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Tom
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Date Joined Jul 2001
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| Posted 11/5/2003 10:30 AM (GMT -6) |   | Laika-
Can you give that to me as a proposition?
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Alan Rhoda
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| Posted 11/5/2003 2:02 PM (GMT -6) | | Tom: I could say, “IF it is indeterminately the case that Tom comes to town next year, THEN it is indeterminately the case that he’ll bring Cohibas with him.” Or I could say, “IF Tom might and might not come to town next year, THEN he might and might not bring Cohibas.” True antecedent, thus no counterfactuals. I could even say “IF it is indeterminately the case that Tom comes to town next year, THEN he 'WOULD' bring Cohibas.” Given the truth of the antecedent, the “would” here is not counterfactual. All I’m trying to say is that not ALL “would’s” and “might’s” are counterfactual. Do you want to deny this?
Alan: I don't deny that there are noncounterfactual "would" and "might" conditionals. But I took you to be arguing a stronger point than that. You have been arguing, if I understand you correctly, that ALL future-tense "might" conditionals are, as such, noncounterfactual. That is the claim I have been challenging.
Tom: Now, logicians and philosophers aside, NOBODY talks like this. Nobody says, “Gee, if Tom might and might not…” or “Hey, if it is indeterminately the case that Tom…” No, your average Joe (like me) simply says, “If Tom comes to town next year…” and we all know what I mean. Now, you’re right that this is technically/logically/philosophically NOT the way to express it.
Alan: I agree with you, Tom, that colloquial speech pattern work just fine in everyday contexts without the anal-retentive precision that logicians strive for. When we move into analyzing the logic of everyday claims, however, we need that kind of precision in order to make logical relations clear. The power of mathematics, for example, is a function of its abstract precision. If "2" and "4" meant approximately 2 and 4, and "equals" meant approximately equal to, then we would no longer be able to say for sure whether "2+2=4" was true or not.
Tom: So what SIMPLE way is there for me to say it?
Alan: I think the closest approximation to logical precision consonant with acceptable colloquial speech is to split the conditional into two claims: I (Tom) will most likely come to town next year. If I do, then I will probably bring Cohibas.
Tom: Hey, could I say ... "If Tom might come to town next year, he might bring Cohibas”? Hey, in this case the “might” of “If Tom ‘might’ come to town next year” doesn't even posit state of affairs (c) with respect to my coming! It's the same “might” of the traditional square.
Alan: I don't see why we would have to interpret "might" either way. The context of the utterance should make clear whether state (c) or the square "might" is intended.
Tom: Why call the square OR our triangle a square or triangle of counterfactuals? This assumes that all conditional proposition to be discussed on the square or triangle have false antecedents. But we admit they don’t. Why talk of "might-counterfactuals" as if this includes all might-propositions?
Alan: No one has claimed that all "might" propositions are counterfactuals. The context that I am operating in when I talk about counterexamples has to do with God's foreknowledge and providence. What could God know sans creation by means of which he could exercise providence over creation? From a sans creation perspective, ALL conditionals in which the antecedent presupposes creation are counterfactuals, for they have false antecedents. 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. In both cases the debate concerns what knowledge is available to God independently of creation. That's why scholars talk of "would" and "might" counterexamples. I don't doubt that you can find noncounterfactual "might" conditionals in other contexts, but with respect to the debates surrounding open theism, it is the counterfactuals that are important.
quote:
I’m still searching ... for a modal operator for Alan’s state of affairs (c) “It is indeterminately the case that” and have come up with nothing. ... Alan, don’t you find it extremely bothersome (I do) that of the three metaphysically distinct states of affairs…
a) Only two can be affirmed as true by all the logics (propositional, modal, tensed, whatever) known to Western academia, and
b) Only these two can be falsified with a proposition (“might” on the one hand, “might not” on the other) that does not also affirm as true any state of affairs?
There is no proposition IN LOGIC that affirms or denies indeterminacy!
Alan: I think you are correct, Tom, that philosophers have traditionally given slight attention to indeterminacy, vagueness, uncertainty, and things like that. Things have been changing over the past couple decades. But I don't think traditional logic is the problem. I think traditional logic is capable of dealing with indeterminacies and such, though maybe not as elegantly as you would like. The most straighforward way to do it is to characterize in-determinacy as the negation of determinacy. Thus, "it is indeterminately the case that p" is equivalent to "it is not determinately the case that p and it is not determinately not the case that p". Another way is in terms of the probability calculus. Thus, "it is indeterminately the case that p" can be interpreted as "The probability that p is between 0 and 1." So it seems quite evident to me that traditional logic can say what needs to be said. The problem that open theists face is not fundamentally one of logic but of metaphysics. It is metaphysical assumptions of thoroughgoing determinacy that lead thinkers to overlook or dismiss the possibilities of indeterminacy inherent within their own logic.
