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Tom
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| Posted Today 5:54 AM (GMT -6) |   | I’d like to branch off in this thread to begin to consider an issue that came up in our discussion of Doc’s article in Philosophia Christi and Werther’s response. In that thread I began to go crazy posting thoughts, most of them somewhat jumbled. It might be best to leave that thread for discussing Doc’s article and commit this thread to considering some of the insights that seemed to be emerging regarding the logic of indeterminacy.
Let me try to recap what my thoughts were regarding the triangle. I can hardly remember what I ranted on about now! After that, I’d like to discuss where open theism goes from here with regard to the logical ‘front’ of the battle. Some open theists may want to argue there is nowhere to go, that we have all we need in the traditional square of opposites to say all that needs to be said, namely, subcontraries “might” and “might not” may both be true, in which case “will” and “will not” are false. The open view is thus ‘logically possible’ (i.e., not self-contradictory) and THAT is all one should expect logic to say.
Two things in response. Granting for the moment that all that logic can say about the open view is that it is not a self-contradictory view, it’s still the case that logic can be understood to say this much more clearly, persuasively, and elegantly than what we get with conjoining “might” and “might not” on the traditional square. Secondly, I personally think there’s more to be said than merely that the open view is ‘logically possible.’ It might (ooh, a “might”!) be the case that we can construct a logical argument that persuades not only regarding the possibility of the open view, but the plausibility, even the probability, of it’s being true.
A Quick Review
Here are the main points that I was arguing for in Doc’s Neo-Molinism thread (some are Doc’s point, others can from I don’t know where; the crazy challenges to the traditional square and the expanded Triangle are mine):
1) Future reality (maybe all reality, but I’m only interested in the future for now) is threefold. That is, there are three possible states of affairs regarding future. Alan expressed them as:- (A) It is determinately the case that p
- (B) It is determinately the case that not p
- (C) It is indeterminately the case that p
2) Given (1) above, there can be only three propositions that can be formed regarding the future. Propositions that affirm (or deny) (A), those that affirm (or deny) (B), and those that affirm (or deny) (C). Like this:
- It is determinately the case that x do y (or “x will do y”)
- It is determinately the case that x not do y (or “x will not do y”)
- It is indeterminately the case that x do y (or formerly, as Doc argued using the traditional square, “x might and might not do y”)
What about “would do” and “would not do”? These are simply the same determinacy behind “will” and “will not” expressed counterfactually.
3) The problem (as I see it) with arguing that “might” and “might not” from the square conjoin to express Alan’s state of affairs (C), or “It is indeterminately the case that x do y,” is simple. Independently true “might” means: “It is not the case that x will not do y” and therefore, it is the case either that “x will do y” or “x might not do y.” Independently true “might not” of the square means: “It is not the case that x will do y” and therefore it is the case either that “x will not do y” or “x might do y.” To conjoin true “might” and true “might not” of the square is to conjoin THESE two meanings: Thus “might and might not” means: “It is not the case that x will do y” and “not the case that x will not do y” (these are the conjoined negations of “might” and “might not”) AND it is either the case “x will do y,” “x might not do y,” “x might do y,” or “x will not do y” (the conjoined possibilities of “might” and “might not” being true). This sounds self-contradictory to me.
Contrary to the rule that subcontraries may both be TRUE, it does not seem to be the case that subcontraries “might” and “might not” may both be true, though it certainly remains the case that contraries “will” and “will not” may both be false. What gives? I suggested several possible reasons why this rule (and perhaps others we're not yet aware of) may not hold here: a) states of affairs described in future tense propositions are not quantifiable, b) the traditional square equivocates on the meaning of “might” (sometimes meaning “merely logically possible that…” and sometimes meaning more than this, i.e., the denial of one of the two determinate states of affairs); “might” and “might not” may be conjointly true if “might” means merely “logically possible that…” but they may not be conjointly true if on the one hand “might” falsifies “will not” and leaves open the question of whether “will” or “might not” is true while “might not” falsifies “will” and leaves open the question of whether “might” or “will not” is true. c) the traditional square conflates epistemology and ontology (this is what the immediately preceding b) does I suppose).
4) The traditional square cannot account for all that can be said about the future, that is, about our three states of affairs. States of affairs (A) and (B) each have separate propositions that affirm and deny their cases. In fact, that is ALL the traditional square has, period. The square offers us NO way to propositionally AFFIRM as true “It is indeterminately the case that p.” And in being incapable of affirming “indeterminately the case,” the square also cannot affirm as true indeterminacy’s contradiction, viz., “It is NOT indeterminately the case that p” so as to leave open the question of the truth values of “It is determinately the case that p” or “determinately the case that not p.”
5) We must recognize the independent truth value of indeterminacy as an equally ontological state affairs alongside “determinately that” and “determinately that not.” I see more and more that this is behind the confusion of even an open theist logician like Prior. I hope to show a few examples of how he conflates epistemology and ontology and so fails to provide a convincing argument for indeterminacy.
