Is the Unexpected Hanging Paradox a Real Challenge for Philosophy?

[apeiron]

Also, I've never seen a paradox derived simply from assuming the law of the excluded middle, can you please give an example?

I did in my initial post. But a trivial standard example is...

An example might be to affirm or deny the statement "John is in the room" when John is standing precisely halfway through the doorway. It is reasonable (by human thinking) to both affirm and deny it ("well, he is, but he isn't"), and it is also reasonable to say that he is neither ("he's halfway in the room, which is neither in nor out"), despite the fact that the statement is to be exclusively proven or disproven.
http://en.wikipedia.org/wiki/Paradox#Logical_paradox

 

[Jamma]

But again, that doesn't seem to be a problem with the law of the excluded middle to me, more a problem with the definition of "being in a room". If we defined "being in a room" to mean you have to totally within the room, then no paradox occurs, the same if only a bit of you has to be in the room. There is no precise statement of what "being in the room" is, and this is emphasised by the statement adding in "by human thinking".

 

[Pythagorean]

in physics boundary problems, we define three different cases. Inside the room, outside the room, and the transient case. This can be seen in more abstract mathematical concepts too, like convolution.

But it's still important to recognize that while your whole arm is inside the room, your hand can't be outside the room at the same time. In quantum, we can talk about a single particle being completely inside and outside the room at the same time, but that's a completely different story.

 

[disregardthat]

The point is not that we haven't defined what it means to physically be in the room, but that we haven't decided that "being in the room" is a logical proposition at all. If you can say p and (not p) you are not dealing with standard logic, but a different logic altogether which will have a different and non-overlapping use, though potentially useful.

Logic is not about physical states of affairs, the proposition "being in the room" does not logically refer to the physical state of "being in the room" (in whatever way it is defined). But we can treat it logically if we decide that it shall be a logical proposition, which for any useful purpose ought to have a well-defined physical counterpart. It is important not to confuse any statement about physical states of affairs with the logical counterpart which may or may not have any use depending on how we treat the statement as a statement of physics. In the case of "being in the room"-if we are inclined to say "both"-we are simultaneously deciding that we are not stating a logical proposition (in standard logic).

 

[apeiron]

But again, that doesn't seem to be a problem with the law of the excluded middle to me, more a problem with the definition of "being in a room". If we defined "being in a room" to mean you have to totally within the room, then no paradox occurs, the same if only a bit of you has to be in the room. There is no precise statement of what "being in the room" is, and this is emphasised by the statement adding in "by human thinking".

You are making my point that paradox can be avoided if you accept that middles don't come excluded. It is an action that has to be performed. And so paradox arises if you have an axiom system where middles do come ready-excluded.

You say this kind of intelligent softening of the formal logic is no big deal, just a pragmatic exercise. I agree, of course. But then much better is also to put that insight itself on a formal basis as "a logic". Which is what a Peircean vagueness approach would be about.

More recently, we've had fuzzy logic and paraconsistent logic. I actually think Peirce's work remains far more radical. But here is discussion that is more orthodox.

This contribution deals with developments in the history
of philosophy, logic and mathematics before and
when fuzzy logic began. Even though the term
“fuzzy” was introduced by Lotfi Zadeh in 1964/65 it
should be noted that older concepts of “vagueness”
and “haziness” have been discussed in philosophy,
logic, mathematics, applied sciences, and medicine.
This paper delineates some specific paths through the
history of the use of these “loose concepts” in science.
The theory of fuzzy sets is a proper framework for
“loose concepts”, that connote the nonexistence of
sharp boundaries.

http://www-bisc.cs.berkeley.edu/BISCSE2005/Abstracts_Proceeding/Saturday/SA3/Rudi_Seising.pdf

Logic is not about physical states of affairs, the proposition "being in the room" does not logically refer to the physical state of "being in the room" (in whatever way it is defined).

No quarrels with your position here. Clearly, what I am interested in are the non-standard approaches that would be formal solutions to the familiar formal paradoxes. And this interest seems justified by QM as well as the developmental perspective that underpins biology and the modelling of life/mind.

 

[fuzzyfelt]

fuzziness might be applied to the truthfulness of the original statements.

 

[Hells]

I don't accept this paradox..

" He begins by concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on Friday"

" He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night"

Friday has been eliminated on a condition, namely that he hasn't been hung before friday, which is a paradoxical statement