Archive for the ‘Paradox’ Category

>Response from James Anderson

July 8, 2009

>I am pleased to report that Mr. Anderson has replied (here) to my previous post (here) regarding irresolvable paradoxes.

Mr. Anderson indicates that he does not accept the following:

(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.

This is, at least to my mind, some progress in our discussion. I view P1 as being representative of a general acceptance of paradoxes.

I do recall that Mr. Manata had characterized Mr. Anderson’s position thus: “James Anderson sets out to show that certain doctrines of the Christian faith are paradoxical, but may be reasonably believed in spite of this feature (if not because of it). Anderson also argues that these doctrines are not actually contradictory, but merely apparent.” (source – emphasis omitted)

I don’t have a problem with merely apparent contradictions. I have a problem with actual contradictions. Given that Mr. Anderson does not appear to subscribe to what I have called the “general acceptance” view of paradoxes, I wonder whether Mr. Anderson would even subscribe to “special acceptance” view?

The special acceptance view would be (continuing the numbering from my previous article:

(P5) 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 a given proposition P iff further condition FC is met.

P5 is not liable to the same critique as P1 if (for P5) FC is met. However, I’m not aware of any good reasons to accept P5. However, again, I’m not sure that Mr. Anderson accepts P5. In fact, Mr. Manata characterized Mr. Anderson’s position as: “If real contradictions could be true, then the desire to preserve orthodox interpretations is gone. Indeed, one could no longer object to heterodox statements.” Assumed, of course, is that the desires to preserve orthodoxy is not gone. I should point out that I do fully agree with Mr. Anderson in this regard.

This makes me think that Mr. Anderson also would not accept P5. I hope he’ll stop by and confirm that he does not accept P5.

In fact, I think part of the issue is that I am using “paradox” in a rather stronger form from Mr. Anderson. Mr. Manata claimed that Mr. Anderson defines paradox with the following:

“X is paradoxical [iff] X amounts to a set of claims which taken in conjunction appear to be logically inconsistent.”

But, of course, I have no problem with apparent logical inconsistencies, so long as they are merely apparent contradictions (MACs). My problem is when apparent logical inconsistencies are also actual logical inconsistencies. Mr. Anderson’s definition is broad enough to include actual and apparent inconsistencies (AACs), which is all that my narrower definition includes. My definition excludes merely apparent contradictions, while his includes them.

Now, I notice that Mr. Anderson’s definition could be made to be completely separate in domain from mine if he were to add the word “only” before “appear.” In other words, with that additional qualifier Mr. Anderson’s definition would no longer include AACs, whereas mine would consist solely of AACs.

(There is a further category we could add to the discussion: non-apparent actual contradictions (NACs). This category isn’t especially useful to our discussion, although it serves to remind us that there may be undiscovered contradictions.)

Furthermore, some of Mr. Manata’s comments in his review of Mr. Anderson’s work appear to reflect an understanding that the “only” that is missing from Mr. Anderson’s definition should be implied. Mr. Manata writes:

Note well the qualifier ‘apparent.’ Thus, a paradox does not entail a logical inconsistency per se, just the appearance of logical inconsistency. This definition “presupposes that a meaningful distinction can be made between apparent and real contradiction.”

(internal quotation apparently from Mr. Anderson)

As a strictly logical matter, the definition quoted above does not presuppose that a meaningful distinction can be made between apparent and real contradictions. It evades that issue, since real contradictions often are also apparent (though sometimes they are secret).

Anyhow, rather than continue to speculate, I’d just pose the question to Mr. Anderson who (I hope) has not given up on reading my comments here.

Is P5 your position?

Did you mean to imply “only” in the definition of paradox that you provided?


Problems with Paradoxes

July 3, 2009

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.


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.


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.


Do Dead Men Bleed?

April 3, 2008

Illustration – the “Dead” Man

Recently, I heard a radio message in which the speaker provided an illustration: whether it is true or not, I do not know. It’s believable, which is all that matters for my purposes.

