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Re: Must equality "=" be defined as the identity relation or as "the same"?
Posted:
Apr 27, 2012 9:52 AM
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On Thu, Apr 26, 2012 at 2:02 PM, kirby urner <kirby.urner@gmail.com> wrote:
> On Thu, Apr 26, 2012 at 9:54 AM, Paul Tanner <upprho@gmail.com> wrote:
>> No.
>>
>> Again: Theorems U, V, W, and X show that no matter how we interpret
>> "=", there are some statements about "=" that we cannot make without
>> contradiction.
>>
>
>
>> One such statement about "=" is any claim that we can view "=" as
>> R&S&T&~I while also claiming that R&S&T&A -> I. This conjunction of
>> claims is always false - it contradicts the above theorems.
>>
>> As I just said in a prior post, that there are some statements about
>> "=" that we cannot make without contradiction no matter how we
>> interpret "=" ultimately rests on the nature of the definition of the
>> antisymmetric property with respect to binary relations - it includes
>> "=" as part of the statement of its definition: The statement of this
>> property is the implication (x#y & y#x) -> x=y.
>
> If "=" is a naming device, used to assign names to objects, then all
> the above theorems are blown out of the water, thanks to disagreement
> on definitions.
>
> Of course no one worries about this, as we're talking about different
> formalisms, different meme pools.
>
> On the other hand, on the ground, we *do* have to worry when "=" is
> blithely assumed to be some kind of boolean operator resulting in True
> of False.
>
Truth-functionality is the key.
The Theorems U, V, W, and X that I stated and proved in the original post of this thread hold for any interpretation of the symbols "#" and "=" such that they are truth functional with possible values T and F. In fact, these theorems can be generalized with this fact in mind. Simply let the symbols "#" and "=" be any truth-functional relations (binary or otherwise) with possible values T and F. We can go further in this generalization of these theorems: We can let R, S, and T in these theorems and proofs be any truth-functional propositions whatsoever with possible values T and F, and in fact, we can simply replace the conjunction R&S&T in these theorems and proofs with any truth-functional proposition P with possible values T and F. Statement A will remain as "(x#y & y#x) -> x=y", where x and y are any variables whatsoever (note that x and y do not need to be truth-functional).
With all this in mind, even with just the understanding that the symbols "#" and "=" are such that they are truth functional with possible values T and F, we have that Theorems U, V, W, and X and then of course any generalization such as the one just given are general theorems indeed, covering essentially all of mathematics throughout all of history. Think about it: Cut out completely all of mathematics throughout all of history involving any claim or sub-claim whatsoever that such and such is true or false. What would then be left? Essentially nothing.
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