Now could we invent a logical system that took indeterminacy as a primitive term? Sure. Doing so may bring to the fore what has often hitherto been overlooked. Here's a start:
I(p) = It is indeterminately the case that p.
D(p) = It is determinately the case that p.
The 3 metaphysical states of affairs can then be represented as: I(p), D(p), and D(~p)
Some basic theorems (these are open to debate):
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)
Anyone wanna help develop this further?
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Laika
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| Posted 11/5/2003 2:24 PM (GMT -6) | | | | |
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Tom
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Date Joined Jul 2001
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| Posted 11/5/2003 4:44 PM (GMT -6) | | Alan: I don't deny that there are noncounterfactual "would" and "might" conditionals. But I took you to be arguing a stronger point than that. You have been arguing, if I understand you correctly, that ALL future-tense "might" conditionals are, as such, noncounterfactual. That is the claim I have been challenging.
Tom: I've been firing this stuff out pretty passionately. Maybe I was unclear. There obviously are false future-tense “might” conditionals. I do think that we are less likely to assume that such a prop is false simply because it has to do with the future and we’re not omniscient. The only way we would know a future-tense "might" conditional was false is if it posited a contradictory state of affairs.
Alan: I agree with you, Tom…
Tom: I needed to hear that! 
Alan: I think the closest approximation to logical precision consonant with acceptable colloquial speech is to split the conditional into two claims: I (Tom) will most likely come to town next year. If I do, then I will probably bring Cohibas.
Tom: Interesting. You used “most likely” to posit what is “indeterminately the case.” Is this just the language of probability logic?
quote:
Tom: Hey, could I say ... "If Tom might come to town next year, he might bring Cohibas”? Hey, in this case the “might” of “If Tom ‘might’ come to town next year” doesn't even posit state of affairs (c) with respect to my coming! It's the same “might” of the traditional square.
Alan: I don't see why we would have to interpret "might" either way. The context of the utterance should make clear whether state (c) or the square "might" is intended.
I was thinking about the anal-retentive logicians! Personally I don’t think the context (we’re talking local yocals, right?) would lead anybody (other than anal-retentive logicians!) to stop to consider the possibility that my “might” means only false that “I will not” and either D(p) or I(p).
quote:
Tom: Why call the square OR our triangle a square or triangle of counterfactuals? This assumes that all conditional proposition to be discussed on the square or triangle have false antecedents. But we admit they don’t. Why talk of "might-counterfactuals" as if this includes all might-propositions?
Alan: No one has claimed that all "might" propositions are counterfactuals…
Fair enough. But as I read this site and articles like the one this thread was started to discuss, the only “might’s” in view are counterfactuals, and as we’ve worked our way around the square discussing this stuff nobody even hinted that there existed “might” and “woulds” that were not counterfactual. I’m glad to hear you say it isn’t so.
quote:
…The context that I am operating in when I talk about counterexamples has to do with God's foreknowledge and providence. What could God know san creation by means of which he could exercise providence over creation? From a sans creation perspective, ALL conditionals in which the antecedent presupposes creation are counterfactuals, for they have false antecedents.
Right. And knowing what is false at such a juncture would be providentially helpful for God. I mean “If…the” clauses with false antecedents. But surely knowing what is antecedently TRUE (true “If…then” conditionals) at such a juncture would be AS providentially helpful for God. That’s what I’m arguing. It's the fundamental point of the infinite intelligence argument. True "might" conditionals are useful in planning and preparing. If a being were to know them all, and sacrifice no intellectual space to process them infinitely, then there is termendous providential advantage. I know Molinists admit divine knowledge of logical possibilities sans creation. But foreknowledge of THIS world they argue, contains no knowledge of what is indeterminately the case.
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.
Tom: I’m close to flushing the category of “middle knowledge.” With regard to the future God’s keeps one account of three states of reality, D(p), D(~p), and I(p). As reality changes, God keeps infallible and immediate track of the changing truth values.
Alan: In both cases the debate concerns what knowledge is available to God independently of creation. That's why scholars talk of "would" and "might" counterexamples. I don't doubt that you can find noncounterfactual "might" conditionals in other contexts, but with respect to the debates surrounding open theism, it is the counterfactuals that are important.