This is the story. I young doctor is working for a large hospital, which includes a facility for those suffering from maladies of the mind. The young doctor is not very experienced, but he’s very idealistic and he hopes to make a difference.

One day, the young doctor meets a patient who swears that he is dead. In fact, he swears that he’s been dead for years. This poor patient is somewhat delusional, obviously, but the young doctor thinks that some cognitive therapy might help. Surely, he could talk this man out of his delusion.

So, he sits the man down and asks him, “You’re dead, eh?”

“Yes,” the young man replies, “been dead for years.”

“Tell me,” the doctor replied, “do dead men bleed?”

The young man answered, “No, I don’t think I’ve ever heard of a dead man bleeding. That wouldn’t really seem possible.”

At this point, the young doctor recognizes he has found the solution. One can almost see the glimmer of hope in his eye as he pulls a sterile hypodermic needle from his kit, requests the man finger, and pricks it with the needle.

Imagine his face, however, when hears the man exclaim, “Well look at that! Dead men DO bleed.”

The story is amusing to most people, because it is so absurd, and yet conceivable. That made me ask why. I realized that the thinking ran this way:

Premise 1: I am a dead man.
Premise 2: Dead mean don’t bleed.

From those premises, the natural conclusion is that I don’t bleed. We might characterize that as the expected conclusion.

Expected Conclusion: I don’t bleed.

But along comes evidence intended to disprove Premise 1 (that was the doctor’s intent). We’ll call this evidence the falsifying datum.

Falsifying Datum: I bleed.

It was hoped that this would cause the “dead” man to recognize that premise (1) was false, but instead the “dead” man instead rejected premise (2).

Parallel – the Scotsman Porridge-Sugaring

Readers may recognize this as similar to what has been called the “No True Scotsman ‘fallacy’,” (“fallacy” gets an extra set of quotes, because it is not strictly speaking a fallacy) in which

Premise (1) Angus puts sugar on his porridge.
Premise (2) No true Scotsman puts sugar on his porridge.
Conclusion (1) Angus is not a true Scotsman.
Conclusion (2) Angus is not a counter-example to the claim that no Scotsman puts sugar on his porridge.

It has occurred to me that both of these examples, the “No Dead Man” and the “No True Scotsman” examples, are simply examples of attempts to falsify, in which something goes wrong.

We could rearrange the NTS example this way:

(P1) Angus is a Scotsman.
(P2) No true Scotsman puts sugar on his porridge.
(EC) Angus doesn’t put sugar on his porridge.
(FD) Angus puts sugar on his porridge.
Selection (by Gourmand): P1 is wrong.

There is a fundamental problem in both cases. In the first case, we’d like the dead man to select P1 as being wrong. In the second case, we’d like the Porridge gourmand to select P2 as wrong. In each case, we feel (intuitively) that the wrong selection has been made, but I respectfully submit to you, the reader, that the error made is not a strictly logical one. Instead, the error is epistemological. I’ll explain that in more detail shortly, but first let’s examine yet a third example (or actually a set of examples).

Roman Catholic Error Examples

(P1) Rome is the true church.
(P2) The true church cannot err.
(EC) Rome does not err.
(FD) Rome errs.
Selection: ?

Let’s knock out the non-Catholic answer right away. The non-Catholic simply says, I’m not surprised by the FD, because I never accepted either P1 or P2. That’s completely uninteresting.

Next, let’s turn to the reaction of someone like Gerry Matatics, who holds a “Traditionalist Catholic” to the point of being labeled by others a “Sedavacantist” and contrast that with the selection of mainstream conservative Catholic (presumably someone like Jimmy Akin or Scott Hahn).

Both of these folks would select not P1 or P2 as false, but would claim that the FD is incorrect. GM would argue that the FD is incorrect because while a mistake has been made, “Rome” is not a correct identification of the errant party. The MCC would argue that the FD is incorrect because, while Rome is the right party, “err” is an incorrect identification.

In other words, using the NDM example, it is as though the “dead” man says, “that’s not my blood” (GM case), or “that’s mine, but it’s not blood.” In the NTS example, it would be as though the gourmand says, “that’s not Angus putting sugar on the porridge” (GM case) or “Angus is putting SALT (or whatever) on his porridge” (MCC case).

