 |
| 
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/13/2004 11:12 PM (GMT -6) |   | Just reading in Prior again:
quote:
2. The Sea-Battle Tomorrow
That there is such a third truth-value [neither true nor false] was suggested by Aristotle himself, in the ninth chapter of the De Interpretatione. This chapter is concerned with propositions in the future tense about matters whose outcome in not yet determined. That there are such matters he is convinced, for otherwise ‘there would be no need to deliberate or to take trouble, on the supposition that if we should adopt a certain course, a certain result would follow, while, if we did not, the result would not follow’. And ‘since propositions correspond to facts’, i.e., their truth or falsehood depends on their relation to facts, ‘it is evident that when in future events there is a real alternative, and a potentiality in contrary directions, the corresponding affirmation and denial have the same character’, i.e., have a potentiality both of being true and of being false, but are not actually either. For, Aristotle argues, it is not really possible to hold at one and the same time (a) that whether or not there will be a sea-battle tomorrow is as yet undetermined, and (b) that it is already either definitely true or definitely false that tomorrow a sea-battle will occur. For what is the case already has by that very fact passed out of the realm of alternative possibilities into the real of what cannot be altered. ‘Once it is, that which is is-necessarily, and once it is-not, that which is-not necessarily-is-not’, i.e., when it passes from the future into the present and so into the past, a thing’s chance of being otherwise has disappeared. This is the original context of the statement which we have already encountered as a subject of discussion by the Schoolmen, and we shall later consider whether this context puts a new complexion on it. In the meantime the moral drawn from it is clear—what is not yet determined cannot be a matter of present fact, by which the truth or falsehood of a proposition uttered now can be measured. Aristotle thinks, all the same, that although neither of its parts is either true or false, the entire disjunction ‘Either there will be a sea-battle tomorrow or there will not’ is definitely true.
If there really are propositions which are neither true nor false but just ‘potentially either’, this clearly means that the logic of truth-functions…cannot be applied without modification to fields in which such propositions arise. Among medieval writers, this was perceived at least by Ockham, who in his commentary on the De Interpretatione tries to work out what consequentiae involving ‘neuter’ propositions Aristotle would have admitted, and what he would have rejected. He takes his examples from theology, medieval writers being interested in the bearing of the three-valued theory on the doctrine of God’s omniscience. What he gives us is in fact a fascinating blend of three-valued logic and theological (or theo-epistemic) modalities. Since whatever is known can be expressed in a true proposition, if there are matters about which it is not at present possible to frame either true propositions or false ones, it would seem that there are questions to which God cannot yet be said to ‘know’ the answer. Consider, then, the two propositions ‘A will occur’ and “God knows that A will occur’, where A is a contingent, i.e., undetermined, future event. Can we say in such a case, on the Aristotelian view, ‘If God knows that A will occur, then A will occur’? Ockham thinks we can, for the antecedent in this case is simply false, and the consequent is ‘neuter’, and it is fitting that we should regard false propositions as implying neuter propositions, just as they imply true ones….And this means…that ‘If God knows that A will occur, then A will occur’ can be laid down as true no matter what sort of event A may be. For if it is already determined that it will not occur, the consequent will be definitely false, and so will the antecedent, for one cannot be said to ‘know’ that A is going to happen when it is in fact not going to happen.
I’d like to reopen the discussion of bivalence. And I’m looking for something more formal or ‘tight’ regarding the issue.
Alan-
Some time ago I argued that there is a difference between noting “that an event will or will not occur” and inquiring into “whether an event will or not.” As you can see, Aristotle appears to support a three-valued logic. Still, he believed that even if a future event S is contingent and no true or false propositions can now be formed about it, it remained true now that ‘Either S will obtain or S will not obtain’. That is…
“…Aristotle held that even when neither ‘There will be a sea-battle tomorrow’ nor its negation [‘There will not be a sea-battle tomorrow’] is definitely true, the alternation ‘Either there will be a sea-battle tomorrow or there will not be a sea-battle tomorrow’ is definitely true….His ground for asserting that ‘Either there will be a sea-battle tomorrow or there will not’ is true was almost certainly not that an alternation is to be automatically counted true if both its components are ‘neuter’. It is not because its components are both neuter, but because they are contradictories, that Aristotle affirms this disjunction. There is thus something non-truth-functional about it, and what it is would seem to be this: Aristotle has here confused two rather different propositions. The first is ‘Either there will or there will not be a sea-battle tomorrow’, taken as a truth-function which is true if and only if at least one of its components is true. In this sense the alternation in question is not true, since ex hypothesi neither of its components is true. The second is ‘It will be the case tomorrow that either there is a sea-battle or there is not’. This is not purely truth-functional, the truth-functional ‘either’ being governed by the non-truth-functional operator ‘It will be the case tomorrow that…’. There is no such operator in Lukaseiwicz’s three-valued system; but that system, as we shall see later, does afford grounds for considering this second proposition as true.”
What Prior is saying is that if tomorrow’s sea-battle is contingent, then we cannot agree with Aristotle that:
Zp v Z~p (where p = a sea-battle tomorrow)
But I think Prior realizes that Aristotle rather meant this:
Qp at t0, therefore p v ~p at t1. There is no third alternative for p at t1. Thus, even if tomorrow’s sea-battle is Q now, it is also now true that “It is determinately the case that the day after tomorrow either ‘There was a sea-battle yesterday’ or ‘There was not a sea-battle yesterday’ will be true.” This brings us back to the difference I noted some time ago between THAT and WHETHER. To note ‘that’ an event will either occur or not, or more popularly, that “Tomorrow, there either will or will not be a sea-battle” is true, is not to assume that either of its components is true, for what is meant here is not Zp v Z~p. But to note ‘whether’ an event will occur of not is to inquire into the ontological status of the present moment with respect to the determinate vs. indeterminate nature of tomorrow’s sea-battle, in which case Zp v Z~p v Qp is now true. ‘That’ and ‘whether’ may not be the best terms to use. I can’t think of anything better off hand though. To say ‘that an event will either occur or not at noon tomorrow’ is to say that at 12:01 it will be the case that the event in question will have either happened or not. It’s simply to note the either/or-ness of actuality. Events are either actual or they are not. S either obtains or it does not obtain. There is no third option. But to ask ‘whether an event will happen or not at noon tomorrow’ is to ask if its occurring is determined or contingent, in which case there are three options (Zp, Z~p, or Qp).
Prior gets complicated, but I posted the important passages from him before (can't remember where!). Too tired to search for it. But I'd like to delve into bivalence again (if anyone else wishes to).
Tom | | Back to Top | | |
 |
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/14/2004 1:39 AM (GMT -6) | | Take a walk down memory lane! http://www.gregboyd.org/gbfront/forum/topic.asp?TOPIC_ID=2801&whichpage=2
Wow.
Just thinking out loud here.
The fundamental reason, it seems, that Prior denied bivalence is because he followed Aristotle in believing that if a proposition is contingent (that is, a proposition describing a future contingent event), then so is its contradictory. If Qp, then Q~p (as Prior notes in his Formal Logic following Aristotle). He reasons that since a proposition and its contradictory by definition cannot both be true (or both false), and since it is the case that both p and ~p’s occurring is contingent, we therefore have to deny that the proposition expressing this contingency is bivalent, viz., that it is true (or false, as the case may be).
Now, it seems to me that Prior is wrong in arguing that Qp and Q~p are contradictories. They are not. These are contrary propositions. The contradictory of Qp is ~Qp, not Q~p. But what seems strange that when you remove the operator Q, p and ~p ARE contradictories. Put the operator back and they’re contraries. Weird.
It seems that Prior is thinking: If p is contingent, then ~p is contingent. That is, if p might and might not happen, then ~p might and might not happen. Kinda redundant, but true. So far so good. But, Prior argues, if both p and ~p and contradictories (and they are) and if both are contingent (are they are), then we cannot affirm bivalence, for a proposition and its contradictory cannot (by bivalence) both be true (or false). So, he concludes, if a proposition p is contingent (p might and might not happen), so is its contradictory (~p might and might not happen) contingent, and the proposition expressing this contingency is neither true nor false. In this way Prior means to avoid breaking the law which states that a proposition and its contradictory cannot both be true or false.