Tom: Well, seems to me that if the infinite intelligence argument is important to this debate (and it seems to be), then nocounterfactual “might” conditionals are important to this debate, for they compose a great percentage of God’s knowledge and thus the basis of God’s providential choices sans creation.
Alan: I think you are correct, Tom, that philosophers have traditionally given slight attention to indeterminacy, vagueness, uncertainty, and things like that.
Tom: See, you’re doing it. Indeterminacy is ontological. There’s nothing vague or uncertain about it. It's no more vague or uncertain than either of the other two metaphysically distinct states of affairs you argue for. Vagueness and uncertainty are psychological/epistemological. And if we’re talking epistemological states of affairs, then hey, we're just as often vague and uncertain about what is determinately the case as we are about what is indeterminately the case. So, there is nothing whatsoever vague or uncertain about it’s being the case that my doing x is indeterminate.
Alan: Things have been changing over the past couple decades. But I don't think traditional logic is the problem.
Tom: Not “logic” itself. Sure, for we can argue logically that indeterminacy “fits,” and that it has been slighted. “Logic” itself is our greatest ally.
Alan: I think traditional logic is capable of dealing with indeterminacies and such, though maybe not as elegantly as you would like.
Tom: Maybe I have more faith in logic. I think it’s capable of dealing with indeterminacies as elegantly as it deals with determinacies. If your three metaphysically distinct states of affairs are the case (and I agree they are), then logic will do just fine in accounting equally for them, IF WE WANT IT TO, and if we let it. Presently it’s not being allowed to do as good a job as it could because we assume our square, for example, is sacrosanct.
Alan: The most straighforward way to do it is to characterize in-determinacy as the negation of determinacy.
Tom: How in the world is THIS the MOST straightforward way to characterize indeterminacy? Because in English we form “indeterminacy” by prefixing “determinacy” with “in”? Duuude! We’re talking about metaphysically distinct states of affairs. There’s no metaphysical prefix to what is determinately the case that gives us indeterminacy. Putting “in” in front of “determinacy” doesn’t mean metaphysically speaking indeterminacy is subservient to determinacy, or that you ought to start with determinacy and then just negate it to get to indeterminacy. Why not START with indeterminacy to get to indeterminacy. Think R-E-F-O-R-M-A-T-I-O-N.
I can think of a far more straightforward way to characterize indeterminacy: Positively affirm indeterminacy. Indeterminacy has every right to be as straightforwardly expressed as determinacy. If the most straightforward way to characterize determinacy is to posit it, to affirm it in its metaphysical distinctness, why think indeterminacy cannot be as straightforwardly characterized? Why think indeterminacy’s chief characteristic is what it negates and not what it affirms? Can we only get to indeterminacy THROUGH determinacy? How does this follow from there being three metaphysically distinct states of affairs? You don’t get to determinacy THROUGH indeterminacy. It’s so bogus, Alan. Forgive my passion. It’s the subject. You don’t treat determinacy this way and it’s no more “metaphysically the case” than indeterminacy is “metaphysically the case.” I might just as justifiably say, “Hey, let’s take determinacy off the square and out of the propositional picture and characterize it as the negation of indeterminacy. That way, the only way you can affirm ‘determinately that…’ is to negate ‘indeterminately that…’.” See how one-sided this is? To agree that the most straightforward way to characterize indeterminacy is as the negation of determinacy is to help maintain the matrix of deterministic metaphysics that has defined how we DO logic for so long. It makes indeterminacy a second-class citizen, the proverbial female in a determinately male world.
Alan: Another way is in terms of the probability calculus. Thus, "it is indeterminately the case that p" can be interpreted as "The probability that p is between 0 and 1."
Tom: There seems to be more equality, and so more hope, in this.
Alan: So it seems quite evident to me that traditional logic can say what needs to be said.
Tom: It CAN be said better, more accurately, more straightforwardly than you have suggested, and more reflective of what you believe is the case about reality.
Alan: The problem that open theists face is not fundamentally one of logic but of metaphysics.
Tom: Yes…but. I haven’t been arguing that logic is our problem. Logic is our ally. Metaphysics is the battle. We have to show that “indeterminately the case” is just as metaphysically distinct a state of affairs as is “determinately the case.” But how we DO logic, TALK logic (the logical machine-work, the apparatus, the squares, the diagrams, the notation, the operators and all) is presently set up to suit and express very well a metaphysics that excludes indeterminacy. If all this is to include indeterminacy, it will have to change some.
Alan: It is metaphysical assumptions of thoroughgoing determinacy that lead thinkers to overlook or dismiss the possibilities of indeterminacy inherent within their own logic.
