Tautologies Then and Now
Lester Zick
Tautologies Then and Now
----------------
The tautology is a law whose rules are exhaustive and exclude no
possibility. Thus it is considered useless by many for providing
information regarding the so called real world. Mathematics is held to
be tautological, for example, in providing exhaustive information on
its subjects. However there is no way to tell to what the information
applies and in the so called real world, scientists are left to divine
the applicability of tautological information for themselves, a
doctrine which, for want of a better name, I refer to as positivism.
We might illustrate simple tautological information in the following
way: a train is either blue or not blue. This gives us an exhaustive
account of all things blue and not blue but yields no useful insight
as to which things are blue or not blue of analytical necessity.
However, the case for the tautology is by no means as bleak as the
conventional perspective makes out. For there is exactly one and only
one form of tautology that is useful in explaining things to which
tautological information applies because it applies of necessity to
everything. And that tautology is P "differences" because alternatives
Q "different from differences" are universally self contradictory.
In other words, a tautology applies universally and if one alternative
is self contradictory, then the other must perforce apply universally
to all things. It makes no difference if we rephrase the tautology as
P "not" and Q "not not" or P "contradiction" and Q "contradiction of
contradiction" or P "negation" and Q "negation of negation".
Thus instead of a useless instrument of exhaustive truth we find in
the tautology an instrument of universal truth applicable to
everything from math to science to the analysis of sentient behavior.
And furthermore this also means that everything from math to science
to the analysis of sentient behavior can only be universally defined
in terms of differences, differences between differences, and so on.
Regards - Lester
Uncle Al
[snip]
> We might illustrate simple tautological information in the following
> way: a train is either blue or not blue.
[snip]
Bullshit. Morning glory blue and photographic film vs.
eyeballs. Dochroic pigments as on currently issued money.
Alexandrite and didymium glass in sunlight or incandescent light.
If you know nothing then your conclusions will be equally valid.
That is why the Frist World has porcelain flush toilets and East
Indians with their 360 million gods and 3000 years of
"civilization" squat and crap in their streets. You know nothing
by empirical counterdemonstration.
More to the point, in all the years you have been trolling Usenet
with your crapola, you have never learned anything. That is the
difference between ignorance (fixable) and stupidity (which is
forever). You are despicable.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf
Sam Wormley
>
> We might illustrate simple tautological information in the following
> way: a train is either blue or not blue. This gives us an exhaustive
> account of all things blue and not blue but yields no useful insight
> as to which things are blue or not blue of analytical necessity.
Andreas Feininger (one of the old Life Magazine Photographers,
among other things) in his books, particularly, "Light and Lighting
in Photography" could photograph a subject from different
perspectives, in different light of the day or season and come
up with just about any color you like.
Your train can be made to appear blue under the right circumstances
whether it is "blue" or not.
Things are not so simple and depend of the object, idea or entity,
the context and surroundings and who is and how are the looking.
Human sense are easily fooled and nature fools us all the time!
Rainer Rosenthal
"Sam Wormley"
>
> Human sense are easily fooled and nature
> fools us all the time!
But remember what Bill Taylor reminded us:
Don't anthropomorphize nature.
She hates that!
Thanks to the one here in sci.math who replied to
my sig-nature :-)
--
Rainer Rosenthal, r.ros...@web.de _____________________
| _ | |
| (_) | Given A, P and a circle. Find B, C on the |
| A P | circle with P on BC and area(ABC)=maximum. |
|__________|___(Ingmar Rubin in de.sci.mathematik) ________|
Lester Zick
in comp.ai.philosophy wrote:
>Lester Zick wrote:
>
>[snip]
>
>> We might illustrate simple tautological information in the following
>> way: a train is either blue or not blue.
>[snip]
>
>Bullshit. Morning glory blue and photographic film vs.
>eyeballs. Dochroic pigments as on currently issued money.
>Alexandrite and didymium glass in sunlight or incandescent light.
>
>If you know nothing then your conclusions will be equally valid.
>That is why the Frist World has porcelain flush toilets and East
>Indians with their 360 million gods and 3000 years of
>"civilization" squat and crap in their streets. You know nothing
>by empirical counterdemonstration.
No, but I know you, Al. Your histrionics are more suited to the stage
and you'd make a better political economist than scientist. Obviously
I would be interested in your counterdemonstration of differences,
negation, contradiction, and science.
>More to the point, in all the years you have been trolling Usenet
>with your crapola, you have never learned anything. That is the
>difference between ignorance (fixable) and stupidity (which is
>forever). You are despicable.
From which I can only conclude that there is nothing to learn from the
positivist-materialist-behaviorist branch of dogmatic philosophy that
you so eloquently represent.
Regards - Lester
Lester Zick
From which I suppose we must conclude that trains are blue or not
blue and particles are waves and/or not waves. Not really a hell of
a lot of information that Al's counterdemonstrative science provides,
but there it is for what it's worth.
Regards - Lester
Uncle Al
>
> On Mon, 02 Aug 2004 18:04:33 -0500, Uncle Al <Uncl...@hate.spam.net>
> in comp.ai.philosophy wrote:
>
> >Lester Zick wrote:
> >
> >[snip]
> >
> >> We might illustrate simple tautological information in the following
> >> way: a train is either blue or not blue.
> >[snip]
> >
> >Bullshit. Morning glory blue and photographic film vs.
> >eyeballs. Dochroic pigments as on currently issued money.
> >Alexandrite and didymium glass in sunlight or incandescent light.
> >
> >If you know nothing then your conclusions will be equally valid.
> >That is why the Frist World has porcelain flush toilets and East
> >Indians with their 360 million gods and 3000 years of
> >"civilization" squat and crap in their streets. You know nothing
> >by empirical counterdemonstration.
>
> No, but I know you, Al. Your histrionics are more suited to the stage
> and you'd make a better political economist than scientist. Obviously
> I would be interested in your counterdemonstration of differences,
> negation, contradiction, and science.
You have been empirically counterdemonstrated, Zick. "Blue or
not blue" is not restictive in at least three trivially observed
unrelated categories:
1) Morning glory blue and photographic film vs. eyeballs.
2) Dochroic pigments as on currently issued money.
3) Alexandrite and didymium glass in sunlight or incandescent
light.
Either you have an answer or you are bullshit, Zick. Shit or
Shinola, Zick? Put up or shut up, Zick.
Do ya have an answer Zick, do ya? Do ya have an answer Zick, do
ya? "Blue or not blue," Zick, what does it mean when your
statement is trashed by multiple empirical counterexamples?
Stick to the subject, Zick. Do ya have an answer Zick, do ya?
You are pinned in Uncle Al's killing jar.
Die.
[snip]
Uncle Al
>
> On Mon, 02 Aug 2004 23:20:47 GMT, Sam Wormley <swor...@mchsi.com> in
> comp.ai.philosophy wrote:
>
> >Lester Zick wrote:
> From which I suppose we must conclude that trains are blue or not
> blue and particles are waves and/or not waves.
Bullshit on both counts - by empirical counterdemonstration.
E.g., Poisson's spot vs. the photoelectric effect. Diffraction
of buckminsterfullerene through a grating,
http://www.quantum.univie.ac.at/research/matterwave/c60/
> Not really a hell of
> a lot of information that Al's counterdemonstrative science provides,
> but there it is for what it's worth.
>
> Regards - Lester
Ineducable diot.
Pspoors
from axioms? becasue we have to start somewhere!! constructivist mathematics
like rajthen and co are working on is the best approach to maths i've ever
seen, has anyone out there given thought to the idea that axioms might be wrong
sometimes?????
elliott spoors
Lester Zick
in comp.ai.philosophy wrote:
>Lester Zick wrote:
>>
>> On Mon, 02 Aug 2004 18:04:33 -0500, Uncle Al <Uncl...@hate.spam.net>
>> in comp.ai.philosophy wrote:
>>
>> >Lester Zick wrote:
>> >
>> >[snip]
>> >
>> >> We might illustrate simple tautological information in the following
>> >> way: a train is either blue or not blue.
>> >[snip]
>> >
>> >Bullshit. Morning glory blue and photographic film vs.
>> >eyeballs. Dochroic pigments as on currently issued money.
>> >Alexandrite and didymium glass in sunlight or incandescent light.
>> >
>> >If you know nothing then your conclusions will be equally valid.
>> >That is why the Frist World has porcelain flush toilets and East
>> >Indians with their 360 million gods and 3000 years of
>> >"civilization" squat and crap in their streets. You know nothing
>> >by empirical counterdemonstration.
>>
>> No, but I know you, Al. Your histrionics are more suited to the stage
>> and you'd make a better political economist than scientist. Obviously
>> I would be interested in your counterdemonstration of differences,
>> negation, contradiction, and science.
>
>You have been empirically counterdemonstrated, Zick. "Blue or
>not blue" is not restictive in at least three trivially observed
>unrelated categories:
I never claimed blue or not blue was restrictive. In fact I said just
the reverse. Are you still there, Al? You're not counterdemonstrating
anything, Al. You're only instantiating your own ignorance. You really
ought to learn to pay attention.
Regards - Lester
Lester Zick
in comp.ai.philosophy wrote:
>Lester Zick wrote:
>>
>> On Mon, 02 Aug 2004 23:20:47 GMT, Sam Wormley <swor...@mchsi.com> in
>> comp.ai.philosophy wrote:
>>
>> >Lester Zick wrote:
>[snip]
>
>> From which I suppose we must conclude that trains are blue or not
>> blue and particles are waves and/or not waves.
>
>Bullshit on both counts - by empirical counterdemonstration.
>E.g., Poisson's spot vs. the photoelectric effect. Diffraction
>of buckminsterfullerene through a grating,
>
>http://www.quantum.univie.ac.at/research/matterwave/c60/
>
>> Not really a hell of
>> a lot of information that Al's counterdemonstrative science provides,
>> but there it is for what it's worth.
>>
>> Regards - Lester
>
>Ineducable diot.
What's your counterdemonstration that I'm a diot, Al, since I never
claimed not to be a diot.
>You have been empirically counterdemonstrated, Zick. "Blue or
>not blue" is not restictive in at least three trivially observed
>unrelated categories:
I never claimed blue or not blue was restrictive. In fact I said just
the reverse. Are you still there, Al? You're not counterdemonstrating
anything, Al. You're only instantiating your own ignorance and your
personal predilection for the positivist-materialist-behaviorist axis.
You really ought to learn to pay attention.
>Either you have an answer or you are bullshit, Zick. Shit or
>Shinola, Zick? Put up or shut up, Zick.
I never claimed blue or not blue was restrictive. In fact I said just
the reverse. Are you still there, Al? You're not counterdemonstrating
anything, Al. You're only instantiating your own ignorance and your
personal predilection for the positivist-materialist-behaviorist axis.
You really ought to learn to pay attention.
>Do ya have an answer Zick, do ya? Do ya have an answer Zick, do
>ya? "Blue or not blue," Zick, what does it mean when your
>statement is trashed by multiple empirical counterexamples?
>Stick to the subject, Zick. Do ya have an answer Zick, do ya?
>You are pinned in Uncle Al's killing jar.
My answer is that you really need new glasses.
>Die.
Counterdemonstrationalist positivist-materialist-behaviorist diot.
Regards - Lester
Aristotle Polonium
At least he's eloquent.
Ari
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
"Lester Zick" <lester...@worldnet.att.net> wrote in message news:410fb592...@netnews.att.net...
mitch perkins
> "Lester Zick" <lester...@worldnet.att.net> wrote in message news:410fb592...@netnews.att.net...
> >
> > positivist-materialist-behaviorist branch of dogmatic philosophy that
> > you so eloquently represent.
> >
> > Regards - Lester
> >
>
> Ari
>
Mitch
Mike
You are an idiot. The Law of Excluded Middle has nothing to do with
your fucking ability to determine a color or your perception of it. It
is a slef evident proposition of the form p V ~p, a tautology.