I’m trying to nail down exactly where we (Doc, Alan, and I) would argue Prior goes wrong in arguing FROM p and ~p are contradictories (which they are) and both are contingent (which they are) TO denying bivalence since both a proposition and its contradictory cannot both be true or false. It seems equally as obvious to me that Qp and Q~p and contraries and not contradictories (as are p and ~p). So what technically is happening when we add the operator Q? How does adding Q in front of p and ~p change p and ~p from contradictories to contraries?
I see Prior’s point in claiming that “Either a sea-battle will occur tomorrow or it will not occur” cannot be true unless one of its component’s is true. But I want (with Aristotle) to claim that if the event is contingent, then both component’s (will and will not) are false. But if both component’s are false, how can “Either component 1 or component 2” be true? What is it in “Either component 1 or not-component 1” that permits it to mean “It will be the case that a sea-battle either happens tomorrow or it does not” (which is true)? See how in the latter the “will” is taken away from the component’s and made to form an operator (“It will be the case that either…”)? What do we CALL this animal?
Now, leaving this aside for a moment, I’m wondering whether deniers of bivalence can really be said to have avoided breaking the law of non-contradiction. That is, the law of non-contradiction tells us that A cannot equal ~A. Propositionally, then, a proposition and its contradictory cannot both be true or false. The denier of bivalence believes that by adding a third truth value (‘neuter’) she can affirm the contingency of both p and ~p without breaking the rule of non-contradiction. But the rule that A cannot be ~A amounts to more than simply that a proposition and its contradictory cannot both be TRUE or FALSE. It means in addition that they cannot both SHARE THE SAME VALUE (however many values there may be!). So the denier of bivalence (or the proponent of three-valued logic, as Prior was) hasn’t really avoided breaking the law of non-contradiction by adding a third value. By the logic of her own system, she adds a third value, yes, but she ends up assigning this ONE value to BOTH a proposition and its contradictory, thus breaking the rule of non-contradiction. It would seem then, that three-valued logic is self-contradictory.
Now, it seems to me that the mistake Prior and others are making is one of DISTRIBUTION. Don’t ask me what it means. It just sounds good! Prior ignores the modal operator HE assigns (Q) to propositions and distributes contradictoriness to only the variables. That is, he first takes p and its contradictory ~p, notes that these are contradictories, and THEN adds Q. This is where the freakiness occurs.
Tom | | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/14/2004 5:22 AM (GMT -6) | | OK, try this on for size.
Prior reasons like this:
Assuming...
P1 p and ~p are contradictories, and
P2 p and ~p are both contingent, and
P3 a proposition and its contradictory cannot both be true or false,
Then...
C1 Therefore, propositions describing future contingent events cannot be bivalent, and
C2 Bivalence does not hold universally.
I don’t think so. If p is defined as a contingent event, then ~p is not also contingent as Prior thinks. It is rather the contradiction of contingent-p, that is, if p is contingent, then ~p falsifies this contingency, i.e., ~p = ~contingent-p. In other words, ~p in this instance = Zp or Z~p. So it is a mistake to say that if p is contingent then both p and ~p are contingent and both are contradictory. On the contrary, if p is contingent then BOTH possibilities (i.e., the possibility that the state-of-affairs described in p will obtain and the possibility that it will not obtain) are present in the one p, so that ~p is the contradictory of BOTH these possibilities, viz., ~p = Zp or Z~p. I think this is on now, but I hope there’s are more formal and commonly recognized laws to appeal to in making this point.
I feel an obsession coming on!
I’d really like to know if my argument that three-valued logic fails to avoid breaking the rule of non-contradiction and is therefore self-contradictory makes sense to anyone.
Hmmmm.
Tom
| | Back to Top | | |
|
Alan Rhoda
Registered Member
Date Joined May 2002
Total Posts : 680
| Posted 6/14/2004 7:08 AM (GMT -6) | | Hi Tom,
This is an interesting issue. I'd be happy to kick it around for a bit.
Tom: “…Aristotle held that even when neither ‘There will be a sea-battle tomorrow’ nor its negation [‘There will not be a sea-battle tomorrow’] is definitely true, the alternation ‘Either there will be a sea-battle tomorrow or there will not be a sea-battle tomorrow’ is definitely true….His ground for asserting that ‘Either there will be a sea-battle tomorrow or there will not’ is true was almost certainly not that an alternation is to be automatically counted true if both its components are ‘neuter’. It is not because its components are both neuter, but because they are contradictories, that Aristotle affirms this disjunction.
Alan: Exactly. A disjunction can be true simply because it's disjuncts are jointly exhaustive (as is the case when they are contradictory) and not because one of the disjuncts is in fact true. For example, "Either 'x < 0' or 'x > 0' or 'x = 0'" is true for any x that is a member of the set of real numbers. But which disjunct is true depends on the value of x. Until that is specified, we cannot say of any one disjunct that it is true or that it is false. But we can know that the disjunct is true BEFORE the value of x is specified because the disjuncts are jointly exhaustive.
Tom: To note ‘that’ an event will either occur or not, or more popularly, that “Tomorrow, there either will or will not be a sea-battle” is true, is not to assume that either of its components is true, for what is meant here is not Zp v Z~p. But to note ‘whether’ an event will occur of not is to inquire into the ontological status of the present moment with respect to the determinate vs. indeterminate nature of tomorrow’s sea-battle, in which case Zp v Z~p v Qp is now true. ... To say ‘that an event will either occur or not at noon tomorrow’ is to say that at 12:01 it will be the case that the event in question will have either happened or not. It’s simply to note the either/or-ness of actuality. Events are either actual or they are not. S either obtains or it does not obtain. There is no third option. But to ask ‘whether an event will happen or not at noon tomorrow’ is to ask if its occurring is determined or contingent, in which case there are three options (Zp, Z~p, or Qp).
Alan: Well-explained, Tom. I agree with you completely on your distinctions. In fact, I had recently, just in the past two weeks, come to the same conclusions myself. It occurred to me that one rejoinder that could be raised against my response to the temporal invariance argument would be to say that, obviously, "Either S obtains at T is true or S does not obtain at T is true". Since this disjunction is always true, so the rejoinder goes, at least one of its disjuncts must be true, and must be true at all times prior to T. Thus, either "S obtains at T" or "S does not obtain at T" must be true at all times prior to T, which means that either "S will obtain at T" or "S will not obtain at T" is true at all times prior to T. Hence, the future is exhaustively settled.
The flaw in this rejoinder, as I came to see, is essentially what you have expressed above: that the present truth of a disjunction does not imply the present truth of any one of the disjuncts, where the disjuncts are jointly exhaustive. My example of "Either 'x < 0' or 'x > 0' or 'x = 0'" is a case in point. Similarly, "Either S obtains at T or S does not obtain at T" is true, not because either of the disjuncts is NOW true, prior to T, but because when T arrives it will either be the case that S or it will be the case that not-S.
God bless,
Alan | | Back to Top | | |
|
JET
Registered Member
Date Joined Jun 2002
Total Posts : 397
| Posted 6/14/2004 10:31 AM (GMT -6) | | | TGB, my knowledge of Prior is limited to what I could find on the internet some time ago. I'm wondering about something . . . maybe you can shed some light . . . specifically, do any of Prior's later writings suggest that Prior ultimately affirmed bivalence? In one internet article (I think the one on the Stanford philosphy webpage), the author seemed to suggest that Prior retreated from three-valued logic. In another online article, A. Freddoso covered various possible responses to fatalism. In so doing, as I recall, he (Freddoso) attributed to Aristotle the idea that affirmative statements about the contingent future are neither true nor false. On the other hand, he (Freddoso) attributed to Prior the idea that affirmative statements about the contingent future are, in fact, false (which seems to imply bivalence). (I may have misread and/or misunderstood either or both authors.) Do you have any of Prior's later works? If, in fact, Prior did come around to affirming bivalence, it would be interesting to see how he got there. | | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/14/2004 3:58 PM (GMT -6) | | Hi JET-
I don't know that Prior came around to bivalence late in life. Peter Geach, who edited tons of Prior's writings after his death, probably would know. That'd be interesting.
