Ben,
Peirce says,
"Very many writers assert that everything is logically possible which involves no contradiction Let us call that sort of logical possibility, essential, or formal, logical possibility. It is not the only logical possibility; for in this sense, two propositions contradictory of one another may both be severally possible, although their combination is not possible." (CP3:527)
Two propositions, "Bs" and "-Bs" may both be possible. ( severally) But, the proposition "pos Bs & poss.-Bs" is not possible. The first two propositions are not contradictory of one another. The proposition resulting from their combination appears to be. They are not Aristotelian (sub) contraries dealing with "some" objects. The so called "failure of contradiction" deals usually with general object indefiniteness in the case of the existential quantifier. That is not what is going on here. Vagueness is just as much the result of considering the two propositions severally.
We started this discussion with a number of examples of "whetherhood" designed to expand and qualify the predicate assertion. I chose possibility becasue of its generality and largely epistemic flavor. I read Peirce as developing the meaning of possibility with "states of information," knowledge and probability in mind. I also favored in the beginning attaching the modal operator to propositions and treating the proposition as a subject. I did this in order to preserve the copula "is" in the subject assertion without qualification. The various ordinary ways of expressing the proposition can be rephrased. I also pointed out the way that identity becomes problematic.
You say,
"Maybe there's a necessary difference at a simple logical level between epistemic and ontological treatments of possibility, but such difference isn't evident to me."
Consider the difference between saying each proposition "Bs' and "-Bs" is indeterminate with respect to truth and saying that it is impossible that both propositions are jointly true. Ontologically, I do not think that the same stove or same women can have contrary qualities. Is this a principle of Being or just an idealization of excluded middle? Further, would you be prepared to say that the truth value of the compound proposition is indeterminate in the sense of "not known to be false?"
Jim W
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Jim,
>[Jim Willgoose] There is a difference between treating possibility epistemically or treating it ontologically. "Possibly black' and "possibly non-black" are (sub) contraries, indeterminate with respect to a state of information. But since we are considering "this stove," and not allowing multiple reference for "this," we know that both statements cannot be true for a definite individual. Particular propositions, for Peirce, obey both the laws of non-contradiction and excluded middle. ( 1st order Form: (poss. Bs & poss -Bs ) Notice that I do not use the quantifier "E" since "this stove" denotes a definite individual. ("s" is an individual variable and "B" is a predicate letter.) These two propositions are not "compossible, although they are severally possible." (Peirce's language) However, 2nd order Form creates a problem. EF(Fs & -Fs) Which property? Here "F" is an indefinite predicate variable. Should not all substitutions for "F" be identical regardless of whether we can identify the property? Maybe not. Peirce said in the gamma graphs that for ordinary purposes, "qualities may be treated as individuals." If there is no definite property, then the proposition is vague rather than false. Identity is critical even for possible states of information.
Maybe there's a necessary difference at a simple logical level between epistemic and ontological treatments of possibility, but such difference isn't evident to me.
You don't provide a reference or a quote, but presumably Peirce is referring to the components of "(Bs & ~Bs)" as non-compossible and as severally (separately) possible, but is _not_ referring to a form like "(poss. Bs & poss. -Bs)" at all. It would be strange, I think, if he did. Yet Peirce's technical conception of propositions and predicates and their treatment differs enough from the contemporary, that, well, who knows? So I ask for a quote from him. Somehow you seem to be thinking that "poss.Bs" is the negative of "poss.~Bs".
The same issues are involved with the "(Fs & ~Fs)" in "EF(Fs & ~Fs)."
I don't know what your assumptions are about the 1st-order syntactical status of "poss.", but it's as if you're treating "poss." in "poss. Bs" as a predicate, whereas one needs to treat it as a functor (like the negative sign) and to treat the resultant "poss. Bs" as function of "Bs" rather than as "Bs" itself with some added predicated description "possible." This is the same as one treats "~Bs" as a function of "Bs" rather than as "Bs" with some added predicated description "negative." The appropriate 2nd-order counterpart is not "EF(Fs & ~Fs)" but "EF(poss.Fs & poss.~Fs). But I'm just guessing at your assumption. However it does seem that, however you're treating "poss.", it's not as a functor like "not".
Best,
Ben Udell,
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