6) My edited triangle is nothing more than the traditional square with a proposition added that posits indeterminacy (Alan’s state of affairs (C)) and its contradictory (the top peak of the triangle). The inner triangle is formed by the three ontological states of affairs. The larger dotted triangle is made up of the contradictories of these three states. I called these EQ’s, or “Epistemological Quantifiers” (still not knowing what else to call them). The only ontological state of affairs they directly speak to is their falsifying one of the three states of affairs. But they each leave open TO OUR KNOWLEDGE (thus, epistemological quantifier) what the truth values of the remaining two states of affairs are.
7) I wondered what was behind the square working very nicely for some propositions but so poorly for others. It seemed to me that the props the square has trouble with are props describing the future. Where the square fouls things up in this regard is in conflating epistemology with ontology (by equivocating on the meaning of “might”) and in disrespecting indeterminacy. I’m wondering if the key to the difference here is that some states of affairs (Grass being all green, for example) are quantifiable while some (determinacy and indeterminacy) are not.
8) About notation. Alan suggested D for “determinately the case that” and I for “Indeterminately the case that.” But both these are already spoken for in modal logic. It now turns out that Q is the established operator for “It contingent/indeterminate that…” and has been so for some time. If we have Fp for “It will be the case that p,” then I’m wondering whether or not we need an operator for “Determinately the case that….” I mean, Fp, “It will be the case that p” means “determinately the case that p”, while F~p, “It will be the case that not p,” means “determinately the case that not p.” Also, we ought to be careful about the difference between using operators (P, G, H, and others, and now Q) with variables (p, q, r) that are themselves not tensed (“x does y,” “Tom comes to St. Paul” etc.) and variables which are tensed propositions (“x will do y,” “Tom will come to St. Paul in 2004”). Prior had, it seems, two separate ‘systems’ for these.
We can at least update Q for I (=indeterminacy operator) as "It is indeterminately that case that...." Qp, rather than Ip, means “It is indeterminately the case that p.”
9) Oh yeah, before any work can be done to develop a formal logic of indeterminacy, somebody’s got to define “determinacy” and “indeterminacy.” Let’s do that FIRST!
Hey, I’m tired. If there’s more to say to recap things, I’ll throw it in later. Right now I have to take somebody to the airport!
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Tom
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| Posted Today 7:25 AM (GMT -6) | | Tom continues to obsess…
Here are some very interesting passages from Arthur Prior’s Formal Logic, the chapter on “The Logic of Modality.” I’ll just post this for me and come back later to comment. Anything in brackets [] is me.
quote:
We may now turn to the more usual meaning of ‘contingent’. In the De Interpretatione Aristotle remarks that the world ‘possible’ in ambiguous; we would sometimes say that ‘It is possible that p’ follows from ‘It is necessary that p’, but sometimes that it is inconsistent with it. In the former sense ‘possible’ means simply ‘not impossible’; in the latter sense, ‘neither impossible nor necessary’. It is for ‘possible’ in the second sense that the word ‘contingent’ is general used. That is, ‘It is contingent that p’ means ‘Both p and not-p are possilbe’, KMpMNp [This is Prior’s notation: K=“and”, M=“might”, so: “p might and not p might”]. Contingency in this sense stands between necessity and impossibility, but in quite a different way from that in which the simply factual stands between the necessary and the possible. It is not that necessity implies contingency, and contingency impoosibility; rather, we have here three mutually exclusive alternatives which divide the field between them—either a proposition is necessary, or it is neither-necessary-nor-impossible (i.e., contingent), or it is impossible…” (p. 188)
I thought Prior hit the nail on the head when he speaks of the ‘necessary’ (our “determinately that p”), the ‘impossible’ (our “determinately that not p”) and the ‘contingent’ (our “indeterminate that p”) as “mutually exclusive alternatives which divide the field between them.” Right on! There’s Alan’s three metaphysically distinct states of affairs regarding the future. BUT I’m uncomfortable with a few other things Prior says. I’ll come back with more on this later. For now, here’s more:
quote:
For example, with respect to this division between the necessary, the contingent and the impossible, Aristotle hints that the contingent may be further subdivided into what is more likely to be so than not, what is as likely to be so as not, and what is less likely to be so than not; and plainly we might go still farther and replace the threefold or fivefold distinction by an infinite series of grades of probability. And quite similarly, cases in which some B’s and A’s and some not may be subdivided into those in which most B’s are A’s, those in which half the B’s are A’s and half of them are not, and those in which most are not, and further subdivided into an infinite series of precise proportions. Both these lines of advance were investigated by [Augustus] de Morgan. He brought probability out of mathematics into logic; ‘the theory of probabilities’, he said, ‘I take to be the unknown God which the schoolmen ignorantly worshipped’ in their study of the properties of modals.