Tom | | Back to Top | | |
|
Alan Rhoda
Registered Member
Date Joined May 2002
Total Posts : 680
| Posted 6/14/2004 4:09 PM (GMT -6) | | Tom: I’m trying to nail down exactly where we (Doc, Alan, and I) would argue Prior goes wrong in arguing FROM p and ~p are contradictories (which they are) and both are contingent (which they are) TO denying bivalence since both a proposition and its contradictory cannot both be true or false. It seems equally as obvious to me that Qp and Q~p and contraries and not contradictories (as are p and ~p). So what technically is happening when we add the operator Q? How does adding Q in front of p and ~p change p and ~p from contradictories to contraries?
Alan: Seems to me that p and Qp have different logical behavior simply because they are different propositions, much in the same way that adding the quantifier "some" onto "roses are red" says something quite different than simply "roses are red". From another perspective, however, the unqualified statement "p" may be ambiguous: By "p" do we mean Zp or Qp? Similarly, if a person says "roses are red" one might legitimately ask for clarification: Do you mean ALL roses? MOST roses? SOME roses? etc. Unlike statements, however, propositions are by definition never ambiguous. By Theorem I of our paper, for all p, either Qp, Zp, or Z~p. So if we see an unqualified proposition p, we should regard it as having a built-in qualifier that is concealed from us. In other words, for all p, there is some proposition q, such that p either equals Qq, Zq, or Z~q. So when you see an unqualified proposition p, you are really looking at a qualified proposition (Qq, Zq, or Z~q). So to be told that p and ~p are contraditions is to be told, for example, that Qq and ~Qq are contraditions. In short, Tom, nothing mysterious is going on here. The source of confusion lies in the fact that logical symbolism can conceal just as easily as it reveals.
Tom: What is it in “Either component 1 or not-component 1” that permits it to mean “It will be the case that a sea-battle either happens tomorrow or it does not” (which is true)? See how in the latter the “will” is taken away from the component’s and made to form an operator (“It will be the case that either…”)? What do we CALL this animal?
Alan: How about a disjunction?  Seriously, though, it is a mistake to think that logic is fundamentally or essentially truth-functional. It is not. Many systems of logic are, and they do a passably good job, but none of them is a fully adequate representation of rational human thinking, as is evidenced by the paradoxes of material and strict implication. Logic is certainly related to truth, but no one has, or can, so far as I can see, give a purely truth-functional and paradox-free analysis of "if-then".
Set aside the various systems of logic for a moment. To affirm a disjunction is, clearly, to affirm that at least one of the disjuncts is true, but it does not affirm of any of the disjuncts that it is true. For example, if I say "Either you are late or you are early" I am claiming that (at least) one of those disjuncts is true, but I am not claiming of either of those disjuncts that it is true. In this case, however, the disjunction is false because the disjuncts are not jointly exhaustive--you might be on time (or you might never show up at all). Note, it is false because the disjuncts are not jointly exhaustive, which means it is false even if you do in fact show up on time. Disjunctions, as we use them in natural language, are not, therefore, truth-functional. The truth of a disjunction is not a function of the truth of the disjuncts. Rather, it is a function of the exhaustiveness of the disjuncts. Or, if we countenance degrees of truth, then it is a function of the probability that (at least) one of the disjuncts is true.
Tom: But the rule that A cannot be ~A amounts to more than simply that a proposition and its contradictory cannot both be TRUE or FALSE. It means in addition that they cannot both SHARE THE SAME VALUE (however many values there may be!). So the denier of bivalence (or the proponent of three-valued logic, as Prior was) hasn’t really avoided breaking the law of non-contradiction by adding a third value. By the logic of her own system, she adds a third value, yes, but she ends up assigning this ONE value to BOTH a proposition and its contradictory, thus breaking the rule of non-contradiction. It would seem then, that three-valued logic is self-contradictory.
Alan: I hate to disagree with you, Tom, but here goes. Noncontradiction does not say that the truth values of A and ~A must be different. All it says is that they cannot both be TRUE at the same time in the same respect. Thus the truth of one implies the non-truth of the other. It does not, however, imply the FALSITY of the other, unless we ASSUME bivalence. For multi-valued logics this assumption does not hold. Thus if A is true then ~A may be false OR it may be neither true nor false. If A is neither true nor false then, according to multi-valued logics, ~A is also neither true nor false. This does not violate noncontradiction, however, for noncontraction simply doesn't apply when neither of the logical opposites is TRUE.
God bless,
Alan | | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/14/2004 5:33 PM (GMT -6) | | Thanks Alan!
quote:
Alan: So if we see an unqualified proposition p, we should regard it as having a built-in qualifier that is concealed from us. In other words, for all p, there is some proposition q, such that p either equals Qq, Zq, or Z~q. So when you see an unqualified proposition p, you are really looking at a qualified proposition (Qq, Zq, or Z~q). So to be told that p and ~p are contraditions is to be told, for example, that Qq and ~Qq are contraditions. In short, Tom, nothing mysterious is going on here. The source of confusion lies in the fact that logical symbolism can conceal just as easily as it reveals.
Yeah, I got that. My point was to say that this is precisely where Prior goes wrong when he agrees (with Aristotle) that if a proposition is contingent that its ‘contradictory’ is also contingent and thus bivalence cannot hold universally. Why is he saying this? Because begins with (as you point out) ‘unqualified’ variables p and ~p. These are contradictories. He THEN qualifies them with Q and gets Qp and Q~p and maintains that these are contradictories. But they’re not. They’re contrary. The contradictory of Qp is not Q~p, it’s ~Qp. Thus it’s false to say that if a proposition is contingent that its contradictory is also contingent.
quote:
Alan: Set aside the various systems of logic for a moment. To affirm a disjunction is, clearly, to affirm that at least one of the disjuncts is true, but it does not affirm of any of the disjuncts that it is true. For example, if I say "Either you are late or you are early" I am claiming that (at least) one of those disjuncts is true, but I am not claiming of either of those disjuncts that it is true. In this case, however, the disjunction is false because the disjuncts are not jointly exhaustive--you might be on time (or you might never show up at all).
But I don’t see how, as you argue, this is an example of how logic is NOT truth-functional. On the contrary, you seem to me to have showed that the disjunction IS truth-functional when you point out that it’s false because neither of its disjuncts is true. Saying logic isn’t always truth-functional sounds like a denial of bivalence. I must be misunderstanding what “truth-functional” means.
quote:
Alan: Note, it is false because the disjuncts are not jointly exhaustive, which means it is false even if you do in fact show up on time. Disjunctions, as we use them in natural language, are not, therefore, truth-functional.
I don’t know what truth-functional means then. Doesn’t it simply mean that the function of a proposition is to convey truth or falsity?
quote:
Alan: The truth of a disjunction is not a function of the truth of the disjuncts.
But it seems to me that it is. You’ve argued that the disjunction is false on the grounds that its neither of its disjuncts is true and they are not conjointly exhaustive. Thus, the truth-value (in this case “false”) of the disjunction follows logically from the truth-values (in this case both “false”) and scope of the disjuncts. How is this not truth-functional? What am I missing?
quote:
Alan: Rather, it is a function of the exhaustiveness of the disjuncts.
Which is the same as saying both its disjuncts are false, which seems to me to be the same as saying logic has ‘functioned’ perfectly in the case of this disjunction in conveying truth-value (in this case “false”). How else did you arrive at the truth value of the disjunction if not logically? What ‘functioned’ you into this conclusion if not logic? See how sick I think? Scary isn’t it?
quote:
Tom: But the rule that A cannot be ~A amounts to more than simply that a proposition and its contradictory cannot both be TRUE or FALSE. It means in addition that they cannot both SHARE THE SAME VALUE (however many values there may be!). So the denier of bivalence (or the proponent of three-valued logic, as Prior was) hasn’t really avoided breaking the law of non-contradiction by adding a third value. By the logic of her own system, she adds a third value, yes, but she ends up assigning this ONE value to BOTH a proposition and its contradictory, thus breaking the rule of non-contradiction. It would seem then, that three-valued logic is self-contradictory.