Tom: Now you’re tawkin like a revolutionary! If you’re going to fight the metaphysical war, if you’re going to argue that indeterminacy is a metaphysically distinct state of affairs alongside determinacy, then you’re going to have to rewrite some of the manuals on logic and redraw some of the paradigms that were developed over time to logically express a metaphysic that excluded indeterminacy.
It's like this. If two people are sitting in a room in chairs face to face and a third person enters, you have to add a third chair AND rearrange the chairs if the three are to situate themselves equally with respect to one another. We could say that "indeterminacy" has been standing outside the room, or just standing in the room without a chair, while “determinately that p” and “determinately that not p” have been sitting comfortably owning the discussion (only occasionally talking ABOUT the possibilty of indeterminacy's existence when one of them happens to be negated). Indeterminacy is presently not even allowed to SPEAK for itself (to affirm itself), it can only be talked ABOUT by the other two (via negation).
If we hope to win the battle to admit “indeterminately the case that p” as a metaphysically distinct state of affairs, we’ll have to add a chair won’t we? That’s what my triangle with THREE equally straightforward affirming props (not just two as is now the case on the square) and three equally negating props with equal implications (not just two as is now the case on the square) does. So, as you and Doc argue with the professionals about an open metaphysic, at least do it with logic that’s expressed in terms and figures and operators that respect your metaphysics. The square doesn’t respect your metaphysics, nor does any other area of our logical apparatus (well, maybe probability logic; gotta check that out!).
Alan: Now could we invent a logical system that took indeterminacy as a primitive term? Sure. Doing so may bring to the fore what has often hitherto been overlooked. Here's a start…
Tom: Not only “could” we, Alan, but we should and must IF we expect to win the argument on metaphysics. Not only “might” this bring to the fore what has been hitherto overlooked, it will certainly bring this to the fore. And you HAVE to bring it to the fore to argue your metaphysics. This is what I’ve been crying out for two pages (and for lots of other pages for months) now. The new operators and notation you suggested are great. I’ve been toying with similar notion for at least a year. What we need, my friend, is a reformer with the credentials to lead the way. And you are the FRIGGIN MAN!
Alan: Anyone wanna help develop this further?
Tom: Lead the way, Luther! I’ll shine your shoes, type your papers, and lick your stamps for you. I’ll also get royally on your nerves.
Our eyes are fixed on YOU!
Doc?
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Tom
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| Posted 11/5/2003 6:39 PM (GMT -6) | | quote:
Alan: Some basic theorems (these are open to debate):
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)
Laika: 1. ~I(p) = D(~p) (see 4)
Tom: Alan's right here. ~I(p) (= "It is not indeterminately the case that p" = the independently true "might" of the traditional square) = D(p) or D(~p) (= It is either "determinately the case that p" or "determinately the case that not p").
quote:
Laika: 2. ~D(p) = D(~p) (see 3)
Again, I think Alan on on the money. ~D(p) (= "It is not determinately the case that p" = the independently true "might not" of the square) = I(p) or D(~p) (= It is either "indeterminately the case that p" or "determiantely the case that not p").
Laika: ~I(p) = ~D(p)
Tom: Again, here you simply have the falsifed "It is indeterminately the case that p." This proposition does NOT exist on the traditional square by the way. There is NO WAY on the square you can falsify "indeterminately the case that p" and leave yourself with an option between "determiantely that p" and "determinately that not p." This is a BIG piece missing from the square and a BIG reason to DUMP IT. But all three falsifiers, however, do exist on my modified triangle above. ~I(p) is the very top EQ (False that C: Either A or B) on that triangle. If all we know is ~I(p) ("It is not indeterminately the case that p"), then what logically follows is NOT ~D(p) ("It is not determinately the case that p"). On the contrary, all that follows is, as Alan states it, D(p) or D(~p). Remember the three states of affairs: D(p), D(~p), and I(p)? If all you posit is the falsity of any ONE of these (which is all ~I(p) does), all that logically follows is that one of the remaining two is true and one is false. But you don't know which.
Did I do OK for an amateur Alan? By the way, Alan (you stinker), look at your theorems and then look at my larger triangle with the three states of affairs and the three falsifiers. Notice anything? All you state in your theorems is what I illustrated in my figure. EXACTLY the same thing. I illustrated a more complete way of accounting for propositions expressing what is determinately or indetermiantely the case. And I've been explaining it for two pages. You expressed it EXACTLY as I've been arguing but just used operators and theorems. Take your theorems and illustrate them and you get my formatted triangle above.
