Re: Tautologies Then and Now
From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 12/12/04
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Date: Sun, 12 Dec 2004 10:33:39 GMT
"paul" <paul8801@on-ramp.nl> wrote in message
news:6qqnr057ch4b6ko2t3f6crtgs50lkjun4m@4ax.com...
> On Sun, 12 Dec 2004 02:51:47 GMT, "Stephen Harris"
> <cyberguard1048-usenet@yahoo.com> wrote:
>
>>> "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
>
>>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.
>
>
> It should be clear to you from numerous quotes you posted previously
> that, with the one exception of Barbara Partee, most authors are
> careful when discussing logics to restrict their use of the term
> "tautology" to propositional/sentential logic. For example your oft
> quoted
This is wrong. Why do you think I posted under Mondadic Pred. Logic:
"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."
SH: This is taken from "Mondadic Predicate Calculus" which is about 11th.
http://aka-ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Logic-IFall2002/Readings/index.htm
This is the online MIT course.
Any time you have truth tables or their logical equivalent that can return
all trues, that is described by the word tautological. It could also be
less precisely described by universally valid. That is the definition, and
it applies to some cases in mondadic predicate logic.
_Universally valid_ is used for Predicate Logic, never tautological.
But you didn't specify predicate logic, you said propositional logic. paul:
"what I said -- that the term "tautology" is not applied outside
propositional logic". I did not say tautology was applied to Predicate logic
but to
monadic predicate logic.
"Every tautology is valid" means that all trues are in sentential logic or
in some cases of monadic logic which have truth tables returning all trues,
can be
described by the word tautological. Or the word valid.
"but not vice versa" means there are cases which are described by "valid"
but not tautological. Every valid is not tautological. But some are, the
ones
which are logically equivalent to truth tables with the possibility of all
trues.
>
> http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
>
> says on top: "Corresponding to tautologies in Sentential Logic are
> analytic sentence schemata in First Order Predicate Logic. You will
> remember that a tautology is a sentence schema which is true under any
> consistent interpretation of its sentential letters;"
>
> Clearly the author indicates the term "tautology" is used uniquely in
> sentential logic. Numerous other quotes you have presented also
> indicate that same limit. Which matches Gamut and Copi:
>
This is just wrong, the author does not do that.
Nor does Gamut nor Copi when they refer to monadic predicate logic.
Monadic predicate logic is part of the foundation of Predicate logic. (PL)
But it is less powerful, I think because it is decidable. The reason PL
is more powerful is because it is undecidable. Undecidable IIRC, means
that it doesn't halt according to Turing's 1936 paper. Not halting means
there is no possibility of a truth table.
> * "In predicate logic ... Formulas @ such that V_M(@) = 1 for all
> models M for the language from which @ is taken are called universally
> valid formulas (they are not normally called tautologies)." L.T.F.
> Gamut. "Logic, Langauge, and Meaning." p. 99.
>
> * "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.
>
> If you google "universally valid" wrt predicate logic you'll find many
> instances of its application to always valid predicate statements
> versus the one example of Partee. My question is why is "tautology"
> not normally used outside sentential logic? There must be a reason.
>
> - paul
Well certainly. Universally valid (or maybe just valid, I'm not sure)
applies to Predicate logic. But I am talking about monadic PL, which
is outside of propositional logic, thus a counter example to your claim,
because you didn't claim that tautology was not part of predicate logic,
you claimed there was nothing outside of propositional logic that could
properly use the term tautology. Both of my quotes, Partee and McGee
showed that. McGee has written numerous books and is connected to
MIT, which is going to be the standard, not eccentric.
Since you have brought up Gamut and "normally" twice now in the
context of (they are not _normally_ called tautologies) I will explain
why normally should be interpreted in the sense of usually, as in most
cases. The exceptions you have been hearing about in monadic PL
have been saying in "some" cases. Emphasis on some by Chris Menzel:
"In *some* cases they do, namely, if you restrict your attention to
arguments consisting of formulas of monadic predicate logic (i.e.,
formulas that involve only 1-place predicates), or if the argument in
question is invalid and has a *finite* countermodel."
>SH: I don't think the author (Gamut) of the book would have used the
> >qualifier "normally" (they are not normally called tautologies) if it were
>strictly
>true that the term "tautology" never applied to predicate logic which is
>"outside propositional logic", the boundary paul claimed, "not applied".
paul responded:
"You're assuming that what is abnormal is inherently also appropriate.