Here I thought it cool that Aristotle anticipated probability math and logic. Interesting eh? You can get De Morgan’s Formal Logic through amazon.com. Can you believe that this book, written in 1847, republished in 1926, and then again recently is still available? I read where Charles Pierce stated his conviction that de Morgan was the greatest logician of all time. Hmmm. And he brought probability out of mathematics into logic. A bit more:
quote:
In his exhaustive study, in the Prior Analytics, of syllogisms with modal primisses, Aristotle regularly uses ‘possible’ in the second sense, i.e. as meaning ‘neither impossible nor necessary’. This complicates his discussion somewhat…and his successor Theophrastus, working in the same field, preferred to take ‘possible’ in the simpler sense. However, the properties of both sorts of ‘possibility’ are worth investigating, and one small but fruitful discovery which Aristotle made about ‘contingency’ is that if any proposition is contingent then so is its contradictory. Using Qp for ‘It is contingent that p’ (KMpMNp), Aristotle’s thesis may be written as CQpQNp [C=”implies that” and so: “‘It is contingent that p’ implies ‘It is contingent that not p”]. The point is simple enough; since ‘It is contingent that p’ amounts to ‘Both p and not-p are possible’, ‘It is contingent that not-p’ amounts to ‘Both not-p and not-not-p are possible’, and ‘Both p and not-p are possible’ implies that by double negation and the controvertibility of conjunction (Qp = KMpMNp --> KMNpMp --> KMNpMNNp = QNp). This law may be used to show that modal functions of propositions are not truth-functions.
Oh my stars and garters! Sorry for the confusing caps! You start to get used to reading them if you take LOTS of drugs first! 
I will argue that Aristotle and Prior are WRONG in claiming that if a proposition is contingent so is its contradictory. Prior actually argues Aristotle’s point by suggesting that QNp (or Q~p, that is, “Indeterminately that not p”) is the contradictory of Qp (“Indeterminately that p”). But even Iiiieee know (now!) that QNp (or Q~p) is not the contradictory of Qp. Looki there! Prior made a mistake! Perhaps writing too quickly? Perhaps one Sherry too many? Aristotle and Prior, I think, are both wrong in this case. If indeterminacy, as Prior even argues in the first quote above, is exclusively the case contrary to ‘necessary’ and ‘impossible’ (remember he said these three divide the field among them?) then its contradictory can’t be true. Indeterminacy cannot imply determinacy, which would be the case if the contradictory of Qp (which is ~Qp and which itself implies either "determinately that p" or "determinately that not p") were believed to be implied by Qp. That’s silly! The contradictory of Qp is not Q~p; it is ~Qp. And Qp cannot imply ~Qp. It DOES imply Q~p (as Prior notes) but that’s what indeterminacy IS! And this doesn't prove what Prior says it proves about the contradicotry of Qp. I think Aristotle, and Prior following him, confuses epistemology (it being indeterminate FOR US which of two props are true) and ontology (it being ‘indeterminately the case that p implying indeterminately the case that not p’ but NOT implying ‘not indeterminately the case that p’.
There’s interesting stuff out there. Thanks Laika for that article. Interesting. I’ve ordered de Morgan’s book on logic in hopes of finding out how he brought probability out of mathematics and into logic.
In the meantime, Alan and Doc, you two finish this all up, OK? We want defintions, axioms, theorems, a full slate of operators by next week. Can you deliver on that? Come on!
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Tom
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| Posted Today 8:43 PM (GMT -6) | | Alan suggested the following as a few basic theorems for indeterminacy logic:
quote:
1. I(p) = I(~p)
2. Either I(p) or D(p) or D(~p)
3. ~D(p) = I(p) or D(~p)
4. ~I(p) = D(p) or D(~p)
5. D(p or q) --> D(p) or D(q)
6. I(p or q) --> I(p) or I(q)
Given that ‘I’ and ‘D’ are both used in other functions, and given that there is already an indeterminacy operator ‘Q’, we can perhaps rewrite Alan’s theorems. But what we don’t yet have is a “determinate the case that” operator. There doesn’t seem to any function assigned to ‘Z’, so hey…let’s use that for “determinacy.” Why? Gee, I don’t know. How about because ‘Z’ is the last of a determinate set of letters known as the alphabet! Good enough. So, using the following for the regular stuff…
Q = indeterminate that, or contingent that
^ = and
v = or
--> = If/Then
~ = not
/ = either/or (placed at the beginning: /pq = either p or q, or maybe it should be /pvp, "Either p OR p." This stuff is beyond me! Arghghg).