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Tom
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| Posted 11/5/2003 11:55 PM (GMT -6) | | I’m reading through Arthur Prior’s Papers on Time and Tense. It was first published in ’68, just one year before his death. It is the third of Prior’s books on temporal logic.
Chapter X, “Recent Advances in Tensed Logic,” has a very interesting section subtitled Strong and Weak Futures (pp 107-108). Let me share just a few paragraphs. Everything in brackets [] is me spouting. There appears a backwards “C” in the notation that is common notation for “If-then” which I can’t type here, so I used a closing parenthesis with a space before and after it (as opposed to parentheses with no space, which are just parentheses). So,
p ) q means “If p then q.” Got it? Close enough. Prior gets complicated towards the end, but the point I make is rather simple. I hope I've typed it in without screwing up Prior's theorems.
quote:
There is one law in Lemmon’s minimal system…which doesn’t sound as noncommittal as all that, namely p ) HFpp, asserting that whatever is now that case has always been going to be the case. By contrast with its mirror image p ) GPp (whatever is now the case will always have been the case), this thesis has for many a flavour of determinism [more than just a “flavour” I’d say]. There are various ways of removing this impression. [Prior, an open theist, now goes searching for a way to affirm “indeterminately that,” but notice how contorted an approach he adopts.] We might say, for instance, that if determinism is false there is, at each moment, no actual future but only a number of alternative possible futures [good! ], so that Fp can only mean ‘It will be the case in some possible future that p’. p ) HFp would then mean merely that if anything has in fact come to pass then it must always have been ‘on the cards’ [How’s that for technical anal-retentive jargon Alan!] that it would do so, and there is nothing deterministic about this. Genuine determinism would be the belief that there is only one possible future, and to express this you really do need to go beyond Kt [temporal logic], and add a postulate for nonbranching of the future, e.g., PFp ) (p v Pp v Fp) [v = “and” ] (‘Whatever has been “on the cards” either is the case or has been the case or is “on the cards” still’).
If, however, in an indeterministic universe we read Fp thus, there must surely also be another Fp for which p ) HFp would have deterministic overtones – an Fp which means that p will be the case in every possible future. ~F~p, i.e., Gp, is not exactly what we want here, [No it’s not; what we want is simply Ip, “indeterminately that p,” don’t you see it Prior?], for this goes too far; for since Fp means that p will be true somewhere in some possible future, ~F~p would mean that it will not be true anywhere in any possible future that not p, i.e., that p will be true throughout every possible future. What we want is something in between, which says that p will be true somewhere in every possible future. [Actually, all Prior needs is a primitive operator for “it is indeterminately that case that…”]
My own past attempts to bring out this distinction have been via metric tense logic. If F(n)p means that it will be the case the interval n hence in some possible future that p, then ~F(n)~p would mean that it will not be the case the interval n hence in any possible future world that no p, i.e. that it will be the case the interval n hence in every possible future that p. In brief, if we read F(n)p as ‘It could be the interval n hence that p’, we should read ~F(n)~p as ‘It is bound to be the case the interval nhence that p’. We may then define the weaker of the plain Fps as…‘It could be the case at some future time that p’, and the stronger of the plain Fps as…‘It is bound to be the case at some future time that p’.
Interesting eh? I kept thinking to myself while reading Prior (and this is his third book on tensed logic and he still hasn’t found a primitive operator for “indeterminately that case that p”), What’s so difficult about this? How could somebody, an open theist already, with such a grasp of modal and tensed logic, not see that the postulate needed to logically remove the “flavor” of determinism is simply: Ip, “indeterminately that p.” Rather than leave Fp, “It will be the case that p” to signify what it already means, viz., “It is determinately the case that p” and come up with another (and Prior even says, “…there must surely be another Fp…”!) operator for indeterminacy, Prior tries to finagle a ‘stonger’ (i.e., deterministic) meaning for Fp and a weaker (i.e., indeterministic) meaning for Fp! I feel like throwing him a life-preserver. It hurts to see him struggle to squeeze indeterminism out of Fp, that is, “It will be the case that p”. Leave Fp alone (it posits “determinately the case that p” and is fine as it is) and just provide another primitive, Ip, for “indeterminately that case that p”). Done, finished, outta there.