Alan: I hate to disagree with you, Tom…
Not at all! Go for it bro!
quote:
…but here goes. Noncontradiction does not say that the truth values of A and ~A must be different. All it says is that they cannot both be TRUE at the same time in the same respect.
That’s what I meant by saying they cannot share the same truth value. A (when it is A and in the precise respects which constitute its being A and not something else) cannot be ~A. Given A in its entirety (time and respects, as you point out), it cannot be equal to its contradictory ~A. These cannot share the same truth value. A can’t be ~A. I’m assuming A’s time and respect in positing A.
quote:
Alan: Thus the truth of one implies the non-truth of the other. It does not, however, imply the FALSITY of the other, unless we ASSUME bivalence.
Quite so. I wasn't suggesting otherwise.
quote:
Alan: For multi-valued logics this assumption does not hold. Thus if A is true then ~A may be false OR it may be neither true nor false.
Precisely. As you just stated, multi-valued logics deny that A and ~A must be one TRUE and the other FALSE (they add a third or more truth values), but (this is my point) they DO agree that (given the law of non-contradiction) A and ~A cannot both share the SAME value (yes, at the same time and in the same respect). As you say, A may be true and ~A may be false OR it may be neither true nor false. But (my point again), ~A may NOT be true just in case A IS true. These may not share the same truth value, which is why Prior (at least) still sins against the law of non-contradiction. He gives p and ~p the SAME truth value (at the same time and in the same respects). He says they are both ‘neuter’. They are both ‘contingent’, both Q. This is not possible. If Multi-valued logics claim that a proposition and its contradictory may share the SAME truth value (among a possible three, four, or a hundred truth values), then they’re to this extend self-contradictory. Wouldn't you agree?
Tom | | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/14/2004 6:03 PM (GMT -6) | | Alan-
Let me try to clarify myself this way. Let’s assume a three-valued logic (as Prior did). The truth values are ‘true’, ‘false’, and ‘neuter’ (or whatever you wanna call the third one). Prior gives ‘true’ a value of 1, ‘false’ a value of 0, and the third value ½. But let’s label ‘em like this:
True = V1
False = V2
Neuter = V3
It still remains the case that these three values are jointly exhaustive and mutually exclusive. If any proposition p is V1, then ~p may be either V2 or V3, but it can’t be V1. Similarly, if p is V3, then ~p may be V1 or V2, but it may not be V3. The law of non-contradiction forbids it.
Prior however, assigns to both p and ~p the value of V3. He says they're both 'contingent' (Q). He doesn't want to say they're both TRUE or both FALSE, because he knows that violates the law of NC. But he thinks it perfectly consistent to assign them both a value of V3. I’m arguing that this won’t fly. It violates the law of NC as well. It denies that the three values are mutually exclusive (which they must be). He doesn’t escape violating the law of non-contradiction by assigning p and ~p V3 (as opposed to assigning them both V1 or V2). See?
Tom | | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/15/2004 1:15 AM (GMT -6) | | In the beginning strides of obsessing again!…
Note:
Law of bivalence: Any proposition is either true or false.
Law of excluded middle: p v ~p, that is, p or not p
Law of non-contradiction: ~(p ^ ~p), that is not both p and ~p.
In De Interpretatione, Aristotle gives his famous example of the contingent sea battle. Imagine P refers to the statement "There will be a sea battle tomorrow." The law of the excluded middle clearly holds: There will be a sea battle tomorrow or there won't be” or p v ~p. But doesn’t this limit us to describing contingent future events with either “will” or “will not”? How can the event be contingent? Answer: Claim that “will” and “will not” are neither true nor false but are indeterminate. Attribute the indeterminacy to the truth value of the propositions.
Here’s where I think people misapply the excluded middle. It’s commonly thought out this way:
A non-open theist proponent of bivalence might argue:
P1: p = the proposition “There will be a sea battle tomorrow.”
P2: p v ~p (excluded middle) holds universally
P3 Therefore, “Either ‘There will be a sea battle tomorrow’ is true and ‘There will not be a sea battle tomorrow’ is false, OR ‘There will be a sea battle tomorrow’ is false and ‘There will not be a sea battle tomorrow’ is true” (from P2)
P4 Bivalence holds universally
C1 The future is exhaustively describable in terms of “will” and “will not”
C2 The future is exhaustively settled.
An open theist denier of bivalence might argue:
P1: p = the proposition “There will be a sea battle tomorrow.”
P2: Assuming bivalence, the if p, p v ~p (excluded middle).
P3 Therefore, “Either ‘There will be a sea battle tomorrow’ is true and ‘There will not be a sea battle tomorrow’ is false, OR ‘There will be a sea battle tomorrow’ is false and ‘There will not be a sea battle tomorrow’ is true” (from P2)
P4 There are future contingent events.
P5 If a proposition is contingent, so is its contradictory.
P6 If p is contingent, then so is ~p contingent.
P7 Bivalence does not permit (p ^ ~p)
P8 Therefore, bivalence does not hold universally
P9 Therefore, both p and ~p are neither true nor false but rather both are neuter
C1 The future is not exhaustively describable in terms of “will” and “will not”
C2 The future is not exhaustively settled.
But BOTH make the same mistake. True, (p v ~p) holds universally, but where p = “There will be a sea battle tomorrow,” ~p = “It is not the case that there will be a sea battle tomorrow,” and not “There will not be a sea battle tomorrow.” That is, where p = “will,” ~p = “might not” and not “will not” (which is p~, another animal altogether). I keep reading Prior’s comments on Aristotle and can’t help concluding that Prior (and Aristotle) missed this simple point. Noting this alone would keep the proponent of an exhaustively settled future from affirming bivalence and denying the future may be open, and it would also keep the proponent of three-valued logic from mistakingly believing that if a proposition is contingent so is its contradictory contingent and so thus denying bivalence (which doesn't really help him).
The upshot is that we ARE able to maintain bivalence, falsify p (= the proposition “There will be a sea battle tomorrow”), and hold to the law of the excluded middle (p v ~p) by correctly noting that the contradictory of p (“There will be a sea battle tomorrow”) is ~p (“It is not the case that there will be a sea battle tomorrow”, which is to say “There might not be a sea battle tomorrow”). This is perfectly consistent with the sea battle’s being a contingent event. There is no need to claim that p is neither true nor false today since the matter has not yet been decided. It is not the case that p and ~p are both contingent. On the contrary, p is contingent; ~p is ~contingent.
If we argue there are three metaphysical states-of-affairs with regard to any future-tense p (those being: Zp, Z~p, and Qp, or “p will obtain,” “p will not obtain,” and “p might and might not obtain”) then the excluded middle, ~(p ^ ~p) still holds true, for our three states-of-affairs are mutually exclusive:
~(Zp ^ Z~p)
~(Zp ^ Qp)
~(Z~p ^ Qp)
Looks to me that in saying “Qp = Possibly p and ~p” (which is incorrect), Prior qualifies p with Q on the left side of the equation but leaves it unqualified on the right side, which “=” will not permit. This led him incorrectly to conclude (with Aristotle) that if a proposition is contingent (i.e., Q) then so is its contradictory contingent. But Prior knew that saying a proposition and its contradictory are both true clearly violated the law of noncontradiction, ~(p ^ ~p), so he abandoned bivalence and claimed that if p is contingent, then p and ~p are both neither true nor false but are rather both ‘neuter’ (a third truth value he introduced). But Prior isn’t out of trouble. For all he has done is redefined contingent-p to (neuter-p ^ ~neuter-p) which is still a violation of the law of non-contradiction, which states ~(p ^ ~p).
Hope I got the notation right. I rushed through it. If I made a mistake, it’s Alan’s and Doc’s fault. You should try to keep up with them! My arguments are not as tight as the could be, but I'm just "pooping them out" as Doc would say I do! 