…let’s look at Alan’s theorems:
1. Qp = Q~p (or perhaps Qp --> Q~p) (This is true. Prior attempted to use this to prove Aristotle’s claim that if any proposition is contingent, then its contradictory is contingent; but that can only be true if you’re talking about epistemic indeterminacy and not ontological indeterminacy)
2. /QpvZpvZ~p (Either Qp or Zp or Z~p; if there’s a better way to notate it, let’s have it). We could formalize the implications of this too:
2a. Zp --> ~Qpv~Z~p
2b. Z~p --> ~Qpv~Zp
2c. Qp --> ~Zpv~Z~p
Alan’s theorems 3 and 4 falsify Zp and Qp respectively, but I noticed there’s no falsification of Z~p. So all three would be:
3. ~Zp --> /QpvZ~p
4. ~Z~p --> /QpvZp
5. ~Qp --> /ZpvZ~p
6. (Alan’s 5) Z(pvp) --> ZpvZq (or Z/pq --> /ZpvZq)
7. (Alan’s 6) Q(pvq) --> QpvQp (or Q/pq --> /QpvQq)
Alan further suggested these:
quote:
7. I(p and q) = Ip or Iq
8. I(p or q) = Ip and Iq
9. I(p --> q) = p --> Iq
Using our new notation we could rewrite this 7) (as 8 too, since we already have seven theorems above) as:
8. Qpq = Qpvp (or Qpq --> QpvQp). (I didn’t see this. I take it that if it’s indeterminately the case that ‘p and q’ (that is, Qpq), then it would follow that Qp^Qq, “It is indeterminately the case that p AND indeterminately the case that q.” Two indeterminacies don’t equal an either/or, they equal a both/and. Right?
9. (Alan’s 8) Q(pvp) = Qp^Qq (or Q/pq = Qp^Qq) (I didn’t see this earlier. It’s confusing. But I now think this is right. It’s just another way of falsifying both Zpq and Z~pq. You’re saying, either p or q is indeterminate and we don’t know which is which. Hmm. I guess that DOES = Qp^Qq!).
Having loads of fun!
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Tom
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| Posted 11/10/2003 6:28 AM (GMT -6) | | Tim: God knows.
Tom: What does God know?
Alan, I just came across the letter ‘A’ used to introduce an “either/or” proposition, so now we might express #2 above, /QpvZpvZ~p (Either Qp or Zp or Z~p), as AQpvZpvZ~p (the ‘A’ introducing an “either/or” proposition and the ‘v’ (= “or”) separating each part.
Re: #3, 4, and 5, the same A can replace the /. So:
3. ~Zp --> AQpvZ~p
4. ~Z~p --> AQpvZp
5. ~Qp --> AZpvZ~p
And on #6: Z(pvp) --> ZpvZq…Alan, isn’t Zpvp (“It is determinate that either p or q”) the same as Apq (“Either p or q”)? Seems so. The determinacy operator ‘Z’ seems not needed here. To assert p (it is the case that p) is to assert Zp (it is determinately the case that p) provided p is not itself a future tensed proposition. I’m still not sure when Z needs to be used. Is it just in cases where p is tenseless and refers to the future (e.g., Tom visits St. Paul in 2004)? I mean, couldn’t we just remove the ‘Z’ operator from all your theorems and get the same results?
Somebody stop me. I’m outta control!
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Tom
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| Posted 11/10/2003 6:32 AM (GMT -6) | | OK OK, I get the message. No responses means, "Give us a break Tom! We're not interested." Since I'm tired talking to myself here (and have stuff to catch up on this week) I'll bow out of this stuff for a while.
Thanks for puttin up with me.
Later all,
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Starkman
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| Posted 11/10/2003 1:33 PM (GMT -6) | | Oh, no response (from my end ... though I'm responding!) means I can't follow these kinds of conversations. I get lost about two breaths into it.
Not too smart in these kinds of things!!! So, I don't respond.
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Tom
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| Posted 11/10/2003 6:20 PM (GMT -6) | | Gee, guys...I, eh, I...I think I'm gonna shed a tear.
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Tom
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| Posted 11/11/2003 1:55 AM (GMT -6) | | Laika-
Those articles on Eastern logic…whoah dude?! Cool. But in the end it seemed to me that by indeterminate they were still just talking about epistemic indeterminacy. I only glanced them though. I was in bed reading them and fell asleep!
Starkman-
Don’t you dare tell me this stuff is beyond you! Take it a step at a time. I haven’t found a user-friendly introduction to logic. I’m still looking for “An Idiot’s Guide to Logic.”
OK…
I introduced:
quote:
Q = indeterminate that, or contingent that
^ = and
v = or
--> = If/Then
~ = not
/v = either/or (/ placed at the beginning and v between each member).
Then I scratched /v and suggested A (which some logicians use for “either/or.” But hey, let’s just use ‘v’ for ‘or’ and leave it at that. Perhaps putting a space in there would make it easier too (Qp v Zp instead of QpvZp for Qp or Zp).
I also think that in working this out, for the time being we ought to stick to using variables p, q, r for propositions that are NOT tensed. There’s no sense in putting Z (“It is determinately the case that…”) in front of p if p is “x will do y.” The latter already posits the determinacy of x’s doing y. Similarly, there’s no sense in placing Q (“It is indeterminately the case that…”) in front of p if p is “x might do y and might not do y.” The latter already posits the indeterminacy of x’s doing y. But if p simply states “x does y” without reference to tense or determinacy, then Z and Q can be used.