I believe that what Gamut means by "not normally called tautologies"
is that some may use the term "tautology" within predicate logic, but,
strictly speaking, it is abnormal and inappropriate. That view is
supported by what I was taught in school, and by reading Gamut, which
finds no exception to the restriction of the term "tautology" to
propositional logic, and as I recall other utterances stipulating the
limitation of the term "tautology" to propositional logic."
SH: I am using "normally" (they are not _normally_ called tautologies)
as most people would when reading this sentence. Normally, means
usually, or most of the time. The opposite of this meaning is not
"abnormal".
The opposite would be unusually, infrequently, or rarely as in a small
proportion when measured agains the whole. The decidable conditions
are after all, only met by "fragments" of predicate logic. They are
uncommon.
You are using abnormal in a sense of incorrect. Gamut could have said
that meaning (incorrect) much more plainly:
"is taken are called universally valid formulas (they are not normally
called tautologies)."
Gamut could have changed this to: is taken are _correctly_ called
universally valid formulas.
Or
'is taken are _correctly_ called universally valid formulas, not
tautologies.'
If Gamut meant to clarify normal in the sense of correctness he could
have just used _correctly_ as I did above and would not have needed
a parenthetical remark. I suppose you could just argue he is a bad writer.
But if you assume he is a good writer, it means the proper interpretation
of Gamut's parenthetical remark is that he is distinguishing what is the
most common usage in terms of scope, while hinting in some instances that
the
term tautologous has some actual properly applied descriptive power.
This interpretation of normally meaning prominece or importance due to
the frequency or liklihood of encountering "some cases" is consistent
with understanding monadic PL ( a sort of subset of PL) as carrying over
some foundational ideas into the larger dominant area of PL.
You, paul, have not normally encountered the idea of monadic PL and
truth tables. That doesn't mean they don't exist but that they are not
usually mentioned, that is why you had not heard of them. Likewise with
the limited usage (occurrence) of tautological in describing some special
"fragment" cases of all trues in mondadic PL. Normally mean usually in
this case, the opposite is unusually or rarely. It explains why you haven't
heard of it perhaps. The opposite of normally does not mean abnormally
in this case--- like in the case of some eccentric professor with deviant
ideas. Barbara Partee would have been blasted if she were presenting
some non-standard view, and not having her book used as a textbook.
Next, Chris Menzel, when explaining the some cases of MPL wrote:
"formulas that involve only 1-place predicates"
Copi wrote:
> * "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.
>
"If an argument contains n different predicate symbols..."
The "n" in Copi's definition can be greater than 1-place. So this definition
includes full predicate logic. It is not limited to (it could be a greater
value n)
"formulas that involve only 1-place predicates"
which is a definition for tautological/truth tabled monadic predicate logic.
Since his definition covers a greater logical range which includes FPL
it is not meet the tautological condition. If the definition were restriced
to MPL with 1-place predicates then I think it would be correct to use
tautological or valid. Because all tautologies are valid, but not the
converse.
Certainly an instructor is going to stress that Predicate logic is correctly
described by "universally valid" (or maybe valid). But to be honest with
you, I think teaching the monadic foundation of predicate logic is going
to be very standard among good instructors in good colleges. That is
done by McGee at MIT, who mentions both truth tables and tautology
in his treatment of monadic PL in the progression to full PL.
One of us quite misunderstands what (I would welcome outside comment.)
McGee wrote in lectures 11 and 12, Monadic PL and Derivations of MPL
http://aka-ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Logic-IFall2002/Readings/index.htm
I think it is you because you think this quote
> says on top: "Corresponding to tautologies in Sentential Logic are
> analytic sentence schemata in First Order Predicate Logic.
SH: supports your position. It doesn't because First Order Predicate Logic
is not Monadic Predicate Logic, where all 'tautologies are valid' is true.
The
quote is relating to FOPL which is not the area of controversy.
First Order Predicate Logic only "corresponds" to sentential logic, because
the comparison of valid vs tautological, is the vice versa case where
'Every valid statement is not tautological'.
Remember that "Every tautology is valid" nearly describes both Sentential
logic and Monadic PL (AFAIK) which is under discussion, the category
of not propositional logic which accepts MPL -- and what is not under
discussion is first order predicate logic which has never been disputed
by me as being anything other than 'not tautological' but valid.
You seem to have made the assumption that if it is not propositional logic,
then it must be full-blown predicate logic which has neither tautologies
nor truth tables. I've been harping on monadic predicate logic which is
decidable. I haven't claimed tautologies or truth tables for pure predicate
calculus(PPL) nor full predicate calculus(FPL), both of which are
undecidable. I think there are at most minor errors in this post.
Why How is When,
Stephen
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