Obviously, you lack the foundational background necessary for
scientific endevor and a possibility of breakthrough. You will always
remain in the strict limits of experimentation and the circuklar
reasoning it is subject to.
Listen idiot, a train is either color X or it is not. your arguement
deserves an F in logic 101. If you have lost you common sense and
ability to think at an abstract level it's your problem. Insulting
other people because you think you're right is a sign of insanity.
Go away crank Al.
Mike
patty
Well when you *apply* logic to the real world, you find that the Law of
the Excluded Middle (LEM) does not always hold. What if your train was
mostly brown and someone put a little dab of blue paint on it; what if
the train was in a 6 hour process of being painted blue; at what point
in that process can we say with no equivocation that the train is blue?
My point is that LEM is frequently impractical for real world analysis.
patty
Lester Zick
in comp.ai.philosophy wrote:
>
>At least he's eloquent.
>
>Ari
>
Yes, and winsome too.
Regards - Lester
Lester Zick
comp.ai.philosophy wrote:
Al is a frustrated science wannabe who hasn't been able to contribute
anything of significance and resents others who have. He's basically a
monkey's uncle not mine.
Regards - Lester
Uncle Al
Hey stooopid, it isn't a law if it is empirically broken. It is
bullshit if it is empirically counterdemonstrated. Hey Zick - do
you know what "empirical" means? It means "by observation." You
can ivory tower fart all you want. Reality says you are an ass.
Ineducable diot.
You have been empirically counterdemonstrated, Zick. "Blue or
not blue" is not restictive in at least three trivially observed
unrelated categories:
1) Morning glory blue and photographic film vs. eyeballs.
2) Dochroic pigments as on currently issued money.
3) Alexandrite and didymium glass in sunlight or incandescent
light.
Either you have an answer or you are bullshit, Zick. Shit or
Shinola, Zick? Put up or shut up, Zick.
--
Lester Zick
in comp.ai.philosophy wrote:
The local weather turns out to be a somewhat less intractable problem
than most think. In Al's neighborhood it's always cold and blustery.
Regards - Lester
Lester Zick
in comp.ai.philosophy wrote:
Yes, well, I'm still waiting for you to produce your empirical
counterdemonstration to my tautology. But please hurry because the end
of time approaches faster than your positivist-materialis-behaviorist
counterdemonstrative empirical guesses envision.
>Ineducable diot.
>
>You have been empirically counterdemonstrated, Zick. "Blue or
>not blue" is not restictive in at least three trivially observed
>unrelated categories:
>
> 1) Morning glory blue and photographic film vs. eyeballs.
> 2) Dochroic pigments as on currently issued money.
> 3) Alexandrite and didymium glass in sunlight or incandescent
>light.
>
>Either you have an answer or you are bullshit, Zick. Shit or
>Shinola, Zick? Put up or shut up, Zick.
Well, Al, unable to make any contribution to physics you now offer to
make pretentious contributions to logic. You can't even read straight.
Regards - Lester
Uncle Al
Hey stooopid, I'm holding a pair of didymium clip-ons from the
glassblower. They are pink under incandescent light and blue
under fluorescent or sunlight. They are simultaneously blue and
not blue under both, jackass. Don't "tautology" me, moron, I'm a
scientist. Reality is my ally - and a powerful ally it is.
Since your are too manifestly stooopid to comprehend four
empirical examples to your bullshit, in three independent
classes, Uncle Al will give you another one. Dye white cotton
cloth with Crystal Violet. Being a metachromatic dye, It is rich
deep blue or brilliant royal purple depending on whether you view
it in sunlight/fluorescent light, or incandescent light. View it
under a mixture and it s simultaneously blue and not blue.
You can get it from your pharmacist as "gentian violet." A
little goes a long way. Now you can be just like the One True
Church, Zick, and refuse to look through Galileo's telescope at
Jupiter's moons not orbiting the Earth against inerrant Chruch
doctrine.
Empirical fool.
David Canzi -- non-mailable address
Lester Zick <lester...@worldnet.att.net> wrote:
>... there is exactly one and only
>one form of tautology that is useful in explaining things to which
>tautological information applies because it applies of necessity to
>everything. And that tautology is P "differences" because alternatives
>Q "different from differences" are universally self contradictory.
On the other hand, Q "different from differences" while R "different
from differences of similarities" is consistent like pancake batter
to peanut butter.
So there!
--
David Canzi "Nonconformists travel as a rule in bunches. You rarely find
a nonconformist who goes it alone. And woe to him inside a
nonconformist clique who does not conform with nonconformity."
-- Eric Hoffer
Lester Zick
in comp.ai.philosophy wrote:
>Lester Zick wrote:
>>
>> On Wed, 04 Aug 2004 11:47:49 -0500, Uncle Al <Uncl...@hate.spam.net>
>> in comp.ai.philosophy wrote:
>> >Either you have an answer or you are bullshit, Zick. Shit or
>> >Shinola, Zick? Put up or shut up, Zick.
>>
>> Well, Al, unable to make any contribution to physics you now offer to
>> make pretentious contributions to logic. You can't even read straight.
>>
>
>Hey stooopid, I'm holding a pair of didymium clip-ons from the
>glassblower. They are pink under incandescent light and blue
>under fluorescent or sunlight. They are simultaneously blue and
>not blue under both, jackass. Don't "tautology" me, moron, I'm a
>scientist. Reality is my ally - and a powerful ally it is.
See, Al, here's the problem. You aren't actually a scientist. You're a
wannabe. There are several solons on the sci.physics newsgroup who
really are scientists, Franz Heymann and Edward Green, come to mind.
And there are probably others. But you are not one of them. You are a
counterdemonstrationalist-positivist-materialist-behaviorist diot.
>Since your are too manifestly stooopid to comprehend four
>empirical examples to your bullshit, in three independent
>classes, Uncle Al will give you another one. Dye white cotton
>cloth with Crystal Violet. Being a metachromatic dye, It is rich
>deep blue or brilliant royal purple depending on whether you view
>it in sunlight/fluorescent light, or incandescent light. View it
>under a mixture and it s simultaneously blue and not blue.
White cotton is not a train, Al. The laws of logic and tautologies do
not apply to white cotton. They only apply to differences.
>You can get it from your pharmacist as "gentian violet." A
>little goes a long way. Now you can be just like the One True
>Church, Zick, and refuse to look through Galileo's telescope at
>Jupiter's moons not orbiting the Earth against inerrant Chruch
>doctrine.
Why should I look through Galileo's telescope, Al, when all he could
have seen were differences. Of course, you might have seen something
different from differences, but then that would make you an oxymoron.
Regards - Lester
Lester Zick
dmc...@remulak.ads.uwaterloo.ca (David Canzi -- non-mailable address)
in comp.ai.philosophy wrote:
>In article <410eb720...@netnews.att.net>,
>Lester Zick <lester...@worldnet.att.net> wrote:
>>... there is exactly one and only
>>one form of tautology that is useful in explaining things to which
>>tautological information applies because it applies of necessity to
>>everything. And that tautology is P "differences" because alternatives
>>Q "different from differences" are universally self contradictory.
>
>On the other hand, Q "different from differences" while R "different
>from differences of similarities" is consistent like pancake batter
>to peanut butter.
>
>So there!
R "different from differences of similarities" may indeed be
consistent with "pancake batter to peanut butter" but it's a little
hard to tell as you don't offer any tautological evidence of their
universal exclusiveness.
On the other hand R "different from differences of similarities" seems
to impute some difference to similarity so it's a little difficult to
tell what point you're trying to make. That differences arise from
similarity, perchance? That's just typical positivist legerdemain
since the only similarity that arises does so from the absence of
difference but differences do not arise mechanically or tautologically
from the presence of similarity or identity.
>David Canzi "Nonconformists travel as a rule in bunches. You rarely find
> a nonconformist who goes it alone. And woe to him inside a
> nonconformist clique who does not conform with nonconformity."
> -- Eric Hoffer
Regards - Lester
David Canzi -- non-mailable address
Lester Zick <lester...@worldnet.att.net> wrote:
>On Wed, 4 Aug 2004 20:13:09 +0000 (UTC),
>dmc...@remulak.ads.uwaterloo.ca (David Canzi -- non-mailable address)
>in comp.ai.philosophy wrote:
>
>>In article <410eb720...@netnews.att.net>,
>>Lester Zick <lester...@worldnet.att.net> wrote:
>>>And that tautology is P "differences" because alternatives
>>>Q "different from differences" are universally self contradictory.
>>
>>On the other hand, Q "different from differences" while R "different
>>from differences of similarities" is consistent like pancake batter
>>to peanut butter.
>
>consistent with "pancake batter to peanut butter" but it's a little
>hard to tell as you don't offer any tautological evidence of their
>universal exclusiveness.
Claims of consistency are not exclusive, but may be weak or strong,
according to how hard they are to stir.
>On the other hand R "different from differences of similarities" seems
>to impute some difference to similarity so it's a little difficult to
>tell what point you're trying to make. That differences arise from
>similarity, perchance?
Cats and dogs are differently similar from catfish and dolphins.
Got a problem with that?
--
Acid Pooh
>
> Hey stooopid, I'm holding a pair of didymium clip-ons from the
> glassblower. They are pink under incandescent light and blue
> under fluorescent or sunlight. They are simultaneously blue and
> not blue under both, jackass. Don't "tautology" me, moron, I'm a
> scientist. Reality is my ally - and a powerful ally it is.
>
This is just a play on words. What does "simultaneously blue and not
blue" mean? It's a mixture of blue and pink--some kind of purple?
Different spots are different colors depending on the angle in which
it is viewed? If the first, that's just a cop out. One can easily
deal with the second: "Each face of the clip on is either blue or not
blue, depending on the circumstance."
'cid 'ooh
Lester Zick
>In article <41114623...@netnews.att.net>,
>Lester Zick <lester...@worldnet.att.net> wrote:
>>On Wed, 4 Aug 2004 20:13:09 +0000 (UTC),
>>dmc...@remulak.ads.uwaterloo.ca (David Canzi -- non-mailable address)
>>in comp.ai.philosophy wrote:
>>
>>>In article <410eb720...@netnews.att.net>,
>>>Lester Zick <lester...@worldnet.att.net> wrote:
>>>>And that tautology is P "differences" because alternatives
>>>>Q "different from differences" are universally self contradictory.
>>>
>>>On the other hand, Q "different from differences" while R "different
>>>from differences of similarities" is consistent like pancake batter
>>>to peanut butter.
>>
>>R "different from differences of similarities" may indeed be
>>consistent with "pancake batter to peanut butter" but it's a little
>>hard to tell as you don't offer any tautological evidence of their
>>universal exclusiveness.
>
>Claims of consistency are not exclusive, but may be weak or strong,
>according to how hard they are to stir.
That's nice. But proof of claims can only be strong or wrong.
>>On the other hand R "different from differences of similarities" seems
>>to impute some difference to similarity so it's a little difficult to
>>tell what point you're trying to make. That differences arise from
>>similarity, perchance?
>
>Cats and dogs are differently similar from catfish and dolphins.
>Got a problem with that?
I can't have a problem with that. Problems are only between claims and
proof and so far I haven't seen proof and hardly much of a claim for
anything responsive to my claim and proof for the universal truth of
P "differences".
Regards - Lester
Roy Jose Lorr
Lester Zick wrote:
>
>
> From which I can only conclude that there is nothing to learn from the
> positivist-materialist-behaviorist branch of dogmatic philosophy that
> you so eloquently represent.
And, the deconsructionist knows what?... that he speaks only to
himself, in language he has no hope of understanding?
--
The last stage of
utopian sentimentalism
is homicidal mania.
Chris Degnen
>
>And, the deconsructionist knows what?... that he speaks only to
>himself, in language he has no hope of understanding?