Alan, I have no idea what I did with your 7, 8, and 9. Part of my post got lost and I mixed theorems up. I sometimes confused v (or) and ^ (and). If I would only stop to breathe in between posts I might not have to correct myself so much. I hope this sets it straight.
Also, Alan, you gave the following theorems:
6. I(p or q) --> I(p) OR I(q)
8. I(p or q) = Ip AND Iq
I capped “or” and “and” to show my point. They’re the same argument, but in one you argue OR and in the second AND. These contradict each other do they not? I think (6) is correct.
So here we are thus far (with ‘em spelled out for Starkman!). Please type 'em in right, Tom, please!
Plus, the numbers are rearranged (formerly 2a, 2b, and 2c are now 3, 4, and 5).
1. Qp = Q~p: (“It is indeterminately the case that p” is equivalent to “It is indeterminatey the case that ~p”). This is unique to indeterminacy, because Zp cannot be equivalent to Z~p. They're contraries.
2. Qp v Zp v Z~p: (EITHER “It is indeterminately the case that p,” OR “It is determinately the case that p,” OR “It is determinately the case that not-p”). This lays down the basic claim upon which all of indeterminacy logic rests, namely, there are three states of affairs regarding future events. This is the inner triangle.
3. Zp --> ~Qp ^ ~Z~p: (“IF ‘it is determinately the case that p’, THEN ‘it is not indeterminately the case that p’ AND ‘it is not determinately the case that not-p’”)
4. Z~p --> ~Qp ^ ~Zp: (“IF ‘it is determinately the case that not-p’, THEN ‘it is not indeterminately the case that p’ AND ‘it is not determinately the case that p’”)
5. Qp --> ~Zp ^ ~Z~p: (“IF ‘it is indeterminately the case that p’, THEN ‘it is not determinately the case that p’, AND ‘it is not determinately the case that not-p’”)
Note: 3, 4, and 5 are each an affirmation, with implications, of one of the three possible states of affairs. Each is located on a point of the inner triangle.
6. ~Zp --> Qp v Z~p: (“IF ‘it is not determinately the case that p’, THEN ‘it is indeterminately the case that p’ OR ‘it is determinately the case that not-p’”)
7. ~Z~p --> Qp v Zp: (“IF ‘it is not determinately the case that not-p’, THEN ‘it is indeterminately the case that p’ OR ‘it is determinately the case that p’”)
8. ~Qp --> Zp v Z~p: (“IF ‘it is not indeterminately the case that p’, THEN ‘it is determinately the case that p’ OR ‘it is determinately the case that not-p’”)
Note: 6, 7, and 8 are the three EQ's of the outter triangle. Each falsifies one of the three states of affairs with implicatoins.
Notice that the implications of each of Zp, Z~p, and Qp (3, 4, and 5), when TRUE, are the falsifying of the other two. But the implications of each (6, 7, and 8), when FALSE, leave us in the dark regarding the truth value of the other true.
9. Z(p v q) --> Zp v Zq: (“IF ‘it is determinately the case that either p or q’, THEN ‘it is determinately the case p’ OR ‘it is determinately the case that q’”). Couldn’t this be abbreviated as Z(p v q) = Zp v Zq?
10. Q(p v q) --> Qp v Qq: (“IF ‘it is indeterminately the case that either p or q’, THEN ‘it is indeterminately the case that p’ OR ‘it is indeterminately the case that q’”). Couldn’t this be abbreviated as Q(p v q) = Qp v Qq? Don’t --> and = do the same thing?
11. Q(p ^ q) = Qp ^ Qq: (“It is indeterminately the case that ‘p and q’ is equivalent to ‘It is indeterminately the case that p’ AND ‘it is indeterminately the case that q’”). Alan had, incorrectly I think, 7) I(p and q) = Ip OR Iq (“It is indeterminately the case that ‘p and q’ is equivalent to ‘It is indeterminately the case that p’ OR ‘it is indeterminately the case that q’”).
12. Q(p --> q) = p --> Qq: (“It is indeterminately the case that if p then q” is equivalent to “IF p, THEN ‘it is indeterminately the case that q’”). Just to clarify. Q(p --> q) posits the indeterminacy of BOTH p and q, and furthermore specifies that q’s indeterminacy is dependent upon p's obtaining (which is also indeterminate). So I read it, “It is indeterminately the case p, and should p obtain it is indeterminately the case that q.” Did I get this much right? OK, all this is equivalent to p --> Qq.
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Tom
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| Posted 11/11/2003 2:11 AM (GMT -6) | | Laika: You mean ^ here don't you?
Tom: I knew I'd miss something. That's it! I'm NOT gonna pursue a career as a logician. Too great a possibility for TYPOS!
Thanks Laika. I changed 'em.
