical conjunction 
where 
is a logical tautology using all the variables in 
The following Table shows how the tacit extensions 
of the propositions 
may be expressed in terms of the extended basis 
![\text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]](https://s0.wp.com/latex.php?latex=%5Ctext%7BTacit+Extension+of%7D%7E+%5BA%5D+%7E%5Ctext%7Bto%7D%7E+%5BA%2C+%5Cmathrm%7Bd%7DA%5D&bg=ffffff&fg=000000&s=1&c=20201002)
![Tacit Extension of [A] to [A, dA]](https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-tacit-extensions.png)
the above analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like 
or 
to considering the collection of outcomes, the outgoing changes or singular dispositions springing or stemming from that state.
Inquiry Into Inquiry 