There seems to be something more mysterious than that going on.
Roy Jose Lorr
Chris Degnen wrote:
Without mystery we'd be God.
Chris Degnen
>
>Chris Degnen wrote:
>
>> Roy Jose Lorr wrote:
>> >
>> >And, the deconsructionist knows what?... that he speaks only to
>> >himself, in language he has no hope of understanding?
>>
>> There seems to be something more mysterious than that going on.
>
>Without mystery we'd be God.
The mysteron agents pitch some uncanny shots but as expected
the deconstruction-ray derealizes these curve-balls upon entry.
Pitch characteristics are under analysis: acknowledged to be _of_
reality they nevertheless shift the envelope as if the D-ray had
been applied not to the incoming but directly to reality. The
shifting envelope is expected to hinder mysteron identification,
although there is speculation that the phenomena may be a form
of communication.
Roy Jose Lorr
Chris Degnen wrote:
"Reason lies with good reason." Reason's intent is not to
expose us to reality but to protect us from it.
Chris Degnen
Don't tell me there's no Santa Claus!
Heigegger describes reason, (or 'ground'), as being 'stewn' in three ways:
"establishing", "taking up a basis", and "the grounding _of_ something."
My understanding of "establishing" is that it is everything that leads up
to a being's being - the 'reason for', as it were. Upon being established
the being (Dasein) takes up its basis amongst beings and begins trying to
ground things for itself.
"Because the transcendence of Dasein, as projectively finding itself, and
as forming the development of an understanding of being, is a grounding
[reasoning] of things; and because _this_ way of grounding is equioriginary
with the first two ways within the unity of transcendence, i.e., springs
forth from the finite freedom of Dasein; for this reason Dasein _can_, in its
factical accounting and justifications, cast "grounds" aside, suppress any
demand for them, pervert them, and cover them over. As a consequence
of this origin of grounding things and thus also of accounting for them, it is
in each case left to the freedom in Dasein how far to extend such grounding
and whether indeed it understands how to attain an authentic grounding of
things."
Seems reasoning is about finding the reasons things are as they are.
(Quotation from "On the Essence of Ground", p. 65, Pathmarks.)
Roy Jose Lorr
Chris Degnen wrote:
> Roy Jose Lorr wrote:
> >
> >Chris Degnen wrote:
> >
> >> Roy Jose Lorr wrote:
> >> >
> >> >Chris Degnen wrote:
> >> >
> >> >> Roy Jose Lorr wrote:
> >> >> >
> >> >> >And, the deconsructionist knows what?... that he speaks only to
> >> >> >himself, in language he has no hope of understanding?
> >> >>
> >> >> There seems to be something more mysterious than that going on.
> >> >
> >> >Without mystery we'd be God.
> >>
> >> The mysteron agents pitch some uncanny shots but as expected
> >> the deconstruction-ray derealizes these curve-balls upon entry.
> >> Pitch characteristics are under analysis: acknowledged to be _of_
> >> reality they nevertheless shift the envelope as if the D-ray had
> >> been applied not to the incoming but directly to reality. The
> >> shifting envelope is expected to hinder mysteron identification,
> >> although there is speculation that the phenomena may be a form
> >> of communication.
> >
> >"Reason lies with good reason." Reason's intent is not to
> >expose us to reality but to protect us from it.
>
> Don't tell me there's no Santa Claus!
Santa Clause is one of reason's entertaining little diversions.
>
>
> Heigegger describes reason, (or 'ground'), as being 'stewn' in three ways:
> "establishing", "taking up a basis", and "the grounding _of_ something."
> My understanding of "establishing" is that it is everything that leads up
> to a being's being - the 'reason for', as it were. Upon being established
> the being (Dasein) takes up its basis amongst beings and begins trying to
> ground things for itself.
It is impossible to know the 'reason' for 'being'. Attempts to discover the
reason is another of reason's entertaining little distractions..
>
>
> "Because the transcendence of Dasein, as projectively finding itself, and
> as forming the development of an understanding of being, is a grounding
> [reasoning] of things; and because _this_ way of grounding is equioriginary
> with the first two ways within the unity of transcendence, i.e., springs
> forth from the finite freedom of Dasein; for this reason Dasein _can_, in its
> factical accounting and justifications, cast "grounds" aside, suppress any
> demand for them, pervert them, and cover them over. As a consequence
> of this origin of grounding things and thus also of accounting for them, it is
> in each case left to the freedom in Dasein how far to extend such grounding
> and whether indeed it understands how to attain an authentic grounding of
> things."
If nothing else 'being' is protective of itself. It follows the prime directive
to 'be fruitful and multiply'. This is only possible in an environment that
doesn't permit the cognizant to interact directly with raw reality. The
prime defense between us and raw reality is reason. Since reason cannot
lie to reality it must lie to us. That is the primary thing that keeps us from
literally approaching the transcendent too closely and immediately
disintegrating in its presence. We, like Sartre's Sysiphus are tasked by
reason to ceaselessly rolling a rock to the top of a mountain, only to have
it fall back of its own weight. This is not however, the existentialists'
dreadful punishment of futile and hopeless labor as is generally reasoned
by disillusioned cynical men but a life enhancing and most necessary
entertainment, a consequence of the natural conflict between our
overbearing vanity and the limitations placed on us by nature.
Wolf Kirchmeir
[...]
> The
> prime defense between us and raw reality is reason. Since reason cannot
> lie to reality it must lie to us. That is the primary thing that keeps us from
> literally approaching the transcendent too closely and immediately
> disintegrating in its presence.
The "primary defense between us and raw reality", if this phrase refers
to anything, is the sensory system, which filters out a great deal of
the data "out there", and shapes it besides - much of it even before it
gets to the other parts of the CNS.
There's also the matter of "raw reality" and "the transcendent." Do you
intend these phrases to refer to the same thing? I suppose you do, else
the conjunction of the two sentence I've quoted makes no sense. Bad
poetry, IMO.
Roy Jose Lorr
Wolf Kirchmeir wrote:
> Roy Jose Lorr wrote:
> [...]
> > The
> > prime defense between us and raw reality is reason. Since reason cannot
> > lie to reality it must lie to us. That is the primary thing that keeps us from
> > literally approaching the transcendent too closely and immediately
> > disintegrating in its presence.
>
> The "primary defense between us and raw reality", if this phrase refers
> to anything, is the sensory system, which filters out a great deal of
> the data "out there", and shapes it besides - much of it even before it
> gets to the other parts of the CNS.
Yes, the senses filter out plenty of reality, even possibly having
a role in shaping some of it, (though I can't imagine their ability
to, or purpose in employing this sort of judgmental behavior).
However, that in no way negates reason taking up the slack.
Further, I believe that much of the sensory input is censored
by reason before we act on it.
>
>
> There's also the matter of "raw reality" and "the transcendent." Do you
> intend these phrases to refer to the same thing? I suppose you do, else
> the conjunction of the two sentence I've quoted makes no sense. Bad
> poetry, IMO.
You 'suppose' correctly, and it wasn't intended as poetry.
Wolf Kirchmeir
[...]
>
> Yes, the senses filter out plenty of reality, even possibly having
> a role in shaping some of it, (though I can't imagine their ability
> to, or purpose in employing this sort of judgmental behavior).
> However, that in no way negates reason taking up the slack.
> Further, I believe that much of the sensory input is censored
> by reason before we act on it.
[...]
The senses don't "shape reality", the shape the data that's trnsmitted
to the rest of the CNS. Read up on how the retianl layer does this, for
example.
The filtering isn't "judgemental behavior" - it's just a consequence of
the physics/chemistry of sensors. Eg, the human eye filters out the UV
band, the bee's eye doesn't.
Roy Jose Lorr
Wolf Kirchmeir wrote:
> Roy Jose Lorr wrote:
>
> [...]
> >
> > Yes, the senses filter out plenty of reality, even possibly having
> > a role in shaping some of it, (though I can't imagine their ability
> > to, or purpose in employing this sort of judgmental behavior).
> > However, that in no way negates reason taking up the slack.
> > Further, I believe that much of the sensory input is censored
> > by reason before we act on it.
>
> [...]
>
> The senses don't "shape reality", the shape the data that's trnsmitted
> to the rest of the CNS. Read up on how the retianl layer does this, for
> example.
In other words, the senses don't filter, they merely sense what they sense.
>
>
> The filtering isn't "judgemental behavior" - it's just a consequence of
> the physics/chemistry of sensors. Eg, the human eye filters out the UV
> band, the bee's eye doesn't.
Er... I believe using the term "filter" is misleading. Theres no evidence
that the eye senses UV, then filters it out. On the human level, to filter
is to judge.
Wolf Kirchmeir
>
> Wolf Kirchmeir wrote:
>
>
>>Roy Jose Lorr wrote:
>>
>>[...]
>>
>>>Yes, the senses filter out plenty of reality, even possibly having
>>>a role in shaping some of it, (though I can't imagine their ability
>>>to, or purpose in employing this sort of judgmental behavior).
>>>However, that in no way negates reason taking up the slack.
>>>Further, I believe that much of the sensory input is censored
>>>by reason before we act on it.
>>
>>[...]
>>
>>The senses don't "shape reality", the shape the data that's trnsmitted
>>to the rest of the CNS. Read up on how the retianl layer does this, for
>>example.
>
>
> In other words, the senses don't filter, they merely sense what they sense.
My comment used the word "shape" - why are you using the word "filter?"
The senses do filter what comes in - that's _how_ a sensor "senses what
it senses."
Do you actually think about what you write, or do you just shoot from
the lip?
Etc.
Roy Jose Lorr
Wolf Kirchmeir wrote:
> Roy Jose Lorr wrote:
>
> >
> > Wolf Kirchmeir wrote:
> >
> >
> >>Roy Jose Lorr wrote:
> >>
> >>[...]
> >>
> >>>Yes, the senses filter out plenty of reality, even possibly having
> >>>a role in shaping some of it, (though I can't imagine their ability
> >>>to, or purpose in employing this sort of judgmental behavior).
> >>>However, that in no way negates reason taking up the slack.
> >>>Further, I believe that much of the sensory input is censored
> >>>by reason before we act on it.
> >>
> >>[...]
> >>
> >>The senses don't "shape reality", the shape the data that's trnsmitted
> >>to the rest of the CNS. Read up on how the retianl layer does this, for
> >>example.
> >
> >
> > In other words, the senses don't filter, they merely sense what they sense.
>
> My comment used the word "shape" - why are you using the word "filter?"
Hmm... you did write... "The "primary defense between us and raw reality", if
this phrase refers to anything, is the sensory system, which filters out a great
deal of the data "out there"... did you not?
>
>
> The senses do filter what comes in - that's _how_ a sensor "senses what
> it senses."
Instead of a filtration process wouldn't it be far less work for the
sensor merely to not notice what it has no capacity to see?
>
>
> Do you actually think about what you write, or do you just shoot from
> the lip?
Hmm.
Wolf Kirchmeir
>
> Wolf Kirchmeir wrote:
>
>
>>Roy Jose Lorr wrote:
>>
>>
>>>Wolf Kirchmeir wrote:
[...]
>>>>The senses don't "shape reality", the shape the data that's trnsmitted
>>>>to the rest of the CNS. Read up on how the retianl layer does this, for
>>>>example.
>>>
>>>
>>>In other words, the senses don't filter, they merely sense what they sense.
>>
>>My comment used the word "shape" - why are you using the word "filter?"
>
>
> Hmm... you did write... "The "primary defense between us and raw reality", if
> this phrase refers to anything, is the sensory system, which filters out a great
> deal of the data "out there"... did you not?
Sure, but that's not the sentence you commented on.
Roy Jose Lorr
Wolf Kirchmeir wrote:
Oh boy
George Cox
>
>... Mathematics is held to
> be tautological, ...
