1. How quaint the ways of Paradox!
At common sense she gaily mocks!
W. S. Gilbert, The Pirates of Penzance
Tuesday, June 23, 2009
2. Contradiction vs. Paradox
A contradiction cannot possibly be true
A paradox cannot possibly be true OR false
Contradictions are useful.
Paradoxes are deadly.
Tuesday, June 23, 2009
3. Contradictions are Useful
If you can derive a contradiction from an assumption, you know the
assumption must be false. Very handy way of reasoning, widely used
(modus tollens, reductio).
P assume P for the moment (temporarily)...
... ...do some reasoning, and derive...
Q & ~Q ...a contradiction!
~P So, the assumption must have been wrong.
Tuesday, June 23, 2009
4. Contradictions are Useful
If you can derive a contradiction from an assumption, you know the
assumption must be false. Very handy way of reasoning, widely used
(modus tollens, reductio).
P assume P for the moment (temporarily)...
... ...do some reasoning, and derive...
Q & ~Q ...a contradiction!
~P So, the assumption must have been wrong.
This final conclusion doesn't depend on the assumption.
Tuesday, June 23, 2009
5. The Problem with Paradoxes
A paradox is a P from which you can deduce ~P, and from ~P you can
deduce P.
But then you can deduce a contradiction from nothing.
Tuesday, June 23, 2009
6. The Problem with Paradoxes
A paradox is a P from which you can deduce ~P, and from ~P you can
deduce P.
But then you can deduce a contradiction from nothing.
P assume
~P paradox
P & ~P
~P reductio
P paradox
P & ~P
Tuesday, June 23, 2009
7. The Problem with Paradoxes
A paradox is a P from which you can deduce ~P, and from ~P you can
deduce P.
But then you can deduce a contradiction from nothing.
P assume
~P paradox
P & ~P
~P reductio
P paradox
P & ~P This is now a theorem.
Tuesday, June 23, 2009
8. The Problem with Paradoxes
A paradox is a P from which you can deduce ~P, and from ~P you can
deduce P.
But then you can deduce a contradiction from nothing.
P assume
~P paradox
P & ~P
~P reductio
P paradox
P & ~P This is now a theorem.
And then you can deduce anything from nothing !
~Q assume
P & ~P theorem
~~Q reductio
Q double negation
Tuesday, June 23, 2009
9. The Problem with Paradoxes
Which is why logicians and philosophers of language
and mathematicians all just HATE paradoxes.
Unfortunately, paradoxes are lurking all over the
place.
Tuesday, June 23, 2009
10. The Liar (ancient Greece)
The ancestor of all paradoxes.
A Cretan said, all Cretans are liars.
In modern form:
This sentence is false.
Tuesday, June 23, 2009
11. The Liar (ancient Greece)
The ancestor of all paradoxes.
A Cretan said, all Cretans are liars.
In modern form:
This sentence is false.
Suppose its true. Then what it says is correct, so it must be false.
Tuesday, June 23, 2009
12. The Liar (ancient Greece)
The ancestor of all paradoxes.
A Cretan said, all Cretans are liars.
In modern form:
This sentence is false.
Suppose its true. Then what it says is correct, so it must be false.
Suppose its false. Well, that's what it says, so then it is true.
Tuesday, June 23, 2009
13. Liar Variations (Avoiding 'this')
A: The sentence labeled 'A' is false
The next sentence is true.
The previous sentence is false.
List all the sentences:
1: Aardvarks are abominable.
...
324567: The three hundred and twenty-four thousand, five hundred
and sixty-seventh sentence in the list is false.
...
Tuesday, June 23, 2009
14. Russell's Paradox (1901)
Frege's neat idea: talk about sets of things. Intuitively, a set is defined by a
property: the set of all camels, the set of all numbers greater than 27,
etc.. It turns out you can reduce all of mathematics to be about sets,
with enough ingenuity.
This was Frege's life's work, his masterpiece.
Bertrand Russell said: consider the set of all sets that are not
members of themselves. Is it a member of itself ?
