A Few Good Modal Paradoxes

No, contrary to popular belief Godel's theorem does NOT say there is a single statement that is undecidable in all sufficiently strong axiomatic systems.

Rather, it says that for each sufficiently strong axiomatic system, there exists a statement in that system, but easily decidable in other systems.

Sure, I can go on to other ones. (And it's a he by the way.) But let me first explain my preferred resolution to Fitch's paradox. I came up with it on my own soon after reading about the paradox a while back, but then I later found out that Joseph Melia thought of more or less the same solution in 1991; see the attached paper. The key idea is this: Fitch's "argument presupposes that we can discover a statement's truth value without affecting that statement's truth value. But this is not so: there exist statements which are true, yet which would have been false had we performed the procedures necessary to discover that statement's truth value."

To illustrate this point, suppose for sake of argument that it were possible for someone to be omniscient (i.e. knowing literally everything) but that no one was actually omniscient. Now consider the statement "No one is omniscient." That would be a true statement. But could it be known? Well, since we're assuming that omniscience is possible, by definition it would be possible for someone to know literally all true statements. But in that case "no one is omniscient" would not be a true statement, so it obviously wouldn't be known. So the thesis "all truths are knowable" doesn't make much sense, only because if the truth value of some true statements were found out they would no longer be true, and thus no longer be in the set of statements people can know (because you can't know a false statement).

So how do we remedy that? Surely "all truths are knowable" does try to capture some sensible and debatable sentiment, namely the belief that there are no limits to human knowledge. A better way of expressing that sentiment, one that does not fall victim to Fitch's paradox, is to say, "all truths are verifiable" or to put it another way "the truth value of any statement is knowable". To put it in more formal language, "For all statements P, either P is knowable or not P is knowable." You might think that that's equivalent to "For true statements P, P is knowable and for all false statement P, not P is knowable." But that's not true. Because your knowledge of the truth value of P may change the truth value of P (e.g. "the statement "there is no light in the room" becomes false if you turn on the light to test whether there's any light in the room!). But the important point is that Fitch's paradox allows for the possibility that you can find out the truth value of any statement, and if that's the case then surely it does not put any limitations on human knowledge.

Does that make sense to everyone? If not, look at the attached paper, and if you still have questions I'm happy to try and clarify matters.

Your statement .. suppose for sake of argument that it were possible for someone to be omniscient (i.e. knowing literally everything) .. seems contradictory. I cannot believe that a person, with a limited brain, intelligence, life span, etc (i.e., finite), can ever literally know everything (i.e., infinite). Can you even imagine someone having infinite knowledge ? He would need an infinite space to put it all in, and probably, an infinite span of time to assimilate it - particularly given that new and further knowledge of infinitely more things and events would be coming up all the time.

Thus, the juxtaposition of 'person' and 'omniscient' in the real world, is to me nonsensical, and if I supposed it, would simultaneously suppose that anything flowing from it would be also.

I personally believe that these paradoxes (certainly the one in question) arise from different folk attributing different meanings to words - a nuance here, an inflection there, a not so subtle leap of faith elsewhere .. before you know it - confusion and chaos.

PS - will read the one on #35 soon.

It was just a hypothetical situation designed to illustrate my point. But if things like infinite knowledge trouble you, just imagine a simpler world in which there were, say, only 50 or a 100 truths to be known. Then surely you can imagine not all of the truths being known but it would be possible for you to learn all of them.

People use fantastical examples to illustrate logical or philosophical points all the time.

That can't possibly be the case, at least not in the sense you're thinking of, because Fitch's paradox can be put into unambiguous symbolic language.

I look forward to your thoughts on it. As I said, it's a pretty simple one, so hopefully we can settle it fairly quickly and move to yet another one.

By the way, did you read the Melia paper I attached in post #33?

Now for another modal paradox. This one is pretty simple and unlike Fitch's paradox, where I heavily insisted that it wasn't just a simple case of word play, this one can more justifiably be called playing with words (although it can still be expressed in symbolic form). It goes as follows: Benjamin Franklin was the inventor of bifocals, glasses that correct for both near-sightedness and far-sightedness. And since he was the inventor of bifocals, e.g. Albert Einstein was not the inventor of bifocals. But we can readily imagine alternate histories in which all kinds of things happened, like the Confederates winning the civil war or Japan not attacking us on Pearl Harbor. Similarly, we can say that although Benjamin Franklin invented bifocals, he did not have to be; someone else could have done it instead. So we can say "It is possible that Benjamin Franklin did not invent bifocals."

It is again nonsensical, and takes us right back to the start. To know there are (say) 100 truths to be known, means that you must know they are truths, else, you couldn't call them truths - could you ? How could you call them truths up front if you didn't know they were that ???

It is again nonsensical, and takes us right back to the start. To know there are (say) 100 truths to be known, means that you must know they are truths, else, you couldn't call them truths - could you ? How could you call them truths up front if you didn't know they were that ???

Yes, using fantastical examples certainly does broadens the options, doesn’t it ? Did you have some examples of using fantastical examples to arrive at logical truths, other than by accident, or by the use of metaphor, parable, simile, etc ?

I think it is very true (my earlier statement about fluid use of language). As an overt example, consider this;

Nothing is better than complete happiness in life. A strawberry ice cream cone is better than nothing. Therefore, a strawberry ice cream cone is better than complete happiness in life. But surely it isn't. So have we stumbled upon some deep metaphysical, paradoxical mystery here, or is it just fluid use of language - in this case, that word singularly least disabused of ambiguity, 'nothing' ?

