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With a correction in the formal logical relations table. Sorry about that.
- Ben
----- Original Message -----
Jim,
>[Jim Willgoose] 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)
Just as I thought, Peirce does not discuss modal propositions in the
passage which you had in mind.
>[Jim] 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.
In the context of oppositions, the contradictory of a proposition is the
_negation_ /of that proposition. "Bs" ("This stove is black") and "-Bs"
("This stove is not black") are contradictory of one another. They can't
both be true and they can't both be false. Thus they fit the form defined in the
logic of oppositions for contradictories.
"Bs" and "-Bs" are both internally consistent but are inconsistent with
each other. That is all that Peirce is implying, nothing more.
You are confusing formal logical properties with logical _expression_ of
modality in just such a way that, ironically, you call impossible the same
modal statement which can be used in order to express the idea that two
propositions are severally possible.
Now, there is nothing that constrains modal expressions to be used in order
solely to characterize formal logical relationships such as contrarity,
subcontrarity, implication, etc. However, they _can_ be used in a
context which confines them to that purpose. Taking 'poss.' as the
1st-order _expression_ corresponding to 2nd-order imputation
of possibility or logical internal consistency to a predicate or
proposition,
"poss. Bs & poss.-Bs" == "'Bs' and '~Bs' are severally possible." ==
"[Logically,] this stove can be black and this stove can be non-black."
"~ 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."
(Note: "this stove", a.k.a. "s", is not, as you called it in an earlier
post, an individual variable, but is instead an individual constant. In
traditional logic, the subject of propositions in the form "Hs" (e.g. "Socrates
is human") is taken as constant across propositions. If "this stove" is not
constant across propositions in a given example, then it is really a variable
and we're no longer talking about an already singled-out stove as in Peirce's
example).
>[Jim] The proposition resulting from their combination appears to be
[contradictory].
It does not appear to be contradictory. The components do not imply each
other's negations.
For instance, "poss.Bs" does not imply the negative of "poss.~Bs".
The negation of "poss.~Bs" is "~poss.~Bs".
"~poss.~Bs" is equivalent to "necess.Bs".
Yet "poss.Bs" does not imply "necess.Bs"
Ergo, "poss.Bs" does not imply "~poss.~Bs".
Ergo, "poss.Bs" is consistent with "poss.~Bs".
QED.
>[Jim] They are not Aristotelian (sub) contraries dealing with
"some" objects.
I said nothing about some specifically Aristotelian kind of subcontraries
that deal only with "some" objects, "all" objects, etc.
The oppositional relationships of subcontrarity, contrarity, contradiction,
etc., are certainly not confined to pertaining to quantificational propositions
about some objects, all objects, etc. The Square of Opposition shows some
oppositional relationships arising between quantificational propositions;
however, one does not need quantificational forms at all in order to define such
oppositional relationships -- indeed, a complete system of such binary formal
logical relationships. The forms or schemata of propositional logic are all
that's needed.
>[Jim] 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.
In the context of logical oppositions, contradiction is the validity of
exclusive alternation, and contradictories are defined as two propositions which
can't both be true and can't both be false. Subcontrarity is positive
alternation's validity conjoined with negative alternation's nonvalidity, and
subcontraries are defined as two propositions which can both be true and can't
both be false.
>[Jim] 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.
Identity is _always_ "problematic" when one considers logic as
signs. There is no index which picks out an object with total precision. Some
vagueness is always involved.
The idea that a difference in modifications automatically means such a
signficant difference in the substantial thing itself as to cause a
break-down in identity logic is both unpragmatic and unfounded in deductive
logic. It amounts to holding that allowing an element of chance ushers in total
chaos. It amounts to saying that one cannot discuss possibilities without
severing into multiple objects the object subject to those possibilities. It
amounts to saying that one cannot discuss a given total population as subject to
alternate events.
>[Jim] You say,
>>[Ben] "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."
>[Jim] 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?"
