[FOM] The liar paradox

[Annatala Wolf]

I went ahead and read the first chapter of your book as well, and I'm
increasingly convinced the problem you are attempting to describe is wholly
linguistic in nature. I've thought a little about how to express this in a
way that will make intuitive sense.

First and foremost, sentences aren't true or false. Period. "This sentence
is true." is not true or false. It is a string of symbols. Words are not
their meanings. The word "true" is neither true or false. It is a word that
represents the idea of truth, and those two things are so different that
they are not directly comparable.

One problem here is that English, as with all natural languages, is
inherently imprecise. That is one of the factors that makes natural
languages powerful: they allow us to say things very quickly because they
permit gross ambiguity. This ambiguity can be remarkably easy to miss when
we try to reason intuitively. The reason that the more complex formulations
of the liar's paradox are not true paradoxes is that they rely on meanings
of sentences where the intended meanings of the sentences themselves are
ill-defined.

Humans are hardwired to reason in large steps with ambiguity in ways where
we unconsciously lift a word from one meaning to another without realizing
it, and frequently consider two meanings of the same word in concert as
though they were one. Stating "This sentence is false." is no more
paradoxical than stating "Paradoxes exist." It is not true that "This
sentence is false." has no meaning; rather, the meaning of that sentence is
an abstract concept which is not directly comparable to the meanings of the
words "true" or "false". This is a lot like a type mismatch error in
programming. "This sentence is false." feels intuitively wrong to us
because our brains naturally map "false" in that sentence to two different
and incompatible meanings.

Now, sentences can indeed express a meaning that may be considered true or
false when analyzed with the context of a system of logical deduction and a
universe (etc.). The meaning of certain expressions like "x = x" should be
true in any universe that has something in it, and we tend to think of the
meanings of sentences which express tautologies like "x = x" as
"universally true". But if you try to reformulate the liar's paradox refer
to the meaning itself rather than confusing the message with the medium,
you get something like, "This sentence's meaning will evaluate to false
when using deductive system X.", which will seem quite familiar. Now it
should be easier to spot the flaw.

And just to be clear, I'm not stating that objective truth doesn't exist.
I'm stating that the act of discussing paradoxes is not in itself
paradoxical. Empirically speaking this should be clear, because I'm doing
it right now. The word "paradox" is not a paradox, and (subtly) neither is
its meaning (which is the concept of paradox, not paradox itself).

On Thu, Feb 16, 2017 at 1:07 PM Annatala Wolf <a.lupine at gmail.com> wrote:

> I've read the paper, and I'm uncertain about the foundations of the
> argument you're making. You argue that there is no axiomatization of truth
> which blocks paradoxes and allows what you refer to as "ordinary
> unparadoxical reasoning". But this seems to suggest you're working from the
> assumption that what you refer to as "ordinary unparadoxical reasoning" is
> itself a consistent deductive system.
>
> Let me put it this way. A (real) paradox is a thing which cannot exist. So
> how can we even converse about such a concept? We can do so because the
> discussion itself is not inconsistent. There's nothing inconsistent at the
> meta-level of thinking or reasoning about, describing, or even formally
> defining a paradox. There is nothing inconsistent about the *sentence*
> "this sentence is false": it exists. I just typed it.
>
> When you use English "intuitively" to reason, it is easy to accidentally
> use a definition of "truth" that changes or lifts in some circumstances and
> is by itself innately inconsistent in others. For one example, in your
> arguments you use "true" both as a symbol, and as the meaning you are
> attempting to assign to that symbol. Those are not remotely the same thing.
> Can you elucidate further about why this approach is logically valid?
>
> On Thu, Feb 16, 2017 at 2:27 AM Nik Weaver <nweaver at math.wustl.edu> wrote:
>
>
> A while ago there was a discussion on this list about the seriousness
> of the liar paradox.  It seemed to be framed as a criticism of my
> book "Truth & Assertibility", which begins with a discussion of this
> topic (the question being, I think, why I would waste time on such a
> triviality).
>
> (Not all reactions to my book were negative.  May I mention that Math
> Reviews hailed my "fresh and productive approach" as "a major
> contribution"?)
>
> This is just to say that I have written a short paper on the subject
> titled "The liar paradox is a real problem".  It is available on the
> arXiv here:
>
> https://arxiv.org/abs/1702.03875
>
> The key takeaway: "whatever simple idea you have for an easy resolution
> of the liar paradox --- we've tried it, and it doesn't work."
>
> Nik Weaver
> Math Dept.
> Washington University
> St. Louis, MO 63130
> nweaver at math.wustl.edu
>
>
> _______________________________________________
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>
> --
>
> /* Annatala Wolf, Lecturer
>  * Department of Computer Science and Engineering
>  * The Ohio State University
>  */
> enum E{A;System s;String t="/* Annatala Wolf, Lecturer%n * Department of
> Computer Science and Engineering%n * The Ohio State University%n */%nenum
> E{A;System s;String
> t=%c%s%1$c;{s.out.printf(t,34,t);s.exit(0);}}";{s.out.printf(t,34,t);s.exit(0);}}
>
-- 

/* Annatala Wolf, Lecturer
 * Department of Computer Science and Engineering
 * The Ohio State University
 */
enum E{A;System s;String t="/* Annatala Wolf, Lecturer%n * Department of
Computer Science and Engineering%n * The Ohio State University%n */%nenum
E{A;System s;String
t=%c%s%1$c;{s.out.printf(t,34,t);s.exit(0);}}";{s.out.printf(t,34,t);s.exit(0);}}
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