Why do some propositions defy logical proof?

[jobsism]

I recently read about the unexpected hanging problem and I was so surprised that logic actually failed in determining the solution!:( Is this just an isolated exception, or are there more paradoxes like this? And more importantly, why does logic fail? Isn't there any way around this? I just can't accept the failure of mathematical logic! :(

 

[wisvuze]

it's not a fault in "logic", it's more to the fault of the use of language and definitions of words.. et c

 

[jobsism]

Can you elaborate, please?

 

[micromass]

You must discriminate between ordinary logic and mathematical logic.

In the unexpected hanging problem, there is a problem of ordinary logic. But there is no problem with mathematical logic. Mathematical logic is built up very rigidly, and you will have to formalize EVERYTHING you do. In particular, in the unexpected hanging problem, there is no way to formalize "unexpected". You cannot give mathematical meaning to it!

You do have an intuition of what unexpected means an you can reason with it, but that is only ordinary logic. In mathematical logic, there is no way to formalize the word "unexpected"

 

[jobsism]

I see...so mathematical logic is different from ordinary logic...and for the former to work, everything concerned has to be strictly defined...thanks, micromass! Whew, that's a big relief...I almost believed that math was imperfect! :D

 

[MathematicalPhysicist]

I see...so mathematical logic is different from ordinary logic...and for the former to work, everything concerned has to be strictly defined...thanks, micromass! Whew, that's a big relief...I almost believed that math was imperfect! :D

Well I don't want to burst your bubble, but there are statements in mathematics which are true and are unprovable, obviously we don't know which ones.

So maths is by far from being perfect.

 

[Leptos]

=
So maths is by far from being perfect.

But what is "perfect" according to mathematical logic?

 

[micromass]

Well I don't want to burst your bubble, but there are statements in mathematics which are true and are unprovable, obviously we don't know which ones.

So maths is by far from being perfect.

There are certainly many problems with mathematics. But these problems are all with which axiom we choose, these problems don't lie in mathematical logic. So mathematical logic is still perfect in my opinion...

 

[SW VandeCarr]

There are certainly many problems with mathematics. But these problems are all with which axiom we choose,

Yes, but that is the issue. We can never be certain that reasoning from a set of axioms, (together with rules of inference and defined and undefined terms), will never lead to a contradiction.

Having said that, mathematics is the closest to a "perfect" system of reasoning we know IMO.

 

[micromass]

Yes, but that is the issue. We can never be certain that reasoning from a set of axioms, (together with rules of inference and defined and undefined terms), will never lead to a contradiction.

Well, certainly that's true for ZF axioms and the like, which cannot be shown to be consistent. But maybe there are other (weaker) axiom systems that can be shown to be consistent. For example, reasoning from the group axioms is perfectly consistent, since we can find models for the group axioms.

I am (by far) no expert on logic, but I do think there are certainly axiom systems out there which can be shown to be consistent...

 

[disregardthat]

There is nothing different between "mathematical logic" and "human logic", except by that we sometimes call the rules of logic as mathematical logic. The rules are nevertheless utilized by means of "human logic". Mathematics is not independent or distant in any way from "human logic", it may simply seem this way when we axiomatize our logical rules. Formal logical axioms are not necessary for doing mathematics, and they are almost never explicitly used in ordinary mathematical discourse.

This paradox is not a case for which "human logic" fails, but rather a case for which the use of the term "knowledge" is ill-used. The argument fails (in my opinion) in the second step. He can conclude logically that he will not be hanged on Friday, as on Thursday he will be certain that this will happen. But why can't he be hanged on Thursday? He can conclude by the same argument that he will know on Wednesday that he will be hanged on Thursday, but this argument assumes the first conclusion: namely that he will be certain on Thursday that he will be hanged Friday. But if he is hanged on Thursday, the premise of the previous argument is violated, namely that he survives Thursday noon. Hence on Monday he cannot use the first inductive step as basis for the second.

What I see from this is that the premise "you will not know the day you will be hanged" is not used correctly in the argument. We switch back and forth from knowing on Thursday to knowing on Monday, and then to knowing on Wednesday and so on.. The thing is simply that he can not use knowledge he will gain on Thursday as a basis for knowing anything on Wednesday.

 

[Hammie]

A few mathematicians tried to reduce math and reality itself to logic. Bertrand Russel comes to mind..

Now they call it naive set theory.

If the barber of Seville is the only one to cut the beards of unshaven men of Seville who do not shave themselves. The barber is clean cut..

 

[pwsnafu]

If the barber of Seville is the only one to cut the beards of unshaven men of Seville who do not shave themselves. The barber is clean cut..

There are too many ways of solving this, I have to wonder why this keeps getting asked

1. The barber is female, or a child
2. There is more than one barber in Seville
3. The barber goes to another town to get shaved
4. The barber has Alopecia areata barbae or even Alopecia areata universalis
5. The barber cuts (i.e. trims) the beards, but shaves his own
6. Everyone in Seville shaves themselves, and the barber does dentistry (yes look it up)
7. The barber does not live in Seville, only visits
8. The barber shaves himself, not as a professional, but as himself (i.e. he does not pay himself to get shaved)
9. The barber shaves people who don't shave themselves and people that do. If you get the barber to shave you on Monday, and shave yourself on Tuesday, you fall in both camps.

 

[disregardthat]

All these solutions ignores the point of the riddle, namely that the assumptions are self-contradictory.

 

[Gingia]

I'm not sure what you mean by 'fail', but mathematics cannot be fully 'logicized' in terms of first-order logic. This means that there are statements of mathematics that cannot be decided in first-order logic; in other words, if you think only within 'logicized' mathematics, you will discover undecidable propositions that are actually decidable outside of your domain of thought, so to speak.

I'd also object slightly to the word 'perfect' being used here. If anything, I'd not say that logic is perfect and mathematics is flawed because of the outcome of Goedel's incompleteness theorem - I'd switch the order around. Of course, this is probably for purely aesthetic reasons (or maybe it's just consolation).

 

[xxxx0xxxx]

If you mean "fail" in the sense of inconsistent (i.e. contradictory) no.

However, some propositions cannot be proved with logic.

for instance, the proposition 'God exists' cannot be proved unless God shows up on your doorstep and strikes you dead.