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Tom
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| Posted 11/6/2003 1:40 AM (GMT -6) | | quote:
Laika: Here's Alan's 3 and 4:
[code]
3. ~D(p) = I(p) or D(~p) <---- original
3. ~D(p) = D(~p) <---- it's an "or" so choose one
AND
4. ~I(p) = D(p) or D(~p) <---- original
4. ~I(p) = D(~p) <---- it's an "or" so choose one
Alan will have to chime in 'cause I'm not an expert on this stuff, BUT...I would take issue with "it's an 'or' so choose one." The either/or posited here is not, "Choose either one you want to be true," but rather "Of the remaining two props, one IS now true and one IS now false and we CAN'T choose which is which based on the mere falsity of Ip." Your choosing doesn't resolve what is indeterminate regarding the truth values of the two remaining props. Rather, bivalence being true, the truth values are what they are before you observe the propositions. We just can't know which is which knowing what we know about the first proposition.
I stink at explaining this stuff.
Tom
By the way, as I read this stuff I'm noticing that people don't use parentheses. It's Ip, and not I(p), for "It is indeterminately the case that p." No big deal I guess. | | Back to Top | | |
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Tom
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| Posted 11/6/2003 4:59 AM (GMT -6) | | The Revolution officially begins with THIS figure:
Thanks to Laika for making these URLs available to us.
Alan, this is my adjusted square with the indeterminacy theorems in place of the propositions. Just wanted you to see that what you had posted and what I had drawn were the same thing.
I'll obsess with this some more tomorrow!
Tom | | Back to Top | | |
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Alan Rhoda
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| Posted 11/6/2003 2:44 PM (GMT -6) | | quote:
I'm doing substitution.
Here's Alan's 3 and 4:
[code]
3. ~D(p) = I(p) or D(~p) <---- original
3. ~D(p) = D(~p) <---- it's an "or" so choose one
3. D(~p) = ~D(p) <---- reverse sides "Commutativity"
Laika, your second step is illicit. You can't simplify an "or" like that. A disjunction does not affirm the disjuncts, so I(p) or D(~p) doesn't imply D(~p). Furthermore, the first statement is an equivalence, so if you change one side, then you have to change the other side as well to maintain the equivalence.
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Alan Rhoda
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| Posted 11/6/2003 3:55 PM (GMT -6) | | quote:
Originally posted by TGB
The Revolution officially begins with THIS figure:
Alan, this is my adjusted square with the indeterminacy theorems in place of the propositions. Just wanted you to see that what you had posted and what I had drawn were the same thing.
I like the triangle, Tom. 
Working out the details of an indeterminacy logic will undoubtedly take some time and require some sophistication. I wouldn't be surprised, though, if somewhere out there someone hasn't already published something on it. But who knows, maybe it all started here?
At any rate, to develop a complete indeterminacy logic will require at least (1) rules defining what counts as a well-formed formula in the system, (2) a set of intuitively obvious axioms, and (3) derivations of important theorems from the axioms to extend the usefulness of the logic.
You are right, Tom, that D(p) can be written as Dp, etc. The parentheses are only necessary where the scope needs to be clearly specified.
Here are some further proposed theorems:
7. I(p and q) = Ip or Iq
8. I(p or q) = Ip and Iq
9. I(p --> q) = p --> Iq
I'm not too sure about these. What d'all think?
Alan | | Back to Top | | |
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Alan Rhoda
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Date Joined May 2002
Total Posts : 680
| Posted 11/6/2003 4:04 PM (GMT -6) | | quote:
So ~D(p) = I(p) or D(~p) is not equivalent to:
[~D(p) = I(p)] or [~D(p) = D(~p)] ?
I'm afraid not.  | | Back to Top | | |
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One Love
Registered Member
Date Joined Jan 2003
Total Posts : 13888
| Posted 11/6/2003 5:18 PM (GMT -6) | | | Yo. This triangle works. I'm seriously feelin' it. Good stuff. | | Back to Top | | |
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Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 11/6/2003 7:50 PM (GMT -6) | | Alan: I like the triangle, Tom.
Tom: I was beginning to wonder whether or not I was a total idiot or on to something.
Alan: Working out the details of an indeterminacy logic will undoubtedly take some time and require some sophistication… At any rate, to develop a complete indeterminacy logic will require at least (1) rules defining what counts as a well-formed formula in the system, (2) a set of intuitively obvious axioms, and (3) derivations of important theorems from the axioms to extend the usefulness of the logic.
Tom: So what do we need, a couple of weeks? You and Doc should be able to punch it out before Christmas, eh?
Alan: I wouldn't be surprised, though, if somewhere out there someone hasn't already published something on it.
Tom: I think a couple of people tried, but they were committed to insane asylums. Seriously, just saying “indeterminacy logic” makes me think of “probability of logic.” If probability logic is up and running, doesn’t it already express this sort of stuff? Can we piggy back on it?