By whom? Anybody other than Wittgenstein?
Owen
"George Cox" <george_...@spambtinternet.com.invalid> wrote in message
news:41B34C6C...@spambtinternet.com.invalid...
Frank Ramsay and others including me, etc. agree.
That (1+1=2) is tautologous, ie. logically true, is clear.
Because of (Frege-Russell)'s analysis.
Chris Menzel
Mind deriving that "tautology" so we can see exactly which logical
axioms is follows from?
Stephen Harris
"Chris Menzel" <cme...@remove-this.tamu.edu> wrote in message
news:slrncr6qa6...@philebus.tamu.edu...
It is very conventional to say that mathematical proofs are tautological.
Axioms are givens, not proven, therefore they prove theorems circularly.
Mitch Harris
>On Sun, 5 Dec 2004 14:00:44 -0500, Owen <oori...@yahoo.com> said:
>> "George Cox" <george_...@spambtinternet.com.invalid> wrote in message
>> >>
>> >>... Mathematics is held to
>> >> be tautological, ...
>> >
>> > By whom? Anybody other than Wittgenstein?
>>
>> Frank Ramsay and others including me, etc. agree.
>>
>> That (1+1=2) is tautologous, ie. logically true, is clear.
>> Because of (Frege-Russell)'s analysis.
>
>Mind deriving that "tautology" so we can see exactly which logical
>axioms is follows from?
Not really knowing, why can't you just use the Peano axioms as assumptions
(to define addition) in some logical formula? Is it the "="? or is it
defining addition that is extra-logical? (the PA induction axiom is not
needed since only constants are referred to.)
Mitch
George Cox
Frege-Russell doesn't settle, e.g., GCH.
The Sophist
It'd be trivial to derive it from the Peano axioms, of course. You
might claim those aren't logical, but there really isn't any clear
standard of what counts as a logical axiom, so I don't think you'll get
very far putting a lot of weight on that line of argument.
--
Aaron Boyden
The main division between the so-called Continental and Analytic
traditions has been disputes over whether the task of being unclear
should be carried out in natural language or in a formal system.
Chris Menzel
> "Chris Menzel" <cme...@remove-this.tamu.edu> wrote:
>> On Sun, 5 Dec 2004 14:00:44 -0500, Owen <oori...@yahoo.com> said:
>>> "George Cox" <george_...@spambtinternet.com.invalid> wrote in
>>> message news:41B34C6C...@spambtinternet.com.invalid...
>>> > Lester Zick wrote:
>>> >>
>>> >>... Mathematics is held to be tautological, ...
>>> >
>>> > By whom? Anybody other than Wittgenstein?
>>>
>>> Frank Ramsay and others including me, etc. agree.
>>>
>>> That (1+1=2) is tautologous, ie. logically true, is clear.
>>> Because of (Frege-Russell)'s analysis.
>>
>> Mind deriving that "tautology" so we can see exactly which logical
>
> It is very conventional to say that mathematical proofs are tautological.
No it isn't. But this is neither here nor there.
> Axioms are givens, not proven, therefore they prove theorems circularly.
The claim was not that *proofs* can be viewed as tautologies (better,
logical truths). It's trivial that any valid formal proof can be
converted into a logical truth. The claim was that the single proposition
"1+1=2" is itself a logical truth, a highly nontrivial thesis, to say
the least.
-cm
Chris Menzel
> Chris Menzel wrote:
>
>> On Sun, 5 Dec 2004 14:00:44 -0500, Owen <oori...@yahoo.com> said:
>>
>>> "George Cox" <george_...@spambtinternet.com.invalid> wrote in message
>>>news:41B34C6C...@spambtinternet.com.invalid...
>>> > Lester Zick wrote:
>>> >>
>>> >>... Mathematics is held to
>>> >> be tautological, ...
>>> >
>>> > By whom? Anybody other than Wittgenstein?
>>>
>>> Frank Ramsay and others including me, etc. agree.
>>>
>>> That (1+1=2) is tautologous, ie. logically true, is clear.
>>> Because of (Frege-Russell)'s analysis.
>>
>> Mind deriving that "tautology" so we can see exactly which logical
>> axioms is follows from?
>
> It'd be trivial to derive it from the Peano axioms, of course.
Of course.
> You might claim those aren't logical,
Certainly I would.
> but there really isn't any clear standard of what counts as a logical
> axiom, so I don't think you'll get very far putting a lot of weight on
> that line of argument.
Surely you don't mean to suggest that because there is a fairly wide
swath of grey between the logical and the nonlogical that we don't have
clear instances of either, do you? PA is about numbers (or perhaps
better, the number structure) the way Newton is about forces and masses
and such. That zero is not the successor of any number is no more logic
than F=ma.
That said, let's also remember where the burden is here -- the OP is the
one who claimed that 1+1=2 is logic. The claim is *at least* not
obvious -- your "lack of a standard" argument *does* show that. So it
is upon the OP to justify his claim, not upon me to argue its denial. I
*did* argue its denial simply because the claim is so obviously false. ;-)
Chris Menzel
paul
>
> "George Cox" <george_...@spambtinternet.com.invalid> wrote in message
>news:41B34C6C...@spambtinternet.com.invalid...
> > Lester Zick wrote:
> >>
> >>... Mathematics is held to
> >> be tautological, ...
> >
> > By whom? Anybody other than Wittgenstein?
>
> Frank Ramsay and others including me, etc. agree.
>
> That (1+1=2) is tautologous, ie. logically true, is clear.
While the values on each side of "=" are the same, I'm not sure that
the same "statement" is on both sides. "1+1" describes a function f
for which "2" is the output f(x). This is what's being said:
f(x) = x + x
Is that a tautology? If not, does it become a tautology by defining an
input value?
-paul
George Cox
For me (am I alone?) a tautology (in the logical sense) is a formula of
propositional calculus which is true for all values of the truth values
of its constituent atomic letters.
The Sophist
You are probably not alone, but it is also frequently used in a more
inclusive sense, for a formula of any logical system which is true on
any assignment of values to the variables involved.
robert j. kolker
George Cox wrote:
>
> For me (am I alone?) a tautology (in the logical sense) is a formula of
> propositional calculus which is true for all values of the truth values
> of its constituent atomic letters.
Theorems of first order logic are also tautologies.
Bob Kolker
paul
They're called "universal truths," not tautologies, in the
first-order predicate calculus.
- paul
robert j. kolker
paul wrote:
>
> They're called "universal truths," not tautologies, in the
> first-order predicate calculus.
They are true under all standard interpretations. That is how truth
tables are extended to the first order logics.
Bob Kolker
paul
Err, make that "universally valid."
>- paul
Chris Menzel
In pretty much any logic text in existence, a tautology is a sentence in
the language of propositional logic that is true regardless of the
assignment of truth values to its atomic components. "Tautology" used
in any other way, in the context of mathematical logic, is, well, wrong.
The more general notion that covers both propositional logic and
first-order (and higher-order) logic is that of a logical truth, i.e., a
sentence of a given language that is true in all interpretations of the
language. So, alternatively, a tautology is a logical truth of
propositional logic.
Chris Menzel
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
Owen
"paul" <paul...@on-ramp.nl> wrote in message
news:j1j9r0hnmc3kb0m15...@4ax.com...
This post may be of interest to you.
Owen
Stephen Harris
These are the definitions:
http://www.earlham.edu/~peters/courses/logsys/glossary.htm
"This glossary is limited to basic set theory, basic recursive function
theory,
two branches of logic (truth-functional propositional logic and first-order
predicate logic) and their metatheory." ...
Tautology.
"A logically valid wff of truth-functional propositional logic. A compound
proposition that is true in every row of its truth table or in every
interpretation. See contingency; contradiction; logical validity; semantic
tautology; syntactic tautology.
Tautology schema (plural: schemata). A formula containing variables of the
metalanguage which becomes a tautology when the variables are instantiated
to wffs of the formal language.
Semantic tautology.
A wff of truth-functional propositional logic whose truth table column
contains
nothing but T's when these are interpreted as the truth-value Truth.
See syntactic tautology.
Syntactic tautology.
A wff of truth-functional logic whose truth table column contains nothing
but
T's when these T's are uninterpreted tokens rather than, say, truth-values.
The rules for generating the truth table column tell us to use one of these
uninterpreted T's in exactly those cases where semantic considerations would
have led us to use the truth-value Truth. See semantic tautology."
paul
<cyberguard...@yahoo.com> wrote:
>
>"paul" <paul...@on-ramp.nl> wrote in message
>news:j1j9r0hnmc3kb0m15...@4ax.com...
>> On Mon, 06 Dec 2004 20:28:57 GMT, "robert j. kolker"
>> <now...@nowhere.com> wrote:
>>
>>>
>>>
>>>paul wrote:
>>>>
>>>> They're called "universal truths," not tautologies, in the
>>>> first-order predicate calculus.
>>>
>>>They are true under all standard interpretations. That is how truth
>>>tables are extended to the first order logics.
>>
>>
>> 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
>>
>
>These are the definitions:
Right, it restricts tautologies to propositional logic, like I said.
OTOH, formulae in first-order predicate logic that are true on all
valuations are called "universally valid."
Owen
"Owen" <oori...@yahoo.com> wrote in message
news:Nfydndlw-eM...@rogers.com...
Search Result 1
From: Owen Holden (oori...@yahoo.com)
Subject: Truth tables for monadic predicate logic
View: Complete Thread (3 articles)
Original Format
Newsgroups: sci.logic, sci.math
Date: 2002-08-10 02:21:50 PST
Given that propositions, p, q, r..etc., have two truth
values (T,F), logical truth (tautology) is confirmed if
the propositional expression has the value T for all
possibilities. It is contradictory if it has the
value F for all possibilities and it's contingent
if it has at least one T and at least one F for all
possibilities. My intension here is to extend the
concepts of truth by calculation to monadic predicate
logic. Truth by calculation for predicate logic in
determined domains reduces to propositional logic via
expansion of the quantifiers 'All' and 'Some'.
In undetermined or infinite domains I suggest the
following method of decision. A propositional form
with one free individual variable x, e.g. 'Fx' , is
monadic. ExFx means 'for some x Fx' . AxFx means
'for all x Fx'. The usual propositional logic operators
(not '~', or 'v', and '&', implies '->', equivalence '<->')
are maintained.
Fx ~Fx ExFx Ex~Fx AxFx Ax~Fx
__ ___ ____ _____ ____ _____
1 0 T 1 F 0 T 1 F 0
2 3 T 2 T 3 F 2 F 3
3 2 T 3 T 2 F 3 F 2
0 1 F 0 T 1 F 0 T 1
The following theorems are valid (true by deduction)
and tautologous (true by calculation).
(1) AxFx->ExFx (2) ~Ex~Fx<->AxFx (3) ~Ax~Fx<->ExFx
__________ _____________ _____________
T 1 T T 1 TF 0 T T 1 TF 0 T T 1
F 2 T T 2 FT 3 T F 2 TF 3 T T 2
F 3 T T 3 FT 2 T F 3 TF 2 T T 3
F 0 T F 0 FT 1 T F 0 FT 1 T F 0
(4)(AxFx & ExFx)<->AxFx (5) AxFx<->((ExFx->AxFx)<->ExFx)
(6)(AxFx v ExFx)<->ExFx (7) ExFx<->((ExFx->AxFx)<->AxFx)
(8)(Ex~Fx->Ax~Fx)<->(ExFx->AxFx)
etc.
Table 1. (these tables include propositional logic)
| ~ | E | A (not, some, all)
______________
T | F | T | T
F | T | F | F
1 | 0 | T | T
2 | 3 | T | F
3 | 2 | T | F
0 | 1 | F | F
Table 2.
v | T F 1 2 3 0 (or)
________________
T | T T 1 1 1 1
F | T F 1 2 3 0
1 | 1 1 1 1 1 1
2 | 1 2 1 2 1 2
3 | 1 3 1 1 3 3
0 | 1 0 1 2 3 0
The other tables result from the definitions.
p->q defined ~p v q (implies)
Table 3.
