Re: Is Truth Mysterious?





David Marcus wrote:

herbzet wrote:

I actually tend to agree with what you've been saying in this
thread, but I don't like to think about the Liar too much
because it just makes me want to hide under the covers and
weep quietly.

Did you read

http://www.davidmarcus.com/SemanticParadoxesAsEquations.pdf

?

Yeah, it seems consonant with what LauLuna is saying. If the liar
has no solution, i.e. cannot be consistently assigned a truth value,
then it would seem to be pseudo-proposition, a non-proposition,
however much it is a grammatically well-formed sentence.

I think "This sentence is true" is also a pseudo-proposition,
being totally void of meaning.

It seems there are a lot of grammatically well-formed sentences
that are bearers of neither meaning nor truth-value:

"A negative square root of Tuesday blushes isomorphically."

Pseudo-propositions have no more meaning than a mirage in
the desert has water. They're an illusion.

===============================================
For a different resolution:

"We now consider the equation

x^2 + 1 = 0.

Transposing, we have

x^2 = -1

and dividing both sides by x gives

x = -1/x.

We can see that this (like the analogious statement in logic)
is self-referential: the root-value of x that we seek must be
put back into the expression from which we seek it.

Mere inspection shows us that x must be a form of unity, or the
equation would not balance numerically. We have assumed only
two forms of unity, +1 and -1, so we may now try them each in
turn. Set x = +1. This gives

+1 = -1/+1 = -1

which is clearly paradoxical. So set x = -1. This time we have

-1 = -1/-1 = +1

and it is equally paradoxical.

Of course, as everybody knows, the paradox in this case is
resolved by introducing a fourth class of number, called
_imaginary_, so we can say the roots of the equation above
are +i and -i, where i is a new kind of unity that consists
of a square root of minus one.

What we do in Chapter 11 is extend the concept to Boolean
algebras, which means that a valid argument may contain
not just three classses of statement, but four: true, false,
meaningless, and imaginary."

"Laws of Form" by G. Spencer Brown, preface to the first
American edition [1972].

In Chapter 11 he goes on to consider Boolean expressions
"of degree < 1" which may be considered as infinitely
extended expressions. Some of those expression can be consistently
assigned the values "true" or "false", others must be
considered to have imaginary roots. This does not prevent
us from manipulating and equating these sorts of expressions.

I'm going to go hide under the covers for a while now.

--
hz
.



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