Posts on: Paradoxes
A famous argument, first proposed in Lucas 1961, supposedly shows that the
human mind has capabilities that go beyond those of any Turing machine.
In its basic form, the argument goes like this.
Let S be the set of mathematical sentences that I accept as true. S
includes the axioms of Peano Arithmetic. Let S+ be the set of sentences
entailed by S. Suppose for reductio that my mind is equivalent to a
Turing machine. Then S is computably enumerable, and S+ is a computably
axiomatizable extension of Peano Arithmetic. So Gödel's First
Incompleteness Theorem applies: there is a true sentence G that is
unprovable in S+. By going through Gödel's reasoning, I can see that G
is true. So G is in S and thereby in S+. Contradiction!
1. Suppose you have strong evidence that L are the true laws of nature, where L is
a system of deterministic laws. You also have strong evidence that the universe
started in the exact microstate P. Your have a choice of either affirming or
denying the conjunction of L and P. You want to speak truly. What should you do?
Intuitively, you should affirm. But what would happen if you denied?
Since L is deterministic, L & P either logically entails that you affirm, or it
logically entails that you don't affirm. Let's consider both possibilities.
There are familiar semantic paradoxes for "truth" and "reference", such as the Liar paradox and Berry's paradox. I would have thought that there should be similar paradoxes for "expression", i.e. for the relation between a sentence S and the proposition expressed by S. A quick duckduckgo search didn't come up with anything. Pointers?
Here is a Liar-style one I came up with myself. Assume propositions are sets of worlds (which is the case I'm interested in). Consider the sentence
E: E expresses the empty set.
If E is true, then the proposition it expresses contains the actual world, in which case E doesn't express the empty set. So E can't be true. Since we've just proved not-E from no empirical assumptions, ~E expresses the set of all worlds. Hence E expresses the empty set. So E is true. Contradiction.
Let [] and <> express alethic necessity and alethic possibility, let @ stand for
'actually', and L for 'it is unalterable that'. We are going to prove that
if something happens, then it is unalterable that it happens.
We need the following principles:
- A <-> <>@A.
Something is the case iff it is possibly actually the case.
- <>A -> L<>A.
If something is alethically possible, one cannot make it
alethically impossible.
- L(A -> B) -> (LA -> LB).
If A -> B and A are both unalterable, then so is B.
- If A is provable then LA.
Logical truths are unalterable.
Here is the proof, with a sea battle for illustration.
What about this much simpler argument for halfing:
As usual, Sleeping Beauty wakes up on Monday, knowing that she will have an indistinguishable waking experience on Tuesday iff a certain fair coin has landed tails. Thirders say her credence in the coin landing heads should be 1/3; halfer say it should be 1/2.
Now suppose before falling asleep each day, Beauty manages to write down her present credence in heads on a small piece of paper. Since that credence was 1/2 on Sunday evening, she now (on Monday) finds a note saying "1/2".
While I'm on the topic of repeating well-known mistakes, here's another idea I'm certainly not the first to come up with. Consider the liar paradox:
| | L := "L is not true" |
| 1) | Suppose L is true. |
| 2) | Then "L is not true" is true (by definition of L). |
| 3) | Then L is not true (by the Tarski Schema). |
| | etc. |
The inference from (1) to (2) is only valid if "... is true" is an extensional or intensional context. So couldn't one block the paradox by declaring "true" hyper-intensional?
Sometimes people say that for logical reasons there can be no examples of unknown or unknowable truths. The logical reason is this: to know that p is an unknown truth requires knowing that p is true, which contradicts the requirement of p being unknown.
Before I give examples of unknown and unknowable truths let me give examples of philosophers who died more than 100 years ago: Hume, Leibniz, Kant, and the philosopher first born in the 16th century. One might have thought that it is impossible for physical reasons to give such examples. After all, a philosopher who died more than 100 years ago just isn't there any more, so he can't be given as an example. But not so. In order to give an example of a dead philosopher it suffices to name or describe one; it is not necessary to dig him out.
Here comes the solution to this year's Christmas puzzle:
First, is the story in the museum true or false? The crucial question is whether the last sentence in it is true. It goes:
*) If the story is true, the oracle finds out that it is.
Under what conditions is (*) false? It is false iff i) the story in the museum is true, but ii) the oracle doesn't find out that it is. On the other hand, since (*) is part of that very story, if (*) is false, the story is also false. So if (*) is false, the story is both true and false. So (*) can't be false.
