Re: Tautologies Then and Now

From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 12/12/04 

Date: Sun, 12 Dec 2004 02:51:47 GMT

"paul" <paul8801@on-ramp.nl> wrote in message
news:po9lr0pb106gntfkf86juk2cknab31eu6n@4ax.com...
> On Thu, 09 Dec 2004 06:32:34 GMT, "Stephen Harris"
> <cyberguard1048-usenet@yahoo.com> wrote:
>
>>>>> - 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.
>
>
> The source you cite
>
> http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
>
> cites Copi, and Copi asserts what Owen shows... that truth tables can
> be used for monadic predicate logic. But in so describing, Copi does
> not refer to monadic predicate sentences that are true in all
> valuations as "tautologies" but as "universally valid." Copi states:
>
> "If an argument contains n different predicate symbols, then if it is
> valid for a model containing 2^n individuals, then it is valid for
> every model or universally valid."
>
> Copi, I.M. "Symbolic Logic." p. 81.
>- Paul
>

Monadic means one, one predicate, not n>1. The definition you give to
support your position, doesn't seem specific to monadic predicate logic.

I think Copi's definition that you quote is a broad definition outside the
range "tautology" applies. Shortly, you have shown how it is correct
to use "universally valid" in describing all of general predicate calculus.
Your quote does not show that it is incorrect to use "tautologous"
for specific decidable fragments. I think Copi's quote is the not the same
as the vice versa that I mention below. 'Every tautology is valid, but not
every valid formulation is tautological.' (the monadic ones can be both)

http://www-unix.oit.umass.edu/~partee/726_04/lectures
/Lecture6%20_Predicate%20Logic_.pdf

2.1. Tautologies, contradictions and contingencies
"As in Statement Logic, some closed formulas of Predicate Logic are
always true, i.e. they are true in every model. These are called
tautologies.
Formulas which are false in every model are called contradictions.
All the other formulas are called contingencies: their truth values
depend on models; they are true in some models and false in others."

Barbara Partee, who wrote 2.1 above, is the principle author of
"Mathematical Methods in Linguistics"** which is used as a textbook:

http://www.ling.gu.se/kurser/mathmeth/
"This course gives a general introduction to various tools from
discrete mathematics that are used in linguistics. It is based largely
(or entirely) on Barbara Partee, Alice ter Meulen, Robert Wall (1990)
Mathematical Methods in Linguistics, Kluwer Academic Publishers. **
------------------------------------------------------------

SH: I think there is something odd about you stating that your instructors
did
not cover this. Lecture 11 pages 20 & 21 "Logic: The Art of Persuasion
and the Science of Truth" I think by Vann McGee

http://aka-ocw.mit.edu/NR/rdonlyres/Linguistics-and-Philosophy/24-241Logic
-IFall2002/BE072366-EA9F-4193-97CC-BB2922C3C7B1/0/lec11.pdf [pgs. 20&21]

Normal Truth Assignment = N.T.A.
Definition: "A sentence is tautological iff it is assigned the value1
by every N.T.A. A sentence is valid iff it is true under every N.T.A.
For the sentential calculus,the words "tautological "and" valid" were
different words for the samething. Now that we've started on the
predicate calculus, we need to distinguish them. Validity is the notion
we're really interested in,but we need the notion of tautology as a
technical notion. Proposition. Every tautology is valid, but not vice
versa. [SH: He provides a proof, and then continues:]

A tautological sentence is a valid sentence whose validity is determined
by the sentence's truth functional structure. If, instead,the validity of
a sentence depends upon the meaning of the quantifiers,the sentence won't
be tautological.

We can test whether a sentence is tautological by the method of truth
tables, examining each possible way to assign a truth value to the
sentence's basic truth functional components."
---------------------------------------------------------

"Derivations in the Mondadic Predicate Calculus" [MIT online course]
"The fact that there is a mechanical procedure for testing whether a
sentence is a tautological consequence of a set of sentences is important.
In order for our derivations to have any probative value, we have to be
able to recognize when a sequence of sentences really is a proof,which
means that we need an algorithm for checking when a rule has been properly
applied." http://ocw.mit.edu/NR/rdonlyres/Linguistics-and-Philosophy
/24-241Logic-IFall2002/8A9852F3-D38E-4A94-9634-9EED32367CCB/0/lec12.pdf

SH: That there is a mechancical procedure for testing (truth table) which
can output all trues is correctly described as tautological by definition.
No matter what the logical structure outside of propositional logic is
named.
That there is a mechanical shortcut, is why it can be done by a computer

More about decidable fragments:
http://www-mgi.informatik.rwth-aachen.de/Publications/Graedel/ #38

If my quotes don't convince you, maybe someone else will do a better job.

Something interesting about artificial/formal languages and natural
language.
Knowledge Representation in Sanskrit and Artificial Intelligence by
Rick Briggs
http://www.aaai.org/Library/Magazine/Vol06/06-01/Papers/AIMag06-01-003.pdf

Logically guarded,
Stephen

Relevant Pages

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    ... "Analytic" refers to sentences whose truth values can be determined from ... brings us back to OR - and a tautology. ... It's logically true even if there is no relationship between Jack ... That inference is one of the most common mistakes in reasoning: ...
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