Tom  | | Back to Top | | |
|
Humphrey Bogart
Registered Member
Date Joined Jun 2003
Total Posts : 205
| Posted 6/15/2004 1:57 AM (GMT -6) | | Tom,
Saying that "there will be a sea battle tomorrow" is false, entails that the contradictory of this, "there might not be a sea battle tomorrow" is true. Similarly, saying that "there will not be a sea battle tomorrow" is false, entails that the contradictory of this, "there might be a sea battle tomorrow" is true. Therefore asserting that both statements are false, entails that "there might be a sea battle tomorrow and there might not be a sea battle tomorrow" is true.
Now it seemed initially to me intuitively that this latter statement would entail "Either there will be a sea battle tomorrow or there will not be a sea battle tomorrow". But on thinking about it more carefully, it doesn't. In fact, obviously, if "there will be a sea battle tomorrow" is false and "there will not be a sea battle tomorrow" is false, then the statement "either there will be a sea battle tomorrow or there will not be a sea battle tomorrow" must be false also. In other words, if X will is false and X will not is false, then "either X will or X will not" must be false also.
This is a problem indeed for my personal position which I've outlined previously, and it is puzzling. I'll have to think about it more. Thanks for pointing this out, it's the first real problem with this that I've come across. The problem seems to relate to the fact that when we say "X will" or "X will not" by themselves, we are asserting something about present conditions (that X is entailed by present conditions, or not X is entailed by present conditions, respectively). But when we say "either X will or X will not", we are not necessarily saying anything at all about present conditions. That statement seems to be true irrespective of any present conditions. It is very puzzling. There seems to be a difference in the meaning of "will" involved.
All the best,
Albert.
P.S. I finish exams in two days (3 down, 1 to go), so I might not be able to respond much until then.
| | Back to Top | | |
|
Humphrey Bogart
Registered Member
Date Joined Jun 2003
Total Posts : 205
| Posted 6/15/2004 2:37 AM (GMT -6) | | To refresh people's memories, here are my definitions of "will", "will not", "might", and "might not", which I've proved previously satisfy all the requirements of the square of opposition.
Comment: X is any logically possible proposition.
(A) "X will" means X is entailed by n propositions Y1 ... Yn, where Y1 ... Yn are true propositions about what is.
(E) "X will not" means not-X is entailed by n propositions Y1 ... Yn, where Y1 ... Yn are true propositions about what is.
(I) "X might" means X is not contradictory to any Y, where Y is EITHER a true proposition about what is OR a true proposition entailed by what is (i.e. a "will").
(O) "X might not" means not-X is not contradictory to any Y, where Y is EITHER a true proposition about what is OR a true proposition entailed by what is (i.e. a "will").
Now it is clear from the above definitions, that the statement "either X will or X will not" can be false. In fact, if the statement "X might and X might not" is true, then this entails that "either X will or X will not" is false. Saying that "either X will or X will not" is false, is just to say that neither X nor not-X are entailed by present circumstances.
The problem: saying that "either X will or X will not" can be false, seems counter-intuitive. In ordinary language, "either X will or X will not" would seem to be always true. At the moment I am inclined to just reject this intuitive, ordinary language claim and stare people down and assert that "either X will or X will not" can be false. But I am somewhat uncomfortable with that. But nowhere near uncomfortable enough to go down the road of rejecting bivalence.
| | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/15/2004 4:38 AM (GMT -6) | | Brothers of Abel: Think of V3 as "unknown".
Tom: Proponents of three-valued logic (like Prior) wouldn’t settled for this I don’t think. It’s not that contingent p has a truth value of true or false but we just don’t KNOW it. That’s an epistemological issue. Instead, it is actually claimed that p is neither true nor false. This is not saying you don’t know. It’s saying you know, but that what can be know about p is more than just that its true or false. It can also be ‘neuter’.
Albert, as far as I can tell I agree with everything you said (would tweak a few things here and there, that’s all). We fundamentally agree.
Albert: This is a problem indeed for my personal position which I've outlined previously, and it is puzzling. I'll have to think about it more. Thanks for pointing this out, it's the first real problem with this that I've come across. The problem seems to relate to the fact that when we say "X will" or "X will not" by themselves, we are asserting something about present conditions (that X is entailed by present conditions, or not X is entailed by present conditions, respectively). But when we say "either X will or X will not", we are not necessarily saying anything at all about present conditions.
Tom: “Either there will be a sea battle tomorrow or there will not be a sea battle tomorrow” can be understood as (where Z = “will” and p = the sea battle tomorrow): Zp v Z~p. That is, it is now either the case that there “will” be a sea battle tomorrow or it is now the case that there “will not” be a sea battle tomorrow. But in common speech we also construe “Either there will be a sea battle tomorrow or there will not be a sea battle tomorrow” to mean: “It will be the case tomorrow that there either is or is not a sea battle.” This sort of claim has nothing to do with the determinacy vs. indeterminacy of there being a sea battle tomorrow. It is merely an admission of the either/or-ness of actuality.
Alan’s comments about noting whether or not disjunctions are exhaustive are very good and I think they fix things, Albert. We can say with regard to any claim at T0 regarding any claim regarding p at T1, “Either p or ~p at T1.” This doesn’t rule out p’s contingency, viz., it’s being true that “p might or might not obtain at T1” and false both that “p will obtain at T1” and “p will not obtain at T1.”
Tom | | Back to Top | | |
|
Alan Rhoda
Registered Member
Date Joined May 2002
Total Posts : 680
| Posted 6/15/2004 10:01 AM (GMT -6) | | Tom: Prior goes wrong when he agrees (with Aristotle) that if a proposition is contingent that its ‘contradictory’ is also contingent and thus bivalence cannot hold universally. Why is he saying this? Because begins with (as you point out) ‘unqualified’ variables p and ~p. These are contradictories. He THEN qualifies them with Q and gets Qp and Q~p and maintains that these are contradictories. But they’re not. They’re contrary. The contradictory of Qp is not Q~p, it’s ~Qp. Thus it’s false to say that if a proposition is contingent that its contradictory is also contingent.
Alan: To my mind if it is contingent that p, then it must also be contingent that ~p because if p might and might not be true (Qp) then ~p might and might not be true (Q~p). In other words, Qp and Q~p are logically equivalent: Qp = Q~p. So they aren't contraries as you suggest.
Now, is the contradictory of a contingent proposition contingent. I think the answer is "yes". To see this we need to distinguish between the proposition "p" and the very different proposition "p is contingent" (Qp). The contradictory of p is ~p. If p is contingent then Qp is true. If Qp is true then so is Q~p, since they are logically equivalent. If Q~p is true then "~p is contingent is true". Therefore, ~p is contingent. So Prior is correct to say that if a proposition is contingent then its contradictory is also contingent.
quote:
Alan: To affirm a disjunction is, clearly, to affirm that at least one of the disjuncts is true, but it does not affirm of any of the disjuncts that it is true. For example, if I say "Either you are late or you are early" I am claiming that (at least) one of those disjuncts is true, but I am not claiming of either of those disjuncts that it is true. In this case, however, the disjunction is false because the disjuncts are not jointly exhaustive--you might be on time (or you might never show up at all).
Tom: But I don’t see how, as you argue, this is an example of how logic is NOT truth-functional. On the contrary, you seem to me to have showed that the disjunction IS truth-functional when you point out that it’s false because neither of its disjuncts is true.
Alan: I didn't say it was false because neither of the disjuncts is true. I said it was false because the disjuncts are not jointly exhaustive of the possibilities. In other words, it is false because it is possible that neither of the disjuncts be true. Thus it would still be false even if one of the disjuncts happened to be true.
Tom: I don’t know what truth-functional means then. Doesn’t it simply mean that the function of a proposition is to convey truth or falsity?
Alan: No. To say that a logical connector like "or" is truth-functional is to say that the truth value of the compound proposition "p or q" it is purely a function of the truth values of the components. Thus, it is typical in elementary logic textbooks to give truth-functional definitions of the various logical connectors by means of truth-tables that run through all the possible permutations of truth values. Thus, "p or q" is often said to be "true" whenever the truth values for p and q, respectively, are (T,T), (T,F), or (F,T). And "p or q" is said to be false when the truth values for p and q are (F,F). On this definition, the truth value of "p or q" is completely determined by the truth values of p and q. That's what it means to say that something is "truth-functional".
