Kathy and Pat, (01)
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. (02)
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? (03)
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. (04)
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. Suppose we decide to translate that sentence to (05)
(Ax)(chunkOf(x,snow) -> isWhite(x)). (06)
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? (07)
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"? (08)
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. (09)
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. (010)
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. (011)
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. 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. (012)
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. 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. (013)
John (014)
___________________________________________________________________ (015)
Source: http://www.jfsowa.com/logic/tarski.htm (016)
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: (017)
The sentence "snow is white" is true if, and only if, snow is white. (018)
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 (019)
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