-> | T F 1 2 3 0
________________
T | T F 1 2 3 0
F | T T 1 1 1 1
1 | 1 1 1 1 1 1
2 | 1 3 1 1 3 3
3 | 1 2 1 2 1 2
0 | 1 1 1 1 1 1
p & q defined ~(~p v ~q) (and)
Table 4.
& | T F 1 2 3 0
_______________
T | T F 1 2 3 0
F | F F 0 0 0 0
1 | 1 0 1 2 3 0
2 | 2 0 2 2 0 0
3 | 3 0 3 0 3 0
0 | 0 0 0 0 0 0
p<->q defined (p->q)&(q->p) (equivalence)
Table 5.
<-> | T F 1 2 3 0
_________________
T | T F 1 2 3 0
F | F T 0 3 2 1
1 | 1 0 1 2 3 0
2 | 2 3 2 1 0 3
3 | 3 2 3 0 1 2
0 | 0 1 0 3 2 1
(8) Ax:Fx->ExFx (9) Ax:AxFx->Fx (10) Ax:(Fx->AxFx) v ExFx
___________ ___________ ____________________
T 1 1 T 1 T T 1 1 1 T 1 1 T 1 1 T 1
T 2 1 T 2 T F 2 1 2 T 2 3 F 2 1 T 2
T 3 1 T 3 T F 3 1 3 T 3 2 F 3 1 T 3
T 0 1 T 0 T F 0 1 0 T 0 1 F 0 1 F 0
(11) Ax:(ExFx->AxFx)->(ExFx->Fx)
(12) Ex:(Fx->AxFx) (13) Ex:(Fx->AxFx)
(14) Ex:((ExFx->Fx)->AxFx)<->ExFx)
(15) Ax:((ExFx->Fx)->AxFx)<->AxFx)
(16) Ax(Fx<->p)->(AxFx<->p)
(17) (AxFx<->p)->Ex(Fx<->p)
(19) Ex(p)<->p (20) Ax(p)<->p
Predicate logic tautologies with pure and prenex forms.
(20) Ex~Fx<->~AxFx (21) Ax~Fx<->~ExFx
(21) Ex(Fx v p)<->(ExFx v p)
_______________________
T 1 1 T T T 1 T T
T 2 1 T T T 2 T T
T 3 1 T T T 3 T T
T 0 1 T T F 0 T T
T 1 1 F T T 1 T F
T 2 2 F T T 2 T F
T 3 3 F T T 3 T F
F 0 0 F T F 0 F F
(22) Ax(Fx v p)<->(AxFx v p)
(23) Ex(Fx & p)<->(ExFx & p)
(24) Ax(Fx & p)<->(AxFx & p)
(25) Ex(p->Fx)<->(p->ExFx)
(26) Ax(p->Fx)<->(p->AxFx)
(27) Ex(Fx->p)<->(AxFx->p)
(28) Ax(Fx->p)<->(ExFx->p)
(29) Ex(Fx<->p)<->((AxFx->p)&(p->ExFx))
__________________________________
T 1 1 T T T 1 T T T TT T 1
T 2 2 T T F 2 T T T TT T 2
T 3 3 T T F 3 T T T TT T 3
F 0 0 T T F 0 T T F TF F 0
F 1 0 F T T 1 F F F FT T 1
T 2 3 F T F 2 T F T FT T 2
T 3 2 F T F 3 T F T FT T 3
T 0 1 F T F 0 T F T FT F 0
(30) Ax(Fx<->p)<->((ExFx->p)&(p->AxFx))
__________________________________
T 1 1 T T T 1 T T T TT T 1
F 2 2 T T T 2 T T F TF F 2
F 3 3 T T T 3 T T F TF F 3
F 0 0 T T F 0 T T F TF F 0
F 1 0 F T T 1 F F F FT T 1
F 2 3 F T T 2 F F F FT F 2
F 3 2 F T T 3 F F F FT F 3
T 0 1 F T F 0 T F T FT F 0
The last two theorems,(29) and (30), do not appear in
most logic texts. I believe they should appear to
complete the pure and prenex forms.
To deal with truth functions of two or more predicates,
which includes Boolean logic and Syllogistic logic,
we need to extend the truth tables. It will then become
evident that all of the axioms of propositional logic
and all of the axioms of general predicate logic are
tautologies.
Truth tables for two monadic predicates. These tables include
the above tables.
| ~ | E | A (not) (Some) (all)
_________________
T | F | T | T
F | T | F | F
15 | 14 | T | F
13 | 12 | T | F
11 | 10 | T | F
9 | 8 | T | F
7 | 6 | T | F
5 | 4 | T | F
3 | 2 | T | F
1 | 0 | T | T
0 | 1 | F | F
2 | 3 | T | F
4 | 5 | T | F
6 | 7 | T | F
8 | 9 | T | F
10 | 11 | T | F
12 | 13 | T | F
14 | 15 | T | F
v | T F 15 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14 (or)
_________________________________________________________
T | T T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
F | T F 15 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14
15 | 1 15 15 13 11 9 7 5 3 1 15 13 11 9 7 5 3 1
13 | 1 13 13 13 9 9 5 5 1 1 13 13 9 9 5 5 1 1
11 | 1 11 11 9 11 9 3 1 3 1 11 9 11 9 3 1 3 1
9 | 1 9 9 9 9 9 1 1 1 1 9 9 9 9 1 1 1 1
7 | 1 7 7 5 3 1 7 5 3 1 7 5 3 1 7 5 3 1
5 | 1 5 5 5 1 1 5 5 1 1 5 5 1 1 5 5 1 1
3 | 1 3 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1
1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 | 1 0 15 13 11 9 7 5 3 1 0 2 4 6 8 10 12 14
2 | 1 2 13 13 9 9 5 5 1 1 2 2 6 6 10 10 14 14
4 | 1 4 11 9 11 9 3 1 3 1 4 6 4 6 12 14 12 14
6 | 1 6 9 9 9 9 1 1 1 1 6 6 6 6 14 14 14 14
8 | 1 8 7 5 3 1 7 5 3 1 8 10 12 14 8 10 12 14
10 | 1 10 5 5 1 1 5 5 1 1 10 10 14 14 10 10 14 14
12 | 1 12 3 1 3 1 3 1 3 1 12 14 12 14 12 14 12 14
14 | 1 14 1 1 1 1 1 1 1 1 14 14 14 14 14 14 14 14
(32) Ax(Fx->Gx)->(AxFx->AxGx)
(33) Ax(Fx->Gx)->(ExFx->ExGx)
(34) Ax(Fx<->Gx)->(AxFx<->AxGx)
(35) Ax(Fx<->Gx)->(ExFx<->ExGx)
(36) Ax(Fx & Gx)<->(AxFx & AxGx)
(37) Ex(Fx v Gx)<->(ExFx v ExGx)
(38) (AxFx v AxGx)->Ax(Fx v Gx)
(39) Ex(Fx & Gx)->(ExFx & ExGx)
(40) (Ax(Fx->Gx)->Ex(Fx & Gx))<->ExFx
(41) ~ExFx->Ax(Fx->Gx) (42) AxGx->Ax(Fx->Gx)
(43) (Ex(Fx & Gx) & Ex(Fx & ~Gx))->AxFx is invalid
(44) (Ex(Fx & ~Gx) & Ex(~Fx & Gx))->Ax(Fx v Gx) is invalid
(45) (Ex(Fx & Gx) & Ex(Fx & ~Gx) & Ex(~Fx & Gx))->Ax(Fx v Gx)
is invalid.
etc.
Proofs of validity and invalidity of monadic predicate logic
expressions with three predicates requires truth tables that
have 64 function values.
Any opinions?
Owen Holden
------------------------------------------------------------------------------
> Owen
>
>
>
>
>
>
Stephen Harris
> On Tue, 07 Dec 2004 00:17:46 GMT, "Stephen Harris"
> <cyberguard...@yahoo.com> wrote:
>
>>
>>"paul" <paul...@on-ramp.nl> wrote in message
>>news:j1j9r0hnmc3kb0m15...@4ax.com...
>>> On Mon, 06 Dec 2004 20:28:57 GMT, "robert j. kolker"
>>> <now...@nowhere.com> wrote:
>>>
>>>>
>>>>
>>>>paul wrote:
>>>>>
>>>>> They're called "universal truths," not tautologies, in the
>>>>> first-order predicate calculus.
>>>>
>>>>They are true under all standard interpretations. That is how truth
>>>>tables are extended to the first order logics.
>>>
>>>
>>> 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
>>>
>>
>>These are the definitions:
>
>
>
> Right, it restricts tautologies to propositional logic, like I said.
> OTOH, formulae in first-order predicate logic that are true on all
> valuations are called "universally valid."
>
>
http://www.carleton.ca/iis/TechReports/files/2004-09.pdf
pages 16 through 19
It wasn't conclusive for me.
Stephen Harris
> On Tue, 07 Dec 2004 00:17:46 GMT, "Stephen Harris"
> <cyberguard...@yahoo.com> wrote:
>
>>
>>"paul" <paul...@on-ramp.nl> wrote in message
>>news:j1j9r0hnmc3kb0m15...@4ax.com...
>>> On Mon, 06 Dec 2004 20:28:57 GMT, "robert j. kolker"
>>> <now...@nowhere.com> wrote:
>>>
>>>>
>>>>
>>>>paul wrote:
>>>>>
>>>>> They're called "universal truths," not tautologies, in the
>>>>> first-order predicate calculus.
>>>>
>>>>They are true under all standard interpretations. That is how truth
>>>>tables are extended to the first order logics.
>>>
>>>
>>> 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
>>>
>>
>>These are the definitions:
>
>
>
> Right, it restricts tautologies to propositional logic, like I said.
> OTOH, formulae in first-order predicate logic that are true on all
> valuations are called "universally valid."
>
>
According to this quote, tautological and valid mean the same
thing: www.math.lsa.umich.edu/~ablass/llc9.ps (page 4)
but it does not say "universally valid".
"...then, remembering that implication is treated as an abbreviation and
negations are to be pushed in to the atomic level, we find that the
universally quantified variables are the y in the antecedent and the x
and z in the consequent. Thus, the Herbrand form (/)H is (if we use
u; v; w as the new variables in the consequent at step 1)
P (x; Y (x); z) --> Q(U; v; W (v)):
In stating Herbrand's Theorem, we use the usual terminology:
"Tautology'' means valid in propositional logic. The statement of the
theorem also involves the notion of a ``closed instance'' of a formula.
Since three other notions of ``instance'' will play a role later in the
paper, we define all four notions here to avoid confusion."
Definition 3.5
A closed instance of a first-order formula (/) is
obtained by substituting closed terms for all the free variables in (/).
A first-order instance of a propositional formula (/) is obtained by
replacing the propositional variables in (/) by first-order sentences.
Stephen Harris
"Stephen Harris" <cyberguard...@yahoo.com> wrote in message
news:KE6td.39426$6q2....@newssvr14.news.prodigy.com...
>
> "paul" <paul...@on-ramp.nl> wrote in message
> news:j1j9r0hnmc3kb0m15...@4ax.com...
>> On Mon, 06 Dec 2004 20:28:57 GMT, "robert j. kolker"
>> <now...@nowhere.com> wrote:
>>
>>>
>>>
>>>paul wrote:
>>>>
>>>> They're called "universal truths," not tautologies, in the
>>>> first-order predicate calculus.
>>>
>>>They are true under all standard interpretations. That is how truth
>>>tables are extended to the first order logics.
>>
>>
>> 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
>>
>
> These are the definitions:
>
>
> http://www.earlham.edu/~peters/courses/logsys/glossary.htm
Logical validity.