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Tom
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| Posted 11/11/2003 4:31 AM (GMT -6) | | Have I ever showed you guys my triangle within a triangle? Or as that weird Anglican priest on "Princess Bride" says, "A dweem wifin a dweem."
Here it is with the new and improved notation!
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Tom
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| Posted 11/11/2003 4:34 AM (GMT -6) | | Laika, thanks for making the figures possible! Can we make this one a little smaller? Seems kinda big.
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Electra
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| Posted 11/11/2003 6:18 AM (GMT -6) | | | | |
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Tom
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| Posted 11/11/2003 6:34 AM (GMT -6) | | Thanks Laika! Hey, nice Arabic!
Electra, I'm on that book! Checking out amazon! That's a title for ME.
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Tom
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| Posted 11/12/2003 1:14 AM (GMT -6) | | (Doc, here’s my response to your last post over on the Neo-Molinism thread)
I totally LOVE this stuff!
quote:
Alan: ALL conditionals in which the antecedent presupposes creation are counterfactuals, for they have false antecedents.
Doc: Its not clear to me why "If God created x, then x might do y and z" is necessarily counterfactual just because God hasn’t created x. For the proposition says "if".
Tom: I see what you mean too, Doc. And you CAN phrase this so that the antecedent expresses a strict conditional, but (as I think Alan’s point was) simply saying “If” doesn’t posit a conditional. It does for normal people. But it doesn’t for logicians. You have to spell it out when speaking anally-retentive logic-lingo (still not sure just why): “If it’s indeterminately the case that God create x, then it’s indeterminately the case that x do y in z.” If you say it this way, THEN you have an antecedent that is presumably TRUE (only God knows) and the proposition is not counterfactual. But just saying “If God creates…” is not enough according to the square.
quote:
Alan: Molinists claim that among this counterfactual knowledge are true "would" counterfactuals of creaturely freedom (CCFs). Open theists say there are no true "would" CCFs, but only true "might" CCFs.
Doc: I argue that there ARE true "would counterfactuals" of creaturely freedom. These are the "woulds" that follow from the use of our libertarian freedom.
Tom: If the “would” in the consequent follows from the use of libertarian freedom, then the antecedent must express that libertarian freedom, that is, it must be a strict conditional, in which case it’s NOT a counterfactual. Hence your example would then be: “If it is indeterminately the case that Peter come to exist (strict conditional), and if it is indeterminately the case that he exercise his freedom in certain ways (strict conditional), he WOULD become….” This is not a counterfactual, for the antecedents are not false. It’s just a plain old conditional. No raisins or blueberries or anything fun in it at all.
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Doc: The "would" counterfactual is predicated on the "might" counterfactual. But then again, why the heck call this COUNTERfactual? I agree with TGB: they are FACTUAL.
Tom: Right. So if it’s not a counterfactual, then Alan is correct: Open theists deny that there are any TRUE counterfactuals of creaturely freedom. In as far as a proposition’s antecedent is TRUE, it is not COUNTERfactual. And in as far as the antecedent is FALSE, the consequent cannot describe libertarian freedom. Hasker is right: There are no true counterfactuals of creaturely freedom.
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TGB: Do we say "Tom might and might not come to town in 2004"? That’s what Doc’s been using. But I already tried to show that the independently true "might" and "might not" of the traditional square, being epistemologically open as to which state of affairs is TRUE, cannot be conjointly true (contrary to the rules of the square). We cannot conjoin what "might" and "might not" mean when independently true. This is the raw deal we have now. The traditional square robbed "might" and "might not" BOTH of the ability to affirm a true state of affairs and the ability to conjoin in affirming a true state of affairs.
Doc: This is an aspect of your manic postings I never quite got Tom. As much as I dislike the square, it seems to me it left open the possibility (rarely if ever explored) that both "might" and "might not" could be true, hence rendering BOTH "would" and "would not" false. Grant it, the square "non-elegantly" does not allow for ONE single prop to affirm this – hence the superiority of the triangle – but it didn’t rule it out. Did it? (This seems to be behind Alan's defense of the square] And in this case, the ONLY case in which "Mights" are "independently true" is when they are "conjoined." By the way, I like the notation Mp for "might and might not p."
OK, pull up a chair and light that Cohiba! I could be wrong on this point, but I haven’t had anybody interact with it till now.
OK, let's conjoin "might" and "might not":
MIGHT: ~Z~p --> Zp v Qp (If “not determinately the case that not-p (= a true ‘might’)”, THEN EITHER “determinately the case that p” OR “indeterminately the case that p.”
MIGHT NOT: ~Zp --> Qp v Z~p (If “not determinately the case that p (= a true “might not”)”, THEN EITHER “indeterminately the case that p” OR “determinately the case that not-p.”
