On Jul 17, 2007, at 10:22 AM, John F. Sowa wrote: (01)
> Kathy and Pat,
>
> JFS> Many people have pointed out that his choice of example was
>>> unfortunate, since that sentence raises two thorny issues that
>>> Tarski did not intend to represent in his introductory paper:
>>> snow as a continuous substance, and the use of a singular noun
>>> for making a generic statement about all snow.
>
> KBL> Not to mention what it means for something to "be" a color.
>> Are we talking physics of light absorption? In what kind of light?
>> Or are we talking about human perception?
>
> Yes, the level of granularity, the assumed technology for sensing or
> measuring the data, the relationship to other theories (e.g., of
> light,
> human sensory organs, etc.) and the intended applications influence
> the
> choice of categories in any ontology.
>
> And to elaborate on that simple sentence "Snow is white", we not only
> have to worry about how to relate the predicate 'isWhite' to reality,
> we also have to consider how to relate the substance term 'snow' to
> chunks of snow. (02)
No, we don't. All this discussion, while fascinating, is entirely
beside the point at issue. One does not have to go into all this
detail in order to apply a Tarskian semantics. (03)
> Suppose we decide to translate that sentence (04)
Well, if you decide to do that, then you have extended your theory,
and now you have to extend your semantics to match. Yes, of course.
But you are not obliged to perform this act of translation. (05)
> to
>
> (Ax)(chunkOf(x,snow) -> isWhite(x)).
>
> What would qualify as a chunk of snow? An individual snowflake?
> A compacted lump, such as a snowball? A layer of newfallen snow
> on the ground? Compacted snow that has been on the ground for
> a week and has partially melted and refrozen? The so-called
> 'corns' of "corn snow" in the late spring after snow has melted,
> refrozen, broken up, and fragmented many times? (06)
Any and all of the above, provided that they satisfy your axioms when
examined. (07)
You seem to be assuming that one is obliged to go into full detail
when describing reality. That is WRONG. That is exactly the fallacy I
tried to describe in an earlier message. One is obliged only to
describe reality with enough granularity to determine its fit to the
theory whose semantics is being described. (08)
> Even then, where is the boundary between snow and the partially
> melted and/or refrozen layer in contact with and/or mixed with
> the ground, vegetation, debris, etc.? Which parts do or do not
> qualify as "snow"? (09)
Why should I care? Any answer you like will do. The Tarskian theory
only requires that a set of items is somehow selected. Reality can no
doubt be carved into sets in any number of ways. Of course. What is
your point? Why do you think that this elementary fact has such
importance? (010)
> PH>> They may not know about them [Tarski-style models], but they
>>> can still be thinking about them. Just as someone who knows
>>> nothing of botany can think about plants.
>
> KBL> I don't think it's correct to say they "really are" thinking
>> about Tarskian models even if they don't know about them.
>
> I don't even think that Tarski was thinking about a Tarski-style
> model when he said "Snow is white". For an actual quotation
> from his 1944 paper, which devotes slightly more attention to
> the issues of relating language to reality than his 1993 paper,
> see the excerpt at the end of this note.
>
> In that paper, he never takes any sentence about any empirical
> subject (e.g., snow) and relates it to an explicitly specified
> domain D and set of relations R. (011)
He does not do this because he perceived, correctly, that there was
no need to do so. Your very way of phrasing this thing he never did
has your false assumption built into it: that by using 'mathematical'
language like D and R, that one must be talking about something other
than reality, that then must be 'related' to it. (012)
> I have read several other papers
> by Tarski, which went into great detail on well-defined mathematical
> issues, but which did not go into any detail about how to select
> the domain D or the set of relations R for any empirical subject. (013)
Of course they didn't. Why should they? What kind of detail would you
want? ANY way of selecting them will serve. The only requirement that
the theory places on them is that D is nonempty. (014)
>
> The issues for selecting D and R for mathematics are indeed
> "blindingly obvious", especially when the mathematicians themselves
> say "Let S be a set such that..." But if the issues for any
> empirical subject were indeed "blindingly obvious", Tarski would
> have given some examples. (015)
And he did. As did Russell, Quine and others. The fact that you
refuse to believe that his example is an example is your problem, not
his. (016)
> I believe that he did not work out the
> examples for a very good reason: it's very hard to do, and anything
> other than a toy example (e.g., six pennies on a table) raises an
> enormous number of sticky problems. (017)
He probably didn't because he thought it was so obvious that it
didn't need to be given any exposition. Quine seems to have taken a
similar view. (018)
Applying set theory to reality raises no problems at all. I have
never understood why you insist that it does. It is very hard to even
conceive of a theory which makes fewer ontological commitments than
set theory does of its ur-elements. (019)
Pat (020)
>
> John
>
> ___________________________________________________________________
>
> Source: http://www.jfsowa.com/logic/tarski.htm
>
> Let us start with a concrete example. Consider the sentence
> "snow is
> white." We ask the question under what conditions this sentence is
> true
> or false. It seems clear that if we base ourselves on the classical
> conception of truth, we shall say that the sentence is true if snow is
> white, and that it is false if snow is not white. Thus, if the
> definition of truth is to conform to our conception, it must imply the
> following equivalence:
>
> The sentence "snow is white" is true if, and only if, snow is
> white.
>
> Let me point out that the phrase "snow is white" occurs on tile
> left
> side of this equivalence in quotation marks, and on the right without
> quotation marks. On the right side we have the sentence itself, and on
> the left the name of the sentence. Employing the medieval logical
> terminology we could also say that on the right side the words
> "snow is
> white" occur in suppositio formalis, and on the left in suppositio
> materialis. It is hardly necessary to explain why we must have the
> name
> of the sentence, and not the sentence itself, on the left side of the
> equivalence. For, in the first place, from the point of view of the
> grammar of our language, an expression of the form "X is true" will
> not
> become a meaningful sentence if we replace in it 'X' by a sentence
> or by
> anything other than a name — since the subject of a sentence may be
> only
> a noun or an expression functioning like a noun. And, in the second
> place, the fundamental conventions regarding the use of any language
> require that in any utterance we make about an object it is the
> name of
> the object which must be employed, and not the object itself. In
> consequence, if we wish to say something about a sentence, for
> example,
> that it is true, we must use the name of this sentence, and not the
> sentence itself.8
>
>
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