Friday, July 03, 2009

Problems with Paradoxes

Over at the Triablogue, in the comments box, Mr. Anderson wrote:
Unless I've badly misunderstood it, which is entirely possible, your argument is designed to show that the claim that there can be irresolvable paradoxes is itself a paradox.

Your premise (i) states your opponents' position, for the second of argument. Your (ii) then apparently tries to deduce some further proposition from (i) (since you say "Given (i)..."). What you deduce from (i) is that the negation of (i) (i.e., that there cannot be an irresolvable paradox) would be "either a paradox or a real contradiction".

But as I've pointed out, this is just a non sequitur. You've given no good reason to think this follows from (i).

Perhaps the idea is that, if irresolvable paradoxes are possible, then for just any proposition p we affirm, we must also be prepared to affirm non-p. But again, this is simply a non sequitur. It doesn't follow from (i) at all. Why think that it does?

One might as well argue that, if irresolvable paradoxes are possible, and we believe that the Earth orbits the Sun, then we should also be prepared to believe that the Earth doesn't orbit the Sun. The problem, of course, is that we have plenty of good reason to affirm the former and no good reason to affirm the latter. So your suggestion (if I read you correctly) that if the irresolvable-paradox view is true then it must (by its own lights) be on a par with the no-irresolvable-paradox view, begs the question entirely.
(source)

Mr. Anderson is no lightweight when it comes to critical thinking, so I've taken a good bit of time to mull over his comments. Nevertheless, I see a few problems with his critique, or at least a few weaknesses. Let's see if I can explain.

Restatement of the Main Argument

The main argument against irresolvable paradoxes is this:

Suppose for the sake of the argument, a first premise

(P1) The situation that both proposition P is true and P is false (at the same time and in the same way) is a possible situation for any given P.

P1 actually combines two ideas: (1) irresolvable paradox is possible, where irresolvable paradox is defined by a given statement being both true and false in the same way and at the same time; and (2) paradoxes are not limited to only certain categories of propositions.

If P1 is accepted, and if we further add a second premise

(P2) P1 is a proposition, i.e. a member of the set of "any given P"

then we may conclude

(C1) It is possible that (P1) is also false.

Or in other words, if we accept the existence of unlimitable paradoxes, we must also be prepared to accept at least the possibility of the nonexistence of unlimitable paradoxes.

Enhancement to the Main Argument


The main argument may be enhanced, however, through simplification. One enhancement is as follows:

(P3) Reasoned thought is present IFF (i.e. if and only if) the law of non-contradictions is not violated;

(P4) Paradoxes violate the law of non-contradiction; and

(C2) Therefore, reasoned thought is not present when paradoxes are present.

Responses to Objections

Mr. Anderson's main objection seems to be to the boundless aspect of P1. Mr. Anderson, if I have understood him correctly, believes in the existence of irresolvable paradoxes, but only within certain bounds. I'm not sure what objection Mr. Anderson would be able to give to the enhancement argument.

Mr. Anderson's main objection does not appear to be sustainable. It is, of course, handy to say that paradox only exists within special, contained boundaries. And if that were strictly true that would seem to address the problem. Unfortunately, we cannot be assured (within a system that accomodates paradox) that the boundaries themselves are strictly true as opposed to merely paradoxically true.

Conclusion

I don't see any good reason to accept the existence of irreconcilable paradoxes. Such things, were they to exist, would seem to be outside the realm of rational discussion. Accordingly, it would be odd to call any basis for accepting them a "reason." Furthermore, I have seen no reason to reject the strongly intuitive position of the universality of the laws of logic and particularly the law of non-contradiction. I also would see no valid reason for setting boundaries on irreconcilable paradoxes if I were to accept them at all. I'm willing to hear arguments for why I should deny the universality of the laws of logic, but so far I haven't seen any that are logical ... and I'm willing to hear reasons to set boundaries on irreconcilable paradoxes but so far, again, I haven't seen anything beyond simple fiat to support the idea that irreconcilable paradoxes only exist within specific boundaries.

-TurretinFan

13 comments:

wtanksley said...

What do you mean by "the existence of unresolvable paradoxes"?

The usual meaning of "paradox" is a statement that if taken as true is actually false, and vice versa. Such statements actually exist, and Godel proved that they will always be expressible in any system powerful enough to reason about the natural numbers.

However, I'm thinking that you actually are referring to the reality to which a statement is supposed to correspond; i.e. you're asking whether something can possibly be both A and non-A at the same time and in the same way. If this is your question, it has a fundamental error: it requires formal logic to correspond to reality precisely, and no system of formal logic is capable of doing that. There will always be either incompleteness or contradiction within every system of formal logic; and that does not show that reality is itself incomplete or contradictory.

