Re: Tautologies Then and Now
From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 12/09/04
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Date: Thu, 09 Dec 2004 06:32:34 GMT
"paul" <paul8801@on-ramp.nl> wrote in message
news:44bar01jlkobaf5g7p7mgvf4c7ntn75buu@4ax.com...
> On Tue, 07 Dec 2004 00:17:46 GMT, "Stephen Harris"
> <cyberguard1048-usenet@yahoo.com> wrote:
>
>>
>>"paul" <paul8801@on-ramp.nl> wrote in message
>>news:j1j9r0hnmc3kb0m15c7jss9n7n5dsm5pl9@4ax.com...
>>> On Mon, 06 Dec 2004 20:28:57 GMT, "robert j. kolker"
>>> <nowhere@nowhere.com> wrote:
>>>
>>>>
>>>>
>>>>paul wrote:
>>>>>
>>>>> They're called "universally valid," not tautologies, in the
>>>>> first-order predicate calculus.
>>>>
Bob:
>>>>They are true under all standard interpretations. That is how truth
>>>>tables are extended to the first order logics.
>>>
>>>
Paul:
>>> Can you cite a text that extends truth tables beyond propositional
>>> logic? My professors always said that doesn't happen, and it certainly
>>> didn't in any of my texts.
>>>
>>> - paul wrote:
"I don't know why you keep posting quotes that support what I said --
that the term "tautology" is not applied outside propositional logic"
>>>
SH: I think the following paragraph could easily be seen to support
Bob's statement, ("That is how truth tables are extended to the first order
logics.") but I wouldn't say you are wrong either. I used _word phrase_
to represent what the author put in bold in the original [underscore].
* is my emphasis.
"In Sentential Logic, we can prove an argument schema to be invalid by
specifying a set of truth assignments to the sentential letters which
results in true premises and a false conclusion; we thereby show that one
line of the argument schema's truth table allows an interpretation having
true premises and a false conclusion. In Predicate Logic, an argument schema
typically consists of sentence schemata which are not truth functional:
quantifiers, not truth functional connectives, are the major operators of
the typical "quantified
*argument schemata." And quantifiers are not truth functional operators
since they may represent an infinite number of individuals; the truth value
of a quantified sentence schema is therefore not a function of the truth
values of any _finite_ number of simple sentence schemata.
*Nevertheless, we can test the validity of a quantified argument schema
_indirectly_ by constructing and testing its _truth functional proxy for
some_ (non-empty) _domain_ of a specified (finite) number of individuals;
each of the premises, and the conclusion, in the original schema will be
equivalent _in that domain_ to its truth functional counterpart in the
proxy.
Because it is comprised of truth functional sentence schemata, a proxy may
be tested for validity by the short-cut method of truth value assignment, or
by *means of a truth table.*
And if a proxy proves to be invalid, it will provide a "recipe" for
constructing an interpretation of the corresponding quantified argument
schema into the same domain which will serve as a counter example, or
refutation, to that argument schema. Thus, if the original quantified
argument schema is valid, then _all_ of its corresponding proxies must also
be valid. If _any one_ of the proxies corresponding to a quantified argument
schema is invalid, then since it is therefore possible for the schema to
have an interpretation into some domain under which its premises are true
while its conclusion is false, the schema itself is invalid. Note that even
though one particular corresponding proxy is valid, the original quantified
argument schema might nevertheless be invalid: to be valid, _every_
corresponding proxy (for _every_ non-empty domain) must be valid."
http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
SH: I think the word tautological (no line of its truth table is false)
could be applied in this rather technical and specific connection to a truth
table in predicate logic as presented in this quote.
"Because it is comprised of truth functional sentence schemata, a proxy may
be tested for validity by the short-cut method of truth value assignment, or
by *means of a truth table.*"
Nevertheless, I think you are typically correct; the limited domain
described
probably does admit a standard treatment albeit an unusual one. I would
probably not have had an inkling about "truth functional proxies" except I
read a lot of papers. I am unsure what practical value constructing truth
functional proxies has. As a speculation: there are common sense AI
programs which accept natural language queries. Perhaps a truth functional
proxy could narrow down the domain of appropriate answers which go through a
series of filters and rules in order to produce a natural language
response. As I said, just a speculation because I think in computer ideas
and philosophy rather than the logical requirement for a truth functional
proxy.
Constructing a Truth Functional Proxy for Multiply-general Expressions:
a recipe pages 46 and 47 of Boardman's Supplement
http://www.lawrence.edu/fast/boardmaw/Logic_prox_pp46-7.html
"The first step in finding a truth functional proxy (for a domain of your
choosing) for a quantified argument schema is to find the truth functional
counterpart (in that domain) of each of the premises and the conclusion.
But these may be multiply-general expressions."....
Syntax and Semantics
"Meanings can become objects or targets of special types of reflective act;
it is acts of this sort which make up the science of logic. Logic arises
when
we treat those species which are meanings as special sorts of proxy objects
(as 'ideal singulars'), and investigate the properties of these objects in
much the same way that the mathematician investigates the properties of
numbers or geometrical figures."
http://www.nickbostrom.com/old/quine.html
Back to Quine's ontological relativity.
"We may now ask what may be the philosophical significance of the thesis
of indeterminacy of reference, interpreted so as to be proved by the
presentation of any proxyfunction. Does it show (a) that there are
incompatible theories of reference, all of which are equally adequate,
albeit we have happened to choose one particular theory by opting for
our actual notion of reference? Or does it show (b) that there is no fact
of the matter as to which sense of "reference" we are using?
I think it does not show (b) because in order to prove that the purported
notion of "reference" is merely purported, it is not enough to show that
there can be divergent denotation assignments conserving the truth values
of all sentences; one would also have to show that conserving truth value
of all sentences is sufficient for an assignment to be correct in the
intuitive sense, as far as this is naturalistically scrutable. That has not
been shown. There is no obvious reason why there may not be determining
criteria hooked directly onto words.
I think it does not show (a) either. (a) would in effect be a case of
underdetermination of theory by data, underdetermination of the countless
possible theories of reference. I will say more about the relation between
indeterminacy and underdetermination."
SH: The reason I quoted this is to show their are other angles of approach
to the truth functional proxy idea. In this case it seems possible for the
domain to be sufficiently determined. "There is no obvious reason why there
may not be determining criteria hooked directly onto words." ties into my
speculation about mechanical machine translation of natural language.
Regards,
Stephen
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