Re: Binary Logic is Insufficient
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 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
Binary Logic is Insufficient
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 contai
Binary logic is insufficient
*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 con