For a wff, to be true for every interpretation of the formal language; to
have every interpretation be a model. "Every interpretation" here is
understood to mean all, but only, those interpretations in which the
connectives and/or quantifiers take their standard meanings. In
truth-functional propositional logic, logically valid wffs are also called
tautologies. In standard predicate logic, logical validity is limited to
interpretions with non-empty domains. Logical validity is also called
logical truth.
G. Frege
> >
> > This post may be of interest to you.
> >
----------------------------------------------------------------------
The following text is based on a proposal by Owen (oori...@yahoo.com)
----------------------------------------------------------------------
MONADIC PREDICATE LOGIC WITH TRUTH-VALUES
Introduction
------------
Given that propositions, p, q, r,... etc., have two truth
values {T,F}, logical truth (tautology) is confirmed if
the propositional expression has the value T for all
possibilities [...]. It is contradictory if it has the
value F for all possibilities [...] and it's contingent
if it has at least one T and at least one F for all
possibilities [...]. My intention here is to extend the
concepts of truth by calculation to monadic predicate
logic. Truth by calculation for predicate logic in
determined domains reduces to propositional logic via
expansion of the quantifiers 'Ax' and 'Ex'. In undeter-
mined or infinite domains I suggest the following
method of decision.
Some Stipulations
-----------------
A propositional form consisting of predicate letter
and a free individual variable, e.g. 'Fx', is monadic.
ExFx means 'for some x, Fx'. AxFx means 'for all x, Fx'.
The usual connectives of propositional logic (not '~',
or 'v', and '&', implies '->', equivalence '<->') are
maintained.
We now introduce the following set of formal truth-values
for monadic predicates: {T, t, f, F}. T means that the
predicate holds for any value in our domain, t that the
predicate holds for some values (but not all), f that the
predicate doesn't hold for some values (but at least for one),
and F that there's no value in our domain such that the
predicate holds.
Note: If the truth-values T,F appear in connection with
statements (i.e. closed formulas), they are meant to denote
the usual "standard" truth-values of propositional logic.
Basic Truth-Tables
------------------
Fx | ~Fx ExFx Ex~Fx AxFx Ax~Fx
-- | --- ---- ----- ---- ----- (where Fx is atomic)
T | F T T F F T T F F
t | f T t T f F t F f
f | t T f T t F f F t
F | T F F T T F F T T
Basic Theorems
--------------
The following statements are theorems (true by deduction)
as well as tautologies (true by calculation).
(1) AxFx->ExFx (2) ~Ex~Fx<->AxFx (3) ~Ax~Fx<->ExFx
---------- ------------- -------------
T T T T T TF F T T T TF F T T T
F t T T t FT f T F t TF f T T t
F f T T f FT t T F f TF t T T f
F F T F F FT T T F F FT T T F F
(4)(AxFx & ExFx)<->AxFx
(5) AxFx<->((ExFx->AxFx)<->ExFx)
(6)(AxFx v ExFx)<->ExFx
(7) ExFx<->((ExFx->AxFx)<->AxFx)
(8)(Ex~Fx->Ax~Fx)<->(ExFx->AxFx)
etc.
Compound statements
-------------------
a) Compound statements with at most one predicate
The following tables include propositions.
Table 1.
A | ~A | ExA | AxA (not, some, all)
--------------------
T | F | T | T
t | f | T | F
f | t | T | F
F | T | F | F
Table 2.
v | T t f F (or)
-------------
T | T T T T
t | T t T t
f | T T f f
F | T t f F
The other tables result from the usual definitions.
A->B =df ~A v B (implies)
Table 3.
-> | T t f F
--------------
T | T t f F
t | T T f f
f | T t T t
F | T T T T
A & B =df ~(~A v ~B) (and)
Table 4.
& | T t f F
-------------
T | T t f F
t | t t F F
f | f F f F
F | F F F F
A<->B =df (A->B)&(B->A) (equivalence)
Table 5.
<-> | T t f F
---------------
T | T t f F
t | t T F f
f | f F T t
F | F f t T
Theorems with at most one predicate.
(9) Ax(Fx->ExFx) (10) Ax(AxFx->Fx) (11) Ax((Fx->AxFx) v ExFx)
----------- ----------- --------------------
T T T T T T T T T T T T T T T T T T
T t T T t T F t T t T t f F t T T t
T f T T f T F f T f T f t F f T T f
T F T F F T F F T F T F T F F T F F
(12) Ax((ExFx->AxFx)->(ExFx->Fx))
(13) Ex((Fx->AxFx))
(14) Ax((Fx->AxFx))
(15) Ex(((ExFx->Fx)->AxFx)<->ExFx))
(16) Ax(((ExFx->Fx)->AxFx)<->AxFx))
(17) Ax(Fx<->p)->(AxFx<->p)
(18) (AxFx<->p)->Ex(Fx<->p)
(19) Ex(p)<->p
(20) Ax(p)<->p
Monatic predicate logic tautologies with pure and prenex forms.
(21) Ex~Fx<->~AxFx
(22) Ax~Fx<->~ExFx
(23) Ex(Fx v p)<->(ExFx v p)
-----------------------
T T T T T T T T T
T t T T T T t T T
T f T T T T f T T
T F T T T F F T T
T T T F T T T T F
T t t F T T t T F
T f f F T T f T F
F F F F T F F F F
(24) Ax(Fx v p)<->(AxFx v p)
(25) Ex(Fx & p)<->(ExFx & p)
(26) Ax(Fx & p)<->(AxFx & p)
(27) Ex(p->Fx)<->(p->ExFx)
(28) Ax(p->Fx)<->(p->AxFx)
(29) Ex(Fx->p)<->(AxFx->p)
(30) Ax(Fx->p)<->(ExFx->p)
(31) Ex(Fx<->p)<->((AxFx->p)&(p->ExFx))
----------------------------------
T T T T T T T T T T TT T T
T t t T T F t T T T TT T t
T f f T T F f T T T TT T f
F F F T T F F T T F TF F F
F T F F T T T F F F FT T T
T t f F T F t T F T FT T t
T f t F T F f T F T FT T f
T F T F T F F T F T FT F F
(32) Ax(Fx<->p)<->((ExFx->p)&(p->AxFx))
----------------------------------
T T T T T T T T T T TT T T
F t t T T T t T T F TF F t
F f f T T T f T T F TF F f
F F F T T F F T T F TF F F
F T F F T T T F F F FT T T
F t f F T T t F F F FT F t
F f t F T T f F F F FT F f
T F T F T F F T F T FT F F
The last two theorems, (31) and (32), do not appear in
most logic texts. I believe they should appear to
complete the pure and prenex forms.
b) Compound statements with more than one predicate
To deal with truth functions of two or more predicates,
which includes Boolean logic and Syllogistic logic,
we need to extend the truth tables. It will then become
evident that all of the axioms of propositional logic
and all of the axioms of monadic predicate logic are
tautologies.
Truth tables for two monadic predicates.
A | ~A | ExA | AxA (not) (some) (all)
----------------------
a | a' | T | F
b | b' | T | F
c | c' | T | F
d | d' | T | F
e | e' | T | F
f | f' | T | F
g | g' | T | F
T | F | T | T
F | T | F | F
g'| g | T | F
f'| f | T | F
e'| e | T | F
d'| d | T | F
c'| c | T | F
b'| b | T | F
a'| a | T | F
v | a b c d e f g T F g' f' e' d' c' b' a' (or)
----------------------------------------------------
a | a b c d e f g T a b c d e f g T
b | b b d d f f T T b b d d f f T T
c | c d c d g T g T c d c d g T g T
d | d d d d T T T T d d d d T T T T
e | e f g T e f g T e f g T e f g T
f | f f T T f f T T f f T T f f T T
g | g T g T g T g T g T g T g T g T
T | T T T T T T T T T T T T T T T T
F | a b c d e f g T F g' f' e' d' c' b' a'
g'| b b d d f f T T g' g' e' e' c' c' a' a'
f'| c d c d g T g T f' e' f' e' b' a' b' a'
e'| d d d d T T T T e' e' e' e' a' a' a' a'
d'| e f g T e f g T d' c' b' a' d' c' b' a'
c'| f f T T f f T T c' c' a' a' c' c' a' a'
b'| g T g T g T g T b' a' b' a' b' a' b' a'
a'| T T T T T T T T a' a' a' a' a' a' a' a'
& | a b c d e f g T F g' f' e' d' c' b' a' (and)
----------------------------------------------------
a | a a a a a a a a F F F F F F F F
b | a b a b a b a b F g' F g' F g' F g'
c | a a c c a a c c F F f' f' F F f' f'
d | a b c d a b c d F g' f' e' F g' f' e'
e | a a a a e e e e F F F F d' d' d' d'
f | a b a b e f e f F g' F g' d' c' d' c'
g | a a c c e e g g F F f' f' d' d' b' b'
T | a b c d e f g T F g' f' e' d' c' b' a'
F | T F F F F F F F F F F F F F F F
g'| T g' F g' F g' F g' F g' F g' F g' F g'
f'| T F f' f' F F f' f' F F f' f' F F f' f'
e'| T g' f' e' F g' f' e' F g' f' e' F g' f' e'
d'| T F F F d' d' d' d' F F F F d' d' d' d'
c'| T g' F g' d' c' d' c' F g' F g' d' c' d' c'
b'| T F f' f' d' d' b' b' F F f' f' d' d' b' b'
a'| T g' f' e' d' c' b' a' F g' f' e' d' c' b' a'
etc.
Theorems with two predicates.
(33) Ax(Fx->Gx)->(AxFx->AxGx)
(34) Ax(Fx->Gx)->(ExFx->ExGx)
(35) Ax(Fx<->Gx)->(AxFx<->AxGx)
(36) Ax(Fx<->Gx)->(ExFx<->ExGx)
(37) Ax(Fx & Gx)<->(AxFx & AxGx)
(38) Ex(Fx v Gx)<->(ExFx v ExGx)
(39) (AxFx v AxGx)->Ax(Fx v Gx)
(40) Ex(Fx & Gx)->(ExFx & ExGx)
(41) (Ax(Fx->Gx)->Ex(Fx & Gx))<->ExFx
(42) ~ExFx->Ax(Fx->Gx)
(43) AxGx->Ax(Fx->Gx)
Invalid statements with two predicates.
(44) (Ex(Fx & Gx) & Ex(Fx & ~Gx))->AxFx is invalid
(45) (Ex(Fx & ~Gx) & Ex(~Fx & Gx))->Ax(Fx v Gx) is invalid
(46) (Ex(Fx & Gx) & Ex(Fx & ~Gx) & Ex(~Fx & Gx))->Ax(Fx v Gx)
is invalid.
etc.
Proofs of validity and invalidity of monadic predicate logic
expressions with three predicates requires truth tables that
have 64 function values.
--------------
F.
Chris Menzel
>> http://www.earlham.edu/~peters/courses/logsys/glossary.htm
>
> Logical validity.
> For a wff, to be true for every interpretation of the formal language; to
> have every interpretation be a model. "Every interpretation" here is
> understood to mean all, but only, those interpretations in which the
> connectives and/or quantifiers take their standard meanings. In
> truth-functional propositional logic, logically valid wffs are also called
> tautologies. In standard predicate logic, logical validity is limited to
> interpretions with non-empty domains. Logical validity is also called
> logical truth.
Is there an echo in here?
;-)
paul
>>
>> http://www.earlham.edu/~peters/courses/logsys/glossary.htm
>
>Logical validity.
>For a wff, to be true for every interpretation of the formal language; to
>have every interpretation be a model. "Every interpretation" here is
>understood to mean all, but only, those interpretations in which the
>connectives and/or quantifiers take their standard meanings. In
>truth-functional propositional logic, logically valid wffs are also called
>tautologies. In standard predicate logic, logical validity is limited to
>interpretions with non-empty domains. Logical validity is also called
>logical truth.