God exists. God cannot be comprehended by any system of formal logic.

Turretinfan said...

"What do you mean by "the existence of unresolvable paradoxes"?"

I'm not sure what's unclear about it ... perhaps the confusion is over word "paradox"?

"The usual meaning of "paradox" is a statement that if taken as true is actually false, and vice versa. Such statements actually exist, and Godel proved that they will always be expressible in any system powerful enough to reason about the natural numbers."

I'm not sure if that's the "usual" meaning, but no - that's not the meaning I'm using here.

"However, I'm thinking that you actually are referring to the reality to which a statement is supposed to correspond; i.e. you're asking whether something can possibly be both A and non-A at the same time and in the same way."

There is at least some aspect of correspondence involved in the issue. But that's not really the primary issue.

The primary issue is whether both "the world is a sphere" and "the world is not a sphere" can be true at the same time and in the same way.

That then leads us at least in the general direction of your next issue ...

"If this is your question, it has a fundamental error: it requires formal logic to correspond to reality precisely, and no system of formal logic is capable of doing that."

Propositions can correspond to reality. It is not clear what other reality formal logic is supposed to correspond to (after all, logic is something real, although not tangible).

"There will always be either incompleteness or contradiction within every system of formal logic; and that does not show that reality is itself incomplete or contradictory."

If that were true, it would seem to be unknowable. Why do you think it is true?

"God exists. God cannot be comprehended by any system of formal logic."

a) If by "comprehended" you mean "fully understood," of course that is true with one qualification;

b) The qualification is that one excludes God himself, who does fully comprehend himself; and

c) He it is who gives us logic to understand himself and the world.

Turretinfan said...

Incidentally, on the issue of correspondence - check this timely link (link).

Coram Deo said...

I suppose one could always take the position articulated by J.I. Packer which is to create a whole new category called "antinomies" and then simply redefine the argument.

I'm personally troubled by Packer's thinking in this area as it strikes me as being irrational to the point of absurdity. In essence, as I understand his definition, his appeal to antinomy is nothing less than an appeal to simply "embrace the mystery" that seemingly paradoxical statements in the Bible are both equally true at the same time.

To me this sort of thinking inevitably spirals downward into accepting - even embracing - real and/or perceived contradictions, which thing is wholly unacceptable.

The Triune One True and Living God is not self-contradictory. Certainly His thinking is much higher than ours, and from our finite perspective seems frequently counterintuitive, but He doesn't contradict Himself.

Just my two cents worth.

In Christ,
CD

natamllc said...

CD, for my edification please cite two examples of Scripture as a paradox.

thanks
natamllc

PuritanReformed said...

TF:

Excellent reductio ad absurdum argument. Those who believe in the existance of irreconciliable paradoxes should be willing to believe in their non-existance too.

Anonymous said...

Fortunately, unresolvable paradoxes aren't found in the religion of christianity.

-Another Anonymous On American Soil

Coram Deo said...

natamllc,

I'm not advocating nor attempting to advance Packer's view on antinomy/paradox, sorry if I confused you.

But from reading his works he'd respond to your inquiry by pointing towards the antinomy (code language for "apparent paradox") of the biblical concept of God's Sovereignty as King and Judge and how that works out over and against the biblical concept of human responsibility.

Again, I'm not taking Packer's position, I was simply pointing out that there's more than one variation on the theme of paradox, apparent paradox, contradiction and seeming contradiction running around out there in various theological circles.

Insofar as I can tell Packer would never allege an actual contradiction or paradox, he'd demure and say that such things are only problems in human minds because they're finite, and they're no problem at all in God's mind. He would suggest that we "embrace the mystery", not unlike orthodox Confessional Lutherans who say Christ atoned for all the sins of all humanity from Adam to the last mortal who will ever live (universal atonement), but that only some actually come to saving faith and that's a divine mystery to which the human mind can only say "yea and amen, Lord".

Thanks for asking.

In Christ,
CD

natamllc said...

CD,

you wrote: "....sorry if I confused you...."


I would demure also and so should you and take to heart that I was confused already. No harm done under those circumstances by you then, right? :)

As for your words after, it seems to me the paradox is that some also become a paradox from reading the Scriptures?

Would you sign on to this then about Peter about Paul and himself and himself about us, too?

::::> 2Pe 3:15 And count the patience of our Lord as salvation, just as our beloved brother Paul also wrote to you according to the wisdom given him,
2Pe 3:16 as he does in all his letters when he speaks in them of these matters. There are some things in them that are hard to understand, which the ignorant and unstable twist to their own destruction, as they do the other Scriptures.
2Pe 3:17 You therefore, beloved, knowing this beforehand, take care that you are not carried away with the error of lawless people and lose your own stability.
2Pe 3:18 But grow in the grace and knowledge of our Lord and Savior Jesus Christ. To him be the glory both now and to the day of eternity. Amen.

