Re: mathematical language
From: Lester Zick (lesterDELzick_at_worldnet.att.net)
Date: 12/12/04
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Date: Sun, 12 Dec 2004 16:14:03 GMT
On 11 Dec 2004 20:34:30 GMT, harrisq@tcs.inf.tu-dresden.de (Mitch
Harris) in comp.ai.philosophy wrote:
>Lester Zick <lesterDELzick@worldnet.att.net> wrote:
>>On Fri, 10 Dec 2004 13:28:30 +0100, Mitch Harris
>><harrisq@tcs.inf.tu-dresden.de> in comp.ai.philosophy wrote:
>>
>>>Lester Zick wrote:
>>>> On Thu, 09 Dec 2004 17:12:46 +0100, Mitch Harris
>>
>>[. . .]
>>
>>>> Let me see if I can explain why the idea of empirical observations
>>>> works out the way it does. Let's suppose we have something we call an
>>>> empirical observation based on conventional ideas of experience,
>>>> observation, or whatever. Then we have something else called a
>>>> tautology according to conventional ideas on tautologies of the
>>>> general form t:[subject][not subject].
>>>
>>>As before, you lost me right there because your special syntax
>>>"t:[blah1][blah2]" conjures up no connections for me whatsoever. You
>>>might have a lot of thought behind that compressed syntax, but I have
>>>not been able to figure that out from your other words.
>>
>>A tautology (t) is merely a statement which excludes no logical
>>possibility such as "it is a car or not a car"
>
>Hm. Your example corresponds to what I think is a tautology. But "a
>statement which excludes no logical possibility" took me a while to parse,
>and now that I have, I stil find it difficult to handle. TO remove one
>level of negation and still maintain , could it mean:
>"a statement enjoining all logical possibilities" or a statement which is
>true under all logical possibilities? or a statement which is true under
>every logical possibility of its logical variables? (this latter one is
>closer to the wording that I am familiar with)
The definition I used is just the dictionary definition. I don't see
that adding terms like logical variables enhances the meaning, but you
seem to have the basic idea.
>>and so is considered
>>always true when both halves ([subject][not subject]) are taken
>>together.
>
>this is way out of my experience. what are the two halves halves of? what
>is "subject"? Are the tautologies you are talking about sentences in some
>language (propositional calculus) or something else? Do you also have an
>example of something that is not a tautology?
Intended as just plain language. Given a particular proposition, p, is
p true or false? If we take a proposition such as p "car", meaning "it
is a car", is the proposition true or false?If we combine propositions
to form t:[car][not car] we form a tautological proposition which
covers all possibilities. The halves referred to are just parts of the
tautological proposition, positive and negative parts. I call the
positive half or part an empirical observation.
>>(Conversely, circular logic such as "it is a car because it
>>is a car" includes no logical possibility and is considered always
>>false for this reason.)
>
>the commonly accepted way to rewrite this sentence is as "P->P"
>which is (also commonly accepted to be) a tautology.
For what it's worth, I'm trying to avoid specialized notation whether
commonly accepted or not. I realize the notation I've adopted seems
pretty specialized, but it's easy enough to explain in plain language
without resort to commonly accepted formalisms.
As to whether circular logic is tautological, I can only say that a
proposition such as p "car" is non inclusive of anything that I can
tell. People may want to consider such a thing tautological; however,
I think they're just being sloppy. Maybe they define tautologies as
useless and anything which is useless is a tautology and circular
logic is certainly useless. The difference is that circular logic is
never true because it contains no proposition. It just contains an
observation. And there is no way simple observations are always true.
They certainly can be false or we wouldn't be investigating them
logically. In any event tautologies and circular logic represent polar
opposites in terms of meaning apart from being considered useless.
>>>> And, further, according to conventional ideas we find that tautologies
>>>> are considered always true.
>>>
>>>I would not be going out on a limb by saying that the word "tautology"
>>>refers to a thing that -is- true (for a particular definition of
>>>truth, i.e. consideration has nothing to do with it).
>>
>>Well, a statement which excludes no logical possibility would always
>>be true when considered in its entirety because there can be no true
>>alternatives.
>
>That just doesn't follow.
Well, if a statement or proposition includes all possibilities and
excludes no possibilities it's difficult to see how it could be false.
>...
>>>> And this in turn makes every positive
>>>> part of any tautology empirical whatever its source.
>>>
>>>And here I'm lost. what does anything that came before have to do with
>>>"positive" what does positive mean for you? For all the reasonable
>>>meanings I can imagine for that word, none of them bridge the gap.
>>
>>Well, here, admittedly I'm just retrofitting the idea of empirical to
>>the positive half of a tautology because it fits very conveniently.
>>Positive just means not negative or not the not half of a tautology.
>>
>>What I'm saying is that here we have this thing called a tautology
>>that is always true.
>
>Yes.
>
>>And we have observations called empirical which
>>conform exactly to the non negative part of tautologies
>
>What? What is the nonnegative part of a tautology? how does that conform
>to being empirical observations (as opposed to nonempirical observations?)
I'm not sure what the objection is here. For a given tautology of the
form t:[subject][not subject] there is a negative part [not subject]
and a non negative or positive part [subject]. I consider the non
negative or positive part to be the same as what we call an empirical
observation and the negative part to be a logical observation.
(Technically there is also a third part to particular tautologies
consisting of a self contradictory part as well, t:[subject][not
subject][subject not subject]. But I have omitted consideration of it
because this third part is always false.)
>...
>
>>>2) a community has already explored there, found what works well (for
>>>lots of people in the community) and named things for themselves, and
>>>since they were there first, they get dibs.
>>
>...
>>The math community can have any dibs they want as
>>long as they get it right, and when they don't definitions like any
>>other analytical situation need to be corrected. I don't see working
>>with inexact or incorrect definitions as of any merit regardles of
>>pedigree.
>
>Sure. You just happen to be in a place were there's little or no room for
>correction/inexactness. I have no justification for that other than my
>fallible memory which (my memory tells me) is similar to many other
>people's observations.
>
>>>3) mathematical definitions are stipulative in that a definition is a
>>>matter of -arbitrarily- naming a concept, such that the name now
>>>refers only to such a concept, no matter what connotations and prior
>>>associations anyone might have had for that name.
>>
>>But stipulation is only a provisional substitute for correctness and
>>truth whether in student problems or the foundations of mathematics.
>
>Sure. But it makes things easier to discuss. Everybody knows what things
>are supposed to mean.
I agree if we are discussing the consequences of certain definitions
instead of their causes. Then we need to understand exactly what it is
we are discussing so when we say certain conclusions follow from
certain definitions, we're all on the same page. My objections to
conventional definitions aren't necessarily intended to impact math
conclusions in general, only to clarify the basis on which definitions
have to be drawn in terms of essential or primary characteristics.
In the case of cardinality, I don't see how we can get away from equal
differences because the alternative would be unequal differences. Just
as in the case of the term "frame of reference" in Special Relativity
we can't get away from common velocity as the defining determinant.
Regards - Lester
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