Peirce’s Categories • 14

RM:
What do you think of the presuppositions between the levels?
Do they make sense to you?

At this point I have mostly questions, which would take further research to answer, not to mention unpacking many books still in boxes from our move a year and a half ago, none of which I’m at liberty to do right now.  So, just off the cuff …

Presupposition is one of those words I tend to avoid, as it has too many uses at odds with each other.  There are at least the architectonic, causal, and logical meanings.  It it were only a matter of logic, I would say $P ~\mathrm{presupposes}~ Q$ means $P \Rightarrow Q.$  But usually people have something more pragmatic or rhetorical in mind than pure logic would require, something like enthymeme.

It’s also common for people to confound the implication order $P \Rightarrow Q$ with the causal order $P ~\mathrm{causes}~ Q,$ whereas it’s more like the reverse of that.  In more complex settings we usually have the architectonic sense in mind, and that is what I sensed in the case of the normative sciences.  Viewed with regard to their bases, logic is a special case of ethics and ethics is a special case of aesthetics, but with regard to their level of oversight, aesthetics must submit to ethical control and ethics must submit to logical control.

Early on, Peirce used involution with the meaning it has in arithmetic or number theory, namely, exponentiation, where $x^y$ means $\text{taking}~ x ~\text{to the power of}~ y.$  See the following passage and commentary.

As far as the boolean or propositional analogue goes, $x^y ~\text{for}~ x, y ~\text{in}~ \{ 0, 1 \}$ means the same thing as $x \Leftarrow y,$ as one can tell by comparing the following two operation tables.

I haven’t looked into whether Peirce uses “involution” or “involvement” with that sense in his later writings.

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