Alan: 7. I(p and q) = Ip or Iq
Tom: Ooo, I don’t see this. If it’s indeterminately the case that ‘p and q’ (that is, Ipq), then wouldn’t it follow that it’s Ip AND Iq (it is indeterminately the case that p AND indeterminately the case that q)?
Alan: 8. I(p or q) = Ip and Iq
Tom: Same thing here. I(p or q), I take it means, “It is indeterminately the case that either p or q,” or “It is either indeterminately the case that p or indeterminately the case that p.” Here it seems you’re using “I” as an epistemological quantifier only, just to falsify ~Ipq. I think we want to use I for positing ontological categories. Or am I missing something here?
Alan: 9. I(p --> q) = p --> Iq
Tom: Seems right on.
Alan, reading some tense-logic stuff yesterday I ran across a modal operator (or functor?, what d’ya call ‘em?) “I” that’s used in tense logic and means something like “Instant” where Ipq means “At the instant of p, q” or “When p, then q.” So we’ll have to find some other letter or some symbol for “indeterminately the case that….” “I” is already spoken for. How about the capital E, for “ean,” the Greek particle meaning “if”? But that could be confused with the backwards capital E (can’t type it, the three forks point to the left) which is a quantifier (or something) meaning “For every x, y” or something like that.
How about a capital M for (the ontological) “Might” (in the strongest sense)? That’ll work. I don’t think M is spoken for in modal logic. Mp would mean ‘It is indeterminately the case that p.”
Here’s another important question I have.
Prior, for example, had more than one “system” of tense logic. One used what he called “instant variables” (a, b, c, etc.) that were sentences that were themselves not tensed, like “Tom comes to St. Paul” or “Alan becomes the father or indeterminacy logic,” sentences that were stripped, you might say, of tense. Then Prior used the operators P, F, G, H, and others to posit the tense. So Fp means “It will be the case that p” and Pp means “It has always been the case that p” (I think that’s what the P operator means). Anyhow, you see what I mean. When the notation is spelled out in words, the tense inflection “will be,” “has always been,” etc., comes from the operator and not from the variable p.
But Prior developed another system to deal with variables (like, p, q, and r) that were themselves tensed propositions, like “Tom will come to St. Paul in 2004.” In this case the tense operator F (“It will be the case that…”) wouldn’t be used. A system one Fp (where p = “Tom comes to St. Paul in 2004”) would be equivalent to a system two p, “It is the case that p” (where p = “Tom will come to St. Paul in 2004”). It would be tautological to say Fp where p is already tensed. That would give us “It will be the case that Tom will come to town.”
This holds for indeterminacy too. If M means “indeterminately the case that…” and our p is, like Prior’s system two, already tensed and indeterminate, then we don’t need M. Question is, we have to come up with some consistent way of expressing “indeterminate that” without the operator M. 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 indepdently 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. So when we’re talking anal-retentive logic, how do we spell out the tensed p where p posits “It is indeterminately the case that Tom comes to town in 2004.” Hey, there it is! I just said it! We’ll just spell out M and say “It is indeterminately the case that Tom comes to town in 2004.”
See the difference in Prior’s two systems? On the one hand you can limit your variables (p, q, and r) to propositions void of tense, like “Tom comes to St. Paul,” and leave it to your operators P, F, G, and our new M, etc. to invest these variables with tense. On the other hand you can invest the variables themselves with tense (“Tom will come…” or “Tom came…” or “It is indeterminately the case that Tom comes” and not use the operators. Well, you could use some operators. If p = “Tom will come to St. Paul” you could say Gp, for “It has always been the case that Tom will come to St. Paul” and thus posit my coming as a necessary truth or a predetermined event.
What are we wanting to say with an indeterminacy operator M? Is this by definition tensed already? Are “determinacy” and “indeterminacy” by definition already invested with tense? One might argue they are. What is ‘determinately the case’ is determinately the case NOW. That doesn’t mean the state of affairs of p, “Tom’s coming to town,” is actual now. But the determinacy is a NOW thing. Whatever “was” is determinate, for it cannot be other than it is. Whatever “is” is determinate, for it cannot be other than it is. And whatever “will be” or “will not be” is determinate, for it cannot be other than it will or will not be. The past and present cannot by definition be indeterminate, for they cannot be other than they are. So whatever our indeterminacy operator means, it would seem to be limited to the only category of propositions which are not by definition determinate, and that is the FUTURE.
Indeterminacy then describes a state of affairs that is necessarily future, that is both actualizable and preventable (avoidable), neither inevitable nor—what’s the stinkin contrary of inevitable?—out of the question? impossible? You know what I mean.
