A Few Good Modal Paradoxes

alt

I did - again, you said;

The conclusion to this 'paradox' is that "it is possible that Benjamin Franklin did not invent bifocals."

Wrong conclusion. It WAS possible that BF did not or would not invent bifocals (before he did so) but when he did invent bi focals, he invented them.

Therefore, it is now IMPOSSIBLE that BF did not invent bifocals.

Glad we got that sorted.

Next !

alt

Underlined .. yep - that's part of what I've been trying to say all along. I think it arises from a degree of word play.

Office_Shredder

You're interchanging "asserting P to be true" with "knowing P to be true" here

lugita15

Sorry, a lot of your confusion is because I didn't word things well enough. The "you" that I'm discussing the argument with is different from the "you" whose knowledge we're discussing. So instead of using "you", let me call the individual John, and let me restate the argument in that way.

Assume that the truth P is unknown to John. Let Q be the statement "P is a truth unknown to John." The question is, is Q knowable by John? Well, suppose Q were known to John. Then John would know that P is a truth unknown to him. But if he knew that P is a truth, that's the same as knowing P. So he would know P and he would know that P is a truth unknown to him. But if he knew P, then it would be incorrect to say that P is a truth unknown to him, so Q would be false, and you can't know a false statement. Thus from the supposition that Q were known to John we get a contradiction, so it must be impossible for John to know Q. Hence Q is a truth unknowable by John, and therefore not all truths are knowable to John. Now do you get it?

lugita15

You literally don't think there are any true statements that are unknown? Don't you think the results of the 2016 US Presidential election are unknown? When you flip a coin in the air, don't you think it's unknown which side it will land on?

lugita15

Perhaps you'll know it after you check each one of your 52 guesses, but do you agree that right when your handed the deck you don't know which place the Ace of Spades is? So if the Ace of Spades is in the 10th place, then at that moment wouldn't "The Ace of Spades is in the 10th place" be an unknown truth?

lugita15

How many times do I have to say this? You can change "it is possible" to "it was possible" if you want. That's not the important part of the logic. Let me change it myself, so there's no confusion.

Benjamin Franklin invented the bifocals. But we can imagine alternative histories, so we can say "It was possible for Ben Franklin to not have been the inventor of bifocals". And for any person X, we can say "It was possible for X to not have been the inventor of bifocals." For instance, X can be "William Shakespeare" or "The inventor of special relativity" or "The eighth president of the United States", etc. Thus we can let X = "the inventor of bifocals" and thus we reach the conclusion "It was possible for the inventor of the bifocals to not have been to inventor of the bifocals." But that seems absurd, because obviously the inventor of bifocals had to be the inventor of bifocals. How can you have the inventor of bifocals not be the inventor of bifocals?

Now do you understand the paradox?

Office_Shredder

My gut feeling says that the resolution of the paradox is that the bifocals don't have to be invented, so assuming there is an inventor is a fallacy. However the way that it's worded makes the whole premise absurd

If the statement "it was possible for the inventor of the bifocals to not invent the bifocals" is a paradox, then the statement "it was possible for Benjamin Franklin to not invent the bifocals" is the exact same paradox, because Benjamin Franklin IS the inventor of the bifocals (so you can freely substitute 'inventor of the bifocals' for him).

Hurkyl

"The inventor of bifocals" is a well-defined^* 'variable constant'.

The informal argument that "it is not necessary that X invented bifocals" makes critical use of X being a non-'variable' constant. Roughly speaking, it boils down to observing "X does not vary with 'the inventor of bifocals'", and therefore it is possible for "X" and "the inventor of bifocals" to be different.

The argument, of course, doesn't work if X is a constant that does vary along with 'the inventor of bifocals'.


*: Ignoring the technicalities of whether there is an inventor and it is unique

lugita15

Congratulations Hurkyl, you solved it! The technical way to say this is that in modal logic, you can only freely subsitute "rigid designators" into the modal operators.

alt

No .. I gets me P's & Q's mixed up. Can you try it again with a real example ?

Will read the rest of your posts soon. All look very interesting and worthwile, mind.

lugita15

Sure. John's coat is in his closet, but he doesn't know it. So "the coat is in the closet", which we'll call P, is a truth unknown to John. So let Q="P is a truth unknown to John". Then Q is certainly a true statement, because in fact P IS a truth unknown to John. With me so far?

Now the question, is Q knowable by John? Well, that's the same asking, can John know that P is a truth unknown to John? And more concretely, that's the same as asking, can John know that the statement that the coat is in the closet is a truth unknown to him? Or equivalently, can John know that it is true that the coat is in the closet and know that the statement that the coat is in the closet is unknown to him? Or in other words, can John know that the coat is in the closet and know that he does not know that the coat is in the closet? And the answer to the last rephrasing of the question is obviously No, because if he knew the coat was in the closet, then it would be wrong to say that he does not know that the coat is in the closet, so the statement "John does not know that the coat is in the closet" is false, and it's impossible to know a false statement. Thus the answer to the initial question, is Q knowable by John, is also No. So not all truths are knowable by John.

When I write it out like that in words, I'm afraid it will sound too confusing, which is why I wrote it using lots of P's and Q's before. I hope this helps.

I_am_learning

hi lugita,
Thanks for the clarification.
Lets assume john doesnt know sun rises in the east.
Now , can john know that "he doesn't know that sun rises in the east"?
No. Because that is a paradoxical sentence. Ah! I see it now.
P = Sun rises in east.
Q = I/jhon dont know P.
Jhon dont know thousands of truth. He may know lot of them later. But for now Q is truth. Suppose Jhon searches really hard to know all truths. He may eventually know P but not Q because knowing P (or trying to know Q) makes Q false.
So Q will rather be destroyed than being known to john.
But jhon may know R = "Q was a truth."
But its not same as knowing Q.
Things are making sense. So, where is the paradox?

lugita15

Exactly, you see the solution! By knowing P, John makes Q false, so he cannot know that Q is true, because it's not. But he can know that Q is false. So the only reason Fitch's paradox says there are unknowable truths is that there are some truths that become false as soon as you find out their truth value, so you can never know them to be true, but you can know them to be false.

If you look at my post #33, you'll see that I discussed exactly this resolution to Fitch's paradox. See the attached paper by Joseph Melia in that post.

alt

Earlier in a post you said;
People use fantastical examples to illustrate logical or philosophical points all the time.

Very well. Let's do a bit of that.

The election might never happen as you think it would. The anarchist party might take over, or any other political upheaval might occur that would prevent them.
(In fact, this is not nearly so fantastical)

The coin might fall down a drain in the pavement - or any other similar possibility.
(Not too fantastical, either)

The ace of clubs - before you examined the deck, it might be vapourized by some explosion - imagine you were in the Twin Towers just as you were about to examine it.
(A little more fantastical, but hey, I'm no 911 denier)

All above three, therefore, cannot be called unknown truths with 100% certainty.

I still maintain that 'unknown truth' is an oxymoron - to the one observer.

Sure, my 2 year old niece doesn't know I have a 4 litre motor in my car, but if we go down that path, we are again skating on the trivial, which I can reduce to absurd infinities - as discussed earlier.

alt

See my above post.

alt

But you do, in order to make the statement. Therefore, it is a known truth, and all John has to do is ask you.

But to use another fantastical example to illustrate a logical point - as you have allowed earlier, it might not even be there when he opens the closet. Someone might have stolen it. Or the cat might have pulled it through a crack in the floor. So YOU might even be wrong in your assumption of a KNOW truth, never mind Johns unknown truth.