Tom: You’ve argued that the disjunction is false on the grounds that its neither of its disjuncts is true and they are not conjointly exhaustive. Thus, the truth-value (in this case “false”) of the disjunction follows logically from the truth-values (in this case both “false”) and scope of the disjuncts. How is this not truth-functional? What am I missing?
Alan: Again, I didn't say it way false because neither of its disjuncts is true. But even if I had it still wouldn't be truth-functional because the "scope" of the disjuncts is not a function of their truth values.
Tom: Multi-valued logics deny that A and ~A must be one TRUE and the other FALSE (they add a third or more truth values), but (this is my point) they DO agree that (given the law of non-contradiction) A and ~A cannot both share the SAME value (yes, at the same time and in the same respect).
Alan: No. The law of non-contradiction does NOT say that a proposition and it contradictory can't have the same truth value. What is says is that they cannot both have the value TRUE. You might not buy that, but this is how the defender of multivalued logic would argue. So if p has the truth value "I" (for indeterminate) and ~p also has the truth value "I" non-contradiction is not violated because it is not the case that both propositions have the value "T" (for true). To get your argument to work you have to take the law of non-contradiction in a stronger sense than the non-bivalentist will grant. So your argument begs the question.
Make sense?
Alan | | Back to Top | | |
|
JET
Registered Member
Date Joined Jun 2002
Total Posts : 397
| Posted 6/15/2004 1:24 PM (GMT -6) | | Other than for purely "political" reasons (i.e., making indeterminancy logic more palatable to those inclined to hold onto the classical law of bivalence), does retaining bivalence, in and of itself, afford any real advantage in terms of advancing indeterminancy logic?
If I understand correctly, we have been discussing two options for indeterminancy logic, as follows:
Per bivalence: Qp is either True or False
Per 3-valued logic: p is either True or False or Indeterminate
One option affirms bivalence (by adding an indeterminancy qualifier, Q). The other option abandons bivalence in favor of a 3-valued logic.
It seems to me that the foregoing statements represent two different ways of saying exactly the same thing.
I don’t know if this illustration works, but lets try something:
First, lets assume bivalence:
1. Qp is either True or False (assumed)
2. If Qp is True, then p is Indeterminate
3. If Qp is False, then p is either True or False
4. Therefore, p is either Indeterminate or True or False
(I.e., you start with 2 values and arrive at 3 values)
Now, lets assume a 3-valued logic:
1. p is either True or False or Indeterminate (assumed)
2. If p is True then Qp is False
3. If p is False then Qp is False
4. If p is Indeterminate, then Qp is True
5. Therefore, Qp is either True or False
(I.e., you start with a 3-valued logic and arrive at bivalence)
| | Back to Top | | |
|
Alan Rhoda
Registered Member
Date Joined May 2002
Total Posts : 680
| Posted 6/15/2004 1:32 PM (GMT -6) | | quote:
Originally posted by JET
Other than for purely "political" reasons (i.e., making indeterminancy logic more palatable to those inclined to hold onto the classical law of bivalence), does retaining bivalence, in and of itself, afford any real advantage in terms of advancing indeterminancy logic?
I tend to agree with you, JET. It seems to me that open theists who affirm bivalence and open theists who deny bivalence want to say pretty much the same thing--that the future is partly indeterminate and God knows it as such. Where they differ is in how best to express that conviction in logical terms.
Alan | | Back to Top | | |
|
JET
Registered Member
Date Joined Jun 2002
Total Posts : 397
| Posted 6/15/2004 1:41 PM (GMT -6) | | quote:
I tend to agree with you, JET. It seems to me that open theists who affirms bivalence and open theists who deny bivalence want to say pretty much the same thing--that the future is partly indeterminate and God knows it as such. Where they differ is in how best to express that conviction in logical terms.
Yes, but do you think indeterminancy can really be expressed better in terms of bivalence or do you think you could (if you wanted to) express indeterminancy logic as persuasively using three values. I know that some OV'ers who deny bivalence don't do a very good job expressing the relation between indeterminancy and omniscience. But couldn't YOU, if you wanted to, express it equally well using either approach (assuming no political need to use one or the other)? | | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/15/2004 6:59 PM (GMT -6) | | Alan-
On non-contradiction, I see your point. Like I said, I was floating that out as an argument against non-bivalent approaches. So I’ll give that up. Thanks.
On what it means to be “truth-functional,” thanks for the help. I follow you. My problem is with Prior’s claim that Q is NEVER truth-functional. Sounds to me like you’re saying functors (or truth operators like “and” “or” “if…then” ) are only SOMETIMES not truth-functional. It depends on whether or not the proposition’s components embrace all the possibilities (i.e., are exhaustive or not). Thanks again.
But on the claim that if a proposition is contingent so is the contradictory proposition contingent, I’ll hold my ground and suggest why I think you’re wrong on this score.
quote:
Alan: To my mind if it is contingent that p, then it must also be contingent that ~p because if p might and might not be true (Qp) then ~p might and might not be true (Q~p). In other words, Qp and Q~p are logically equivalent: Qp = Q~p. So they aren't contraries as you suggest. Now, is the contradictory of a contingent proposition contingent. I think the answer is "yes". To see this we need to distinguish between the proposition "p" and the very different proposition "p is contingent" (Qp). The contradictory of p is ~p. If p is contingent then Qp is true. If Qp is true then so is Q~p, since they are logically equivalent. If Q~p is true then "~p is contingent is true". Therefore, ~p is contingent. So Prior is correct to say that if a proposition is contingent then its contradictory is also contingent.
Alright, let’s start where you start, with “if ‘it is contingent that p’.” That’s a qualified p. Prior posits a contingent PROPOSITION p. He defines p AS Qp. Let’s leave unqualified p (i.e., p without the operator “It is contingent that…”) as a small p and make qualified p (i.e., the proposition “It is contingent that p” that Prior posits) a capital “P", OK? That’ll keep us from confusing them, which is what you appear to me to be doing. Thus,
p = the component (not the proposition): “...a sea battle tomorrow”
P = the proposition Qp (containing the component p): “It is indeterminately the case that a sea battle obtains tomorrow.”
Thus “p is contingent” = Qp = P. And the contradictory of THIS (i.e., the contradicotry of Prior's P) is not Q~p, but ~Qp (or ~P). If you qualify a component (“a sea battle tomorrow”) with an operator to form a propositional truth claim and THEN form the contradictory of the proposition, i.e., the contradictory of the truth claim, the “~” is placed before the qualifying operator, that is, before the entire proposition, and not between the operator and the variable. Thus, the contradictory of “It is contingent that p,” or Qp, is ~“It is contingent that p,” or ~Qp. You argue this is the paper dude!
Moving on, it seems to me a mistake to say Qp and Q~p are logical equivalents if p is a propositional truth claim and not merely a component in a proposition. By “Qp ^ Q~p” you mean to say “p might and might not occur [that's what Qp means] and ~p might and might not occur [that's what Q~p means].” This if perfectly fine, because here p and ~p are components that contribute to forming their respective propositions (p in "p might and might not obtain" and ~p in "~p might and might not obtain") and not independent propositions making truth claims (as Prior takes unqualified p to BE qualified Qp and then claims that the contradictory is also contingent). The entire conjunction Qp ^ Q~p (where p is just a component) is formed by conjointly true SUBCONTRARY PROPOSITIONS and not conjointly true CONTRADICTORY PROPOSITIONS (as Prior claims). Qp already says everything about component p’s contingency that one could say, for it embraces the possibility of conjointly contingent components p and ~p (that is Qp and Q~p) and not the possibility of conjoined propositions Qp and its contradictory ~Qp. You could add Q~p to Qp (where p is just a component) if you wanted to, but Q~p doesn't add anyting that's not already said in Qp. But these are not contradictories as Prior claims.