So, what about our D, “determinately the case that…”? If the variable p is tensed, as in “Tom will come to town in 2004” then we don’t need D in front of p, for p already claims ‘determinately the case’ in saying Tom “will” come. But if p is stripped of tense (“Tom comes to town in 2004”) then you’re dealing with what Prior calls a system one claim: Dp. This would be equivalent to a system two p (where p = the tensed proposition “Tom will come to town in 2004” which posits its own determinacy without the operator D). To put the operator D (“determinately the case that…”) in front of the tensed variable p (“Tom will come to town in 2004”) is tautological. It posits determinacy twice.
It can become very confusing very quickly. We can stick to atemporal variables and leave it to our operators to posit tense, but we’ll have to deal with tensed propositions too.
So the triangle is system one tense logic (if you wanna call it that). The variable p in each case is tenseless, otherwise we wouldn’t need the tense operators. And we’ll have to dump “I” for another cap. M is fine unless you want to go with some other cap.
I think I’m out of my league now. The experts will have to take it from here.
I really appreciate everybody’s patience with my obsessing over this stuff. I think I’m done for a spell. Well, my wife certainly thinks so. Gotta be outta town for the weekend. Hopefully Doc will chime in and tell us what he thinks of our little revolution.
Tom | | Back to Top | | |
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Tom
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Date Joined Jul 2001
Total Posts : 10353
| Posted Yesterday 12:10 AM (GMT -6) | | I'm on vacation this weekend with the family and can't say much. Just wanted to give you a head's up, Alan. When I get back to the office I want to post some things I'm reading in A. Prior's Formal Logic. Unbelievable. There already IS an indeterminacy operator! Can you believe that? Its' the capital "Q". So, Qp means "It is contingently (or indeterminately) the case that p." Prior actually argues for a triangle of "determinately that p," "determinately that not p," and "indeterminately that p." He doesn't use those words, but it's there. He gets a little confused (listen to me! Sheesh) about it by conflating epistemology with ontology.
The "M" I suggested is already taken too. You won't believe how Prior and others use Q. Very interesting. But it's all there. Our triangle and everything! I almost shot through the ceiling when I read it.
Another guy Prior refers to is Augustus de Morgan (1806-1876). He brought probability out of mathmatics and into logic. He also developed logics for indeterminacy. Gotta find his books. Hmmm. De Morgan developed theories on the probabilities of life events that are still used today by insurance companies! I think I hit the jackpot!
See ya in a few days!
Tom | | Back to Top | | |
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Matthew Timothy Short
Huh?
Date Joined Apr 2002
Total Posts : 635
| Posted Yesterday 6:26 AM (GMT -6) | | It all sounds vague and not leading to any definite end worth the result; such a proposition has an infinite number of solutions?
Anyway, it’s not all together that clearly defined, grasped, or understood by me, that is what your trying to get at; I truly find the whole decision to discuss the matter INDISTINCT, HAZY, and OBSCURE!
Having P(F)un intended . . .
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Tom
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Date Joined Jul 2001
Total Posts : 10353
| Posted Today 3:29 AM (GMT -6) | | Hi Tim-
Believe me, nobody shakes their head at this stuff more than I do. I know how you feel (I think!).
But in asking, "...such a proposition has an infinite number of solutions?" I wonder if you're misunderstanding what a contingent proposition claims. For example, "It is indeterminately the case that Tom comes to St. Paul next year" doesn't have an infinite number of solutions that ground the truth of it's being indeterminately the case that I come to St. Paul. It only has two: Tom's coming and Tom's not coming. By indeterminacy we don't mean there are an unlimited number of solutions to how a contingent proposition could resolve itself. It has only those solutions specified in the proposition itself.
Sorry it's so hazy and unclear!
Tom | | Back to Top | | |
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Matthew Timothy Short
Huh?
Date Joined Apr 2002
Total Posts : 635
| Posted 11/10/2003 1:03 AM (GMT -6) | | Hey there Tom,
I am simply making fun of my own understanding of things uncertain, settled, fixed or not definite . . . humor is (not) the only thing intended? Think about it!
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Tom
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Date Joined Jul 2001
Total Posts : 10353
| Posted 11/10/2003 6:17 PM (GMT -6) | | Nice to have you back Doc!
Doc: But its worth mentioning at least that the triangle within a triangle that Alan drew...
Tom: Yeah, eh, great triangle figure there Alan!
Tom | | Back to Top | | |
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Tom
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Date Joined Jul 2001
Total Posts : 10353
| Posted 11/11/2003 8:30 AM (GMT -6) | | Even the ancient Egyptians understood logic better than we do. See?
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