Suppose it is: then it has to satisfy the defining property, so it isn't.
Suppose it isn't. Then it does satisfy the defining property, so it must be in
the set.
Frege was seriously bummed.
Much work since has been devoted to finding a precise theory of sets that
avoids this paradox. It can be done, but there is no single obvious best
way.
Tuesday, June 23, 2009
15. Avoiding Paradoxes
(A) Make the paradoxical forms illegal. (This usually
results in a layered hierarchy; things, sets of things, sets of sets of
things, etc.;1-sentences, 2-sentences about 1-sentences, 3-sentences
about 2-sentences, etc.)
(B) Have an explicit axiomatic theory of what exists,
and stay inside it. (Modern set theories: axiom of choice,
axiom of foundation, ... )
(C) Use 3-valued logic? (Naaah.)
Tuesday, June 23, 2009
16. Gödel's Incompleteness Theorem
(1931)
1: Aardvarks are abominable.
...
324567: The three hundred and twenty-four thousand, five hundred
and sixty-seventh sentence in the list is unprovable.
...
This isn't paradoxical any more, but it is unprovable; which is
why it is true.
Gödel showed that any formal system for arithmetic has such a
sentence, which is unprovable (in that system) but is also true. So the
system doesn't prove all truths. In other words, the system is incomplete.
So arithmetic cannot be formalized !
Tuesday, June 23, 2009
17. Tarski's Truth Theorem (1933)
Can a formal system contain an accurate
(meta) theory of its own truth?
IsTrue(<name of sentence>)
<=>
<sentence>
(Convention T)
Theorem: No.
Tuesday, June 23, 2009
18. George & Bill's Contingent Paradox
A play in one act, by Saul Kripke, 1972
Two men are standing together. George has his arm on Bill's shoulder.
George: (to audience) This is Bill. He has never spoken. One
day he will speak, and lie.
Bill: That's all true.
As George turns to Bill in astonishment, Bill clutches his chest and collapses.
George kneels. He listens to Bill's heart, then looks up in horror.
George: (to audience) Alas, he will never speak again...
(Curtain)
If Bill has in fact spoken before, no problem. (George and Bill were
both wrong.) If not, it's a paradox.
Tuesday, June 23, 2009
20. The IKRIS Project (2005-6)
The IKRIS project's aims were to allow a variety
of logic-based systems to interoperate, sending
information to one another.
They all used different logics. Not just different
notations, but really different logics.
So we had to invent a 'universal' logic they could
all be mapped into. We called it IKL.
Tuesday, June 23, 2009
21. IKL
IKL is Common Logic (CL, an ISO standard version
of first-order logic which a bunch of us invented) plus
the idea of proposition names.
Which turns out to be a very powerful idea (more
than we knew).
Tuesday, June 23, 2009
22. IKL
The idea is that logical sentences indicate propositions,
which are real things that can be described in the
logic. So you can say things like
<sentence> in logic X means <proposition>
Harry believes <proposition>
<proposition> is true in <context>
all in the same logic.
In IKL, it's the propositions (rather than the
sentences) which are true or false. Which turns out to
be the key.
Tuesday, June 23, 2009
23. Captain's Notice.
I now have to show you some actual logic.
Please fasten your safety belts in case of
unexpected turbulence.
Thank You.
Tuesday, June 23, 2009
24. CL/IKL notation
CL/IKL uses a Lisp-ish notation where operator
names are always written first and enclosed in
brackets. So for example
3=1+2
in CL/IKL notation looks like this:
(= 3 (+ 1 2))
This also applies to things like and and or and
forall.
Tuesday, June 23, 2009
25. CL/IKL notation
Example (Spencer Williams, 1924).
Everybody loves my baby
(forall ((x Person))(Loves x MyBaby))
my baby don't love nobody but me
(forall ((y Person))(if (Loves MyBaby y)(= y Me)))
Tuesday, June 23, 2009
26. CL/IKL notation
i.e.
Example (Spencer Williams, 1924). x Loves MyBaby
Everybody loves my baby
(forall ((x Person))(Loves x MyBaby))
my baby don't love nobody but me
(forall ((y Person))(if (Loves MyBaby y)(= y Me)))
Tuesday, June 23, 2009
27. CL/IKL notation
i.e.
Example (Spencer Williams, 1924). x Loves MyBaby
Everybody loves my baby
(forall ((x Person))(Loves x MyBaby))
my baby don't love nobody but me
(forall ((y Person))(if (Loves MyBaby y)(= y Me)))
i.e.
y = Me
Tuesday, June 23, 2009
28. that is a proposition name
IKL adds that:
(that <sentence> )
which makes any sentence into the name of a proposition.
(that (forall ((x Person))(Loves x MyBaby)) )
Tuesday, June 23, 2009
29. that is a proposition name
IKL adds that:
(that <sentence> )
which makes any sentence into the name of a proposition.
(that (forall ((x Person))(Loves x MyBaby)) )
sentence
Tuesday, June 23, 2009
30. that is a proposition name
IKL adds that:
(that <sentence> )
which makes any sentence into the name of a proposition.
(that (forall ((x Person))(Loves x MyBaby)) )
sentence proposition name
Tuesday, June 23, 2009
31. that is a proposition name
IKL adds that:
(that <sentence> )
which makes any sentence into the name of a proposition.
(that (forall ((x Person))(Loves x MyBaby)) )
sentence proposition name
So IKL can say wicked things such as
(Believes John
(that(forall((x Irishman))(< (IQ x) 45)) )
)
Tuesday, June 23, 2009
32. So?
IKL can talk about its own propositions. So?
So, it turns out when you crank the semantic
machinery, that IKL can define its own truth predicate:
(isTrue (that <sentence>))
is true exactly when
<sentence>
is true.
Tuesday, June 23, 2009
33. So?
IKL can talk about its own propositions. So?
So, it turns out when you crank the semantic
machinery, that IKL can define its own truth predicate:
(isTrue (that <sentence>))
is true exactly when
<sentence>
is true. For any IKL sentence.
Tuesday, June 23, 2009
34. So?
IKL can talk about its own propositions. So?
So, it turns out when you crank the semantic
machinery, that IKL can define its own truth predicate:
(isTrue (that <sentence>))
is true exactly when
<sentence>
is true. For any IKL sentence.
Hmmm. But didn't Tarski prove that was impossible??
Tuesday, June 23, 2009
35. Remember...
Tarski's Truth Theorem (1933)
Can a formal system contain an accurate
(meta) theory of its own truth?
IsTrue(<name of sentence>)
<=>
<sentence>
(Convention T)
Theorem: No.
Tuesday, June 23, 2009
36. Did we just top Tarski??
Q: But didn't Tarski prove that was impossible??
A: Actually, not quite. What Tarski proved is that you
can't do this for sentences. IKL does it for propositions.
Tuesday, June 23, 2009
37. Did we just top Tarski??
Q: But didn't Tarski prove that was impossible??
A: Actually, not quite. What Tarski proved is that you
can't do this for sentences. IKL does it for propositions.
Q: But there is a proposition name corresponding to
every sentence, so what's the difference?
A: The key difference is that isTrue applies to
proposition names.
Tuesday, June 23, 2009
38. Did we just top Tarski??
Q: But didn't Tarski prove that was impossible??
A: Actually, not quite. What Tarski proved is that you
can't do this for sentences. IKL does it for propositions.
Q: But there is a proposition name corresponding to
every sentence, so what's the difference?
A: The key difference is that isTrue applies to
proposition names.
Q: What difference does that make?
A: Glad you asked. Lets look at an example...
Tuesday, June 23, 2009
39. The Liar in IKL
The liar in IKL would be a proposition which asserts
that it, itself, is false. That would a p which satisfied
this equation:
(= p (that (not (isTrue p))) )
Tuesday, June 23, 2009
40. The Liar in IKL
The liar in IKL would be a proposition which asserts
that it, itself, is false. That would a p which satisfied
this equation:
(= p (that (not (isTrue p))) )
So, is this an IKL paradox? No, its just a harmless
contradiction in IKL because there is no such p in the
universe. This is an equation with no solution.
Tuesday, June 23, 2009
41. The Liar in IKL
The liar in IKL would be a proposition which asserts
that it, itself, is false. That would a p which satisfied
this equation:
(= p (that (not (isTrue p))) )
So, is this an IKL paradox? No, its just a harmless
contradiction in IKL because there is no such p in the
universe. This is an equation with no solution.
Notice, you don't get this option with sentences.
The Liar sentence certainly exists (you were looking
at it a while ago.)
Tuesday, June 23, 2009
42. The Liar in IKL
(= p (that (not (isTrue p))) )
The key is that this looks like a definition (of a
proposition p):
(= p (that <sentence which defines p>) )
but it can't be, because it's a contradiction.
If you insist that it really is a definition, you will get
the paradox back. But IKL doesn't have strict
definitions, it only has descriptions with the form of
definitions. A contradictory description doesn't
describe anything.
Tuesday, June 23, 2009
43. Russell in IKL
Russell's paradox works similarly (this doesn't even
need the proposition names, you can do it in ordinary
logic):
(forall (x)(iff (R x)(not (x x)) ))
is a contradiction, which also has a definitional form:
(forall (x)(iff (R x)
<expression defining R>
))
Tuesday, June 23, 2009
44. Kripke in IKL
The 'pragmatic paradoxes' (eg George & Bill) also
appear in IKL but are more complicated. For
example:
(forall (x)(iff
(Bill x)
(= x (that (forall (y)(if (Bill y)
(not (isTrue y))
))
))
))
This is an IKL contradiction too: there isn't any such
Bill property, it's impossible.
Tuesday, June 23, 2009
45. Remember...
George & Bill's Contingent Paradox
A play in one act, by Saul Kripke, 1972
Two men are standing together. George has his arm on Bill's shoulder.
George: (to audience) This is Bill. He has never spoken. One
day he will speak, and lie.
Bill: That's all true.
As George turns to Bill in astonishment, Bill clutches his chest and collapses.
George kneels. He listens to Bill's heart, then looks up in horror.
George: (to audience) Alas, he will never speak again...
(Curtain)
Tuesday, June 23, 2009
46. An interview with IKL
Interviewer: This short Kripke play that everyone is talking about, George & Bill, I gather
you have strong views about it?
IKL: It is complete nonsense, or the work of the devil.
Int: Strong views indeed! Can you say why?
IKL: Because the words spoken by the characters don't mean what they seem to mean.
Int: Yet other critics have applauded the stark poetry and the minimalist clarity of the
language, and compared it to Beckett...
IKL: It seems to be meaningful, and each piece taken alone is, but when the entire play is
considered, the meaning evaporates. In fact, I have shown quite rigorously that it is
impossible to interpret it with a coherent meaning.
Int: I see. But you will agree that the words as written do give the impression of
something which makes sense, and indeed seem to pose an ingenious puzzle?
IKL: Oh, indeed. But that puzzle has no solution, you see. That is my point. The play has
been constructed so that the only words that Bill utters cannot possibly have the meaning that they
seem to have. That is why it may be the work of the devil, or perhaps of an ancient secret
society which lurks in the bosom of the Catholic church ...
Int. Yes, I see, I see. Aha. Well thank you very much for sharing your views with us. And
now, back to our main program...
Tuesday, June 23, 2009
47. So???
All great fun, but does any of this really matter to anyone in the
real world?
Yes, because the Intelligence community (for example) need to
have richly expressive languages containing their own
metalanguage (to express security levels, track provenances, etc.)
and they need to able to put together information from many
sources. It is easy for paradoxes to arise in such settings.
The traditional solution is to restrict the logic in some way to be
safe. But restricted notations are a major source of interoperation
problems and complexities.
So although we started today with the Pirates of Penzance, this
work actually started closer to Al-Qaeda.
Tuesday, June 23, 2009