I don't know why you keep posting quotes that support what I said --
that the term "tautology" is not applied outside propositional logic
(Owen's analysis notwithstanding) -- as if they were counter examples
to what I said. Here's a quote from a chapter on predicate logic:
"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: Volume 1, Introduction to
Logic." The University of Chicago Press. p. 99.
-paul
Stephen Harris
> On Tue, 07 Dec 2004 23:58:40 GMT, "Stephen Harris"
> <cyberguard...@yahoo.com> wrote:
>
>>>
>>> http://www.earlham.edu/~peters/courses/logsys/glossary.htm
>>
>
>
> that the term "tautology" is not applied outside propositional logic
> (Owen's analysis notwithstanding) -- as if they were counter examples
> to what I said. Here's a quote from a chapter on predicate logic:
"But some sentence schemata in Predicate Logic--like the tautologies of
Sentential Logic--are true no matter what their interpretation--that is,
they are logically true, or analytic . (Tautologies are a specific sub-class
of analytically true sentence schemata--ones which are truth functional.)
Like tautologies, the analytic sentence schemata of Predicate Logic are made
true by their structure and the definitions of their logical connectives and
quantifiers rather than their content."
>
>> 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.
>>
"(Tautologies are a specific sub-class of analytically true sentence
schemata--ones which are truth functional.) Like tautologies, the analytic
sentence schemata of Predicate Logic are made true by their structure and
the definitions of their logical connectives and quantifiers rather than
their content."
>> - paul
>>
SH: It would seem there is conceptual similarity between tautologies
and the validity used in predicate logic. Though this seems not to extend
to truth tables. I wondered about this statement:
> "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)."
SH: Does this mean they can be called tautologies in some instances?
"But some sentence schemata in Predicate Logic--like the tautologies of
Sentential Logic--are true no matter what their interpretation--that is,
they
are logically true, or analytic."
SH: In terms of decidability it seems you are right. Bob did not defend.
I think you are right about truth tables (unless there is something
technical),
but I'm not sure about this claim:
> that the term "tautology" is not applied outside propositional logic
does not seem to be fully supported by your own quote:
> formulas (they are not normally called tautologies)."
SH: Should read they are not ________ called tautologies, without the
"normally" qualifier which tends to mean there are exceptions.
The larger context of the quote used above.
http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
Analytic Sentences & Valid Arguments in 1st order Predicate Logic
from Supplement for Symbolic Logic (c) Boardman
"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; no line of its truth table is false, each line
sketching a class of interpretations (classified according to the
combinations of truth values of the sub-sentences represented by the
sentential letters). For a tautology is true independently of the content
which its sentential letters might represent: its structure and the
truth-functional definitions of its logical connectives are what make it
true.
The sentence schemata in Predicate Logic are used to represent sentences
which make claims about the individuals inhabiting some universe of
discourse--the subset of the world which we choose to represent--and their
properties and relations to one another. The typical sentence schema is
contingently true (or false), its truth value depending upon whether the
assertion the interpretation assigns to it happens to be true in the chosen
universe of discourse. But some sentence schemata in Predicate Logic--like
the tautologies of Sentential Logic--are true no matter what their
interpretation--that is, they are logically true, or analytic . (Tautologies
are a specific sub-class of analytically true sentence schemata--ones which
are truth functional.) Like tautologies, the analytic sentence schemata of
Predicate Logic are made true by their structure and the definitions of
their logical connectives and quantifiers rather than their content."
Stephen Harris
>
> "paul" <paul...@on-ramp.nl> wrote in message
> news:r58er0t5vm620c96l...@4ax.com...
>> On Tue, 07 Dec 2004 23:58:40 GMT, "Stephen Harris"
>> <cyberguard...@yahoo.com> wrote:
>>
> I think you are right about truth tables (unless there is something
> technical),
> but I'm not sure about this claim:
>
>> that the term "tautology" is not applied outside propositional logic
>
> does not seem to be fully supported by your own quote:
>> formulas (they are not normally called tautologies)."
>
> SH: Should read they are not ________ called tautologies, without the
> "normally" qualifier which tends to mean there are exceptions.
>
For tautologies there is a general method for showing intrinsic truth,
a truth table. There is no general method for showing the intrinsic
truth of valid statements. A general algorithm for proving a formula to
be valid is not possible. But I'm wondering if in a particular class of
cases, if there is a specific algorithm for proving formulae to be valid,
which would function in principle like a constrained truth table.
Stephen Harris
>> "paul" <paul...@on-ramp.nl> wrote in message
>> news:r58er0t5vm620c96l...@4ax.com...
>>> On Tue, 07 Dec 2004 23:58:40 GMT, "Stephen Harris"
>>> <cyberguard...@yahoo.com> wrote:
>>>
>> I think you are right about truth tables (unless there is something
>> technical),
There appears to be "truth functional proxies".
> For tautologies there is a general method for showing intrinsic truth,
> a truth table. There is no general method for showing the intrinsic
> truth of valid statements. A general algorithm for proving a formula to
> be valid is not possible. But I'm wondering if in a particular class of
> cases, if there is a specific algorithm for proving formulae to be valid,
> which would function in principle like a constrained truth table.
Truth functional proxies.
>
As to using logic for reasoning about natural language and common sense.
Talking about Trees and Truth-conditions
Reinhard Muskens http://www.illc.uva.nl/j50/contribs/muskens/muskens.pdf
Abstract
"An attractive way to model the relation between an underspecified
syntactic representation and its completions is to let the underspecified
representation correspond to a logical description and the completions to
the models of that description. This approach, which underlies the
Description Theory of (Marcus et al. 1983) was integrated with a pure
unification approach to Lexicalized Tree-Adjoining Grammars (Joshi et al.
1975, Schabes 1990) in (Vijay-Shanker 1992) and was further developed in
the `D-Tree Grammars' (DTG) of (Rambow et al. 1995). We generalize
Description Theory by integrating semantic information, that is, we propose
to tackle both syntactic and semantic underspecification using descriptions.
Our focus will be on underspecification of scope. We use a generalized and
completely declarative version of the D-Tree formalism. Although trees in
our set-up have surface strings at their leaves and are in fact very close
to ordinary surface trees, there is also a strong connection with the
Logical
Forms (LFs) of (May 1977). We associate logical interpretations with these
LFs using a technique of internalising the logical binding mechanism
(Muskens
1996). The net result is that we obtain a Description Theory-like grammar in
which the descriptions underspecify semantics. Since everything is framed in
classical logic it is easily possible to reason with these descriptions.
Internalising Binding
How can we assign a semantics to the lexical descriptions in fig. 1? We must
e.g. be able to express the semantics of n1 in terms of the semantics of n2,
whatever the latter turns out to be, i.e. we must be able to express the
result of quantification into an arbitrary context. In mathematical English
we can say that, for any @, the value of allx@ is the set of assignments a
such that for all b differing from a at most in x, b is an element of the
value of @. We need to be able to say something similar in our logical
language. The language must talk about meaning; it must talk about things
that function like variables and constants, things that function like
assignments, etc. The first will be called registers, the second states. Two
primitive types are added to the logic: Pi and s, for registers and states
respectively. We shall have variable registers, which stand proxy for
variables and constant registers for constants. ...
We have essentially mimicked the Tarski truth conditions for predicate logic
in our object language and in fact it can be proved that, under certain
conditions, we can reason with terms generated in this way as if they were
**the predicate logical formulas they stand proxy for (see Muskens 1998).
It should be stressed that the technique discussed here can be used to
embed any logic with a decent interpretation into classical logic. For
example, (Muskens 1996) shows that we can use the same mechanism to embed
Discourse Representation Theory (DRT, Kamp & Reyle 1993). In a full version
of this paper we shall also present a version of our theory based
on DRT."
SH: This indicates to me that if predicate logic can be embedded, then
so can a simpler aspect, truth functional proxies, akin to propositional
logic which would help logical reasoning about natural language input.
I found researching this thread from vague memories quite interesting.
Regards,
Stephen
Stephen Harris
> "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
> Regards,
> Stephen
>
>
Stephen Harris
"Chris Menzel" <cme...@remove-this.tamu.edu> wrote in message
news:slrncr9hto....@philebus.tamu.edu...
> On Mon, 06 Dec 2004 19:43:18 GMT, robert j. kolker <now...@nowhere.com>
> said:
>> George Cox wrote:
>>
>>> For me (am I alone?) a tautology (in the logical sense) is a formula of
>>> propositional calculus which is true for all values of the truth values
>>> of its constituent atomic letters.
>>
>> Theorems of first order logic are also tautologies.
>
SH: Not the theorems, it seems like.
> In pretty much any logic text in existence, a tautology is a sentence in
> the language of propositional logic that is true regardless of the
> assignment of truth values to its atomic components. "Tautology" used
> in any other way, in the context of mathematical logic, is, well, wrong.
> The more general notion that covers both propositional logic and
> first-order (and higher-order) logic is that of a logical truth, i.e., a
> sentence of a given language that is true in all interpretations of the
> language. So, alternatively, a tautology is a logical truth of
> propositional logic.
>
> Chris Menzel
>
Would you comment on these quotes? * is my emphasis.. Particularly
"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.*"
Paul wrote:
>>> 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.
http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
"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."
SH: I quoted Peter Suber because your fame has not preceded you to
my limited knowledge of who is a quotable authority in logic.
Regards,
Stephen
Stephen Harris
> On Tue, 07 Dec 2004 00:17:46 GMT, "Stephen Harris"
> <cyberguard...@yahoo.com> wrote:
>
>>
>>"paul" <paul...@on-ramp.nl> wrote in message
>>news:j1j9r0hnmc3kb0m15...@4ax.com...
>>> On Mon, 06 Dec 2004 20:28:57 GMT, "robert j. kolker"
>>> <now...@nowhere.com> wrote:
>>>
>>>>
>>>>
>>>>paul wrote:
>>>>>
>>>>> first-order predicate calculus.
>>>>
Bob:
>>>>They are true under all standard interpretations. That is how truth
>>>>tables are extended to the first order logics.
>>>
>>>
>>> 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.
>>>
"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
Chris Menzel
> ...
>> In pretty much any logic text in existence, a tautology is a sentence in
>> the language of propositional logic that is true regardless of the
>> assignment of truth values to its atomic components. "Tautology" used
>> in any other way, in the context of mathematical logic, is, well, wrong.
>> The more general notion that covers both propositional logic and
>> first-order (and higher-order) logic is that of a logical truth, i.e., a
>> sentence of a given language that is true in all interpretations of the
>> language. So, alternatively, a tautology is a logical truth of
>> propositional logic.
>>
>> Chris Menzel
>
> Would you comment on these quotes? * is my emphasis..
No, I'll comment on the fact that you want me to comment on them.
Apparently you think these quotes show that truth tables applied to
"truth-functional proxies" of quantified argument schemas provide a
*general* method for testing first-order validity. They don't. 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. This information
is pretty much in the link you provide:
> http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
Have a look at the penultimate paragraph and footnote 4.
> SH: I quoted Peter Suber because your fame has not preceded you to
> my limited knowledge of who is a quotable authority in logic.
Shows what you know. ;-)
Chris Menzel
Stephen Harris
> On Thu, 09 Dec 2004 11:38:06 GMT, Stephen Harris said:
>> ...
>>> In pretty much any logic text in existence, a tautology is a sentence in
>>> the language of propositional logic that is true regardless of the
>>> assignment of truth values to its atomic components. "Tautology" used
>>> in any other way, in the context of mathematical logic, is, well, wrong.
>>> The more general notion that covers both propositional logic and
>>> first-order (and higher-order) logic is that of a logical truth, i.e., a
>>> sentence of a given language that is true in all interpretations of the
>>> language. So, alternatively, a tautology is a logical truth of
>>> propositional logic.