You're saying that when we conjoin ~Z~p ^ ~Zp (“might” AND “might not”), all we get is Qp (“indeterminately the case that p”). You're arguing that:
~Z~p ^ ~Zp --> Qp
Can't happen I don't think. You HAVE to conjoin EVERYTHING on the right side of BOTH “if/then” equations above as well, that is, ALL that they conjointly IMPLY (what they hold out as possibly the case). You have to conjoin their “either/or” options is what I’m saying. When you do THIS what do you get?
Here are "might's" entailed options: Zp v Qp (“determinately the case that p” OR “indeterminately the case that p”).
Here are "might not's" enttailed optiosn: Z~p v Qp (“determinately the case that not-p” OR “indeterminately the case that p”).
To conjoin them is to conjoin what they falsify and what they imply. What they falsify is how they are defined via notation (that's the "if" clause). What they conjointly imply is the "then" clause.
~Z~p ^ ~Zp --> Zp v Qp v Z~p v Qp
In other words:
If "might and might not" then Eihter "determinately that p" OR "indeterminately that p" OR "determinately that not-p" OR "indeterminately that p".
Affirming Qp twice as an option isn’t a problem. But what about the other conjoined options? Just forget ‘em? Can’t do that. And remember, to be a “possible option” is to be possible, that is, NOT falsified.
So you have “might and might not” falsifying and affirming as an option the same states of affairs. What one falsifies, the other introduces as an option! “Might” falsifies Z~p (“will not”), but “might not” affirms "will not" as an option! One says Z~p is not an option and the other tells you it could be true. On the other hand “might not” falsifies Zp ("will"), but “might” affirms "will" as an option. One says Zp is false, the other tells you it could be true.
This is why I argued that the square not only cannot affirm Qp (“it is indeterminately the case that…”) in such as way as to leave open the question as to whether Zp or Z~p were true. The square also fails to affirm Qp, and ONLY Qp, in conjoining “might” and “might not.”
Lastly, this is also why I argued on the revised square that NO TWO EQ's (the epistemological quantifiers) can be symultaneously true.
It’s wacko I know. But it makes perfect sense to me. I don’t know how we can conjoin what “might” and “might not” deny without conjoining what they hold out as possibly true. If you have a proven rule that’ll get us out of having to conjoin what “might” and “might not” entail as options, I’m all ears.
Convinced?
quote:
Doc: ...given his formal training, Alan should bear the primary life time responsibility of forging this revolution into the philosophical academy!
Most definitely. Mucho mucho! We just want a percentage of all his royalities.
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Tom
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| Posted 11/12/2003 1:18 AM (GMT -6) | | I was thinking some more on the question of whether “might” and “might not” from the square can be conjointly true; and after looking at our revised triangle within a triangle, I’m even MORE convinced that these cannot be conjointly true. Sorry to post it again, but take a look at the square just six or seven posts up from this.
Look at the three EQs. Those are you quantifiers (or whatever they’re called). It wood seem to me that no two of these EQs can be simultaneously true. Each falsifies it’s contradictory and leaves open the question of the truth values of the other two. In a sense, the EQs function in two ways, one ontological and the other epistemological. To conjoin them involves conflating these functions are screws up our minds (at least mine!). In falsifying their contradictories they come down on one side of the fence with regard to an ontological state of affairs, but in not coming down on one side of the fense regarding the truth value of the other two states of affairs they express epistemological uncertainty regarding what the truth of the matter is in the case of the other two.
Given this dual function, any two conjointly TRUE EQs would by definition falsify and affirm as possible two states of affairs. And you can’t do that. Look closely at what each of two conjointly true EQs entail. Your “might” and “might not” quantifiers are the bottom left and right props on our triangle. If “might” and “might not” are both true, then “will” and “will not” are both falsified and affirmed as possibly the case. Hmm?
And if these can both be simultaneously true, there’s no reason why any two EQ’s cannot be true. So what about the top EQ ~Qp (“not indeterminately the case that p”)? Why couldn’t this be conjointly true with EITHER ~Zp (“not determinately the case that p”) or ~Z~p (“not determinately that not-p”)? That would be incoherent. It’s futile Doc. Any two EQ’s would end up falsifying and affirming as possible the same propositions.
What I need to do is come up with theorems that govern the relationship of the three EQs. Now I have something new to obsess over!
You just have to s-t-a-r-e at the triangle long enough. If you obsess with it, that’ll help too! If we’re abandoning the square for the circle with regard to future tense claims about what is or is not determinately the case, then say good-bye to the square, period. Don’t look to it for help in expressing indeterminacy. It’s wired against this by virtue of being by definition an expression of a metaphysic that excludes ontological indeterminism.
Toss the square out. Bury it with a shovel, and then bury the shovel.
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Tom
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| Posted 11/12/2003 2:08 AM (GMT -6) | | You guys are gonna FREAK!