The Museum of the Myth is not very comprehensive. In fact, it only contains a single story:
The Museum of the Myth is not very comprehensive. In fact, it only contains a single story. The story is not particularly exciting. Moreover, some people wonder whether it is actually false. If not, it would of course be incorrectly classified as a myth. So one day, the oracle is asked about the story. Luckily, the oracle is quite reliable: if the story is true, it undoubtedly finds out that it is.
The story is not particularly exciting. Moreover, some people wonder whether it is actually false. If not, it would of course be incorrectly classified as a myth. So one day, the oracle is asked about the story. Does it find out whether it is true?
There's something odd about Albert's reasoning:
If that stranger's predictions are true, he probably is a time
traveler. I want him to be a
time traveler. Therefore I should try to make his predictions
come true.
The problem is that by trying to make the predictions come true,
Albert decreases the evidential support their truth lends to the
claim that the stranger is a time traveler.
Albert is a time traveler. In 2015 he travels back to 1995. There he meets his younger self and tells him in great detail what he, the younger Albert, will do in the next 20 years: that he will quit smoking, be injured in a traffic accident at a certain date and location, that he will work very hard in a physics lab to build a time machine, and so on. All these predications come true.
Isn't that puzzling? For example, on the day of the predicted traffic accident, why did Albert, who knew about the prediction, not avoid getting to that particular location? Why does he always behave exactly as he was predicted to do? This is certainly not what ordinary people would do. If you claimed to know that I will raise my left hand in a minute and told me so, I would try not to raise my left hand. Does Albert never try to make the predictions false? Or does he, but always fails? That seems unbelievable. How can you try not to work hard in a physics lab but fail? In fact, we may assume that Albert is told by his older self that he will never even try to make the predictions false. Then he never tries and fails because he just never tries. How strange. And how stupid: Albert knows since 1995 that he will eventually travel back in time with a time machine. For he has already met his older self. So why does he work hard at the lab? Why not lie in bed and watch TV instead? No matter what you do, you can't change the past. So no matter what Albert does in 2003, he can't change the fact that in 1995, he arrived as a time traveler from the future. So he's a fool when he's working hard to make it happen (or rather, to make it have happened).
On Friday, I wrote:
Conclusion 2: If we want to avoid Bradley's regress, there is
no reasonable way to defend the principle that every meaningful expression
of our language has a semantic value. (Russell's paradox is an independent
argument for the same conclusion.)
Today, I was trying to prove the statement in brackets. This is more
difficult than I had thought.
Semantic paradoxes usually (always?) arise out of an unrestricted
application of schemas like
Today I found Montague's paper, and it turns out that I was
wrong. Well, Field's presentation was not entirely correct: We
shouldn't take Robinson arithmetic itself as R, but some extension of it
that contains an additional primitive predicate "True" (T, for short). The extension need
not say anything about this predicate. This is why T needn't represent
truth in R. (If R says nothing about T, T either represents nothing at
all or the inconsistent property, depending on how precisely we define
representation.) Montague then shows, very much like Field, that any
theory that contains R -- no matter if it's axiomatizable or not --,
as well as every instance of
So I've started to actually read Field's papers. Unfortunately I already
got stuck on page 4 of "The Semantic Paradoxes and the Paradoxes
of Truth". Field there discusses the following restriction of the
naive truth schema:
T**) If True(p) then p.
He notes that this is rather weak, since it doesn't even imply that there
are any truths at all. Hence, he says, one would presumably add principles like
In my last posting, I argued that to escape the cardinality problem
for thoughts Frege perhaps has to give up
1) For any things there is at least one concept under which all and only
those things fall.
Now (1) is clearly false if, as I think, all there is are objects --
that is, if it makes sense to quantify over absolutely everything. But if
not, as Frege thinks, denying (1) is not an option. A concept is a
function from things to truth values. Given that functions are not
themselves things, how could there fail to be such functions?
A while ago, I was discussing Adam Rieger's alleged paradox in Frege's
ontology (here, here, and here). I'm still confident that the Russellian
version of the paradox can be blocked. But on second thought, the
cardinality version of the paradox appears to be much more difficult. Here
it is again.
1) For any things there is at least one concept under which all and only
those things fall.
2) For each of these concepts, there exists the thought that Ben Lomond
falls under it.
3) All these thoughts are different.
4) All thoughts are objects.
From (1)-(3) it follows that there are more thoughts than objects (2^k
if k is the number of objects), contradicting (4).
When I take a break from philosophy I often find myself creating utterly useless
computer programs. Today, for example, I've spent some hours on Quines.