As you can see, in the GM case, it is P1 that is – in essence – favored, whereas in the MCC case, it is P2 that is – in essence – favored. Perhaps “favored” could be alternatively expressed as “emphasized.” GM emphasizes that Rome is the true church, whereas the MCC emphasize that the true church does not err.

Explaining the Outcomes

What dictates the result? Aren’t any of those escapes as validly logically as any other? Apparent contradictions require resolution, and there are lots of ways to resolve them. One can deny one or another previously held premise, or one can reject the new datum, either favoring one premise or the other. There’s one other option, which we occasionally see from irrationalists, which is to accept all the data, but throw out reason (criticizing rational thought as an “either/or mentality”).

Each of these outcomes reject something:

NTS => reject first premise
NDM => reject second premise
GM => reject falsifier for first premise reason
MCC => reject falsifier for second premise reason
Irr => reject logic

The Irrationalist position is the oddball, but I think we’ll see that it can be fit within an overarching scheme. There are basically two intuitive ways to group the remaining four, either by which premise they favor (NTA and GM vs. NDM and MCC) or by whether they reject a premise or the falsifier (NTS and NDM vs. GM and MCC). Neither way is necessarily incorrect, as will be seen.

Ultimately, the answer to the question as to which outcome gets selected, is “what is the mostly tightly held view?” In other words, is it the first premise (the major premise), the second premise (the minor premise), premises as opposed to new data, or data as opposed to logic.

The Irrationalist falls in the last category. He holds logic the least strongly of all the items. Thus, he’s willing simply to accept contradiction, and throw out logic.

Those who favor the first premise simply interpret the FD in light of that premise, and vice versa for those who favor the second premise.

Finally, those who favor the premises over the FD are those who are not willing to be persuaded.

Judging the Processes

We intuitively recognize in the NTS and NDM examples that the person ought to accept the FD and ought to alter one of the premises. That’s partly because we know that one of the premises is suspect. In the NDM example, we’re pretty sure the guy is alive, and in the NTS example, we think that the broad claim about Scotsmen is too much.

We, Reformed Christians, view the GM and MCC situations as problematic for much the same reason: we believe that both the premises are false, and consequently we think that the FD should persuade those groups to reject the premises. Unfortunately, their minds prefer their premises over the new data.

We run that risk too. Any time something appears that facially contradicts an expected conclusion of our systems of thought, we need to ask ourselves how our premises are grounded. Indeed, that’s what we’d counsel the “dead” man and the gourmand.

“Why do you accept the premise that you are dead?” “Why are you so sure that Scotsmen don’t sugar their porridge?”

To the Catholics, we ask the same questions: “Why do you think that Rome is the true church, but more importantly, why do you think that the true church cannot err?”

Conclusion / Application

I respectfully submit that there is not a valid epistemological basis for the view that the true church cannot err. But trying to prove that to someone who tightly holds that as a premise is quite difficult, because mens minds seek to compromise that which they hold less tightly.

I sincerely think that there are many Catholics (and Orthodox and so forth) who hold to the premises that their church is the true church, and that the true church cannot err so tightly that when an error in the teaching of their church is presented to them they will either deny that it is an error (the majority reaction in the long run), or deny that it is a teaching of their church (the minority reaction in the long run, although sometimes the majority reaction in the short run).

That’s one reason that we need to be careful to limit our premises to things which cannot fail us. By God’s revelation, we are aware that this includes the Word of God in the Scriptures of the Old and New Testaments. By keeping our presuppositional acceptance of Scripture as a minimal set of tightly held premises, we can avoid the various errors mentioned above.

Likewise, I hope that Catholics will consider whether an approach in which they presuppositionally accept the premise that Rome is the true church or (more importantly) the premise that the true church cannot err, is really the best hope for their discernment of the truth of the matter. I respectfully submit to them that they ought to reconsider those premises, as we have good reason to believe that both are incorrect.

May God give us grace to discern our errors,


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