Watch closely. Take theorem #5 from the final list of indeterminacy theorems above:
5. Qp --> ~Zp^~Z~p
What’s this say? It says: “IF ‘it is indeterminately the case that p’, THEN ‘it is not determinately the case that p’, AND ‘it is not determinately the case that not-p’”)
Spell it out simpler: “If it’s the case that p is indeterminate, then it’s the case that p ‘might’ and ‘might not’ obtain.” Isn’t this what Doc is saying? Shouldn’t we end up with Qp if we START with ~Zp^~Z~p? That is, if Qp --> ~Zp^~Z~p holds, then shouldn’t ~Zp^~Z~p --> Qp (which is what Doc argues) hold? I don’t think so. There’s something else freaky going on.
For example, if our theorem, Qp --> ~Zp^~Z~p, is correct, then should not Qp imply or entail what necessary follows from both ~Zp and ~Z~p? ~Zp entails ~Z~p v Qp, while ~Z~p entails Zp v Qp. Shouldn’t Qp --> ~Zp^~Z~p then be equivalent to Qp --> (~Z~p v Qp) ^ (Zp v Qp)? That’s NUTS!
I mean, if a --> b and b -->, then should not a --> c? If Qp implies ~Zp^~Z~p, then shouldn’t Qp imply what ~Zp and ~Z~p each implies? It can’t. Starting with Qp and saying Qp --> ~Zp^~Z~p (If ‘indeterminacy’ then not ‘determinacy’) isn’t the same as starting with ~Zp^~Z~p and saying ~Zp^~Z~p --> Qp. That is, though ~Zp follows from Qp, Qp doesn’t imply what ~Zp implies when one begins with ~Zp.
We HAVE to determine what the relationship is between the EQs on our new triangle. Are they contrary or what? Or is the fact that they affirm both what is ontologically and merely epistemologically the case screwing with my mind? And I’m banging my head against the wall right now ‘cause I can’t figure it out!
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| Posted 11/12/2003 3:52 AM (GMT -6) | | Blaaaaaah-
Is anybody even paying attention to this stuff any more? Yeeeeow!
I have a question:
Is a --> b = b --> a a valid argument? That is, is it true that “If a then b = If b then a”? If this is true, then Doc is probably right about “might and might not.” If it’s not true, then I’m probably right about “might and might not.”
OK, I think Doc just MIGHT (ha!) be correct about conjoined “might and might not.” Here’s why.
Take our three metaphysically distinct states of affairs:
A = determinately the case that p
B = determinately the case that not-p
C = indeterminately the case that p
Now, run the numbers on this:
First—
This is definitely true: A --> ~B ^ ~C (“If A, then not-B and not-C”)
But does it follow that: ~B ^ ~C --> A (“If not-B and not-C, then A”)
Again—
This is definitely true: B --> ~A ^ ~C (“If B, then not-A and not-C”)
But does it follow that: ~A ^ ~C --> B (“If not-A and not-C, then B”)
Again—
This is definitely true: C --> ~A ^ ~B (“If C, then not-A and not-B”)
But does it follow that: ~A ^ ~B --> C (“If not-A and not-B, then C”)
If the argument I started with above (a --> b = b --> a) is valid, the second in each of the three examples does follow from the first and Doc is right about “might and might not” being equivalent to “indeterminately the case.” In fact this is the third example immediately above. The only way I can explain it would be to say that though ~Zp and ~Z~p each respectively falsify Zp and Z~p but respectively leave us epistemologically in the dark regarding Z~p and Qp (in ~Zp’s case) and Zp and Qp (in ~Z~p’s case), what is falsified and affirmed as possible by “might” and “might not” gets cancelled out and we’re left with just Qp. That is, ~Zp and ~Z~p each falsify their contradictories, Z~p and Zp respectively. So those two are out the window. But ~Zp and ~Z~p also each hold out as possibly true Qp and Z~p (in “might not’s” case) and Qp and Zp (in “might’s” case). But Zp and Z~p are already cancelled out. What’s left? Qp.
So, any two EQ props on the revised triangle MAY be conjointly true, in which case the state of affairs which is TRUE is that which lies in between the two EQ props.
Congratulations Doc. Friggin Yeah! I still wish somebody smarter than me would formalize the relationships. Or did I just do that?
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| Posted 11/12/2003 4:34 AM (GMT -6) | | Laika: Tom, from a definition of "Three-Valued Logic": "A logical structure which does not assume the law of the excluded middle. Three truth values are possible: true, false, or undecided."
Tom: Here's my problem. This seems to deny bivalence. If it doens't deny bivalence, then it would seem that the "undecided" truth value spoken of here in fact does not attach to propositions (bivalence would demand that all props are either true or false), but rather to our knowledge of propositions. And certainly WE can be undecided about the truth value of a proposition and a proposition describing this uncertainty would be true. But for a proposition to be neither true nor false but "undecided"? I don't know what that means.
If "undecided" means "indeterminately the case that..." then we in fact only only two truth values. It's either true or false that it's indeterminately the case that p.
I definitely need more education on all this, but that's where I am right now. Thanks for the info too!
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