Hmmmmm?

My guess is the reason Peter wrote that about Paul was because Paul wrote and lived both the false and the true Law by his own experiences as a Pharisee and as an Apostle of Christ to His Church:::>

Rom 3:31 Do we then overthrow the law by this faith? By no means! On the contrary, we uphold the law.

Derek Ashton said...

TF,

I think the point is being missed here.

Let's take the classic example of divine sovereignty and human responsibility (indeed, this is Packer's famous "antinomy"). It's clearly not a case of "A is both A and non-A." It's more like this:

A implies non-B, and B implies non-A. Yet both A and B are Biblical propositions (and therefore true).

A. God is completely sovereign.
(this IMPLIES that created beings are not responsible)
B. Man is fully responsible for his choices.
(this IMPLIES that God is not completely sovereign)

The two propositions do not directly contradict one another. They only appear to contradict on the basis of what they imply. Yet the implications are natural and reasonable (some might even say they're necessary, but I wouldn't).

Since Scripture does not give us the resolution for this, we have just grounds to affirm that there is an area of mystery between these two Biblically TRUE propositions. The paradox arises when human beings try to understand HOW the two propositions relate to one another, and as our minds interpret their implications. There is no ACTUAL CONTRADICTION, but in our minds there appears to be one (apparent contradiction = paradox).

If it was simply a matter of "A cannot be both A and non-A," the whole matter would be simple. But it's more complicated than that.

If Scripture sets forth two propositions with apparently contradictory implications, but it remains silent on how the implications can be reconciled, then we are left with unresolved paradoxes. Our various attempts to explain them are mere man-made theories which may or may not be true.

In the end, don't we have to say Scripture is more weighty than human reason? Only by elevating human reason to the level of Scripture can one affirm categorically that all Biblical paradoxes are resolvable.

Stated formally, my argument looks like this:

P1) A and B are Biblically warranted propositions
P2) A implies that B cannot be true
P3) B implies that A cannot be true

Conclusion 1 (based on P1 alone) Both A and B are true

Conclusion 2 (based on P2 & P3) The relationship between A and B is an unrevealed mystery, and the implications of A and B form an unresolved paradox.

As long as the information which is needed to resolve the paradox is unrevealed (or if, as some would argue, we are incapable of receiving it), the mysterious relationship and the paradoxical implications remain.

This has nothing at all to do with "A cannot be both A and non-A." For me, belief in that rule is what leads to an affirmation the existence of humanly unresolvable paradoxes.

I wonder if the real reason so many people chafe at this is that our proud hearts simply can't tolerate not knowing - and not being able to know - the divine secrets. Wasn't that part of the motivation that led to the fall of our race?

Grace & peace,
Derek Ashton

Turretinfan said...

Mr. Ashton,

I cannot really tell if you bothered to read my post. If so, I am at a loss as to how you think your comments address the post. It looks like you're simply saying you'd like to discuss something other than what the post is about.

At least I can agree that the point is being missed.

-TurretinFan

Derek Ashton said...

Turretinfan,

YOU SAID:
"The main argument against irresolvable paradoxes is this:

Suppose for the sake of the argument, a first premise

(P1) The situation that both proposition P is true and P is false (at the same time and in the same way) is a possible situation for any given P.

P1 actually combines two ideas: (1) irresolvable paradox is possible, where irresolvable paradox is defined by a given statement being both true and false in the same way and at the same time; and (2) paradoxes are not limited to only certain categories of propositions."

MY RESPONSE: In the above quotation, you are defining paradox in a way that is far afield from Anderson's definition. He doesn't define paradox as a violation of the law of non-contradiction, but as an APPARENT violation of that law. For him, a paradox is never formed by the assertion that a proposition and its opposite are both true. He doesn't propose that "A can be both A and non-A, in the same way and at the same time," or that a proposition can be "both true and false, in the same way and at the same time."

I may have misunderstood your argument, to some extent. Ultimately, though, your P1 is not remotely close to James Anderson's definition of a paradox. You're effectively arguing against (real) contradiction, not paradox, so maybe your post is mis-titled and should have been called "Problems with Contradictions."

Thanks,
Derek

Turretinfan said...

Mr. Ashton,

I viewed myself as defending my own comment against Mr. Anderson's criticism, not as trying to criticize Mr. Anderson's book. If you have not seen my response to Mr. Anderson (link to that response) then please check it out.

I'd love to hear your answers to the two questions I've asked Mr. Anderson there.

As to the definition of paradox, yes - I think Mr. Anderson's definition is not the best one, and that he should use a different word for what he calls a "paradox."