Ben,
I am playing at trying to reject it. ("poss.Bs & poss.~Bs") I have accepted it more often than not. I also understand the difference between discussing formal properties that hold between propositions (modal or non-modal) and forming a "1st order" proposition out of the discussion of contingent propositions and the formal properties. This could be made clearer by noting the following:
"P" is a contingent proposition
"P" & "-P" are feasible.
"&" and "feasible" are part of the metalanguage used to discuss contingent propositions.
"feas. P" & "feas.-P" are ill-formed.
Explanation: "Feasibility" is a 2nd order predicate used to discuss....
This looks strangely similar to where we started in terms of rephrasing "she is possibly pregnant" as 'it is possible that "she is pregnant"' or ':she is pregnant" is possible.' But then, 2nd order assertions obey the theory of NLC cognition except we talk of feasibility, optimality, possibility and even assertibility. "Assertibility" would be an even higher-order predicate. Wasn't your initial concern with certain epistemic predicates that qualify "1st order" predicates?
Jim W
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Jim,
>[Jim Willgoose] Well, I guess the passage doesn't discuss modal propositions if you disallow rephrasing "this stove is possibly black" with 'It is possible that "this stove is black."' There is certainly a logic of possibility at work. Why aren' t these modal propositions? It is just that the possibility operator is outside of the proposition. I took it that Peirce is saying that "this stove is black" and this "stove is not black" are formally possible. What would a "logical _expression_ of modality" be? The operator is a unary connective much like negation. ( 'it is not the case that "this stove is black."')
Peirce makes some assertions themselves modal in character about some non-modal propositions. This can be translated into modal propositions or assertions but it is not the same thing as discussing modal propositions. To say that "Bs" & "~Bs" are incompossible is to say "~poss.(Bs & ~Bs)" and isn't to say "~(poss.Bs & poss.~Bs)" or "~poss.(poss.Bs & poss.~Bs)". Peirce was not implying either "~(poss.Bs & poss.~Bs)" or "~poss.(poss.Bs & poss.~Bs)" in any way, shape, or form. He was implying that "Bs" & "~Bs" are severally possible, "distributively" possible, each in its turn possible -- "poss.Bs & poss.~Bs" -- but not compossible, not collectively possible -- "~poss.(Bs & ~Bs).
>[Jim] You say,
"~ poss.(Bs & ~Bs)" == "'Bs' and '~Bs' are incompossible." == "[Logically,] it can't be both that this stove is black and that this stove is not black." (END)
>[Jim] I like this alot and have read it this way too. (at times) My mistake with respect to mixing contrary and contradiction up. It is easy to get in the habit. What is the other sort of possibility Peirce refers to? I have always looked for the supposed vague possibility. Maybe this is not the right passage from Peirce. Yet, If we accept the proposition "poss. Bs & poss.-Bs", then the point of the passage might be that besides formal possibility, there is vague possibility. In the other mode of possibility, contradiction is inapplicable. Thus, the proposition "poss. Bs & poss.-Bs" is not a contradiction. But I reject this for the example "this stove is possibly black and this stove is possibly not black."
>[Jim] I think I know my problem. In the context where "this stove" is a definite, actual individual and I assert this stove is black, every state of affairs is restricted to this stove and blackness. Thus, necessarily this stove is black and what does not occur is impossible or vice versa. This is an extreme form of actualism. But, I can make some sense of the claim that -poss.( poss.Bs & poss-Bs) The confusion and irony, however, doesn't lie with the possibility operator or where possibility appears in an ordinary proposition. It is all modal logic.
What's happening is that you're simply refusing to accept definitions of modal logic going back to Aristotle such that "necessary to do X" = "impossible not to do X" and "possible to do X" = "unnecessary not to do X" and "necessary to do X" implies but is unimplied by "possible to do X" and so forth. Instead, for you "poss." = "necess." = straightforward affirmation, and "~poss." = "~necess." = straightforward negation. The sense that you're making of " -poss.( poss.Bs & poss-Bs)" is your interpreting it as being practically no different from "~(Bs & ~Bs)." Yet 2nd-order logic itself offers a model for ideas of possibility and necessity in the ideas of consistency and validity, and furthermore allows for the distinction between contingently true and necessarily true -- which is a distinction which you don't accept.
Even when it is a premiss that the stove is black, it does not become formally true, in further inference, that the stove is black. *_That is the difference between a premiss and an assumption._* It's been said that a true proposition implies all true propositions and that a false proposition implies all propositions -- but that "implies" refers to _material_ implication, nowadays oftener called "the conditional" and not to _formal_ implication. It's true that I'm writing this post, but that doesn't formally imply that I'm in my apartment, though that's true too. But it _is_ true that either I'm not writing this post or I'm in my apartment or both. "~p v q" == "p-->q" -- material implication. Meanwhile, we do assume the rules of formal implication. So, if the premiss is that the stove is black, such that the schema is "Bs," then the schema is consistent and nonvalid -- possible and non-necessary. Hence, logically it is possible but non-necessary that the stove is black, even when it is true that the stove is black. The possibility and nonnecessity are "relative" to the choice of rules whereby we attribute possibility and necessity.
If you don't have a problem with that, then why should you have a problem with attributing necessity and possibility to things in virtue of more complicated and empirically anchored "formalisms" and norms and patterns and laws which we find in the world?
Best, Ben Udell
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