Missives from a fly bottle
barang dot sg
Last revised 18 October 2017
What’s a logical paradox?


This was written in 2009. It explains the notion of a logical paradox in a way that comes very naturally to me. The conclusion drawn at the end is probably the most important thing for me (“We are blinder than we can imagine in matters of straightforward logical reasoning”) but I find that it is rarely emphasized by others.
8. The bomb

Suppose you wagered yesterday that the Irish would win a certain football match and are anxious to know the result.

The match itself is long over and it’s only the result that you don’t know.

Despite your anxiety, you decide to wait one full day before checking the result, because you believe that if you behave as if the result is not urgent, it’s more likely to turn out in your favour! – You got this curious belief from your father, say.

Many would call this superstition, of course, since whether you check the result now or later can’t possibly make a difference, given that the match is already over.

In fact, there’s a simple proof of this, isn’t there?

Either Ireland already won the match, or they didn’t.
If they already won, then waiting one day won’t change this fact, and so the wait will be unnecessary.
But if they didn’t win, then waiting one day won’t change this fact, and so the wait (in this case) will be ineffective.
So waiting one day is either unnecessary or ineffective, and serves no purpose either way.

Many would consider this piece of reasoning to be completely watertight, indeed entirely trivial. We might call them fatalists about the past.

But the English philosopher Michael Dummett pointed out that if we apply the same style of reasoning to the future, it starts to look a little silly!

In Dummett’s example, it’s wartime London, the bomb siren goes off, and the whistle of an approaching bomb is heard. Everyone else rushes at once for the bomb shelter, whereas you find yourself calmly entertaining the following piece of reasoning:

Either the bomb will end up killing me, or it won’t.
If it will, then nothing I do now will change this fact, and so any precautions I take will be ineffective.
But if it won’t, then nothing I do now will change this fact, and so all precautions (in this case) will be unnecessary.
So all precautions against the bomb are either ineffective or unnecessary, and serve no purpose either way.

This is exactly the same style of reasoning, but hardly anyone would endorse it this time!

It’s not easy to say where this perplexing piece of reasoning goes wrong, but it must surely be flawed because it yields the outrageous conclusion that it’s pointless to take precautions against the bomb. For all that, the reasoning seemed impeccable in the previous case, where the conclusion happened to look agreeable. So this is another case where a subtle logical error goes unnoticed if the conclusion looks innocuous, and is brought to light only when the conclusion seems crazy.

Dummett’s paradox (which was also known to the ancient Greeks) is widely known as the paradox of fatalism, and while we won’t try to diagnose the exact flaw in his reasoning here, it seems that Dummett is right that fatalism about the past cannot be established all that easily, otherwise fatalism about the future could just as easily be established.

So, incredibly, in the football case, it remains an open question whether waiting one day to check the result can make a difference. At least, the reasoning shown at the top does not establish that it can’t!

The larger point is that the mistake in the reasoning, whatever it might be, is potentially hazardous, because something more than football might be at issue.

Imagine that a ship has sunk and a loved one of yours was on board. (This example is also from Dummett.) There was only one survivor, rescued yesterday, but the authorities have yet to release his or her name.

While waiting anxiously for the announcement, you wonder if you could possibly do anything to increase the odds that your beloved’s name will be the one that is announced. Cross your fingers? Hold your breath? Say a prayer? Surely not. What would be the point? If she was the one who survived, then anything you do will be unnecessary. But if she wasn’t, then anything you do will be ineffective ...

... right?

Needless to say, a can of worms is being opened here, relating to the question of time. Thus, we tend to believe that the future is “open,” whereas the past is “closed.” So while we can seek to influence what has yet to happen, we have no control over what has already happened.

We don’t normally think to ask why we believe this, but the effect of the foregoing is to force that question upon us. The belief may turn out, in the end, to be justified, only that we tend to articulate its justification improperly. So the can of worms may yet be closed.

But, then again, like our naive belief in the “absoluteness” of space and time, long overturned by Einstein, it may also be a metaphysical prejudice for which we have no good reason, after all. (The matter is not fully settled.)

We were looking for cases where it has not yet dawned upon us that a certain style of reasoning may be flawed, even though we employ it all the time, perfectly convinced of its validity. For those who have never encountered the paradoxes of fatalism, the style of reasoning shown above is likely to be an example, especially as applied to the past.

For a final example, we should consider a bewildering little conundrum known as Newcomb’s paradox. This too involves a “compelling” line of reasoning that culminates in an innocuous-looking conclusion. So there is no real indication that the reasoning contains an error, although there may be a very slight hint.

The question is whether we should dismiss the hint and endorse the “compelling” line of reasoning. Many people do!