Brian,
Not a bad argument, but here are a few possible objections I thought of.
--Is D *really* an object of thought? Some would say no, based on the
paradoxical conclusions that can be reached by considering D. "How is
it an object of thought if we cannot consistently reason about it?"
(Treating "not true and not false" as a contradiction)
--Since the argument is questioning one of the basic properties of
logic, ie, 2 truth values, it seems impertinent to blindly assume that
other typical properties of logic are as usual; in particular, the
arguments for "D contains D can't be true" and "D contains D can't be
false" appear to rely on classical logic.
Are you inclined to add just a third value, or more? What sort of
third value are you inclined towards?
On Mon, Jan 12, 2009 at 1:03 PM, Brian Tenneson <tennesb.domain.name.hidden> wrote:
>
> The universe is not just black and white...
> Or another way to state that is that two truth values (true and false)
> are insufficient to describe all propositions.
>
> I propose the following:
> If the universe exists and if for all things X and Y, the utterance "X
> contains Y" is "proposition," then the universe must "operate" with
> more than the two usual truth values, true and false.
>
> Consequently, since the universe does exist, and if we assume that "X
> contains Y" is decidable, then at least three truth values are
> necessary to describe the state of all "propositions."
>
> First the definitions and then the argument for the above proposition.
> IMHO, the only room you might have for disagreement is in the
> definitions, as the argument is valid.
>
> Definitions
> Proposition
> Define a proposition to be something that can be decided, given enough
> resources (such as computational power), and mapped to a single truth
> value. An example of a proposition is "I perceive the sky to be blue"
> as that is decidedly true (in good weather). Another is "I perceive
> the sky to be green," which is decidedly false. A proposition is a
> statement that is decidable, meaning there is a "best" truth value to
> assign to that statement. If there are only two truth values then an
> example of a non-proposition is "this statement is false," the liar's
> paradox. Later, we will see that "If the universe exists then it
> operates on more than two truth values" is a proposition as well
> because it will be decidedly true.
>
> TV
> Let TV be a set, to be determined, consisting of truth values,
> possibly such as TV = {true, false}, that represents all truth values -
> sufficient- to allow for -all- propositions to have a unique,
> assignable truth value, i.e., sufficient to decide all propositions.
>
> The universe
> For the purposes of this argument, the universe is the totality of all
> that exists. Remark: what exists isn't completely clear and people
> disagree on what exists; some, for example, believe that nothing
> exists save themselves; this is called solipsism. Nevertheless, the
> definition of universe stands as whatever that totality of all that
> exists is.
>
> Thing
> X is a thing if, and only if, X is or can be an object of thought.
> (Slightly modified version of definition 3 from dictionary.com.)
>
> Containment
> One thing contains another thing (where the 'another thing' is allowed
> to be the first thing) if and only if the first thing has all of the
> second thing's contents or constituent parts. In other words, all
> content and/or constituent parts within the second thing is also
> content of the first thing. (Slightly modified version of definition 3
> from dictionary.com.) Examples: the solar system contains the planet
> earth and water molecules contain hydrogen. The primary example is the
> universe: the universe contains -all- things.
>
>
>
>
> Argument
> Overview
> The argument is an augmented form of Russell's theorem, sometimes
> referred to as Russell's paradox, which proves that in Zermelo Frankel
> set theory there is no set which contains every other set. The twist
> is, this time, when talking about the universe, we know it exists.
> However, we'll use the fact that a particular statement is a
> proposition except it is neither true nor false. Recall that a
> proposition must have a decidable truth value in order to be a
> proposition; so since this statement is a proposition, there must be
> at least one extra truth value that this proposition is most
> accurately mapped to in TV.
>
>
> The case for the universe operating on more than two truth values.
>
> Premise 1
> The universe as defined exists.
>
> Premise 2
> For all things X and Y, the utterance "X contains Y" is a proposition.
> (Intuitively, I think the proposition "X contains Y" is 'usually'
> false.)
>
>
> Suppose the universe exists and for all things X and Y, the utterance
> "X contains Y" is a proposition. Consider the thing that contains all
> things that don't contain themselves. Let's denote this thing by the
> letter D. D is a thing because it is now an object of thought. The
> universe, which contains all things, and itself exists by assumption,
> contains D in particular as D is a thing.
>
> Now consider the utterance "D contains D." By assumption, "D contains
> D" is a proposition. (X and Y are both D in this particular case.)
>
> "D contains D" can't be true
> Suppose that "D contains D" is a true proposition. Then, by the
> definition of D, "D does not contain D". Therefore, "D contains D" is
> false. Since "D contains D" can't be both true and false, our original
> assumption that "D contains D" is a true proposition is incorrect.
> Consequently, "D contains D" is not a true proposition.
>
> "D contains D" can't be false
> A similar argument shows that "D contains D" can't be false. If we
> suppose "D contains D" is false, then "D does does not contain D" is
> true. However, by the definition of D, "D contains D" is then true
> since D contains all things that don't contain themselves. This
> contradiction implies that "D contains D" is not false.
>
> Conclusion
> So we've established that "D contains D" is neither true nor false. By
> premise 2, "D contains D" is a proposition; by the definition of
> proposition, "D contains D" must have a decidable truth value. Since
> this truth value is neither true nor false, there must be a third
> truth value which "D contains D" is assigned to. This completes the
> argument.
>
>
>
>
>
> BTW, I apologize if this gets posted twice; I sent it with Thunderbird
> 40 minutes ago and it's still not appearing so I'm "manually" posting
> it.
> >
>
--
Abram Demski
Public address: abram-demski.domain.name.hidden
Public archive: http://groups.google.com/group/abram-demski
Private address: abramdemski.domain.name.hidden
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Received on Wed Jan 21 2009 - 00:27:21 PST