Re: The fallacy of strengthened liar's paradox.


Marshall wrote:

On Dec 29, 3:40 pm, djr...@xxxxxxxxxx wrote:

The statement "1/0" is not syntactically correct.

An intriguing statement! Can you supply further justification?
It appears syntactically correct to me, but perhaps I am not
clear what you mean by that.

Further thoughts:

What is the meaning of the expression

1/(x-1)

When x=2? When x=1? It is the same syntax in both cases,
isn't it?

The expressions [2] and [1],

1/(x-1) [2] (note: x = 2)
1/(x-1) [1] (note: x = 1)

have exactly the same characters in the same order, since
the additional information noted is not part of the
expressions. I am tempted to say that they therefore
have the same syntax.

However, in English, the same expression, character
for character, placed in different contexts can have
clearly different syntaxes. A good example that comes
to mind (Groucho Marx:) "Time flies like an arrow;
fruit flies like a banana." Certainly, the goal
in mathematics is to be completely unambiguous,
very much unlike English, but I can't say, myself,
whether we've arrived at our goal.

In its most common context, the expression "1/(x-1)"
carries an often-tacit assumption that x <> 1.
What does it mean to me to have contradictory
assumptions to deal with? Not much, unfortunately.
If I fall back upon my classroom experience, all I
can remember is that this is a situation to be avoided.

Is there a /syntactic/ difference between having
a contradiction and not having one? I guess I
don't know what is and is not syntax well enough
to be sure. I am willing to be educated on the point.

I think what the poster you're responding to may have meant
that the expression "1/0" can be determined to be
forbidden by purely mechanical means (hence, "syntactic").

Given a definition of syntax that broad, I would stick
my neck out and say that, yes, the syntax of the expression
"1/(x-1)" is different when x = 2 from when x = 1.

Jim Burns
.

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