Also, you mentioned symbolic language before. Do I have to learn a new language to 'grok' with you ? Modern English is a very fine and complex language - as good as any. I know it well, and you seem to be adequate in it :-)
To defer to a more obscure or symbolic language, hints of a dodge to me. Fitch's paradox must stand on it's own two feet as it were .. that being the language in which it's presented. And it still clearly to me, nothing more than word play.

I repeat part of our earlier dialogue;

You said ; .. So to review, we started with the hypothesis that P is an unknown truth .."

I replied .. "But even at the start, that hypothesis seems a little shaky .."

Nothing further to this has really been added, so far as I can discern.

Tell me - what do you really think Fitch's paradox is doing ? You said earlier that you heavily insisted it wasn't just a simple case of word play. So is it revealing some deep metaphysical truth ? Some new science ? Some unknown mystery or secret ? Some undiscovered incongruity in or of human existence, of knowledge... or WHAT ? I'd really like you to give me a specific answer to this question, and in the language we are presently using.

I will dowload it now.

So we can say "It is possible that Benjamin Franklin did not invent bifocals."

Again, some lassitude of, umm, 'crisp' word meaning here.

It WAS possible that BJ did not (or would not) invent bifocals. But he did as it turned out. So it is IMPOSSIBLE that BJ did not invent bifocals, because he DID.

It's easy. As a simple example, if I have a standard deck of cards, I know exactly one of the following statements is a truth:

• The first card is the ace of spades
• The second card is the ace of spades
• The third card is the ace of spades
• ...

And yet, I cannot identify any particular statement as being a truth.

If I didn't know the deck was standard, there is still exactly one truth among those statements, but I wouldn't even know that!

That statement you quoted isn't a paradox at all. It's just an assumption used in the paradox.
We're not talking about you asserting "P is an unknown truth." Here is the logic of the paradox again.

We start with the assumption that there is some truth P which is unknown to you, but perhaps known to others. Now consider the statement Q, which says "P is a truth unknown to you." By assumption, Q is true. Now the question is, can Q be known to you? Well, suppose that Q were known to you. Then you would know the statement "P is a truth unknown to you". But if you knew that, you would know that P is true and that P is unknown to you, or in other words P would be both known to you and unknown to you, which is impossible. Thus the supposition that Q is known to you leads to a contradiction, and thus it is impossible for Q to be known to you, or in other words Q is unknowable to you. Thus we can disprove the thesis that all truths are knowable to you.

assumption that there is some truth P which is unknown to you,but perhaps known to others.

Now consider the statement Q, which says "P is a truth unknown to you." By assumption, Q is true.

Now the question is, can Q be known to you? Well, suppose that Q were known to you.

Then you would know the statement "P is a truth unknown to you"

But if you knew that, you would know that P is true and that P is unknown to you, or in other words P would be both known to you and unknown to you, which is impossible.

I'm not saying that you know that these specific 100 truths were there to be known. I'm just saying, consider a hypothetical world in which there were only a hundred truths to be known. In such a world, it might be easy for someone to know everything, but it might just happen to be the case that they don't know everything. I'm not using a fantastical example to PROVE a logical point. I'm using it to illustrate a logical point. Yes, that is really just playing with words, because the word nothing is ambiguous. But Fitch's paradox is not just playing off of an ambiguity in this trivial sense. But the thing is, English is full of ambiguities and vagaries, so you might assume that Fitch's paradox arose from one of those flaws of the English language. But in fact, Fitch's reasoning can be expressed in the language of symbolic logic, where there is no room for ambiguities or semantic tricks. I'm not asking you to learn the symbolic language (although it's not too hard to learn), just to trust me that the reasoning still works when you translate to the symbolic language, so Fitch's paradox is not as trivial as you might think.But I've given you examples, like the number of hairs on Obama's head, which you've dismissed them as absurd. But the thing is, reductio ad absurdum doesn't mean that anything you feel is absurd should just be dismissed. The "absurd" part in the context of reductio ad absurdum means getting an actual contradiction, like a statement of the form "P and not P". Absurd in this logical context does not just mean anything you find wacky or silly.

And if you don't like my examples, what about Hurkyl's example of the playing cards in post #41? If Fitch's paradox had no resolution, then it would reduce the arguable statement "all truths are knowable", which conveys the sentiment that there are no limits to human knowledge, to the naive statement "all truths are known". Thus from the weak assumption that humans do not know all the truths they could know, Fitch's paradox would somehow be able to place fundamental barriers on the reach of human knowledge.

But at least in my opinion, the reasoning given in Melia's paper (which as I said I thought of independently) satisfactorily resolves Fitch's paradox. So in my view, all Fitch's paradox tells us is that the statement "all truths are knowable" is a bad way of representing the claim that there are no limits to human knowledge.

OK, if that bothers you feel free to change all my instances of "it is possible" to "it was possible". That's not the important part of the reasoning.

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

My response was that it may have been possible at one time, but now it isn't, because manifestly, he did. So the conclusion is wrong.

It is IMPOSSIBLE that BF did not invent bifocals.

Wrt to Fitch's paradox we can just assume that all truths aren't necessarily knowable ... which seems to be a most reasonable assumption.

It's easy. As a simple example, if I have a standard deck of cards, I know exactly one of the following statements is a truth:

• The first card is the ace of spades
• The second card is the ace of spades
• The third card is the ace of spades
• ...

And yet, I cannot identify any particular statement as being a truth.

If I didn't know the deck was standard, there is still exactly one truth among those statements, but I wouldn't even know that!