A Quine is a program that outputs its own source code. (Quines are so called
because Quine, in "The Ways of Paradox" if I recall correctly, introduced the
self-denoting expression "'appended to its own quotation' appended to its own quotation".)
Making Quines is a lot of fun, and also a good training to avoid
use/mention mistakes. I've just written several JavaScript Quines. Here is a particularly
neat one (try it!):
for(i=0;c=[",","'",'"',"for(i=0;c=[",
"][('320202120121023202424').charAt(i++)];)document.write(c)"
][('320202120121023202424').charAt(i++)];)document.write(c)
Back to life. Here is the solution to the Christmas
puzzles:
1. The king said that one day somebody will find a sound proof that he
hasn't always said the truth. Now either this is true or it isn't. If it
isn't, the king hasn't always said the truth. If it is, somebody will find
such a proof, and since the conlusion of any sound proof is true, again the
king hasn't always said the truth. So in any case, the king hasn't always
said the truth.
2. The king had uttered only two sentences. By the above argument we know
that one of them must be false. But we also know that the first one was
true: Somebody really found the requested argument. So the second sentence
must have been the false one. It said that the person who finds the
argument will get the kingdom. Hence it was logically impossible to give the
kingdom to the court jester.
I'm too sick to blog. In the meantime, here is a puzzle I've made up for the second edition of Ansgar Beckermann's Einführung in die Logik. In fact, it's two puzzles.
Once upon a time an old and reticent king made the following announcement: "One day somebody will find a deductively sound argument proving that I haven't always said the truth. To this person I will bequeath my kindom." It was the court jester who first presented such an argument. How did the argument go?
Soon afterwards, the king died, and it came to be known that the above announcement was in fact the only sentences the king had spoken in his entire life. Thereafter, the court jester was refused the kingdom -- for logical reasons. Why?
Brian Weatherson correctly argues that, since
premise 2 of argument Z is analytically true, it
can be simplified to
Argument Z':
1. If the conclusion of argument Z' is true, then argument Z' isn't sound.
Therefore: Argument Z' isn't sound.
The paradox then arises in two different ways. First, for premise 1 to be
false, it must be the case that 'Argument Z isn't sound' is true and argument Z is sound.
Second, and more interestingly, the falseness of premise 1 analytically
implies that argument Z is sound, which in turn analytically implies that
all premises of argument Z are true, which implies that premise 1 is true.
This second paradox can be further simplified to:
Argument Z'':
1. Argument Z'' isn't sound.
Therefore: Snow is white or snow isn't white.
An argument is called sound if it is deductively valid and its
premises are true. Now consider the following argument, which I'll dub
'argument Z':
1. If the conclusion of argument Z is true, then argument Z isn't sound.
2. If the conclusion of argument Z is not true, then argument Z isn't
sound.
Therefore: Argument Z isn't sound.
Is argument Z sound? (If not, which premise is false?)
Let S be the sentence "S contains a quantifier that does not range over everything".
S (and every utterance of S) is contradictory. Interestingly, it is so even if the quantifier in S really does not range over everything. From which it follows that either there are true contradictions, or "S contains a quantifier that does not range over everything" is not true iff S contains a quantifier that does not range over everything.
Okay, as promised here comes the third and last part of my little series on Rieger's paradox. I will first describe a general version of Russell's paradox, of which Rieger's is a special case. Then I'll discuss whether Frege is already prey to the paradox by his admission of too many concepts. Whether he is will depend on whether it makes sense to say that there are entities which are not first-order entities. I'm sorry that there is probably nothing new in all this.
First, the general version of Russell's paradox.
Yesterday, I argued that Frege can escape Rieger's Paradox if it is allowed that the thought that Fb might equal the thought that Gb (briefly, [Fb]=[Gb]) even if F and G are not coextensive.
In particular, to escape the paradox there has to be a concept F, such that [Fb]=[Ob] even though O([Ob]) and 
F([Ob]). O, recall, is defined thus:
O(x) iff 
F(x=[Fb]
Fx)
I did not say how this F might look like. Here is a good candidate:
In the October issue of Analysis, Adam Rieger presents the following paradox in Frege's ontology.
For any object b and first order concept F, there is the thought [Fb] that b falls under F. Let Con and Obj be functions that yield the (referents of the) constituents of such thoughts: Con([Fb])=F, and Obj([Fb])=b. We stipulate moreover that 'b' shall denote the mountain Ben Lomond, and define O ("ordinary") as follows: