How Quaint the Ways of Paradox

How quaint the ways of Paradox!

At common sense she gaily mocks!

W. S. Gilbert, The Pirates of Penzance

Tuesday, June 23, 2009

Contradiction vs. Paradox

A contradiction cannot possibly be true

A paradox cannot possibly be true OR false

Contradictions are useful.

Paradoxes are deadly.


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.


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.


The Problem with ParadoxesA 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.



But then you can deduce a contradiction from nothing.P assume~P paradoxP & ~P ~P reductioP paradoxP & ~P



But then you can deduce a contradiction from nothing.P assume~P paradoxP & ~P ~P reductioP paradoxP & ~P This is now a theorem.



But then you can deduce a contradiction from nothing.P assume~P paradoxP & ~P ~P reductioP paradoxP & ~P

And then you can deduce anything from nothing !~Q assumeP & ~P theorem~~Q reductioQ double negation

This is now a theorem.


Which is why logicians and philosophers of language and mathematicians all just HATE paradoxes.

Unfortunately, paradoxes are lurking all over the place.

The Problem with Paradoxes


The Liar (ancient Greece)

The ancestor of all paradoxes.

A Cretan said, all Cretans are liars.

In modern form:

This sentence is false.





In modern form:


Suppose its true. Then what it says is correct, so it must be false.





In modern form:


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.


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.

...


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.


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.)


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 !


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.


George & Bill's Contingent ParadoxA 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.


Meanwhile, back at the Laboratory...


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.


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).

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.

IKL


Captain's Notice.

I now have to show you some actual logic.

Please fasten your safety belts in case of unexpected turbulence.

Thank You.


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.

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)))

CL/IKL notation







i.e. x Loves MyBaby

CL/IKL notation







i.e. x Loves MyBaby

i.e. y = Me

CL/IKL notation


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)) )



IKL adds that:

(that <sentence> )



sentence



IKL adds that:

(that <sentence> )



sentence proposition name



IKL adds that:

(that <sentence> )



sentence proposition name

So IKL can say wicked things such as

(Believes John (that(forall((x Irishman))(< (IQ x) 45)) ))


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.

So?






<sentence>

is true.

So?

For any IKL sentence.






<sentence>

is true.

So?

Hmmm. But didn't Tarski prove that was impossible??

For any IKL sentence.


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.

Remember...


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.

Did we just top Tarski??




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: 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...


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))) )

The Liar in IKL




The Liar in IKL

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.




The Liar in IKL

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.)



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.

The Liar 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> ))

Russell 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.

Kripke in IKL


George & Bill's Contingent ParadoxA 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)

Remember...


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...


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.


How Quaint the Ways of Paradox

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