Thus, if “p is contingent” is the case, then its contradictory is the denial of THIS CONTINGENCY, or ~(p is contingent) and not (~p is contingent). Remember, “might and might not” are subcontraries, not contradictories. Thus Qp = “p and p~” and not “p and ~p” if you’re talking p and p~ are propositions that make truth claims. But Qp = p and ~p if you're talking components that contribute to forming a single proposition making a truth claim.
If p is not a proposition, that is, if p is just a component within a proposition, then we CAN say that both p and ~p are contingent. For example, where p = “a sea-battle tomorrow at noon” (with no truth operator prefacing it), then ~p = “no sea-battle tomorrow at noon” (with no truth operator prefacing it). Both of THESE can be contingent. That’s what it means to claim Qp (viz., both components p and ~p are possible). Both together would then form the one proposition “ ‘A sea battle tomorrow’ and ‘no sea battle tomorrow’ are both contingent.” This we CAN say IF both p and ~p are components. But this is not what Prior does. He makes p to be more than a component. He makes it a contingent proposition P, He essentially claims two p’s without differentiating them: P (the proposition) = “It is contingent that p (the component).” He conflates P (the proposition) and p (the component) in claiming the contradictory of the proposition P (“It is contingent that p” or Qp) is “It is contingent that ~p” (or Q~p). This is just mistaken. He’s using p in the first instance to mean the proposition Qp, and p in the second instance to mean the mere component p in Q~p. But the contradiction of “It is contingent that p” is ~(It is contingent that p) and not (It is continent that ~p).
One has to define p consistently throughout. If p is held to be a component and not a proposition, then p and ~p can both be contingent (as you point out). This would give us the logically redundant but not contradictory Qp ^ Q~p. But if p is defined as a contingent proposition, that is Qp (which Prior and Aristotle do), then it would be wrong to claim a) that the CONTRADICTORY PROPOSITION of Qp is Q~p (it isn’t) and wrong also to claim b) that that contradiction of contingency is also contingency. Everything in our article argues against this. Pray tell, if Q~p is the contradictory of Qp, what is ~Qp to Qp? Is it not its contradictory? Or does Qp have two contradictory propositions?
So let’s not fail to distinguish between a propositions making truth claims (like Qp does) and components in a proposition (the p in Qp). If you mean p the component (“a sea battle tomorrow”) then you’re right, both components p and ~p are contingent. But these are not, as Prior and Aristotle claimed, contradictory PROPOSITIONS.
Peace,
Tom | | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/15/2004 11:32 PM (GMT -6) | | Alan: The law of non-contradiction does NOT say that a proposition and it’s contradictory can't have the same truth value. What is says is that they cannot both have the value TRUE.
Tom: Can you give me an example, assuming bivalence, of two contradictory propositions being both false?
I’m pretty sure you won’t find one. And that’s because if bivalence holds, then every proposition has a value of either TRUE or FALSE. That’s what bivalent means. In a bivalent system, then, where all propositions have a truth value of either true or false, it is the case that contradictory propositions have contradictory truth values, that is, they cannot share the same truth value, whether true or false.
My point is this, bivalent and non-bivalent systems do not define the law of non-contradiction the same way. But if you ASSUME a multi-valued logic, then you can claim that the law of contradiction doesn’t rule out contradictory propositions both sharing the same truth value. The point remains, however, that you can’t apply the law of non-contradiction as defined and applied in a bivalent system in Prior’s three-valued system. The ball is in Prior's court. He can't assume a three-valued logic and then use a definition of the law of non-contradiction consistent with his assumptions to prove his logic is consistent. He's the one begging the question. You can show I'm the one begging the question by providing us with an example, in a bivalent system, of two contradictory propositions being both false, as you claim is possible.
I wanna learn.
Tom | | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/16/2004 4:28 AM (GMT -6) | | I apologize for obsessing!
Prior once again:
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)
Note that Prior divides the field between Zp, Z~p, and Qp. He admits that “either a proposition is necessary, OR it is neither-necessary-nor-impossible (i.e., contingent), OR it is impossible.” These are “mutually exclusive” [my caps on “or”]. These “divide the field between them.” Thus they are “jointly exhaustive.” We’ve seen this language before, haven’t we Alan? Right. It’s in our paper! Zp, Z~p, and Qp are jointly exhaustive and mutually exclusive. WE argue that this means one has to be true and only one CAN be true. The other two are false. (Digression: The two that are false are not an example of two contradictory propositions being both false, as Alan claimed is possible, for the three propositions are “contrary” and not “contradictory”).
Now, a few pages later Prior claims that if a proposition is Qp, that is, contingent, then Qp is neither true nor false, but ‘neuter’ (or “I” for indeterminate). What truth values do the other two propositions get in Prior’s logic? Remember, Prior claims the three are “mutually exclusive.” Well, he claims that there other two, “p will” and “p will not” (that is Zp and Z~p), are both ‘neuter’ as well. Both are neither true nor false. Thus all three mutually exclusive propositions have the same truth value. But Prior earlier claimed that “EITHER a proposition is Zp, OR it is Z~p, OR it is Qp.” How is this true if all three are “I”? If we want to argue that the law of non-contradiction allows all three to share the same truth value so long as it’s not TRUE, fine. But what meaning is left to “contradictory” or “mutually exclusive” or “or” if all three are jointly exhaustive and mutually exclusive. If Prior didn’t claim that all three “divided the field between them” (i.e., if he didn’t claim that all three are jointly exhaustive) then he might argue that all three could share the same truth value (since there might conceivably be a fourth proposition that IS true). But in claiming that all are jointly exhaustive he argues himself into a corner. All “divide the field between them” and are “mutually exclusive” (“either Z, Z~p, or Qp”) but all are of an indeterminate truth value?
Tom | | Back to Top | | |
|
Humphrey Bogart
Registered Member
Date Joined Jun 2003
Total Posts : 205
| Posted 6/16/2004 5:03 AM (GMT -6) | | Sisters of Cain: "The present king of France is bald."
"The present king of France is not bald."
Albert: I don't have time to post anything much, but I thought I'd respond to this. Isn't the contradictory of "The present King of France is bald", just "it is not the case that the present King of France is bald"? And isn't that latter statement true, unlike "the present King of France is not bald"?
| | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/16/2004 8:08 AM (GMT -6) | | Sisters:
"The present king of France is bald."
"The present king of France is not bald."
Tom: These are contrary propositions and may both be false.
Tom | | Back to Top | | |
|
Alan Rhoda
Registered Member
Date Joined May 2002
Total Posts : 680
| Posted 6/16/2004 10:30 AM (GMT -6) | | quote:
Originally posted by TGB
Alan: The law of non-contradiction does NOT say that a proposition and it’s contradictory can't have the same truth value. What is says is that they cannot both have the value TRUE.
Tom: Can you give me an example, assuming bivalence, of two contradictory propositions being both false?
There are no examples of bivalent contradictory propositions (A and not-A) both being false. If A is true then not-A must be not-true, which in a bivalent system means not-A is false. And if not-A is true then not-not-A is not-true, which in a bivalent system means that A is false. Nor can both A and not-A be false because, in a bivalent system, if A is false then not-A is true, and if not-A is false then not-not-A (which, again, in a bivalent system) equals A is true. So where bivalence is assumed, two contradictory propositions must have opposite truth values.
Tom: My point is this, bivalent and non-bivalent systems do not define the law of non-contradiction the same way.
Alan: On the contrary, they do. Both bivalent and non-bivalent systems construe the law of non-contradiction as saying that A and not-A cannot both be true (at the same time and in the same respect). That contradictory propositions must have different truth values follows only from the additional assumption of bivalence.
Admire your tenacity and ingenuity, Tom. But this attempted refutation of multi-valued logic won't fly.
Later bro,
Alan | | Back to Top | | |
|
Alan Rhoda
Registered Member
Date Joined May 2002
Total Posts : 680
| Posted 6/16/2004 12:04 PM (GMT -6) | | Is the contradictory of a contingent proposition also contingent? I, along with Prior, say Yes. Tom says No.