Peirce says that the same subject can have contrary qualities successively
in time. In fact he says something like, that's just what an event is. >From CP
1. 493. "An event always involves a junction of contradictory inherences in the
subjects existentially the same, whether there is a simple monadic quality
inhering in a single subject, or whether they be inherences of contradictory
monadic elements of dyads or polyads, in single sets of subjects." For more, see
quotations appended to this post.
Of course a sufficiently expressive logical formalism will accommodate that
without getting into trouble over it. I would hold that it is false to say that
the same stove is both black and non-black in the same place, time, and way.
And, for simplicity's sake, we can take "this stove" as "this stove now." Or we
can just pretend, for simplicity's sake, that the problem doesn't arise -- just
let the logic be the low-pixelage thing which it is, and go with its flow.
But simplicity's sake is what's involved, not an ontological or metaphysical
doctrine that the same stove with a different color wouldn't and couldn't be the
same stove.
Best, Ben Udell
Appendix: Quotes from Peirce on contradictory inherences.
CP 1. 493. There are other sorts of events, somewhat more complex because
the characters concerned are not simple monadic qualities. For example, A may
make war upon B, that is, may pass from one sort of relation to B to another
sort of relation to B. But they come to much the same thing. There is a
repugnance between two monad elements. It is hardly for our present purposes
worth while to undertake a long analysis in order to make the very slight
correction of our definition of an event called for on this account. An event
always involves a junction of contradictory inherences in the subjects
existentially the same, whether there is a simple monadic quality inhering in a
single subject, or whether they be inherences of contradictory monadic elements
of dyads or polyads, in single sets of subjects. But there is a more important
possible variation in the nature of events. In the kind of events so far
considered, while it is not necessary that the subjects should be existentially
of the nature of subjects -- that is, that they should be substantial things --
since it may be a mere wave, or an optical focus, or something else of like
nature which is the subject of change, yet it is necessary that these subjects
should be in some measure permanent, that is, should be capable of accidental
determinations, and therefore should have dyadic existence. But the event may,
on the other hand, consist in the coming into existence of something that did
not exist, or the reverse. There is still a contradiction here; but instead of
consisting in the material, or purely monadic, repugnance of two qualities, it
is an incompatibility between two forms of triadic relation, as we shall better
understand later. In general, however, we may say that for an event there is
requisite: first, a contradiction; second, existential embodiments of these
contradictory states; [third,] an immediate existential junction of these two
contradictory existential embodiments or facts, so that the subjects are
existentially identical; and fourth, in this existential junction a definite one
of the two facts must be existentially first in the order of evolution and
existentially second in the order of involution. We say the former is earlier,
the latter later in time. That is, the past can in some measure work upon and
influence (or flow into) the future, but the future cannot in the least work
upon the past. On the other hand, the future can remember and know the past, but
the past can only know the future so far as it can imagine the process by which
the future is to be influenced.
Peirce: CP 1.494 Such, then, is the nature of an event. We can now go
forward to an analysis of the substance of the law of time. It has three
requirements, a monadic, a dyadic, and a triadic. The monadic clause in the law
of time is that whatever fact or dyadic dyad exists, exists during a time, and
in this time. The event is the existential junction of states (that is, of that
which in existence corresponds to a statement about a given subject in
representation) whose combination in one subject would violate the logical law
of contradiction. The event, therefore, considered as a junction, is not a
subject and does not inhere in a subject. What is it, then? Its mode of being is
existential quasi-existence, or that approach to existence where contraries can
be united in one subject. Time is that diversity of existence whereby that which
is existentially a subject is enabled to receive contrary determinations in
existence. Phillip is drunk and Phillip is sober would be absurd, did not time
make the Phillip of this morning another Phillip than the Phillip of last night.
The law is that nothing dyadically exists as a subject without the
diversification which permits it to receive contrary accidents. The
instantaneous Phillip who can be drunk and sober at once has a potential being
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