[a]Alan:[/a] If it is contingent that p, then it must also be contingent that ~p because if p might and might not be true (Qp) then ~p might and might not be true (Q~p). In other words, Qp and Q~p are logically equivalent: Qp = Q~p. ... Now, ... we need to distinguish between the proposition "p" and the very different proposition "p is contingent" (Qp). The contradictory of p is ~p. If p is contingent then Qp is true. If Qp is true then so is Q~p, since they are logically equivalent. If Q~p is true then "~p is contingent is true". Therefore, ~p is contingent.
Tom: "It is contingent that p." That’s a qualified p. Prior posits a contingent PROPOSITION p. He defines p AS Qp.
Alan: Whoa. The claim is not that p = Qp. Those are different propositions. Qp = "It is contingent that p", but p = "p". Plug in any proposition you want for p. Let's say you plug in "It is contingent that q". (We need to pick a different letter to avoid generating a self-referential proposition.) In that case p = Qq. But then "it is contingent that p" does not equal Qq but QQq.
Tom: Let’s leave unqualified p (i.e., p without the operator “It is contingent that…”) as a small p and make qualified p (i.e., the proposition “It is contingent that p”) a capital “P", OK?
Alan: Works for me.
Tom: Thus:
p = the component (not the proposition): “...a sea battle tomorrow”
P = the proposition Qp (containing the component p): “It is indeterminately the case that a sea battle obtains tomorrow.”
Alan: p should read "there is a sea battle tommorow". So it is a proposition and not just a component of a proposition.
Tom: Thus “p is contingent” = Qp = P. And the contradictory of THIS is not Q~p, but ~Qp (or ~P). If you qualify a component (“a sea battle tomorrow”) with an operator to form a propositional truth claim and THEN form the contradictory of the proposition, i.e., the contradictory of the truth claim, the “~” is placed before the qualifying operator, that is, before the entire proposition, and not between the operator and the variable. Thus, the contradictory of “It is contingent that p,” or Qp, is ~“It is contingent that p,” or ~Qp.
Alan: I agree with everything you say here, Tom. But I don't see how it bears on the original question, which was whether a contingent proposition must have a contingent contradictory. If we take p as our contingent proposition, then the question becomes whether ~p is contingent. If we take Qp as our contingent proposition, as you seem to prefer, the question becomes whether ~Qp is contingent.
Tom: Moving on, it seems to me a mistake to say Qp and Q~p are logical equivalents ... By “Qp ^ Q~p” you mean to say “p might and might not occur [that's what Qp means] and ~p might and might not occur [that's what Q~p means].” This if perfectly fine, because here p and ~p are components that contribute to forming their respective propositions (p in "p might and might not obtain" and ~p in "~p might and might not obtain") and not independent propositions making truth claims.
Alan: Whether components or not, p and ~p are propositions. To use the example, p is "there is a sea battle tomorrow" and ~p is "it is not the case that there is a sea battle tomorrow". Now some propositions have components that are also propositions. Qp, for example, says "It is contingent that p", where p is the above-mentioned proposition.
Tom: You could add Q~p to Qp ... if you wanted to, but Q~p doesn't add anyting that's not already said in Qp.
Alan: Agreed. That's why I said that Qp and Q~p are logically equivalent.
Tom: One has to define p consistently throughout. If p is held to be a component and not a proposition, then p and ~p can both be contingent (as you point out). This would give us the logically redundant but not contradictory Qp ^ Q~p. But if p is defined as a contingent proposition, that is Qp (which Prior and Aristotle do), then it would be wrong to claim a) that the CONTRADICTORY PROPOSITION of Qp is Q~p (it isn’t) and wrong also to claim b) that that contradiction of contingency is also contingency.
Alan: Couple points here. First, your distinction between p as a component and p as a proposition is a false distinction. p in Qp is BOTH a component of Qp AND a proposition, namely, "there is a sea battle tomorrow". Second, you are right that the contradictory of Qp is not Q~p, but ~Qp, but this is irrelevant to the question, which is whether contradictories can both be contingent.
To reiterate, what we want to know is whether two contradictory propositions can both be contingent. To ascertain this we start with an arbitrary proposition, p, assume it is contingent, and see if we can show that its contradictory, ~p, is also contingent. The steps of my proof run as follows:
Question: If p is contingent then is ~p also contingent?
1) p is contingent. (assumption)
2) If (1) then Qp.
3) If Qp then Q~p. (Since Qp and Q~p are logically equivalent.)
4) If Q~p then ~p is contingent.
5) Therefore, ~p is contingent.
This is a deductively valid and completely general proof. It assumes nothing about the nature of the propositions in question, p and ~p. Moreover, it is assumed at the outset that p and ~p are propositions. So they cannot be construed as non-propositional components of Qp and Q~p.
The only potentially challengeable premise is (3), but (3) is intuitively obvious. Qp says "there might and might not be a sea battle tomorrow". Q~p says "there might and might not not be a sea battle tomorrow". It is quite clear, I think, that these are two different ways of saying the same thing. So (3) is true, and the proof is sound. If p is contingent, then ~p is also contingent.
God bless,
Alan | | Back to Top | | |
|
Tom
Registered Member
Date Joined Jul 2001
Total Posts : 10353
| Posted 6/16/2004 9:12 PM (GMT -6) | | I just got on quickly to post. Haven’t read your last post yet Alan. Will get to back to you later. Soon!
----------------------
Aristotle defined the principle of non-contradiction this way:
“It is impossible for the same thing to belong and not belong simultaneously to the same thing in the same respect” (Met.)
He noted too that a contradiction is a pair of propositions one of which asserts what the other denies. He considers in On Interpretation whether or not in every contradiction, one proposition must be true and the other false. As an example of an exception he offered the following “indefinite” proposition:
“A man is walking"
and claims that nothing prevents the contradictory “A man is not walking” from being symultaneously true.
But (I’m obssessing again!) Aristotle was mistaken to think these are contradictory propositons. They are subcontrary and may both be true. Put all four propositions of the Sqaure in relation:
All men are walking No man is walking
A man is walking A man is not walking
Aristotle’s propositions are not contradictory, they’re subcontraries equivalent to “Some men are walking” and “Some men are not walking.” How did he miss this? Too much Atic Sea wind I guess!
The other example Aristotle gave of non-bivalent contradictory propositions is:
There will be a sea-battle tomorrow
There will not be a sea-battle tomorrow
He knew if one was true and the other false there would be nothing contingent about the event. And since a) he (mistakenly) limits himself to expresing the future in this case exclusively in terms of “will” or “will not” and b) DOES believe in the contingency of some future events, he either has to deny bivalence (the two propositions are neither true nor false just in case the sea battle is contingent) or deny there are any future contingent events. He does want to deny the latter, so he denies the former.
But there is a way to affirm both bivalence and future contingency IF you don’t describe the future exclusively in terms of “will” or “will not.” If you add a third truth-functor, “might and might not” then all is well.
Back to Sisters’ example of two contradictory propositions both being false:
The present king of France is bald
The present king of France is not bald
But as was noted, these are both contrary propositions and are both FALSE. They occupy the top two places on the Square (top left for “bald” and top right for “not bald”). The subcontraries are:
It is not the case that the present king of France is not bald (bottom left)
It is not the case thte the present king of France is bald (bottom right).
Being subcontraries, these may (and in this case are) both TRUE; there IS NOT present king of France.
In a bivalent system, contradictory propositions ‘deny’ one another. They cannot share the same truth value. Aristotle got that right. Non-contradiction was understood as “It is impossible for the same thing to belong and not belong simultaneously to the same thing in the same respect.” I understand this to entail (assuming bivlance mind ya) that no two contradictory propositions can share the same truth value (not “no two propositions can share the same truth value”).
Sleep tight,
Tom | | Back to Top | | |
| 86 posts in this thread.
Viewing Page : 1 2 3 4 |
| Forum Information | Currently it is Monday, November 09, 2026 1:38 PM (GMT -6)
There are a total of 97,718 posts in 6,040 threads.
In the last 3 days there were 0 new threads and 0 reply posts. View Active Threads
|
Forum powered by dotNetBB v2.42EC SP3
dotNetBB © 2000-2026 |
|
|