>>>
>>> Chris Menzel
>>
>> Would you comment on these quotes? * is my emphasis..
>
> No, I'll comment on the fact that you want me to comment on them.
> Apparently you think these quotes show that truth tables applied to
> "truth-functional proxies" of quantified argument schemas provide a
> *general* method for testing first-order validity. They don't.
No. Read what I wrote and tell me if you think I meant *general*.
Stephen wrote in a previous post:
"For tautologies there is a general method for showing intrinsic truth,
a truth table. There is no general method for showing the intrinsic
truth of valid statements. A general algorithm for proving a formula to
be valid is not possible. But I'm wondering if in a particular class of
cases, if there is a specific algorithm for proving formulae to be valid,
which would function in principle like a constrained truth table."
SH: I used the term "valid" to apply to predicate logic (PL) because
paul stated that tautology was not used in PL, but "universally valid".
I think that from the context it is clear that I'm distinguishing
propositional
logic from PL: "there is a general method" vs. "there is no general method".
I also wrote the above quote before finding the analytic_essay link.
> 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. This information
> is pretty much in the link you provide:
>
>> http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
>
> Have a look at the penultimate paragraph and footnote 4.
>
I read the whole essay. Which is why I changed my mind about Paul being
right about the truth tables; I had doubts about it and kept researching.
Paul wrote:
>>> 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.
SH: I don't think Paul's statement is correct because there is an implicit
"none" asserted and as you and the analytic_essay state, there is "some".
Paul also wrote: ... "...I said -- that the term "tautology" is not applied
outside propositional logic."
SH: But I don't think that can be exactly correct either in part because
Paul posted,
"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: Volume 1, Introduction to
Logic." The University of Chicago Press. p. 99.
SH: I don't think the author 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".
I'm not actually faulting paul, because this exception seems rather
technical.
But I wondered if there were a possible practical use for this 'sometimes'
situation which would elevate my objection from nitpicking to relevancy.
I inkled that this sometime "shortcut method" might be usable in a natural
language translation program. They have appropriate ruled response filters
that are treelike. Maybe I have read this connection, I can't remember.
Do you have a comment about the practical use for constructing "some"
cases using the "short-cut method of truth value assignment, or by means of
a truth table?" I usually read about chance, causation, and counterfactuals
with a whiff of Smullyan, "What is the name of this book?..." By the way,
What is the name of your book?
Stephen
paul
<cyberguard...@yahoo.com> wrote:
>>> http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
>>
>> Have a look at the penultimate paragraph and footnote 4.
>>
>
>I read the whole essay. Which is why I changed my mind about Paul being
>right about the truth tables; I had doubts about it and kept researching.
You mean paul's professors. As I wrote: "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."
Owen's apparently new analysis notwithstanding, can you or someone
cite a logic textbook that uses truth tables in predicate logic? I'm
not refuting that such a text exists, I'd just be interested to know.
>Paul also wrote: ... "...I said -- that the term "tautology" is not applied
>outside propositional logic."
>
>SH: But I don't think that can be exactly correct either in part because
>Paul posted,
>
>"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: Volume 1, Introduction to
>Logic." The University of Chicago Press. p. 99.
>
>SH: I don't think the author 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".
>
>I'm not actually faulting paul, because this exception seems rather
>technical.
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.
You ought to just post a genuine counter example to the limit rather
than every ambiguous description you find. Which of course is not to
say that even some .edu site does not have posted notes that say
something that's normally considered inappropriate. That's why I'm
asking for TEXTBOOK sources, which are most likely less likely to err.
You should not assume you can get the best logic education via google.
- paul
Stephen Harris
> On Thu, 09 Dec 2004 23:40:28 GMT, "Stephen Harris"
> <cyberguard...@yahoo.com> wrote:
>
>>>> http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
>>>
>>> Have a look at the penultimate paragraph and footnote 4.
>>>
>>
>>I read the whole essay. Which is why I changed my mind about Paul being
>>right about the truth tables; I had doubts about it and kept researching.
>
>
> You mean paul's professors. As I wrote: "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."
>
> Owen's apparently new analysis notwithstanding, can you or someone
> cite a logic textbook that uses truth tables in predicate logic? I'm
> not refuting that such a text exists, I'd just be interested to know.
>
>
Boardman teaches logic and provides his class with this supplement.
Menzel teaches logic and agreed with Boardman's description.
Menzel: " In *some* cases they [truth tables applied to "truth-functional
proxies" of quantified argument schemas providing a method for testing
first-order validity] 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. This information
is pretty much in the link you provide:
Boardman: "Because it is comprised of truth functional sentence schemata, a
proxy maybe tested for validity by the short-cut method of truth value
assignment, or by *means of a truth table.*"
> http://www.lawrence.edu/fast/boardmaw/analytic_essay.html
SH: Unless you think both of these professors arrived at the same wrong
conclusion by some unlikely means, it means there should be a graduate
textbook or supplemental text establishing that in "some" cases truth
tables can be applied to "monadic predicate logic". Actually, with a little
research, this idea appears fairly well known.
Decidability and Computability of Logic
------
muxol wrote: http://forums.philosophyforums.com/showthread.php?t=9394
"The propositional calculus/logic is decidable because we have a decision
procedure, truth tables, which tells us whether every formula of the
calculus
is valid or satisfiable, or not.
The predicate calculus is undecidable because there is no such decision
procedure. There are exceptions however. There are subclasses of predicate
logic that are decidable. One of them happens to be the class of all
formulas
in which only monadic (1-place) predicates occur - in other words, monadic
predicate logic.
The question is, why is it that monadic predicate logic is decidable while
polyadic predicate logic isn't? Can someone provide and explain a decision
procedure for monadic predicate logic. Or alternatively, can someone provide
an algorithm?(Decision procedures and algorithms are essentially
equivalent.)"
Weloki replied:
The answer is pretty lengthy, but here is a synopsis given by a professor
Curtis Brown: http://www.trinity.edu/cbrown/logic/decidability.html wrote:
"A yes-or-no question is decidable if there is a procedure that is
guaranteed
to give an answer to the question in a finite amount of time. A logical
system
such as a system of propositional logic or a system of first-order logic is
decidable if, for every argument expressible in the language of the system,
the question whether the argument is valid is decidable. ...
A system of propositional logic is decidable if the question whether an
argument is tautologically valid is decidable; a system of first-order logic
is decidable if the question whether an argument is FO-valid is decidable.)
First-order logic with only 1-place predicates ("monadic" FO logic) is
also decidable, although this is less obvious."
SH: I have a copy of "Introduction to Mathematical Logic" by Mendelson
On page 156: "If we accpet Church's Thesis, then "recursiveley undecidable"
can be replaced everywhere by "effectively undecidable". In particular
Proposition 3.47 asserts that there is no decision procedure for the
_pure predicate calculus_ (emphasis SH) PP, nor for the full predicate
calculus PF. This implies that there is no effective method for determining
whether any given wf is logically valid."
SH: I've already stated that there is no general procedure. More Mendelson
on bottom of page 156:
Exercise: "Show that, in contrast to Church's Theorem, the _pure monadic
predicate calculus_ is effectively decidable. The pure monadic predicate
calculus consists of those wfs of the pure predicate calculus which contain
only predicate letters of one argument."
SH: Then he gives a hint on how to prove this. And concludes:
"The result in this exercise is, in a sense, the best possible. For, by a
theorem of Kalmar [1936], there is an effective procedure producing,
for each wf A of the pure predicate calculus, another wf A* of the pure
predicate calculus such that A* contains only one predicate letter, a
binary one, such that A is valid, if and only if A* is valid."
> 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."
SH: This statement precludes some counter cases, exceptions to the rule.
Perhaps I should re-establish the relationship between an algorithm,
an effectively decidable procedure, and the functions of a truth table.
http://www.cs.duke.edu/~mlittman/courses/cps271/lect-03/node9.html
The most basic algorithm for categorizing formulae is the truth-table
algorithm.
http://users.telenet.be/TaoWeb/Language/TruthTable.htm
"A proofing algorithm (truth table) for Boolean logic
To prove that a valid conclusion e_n+1 can be derived from a number of
premises e1 ..., e_n:
1. Construct a table T1 with each premise and the conclusion as a
column.
2. Add all possible values for each primitive expression as a row.
3. Calculate the value for each unary operation using the following
rules:"
Regards,
Stephen
Owen
> On Sun, 5 Dec 2004 14:00:44 -0500, Owen <oori...@yahoo.com> said:
>>
>> "George Cox" <george_...@spambtinternet.com.invalid> wrote in
message
>> news:41B34C6C...@spambtinternet.com.invalid...
>> > Lester Zick wrote:
>> >>
>> >>... Mathematics is held to
>> >> be tautological, ...
>> >
>> > By whom? Anybody other than Wittgenstein?
>>
>> Frank Ramsay and others including me, etc. agree.
>>
>> That (1+1=2) is tautologous, ie. logically true, is clear.
>> Because of (Frege-Russell)'s analysis.
>
> Mind deriving that "tautology" so we can see exactly which logical
> axioms is follows from?
Russell and Whitehead have already done it. At what point in their proof,
*110.643 page 83 Volume 2, do you have a problem?
What axiom of Principia (except the axiom of infinity) is not logical?
paul
<cyberguard...@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
Stephen Harris
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
paul
<cyberguard...@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
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:
* "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
Chris Menzel
> 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.
Because it usefully picks out a certain class of logical truths, viz.,
those that are true simply in virtue of their truth functional
structure. If its meaning were broadened to include the logical truths
of predicate logic, it would serve no purpose. We already have "logical
truth" and "universal validity" (though the latter is rather less
common; indeed, I can only recall seeing it in the LTF Gamut text --
whose actual authors, BTW, are the frighteningly prolific Dutch logician
Johan van Benthem and a couple of his colleagues).
Stephen Harris
> On Sun, 12 Dec 2004 02:51:47 GMT, "Stephen Harris"
> <cyberguard...@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
Stephen Harris
"Logic: The Art of Persuasion and the Science of Truth"
I think by Vann McGee
"Monadic Predicate Calculus
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 same thing. 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."
---------------------------------------------------------
SH: You have provided similar statements to McGee who is
teaching an online class at MIT. He apparently considers it
important to mention truth tables and tautologies as part of
the background information leading into discussing FOPL.
I have found several references to Monadic Predicate logic
on the net which mention truth tables.
When you teach (or taught) this logic class which goes into PL,
did you provide the same background as McGee? Or did you
perhaps omit it (full discussion of MPL) because you thought it
might be confusing? What I'm trying to get at, is McGees
decision to include the material I quoted a fairly standard decision,
or would some qualified instructors choose to dismiss the material
that McGee deems important enought to include in his lectures, re MPL?
http://aka-ocw.mit.edu/OcwWeb/Linguistics-and-Philosophy/24-241Logic-IFall2002/Readings/index.htm
Introduction. The Place of Logic Among the Sciences (PDF)
Sentential Calculus Introduction (PDF)
Sentential Calculus Semantics (PDF)
Extension Theorem (PDF)
State Descriptions, Disjunctive Normal Form, and Expressive Completeness
(PDF)
SC Substitutions (PDF)
The Search-for-Counterexample Test for Validity (PDF)
Compactness Theorem (PDF)
SC Translations (PDF)
Trouble with "If"s (PDF)
Monadic Predicate Calculus (PDF) ***** lecture 11
Derivations in the Monadic Predicate Calculus (PDF) ***** lecture 12
Completeness in the Monadic Predicate Calculus (PDF)
Predicate Calculus (PDF)
Predicate Calculus Derivations (PDF)
Identity (PDF)
Russell's Theory of Definite Descriptions (PDF)
Sense and Reference (PDF)
Function Signs (PDF)
Sentential Calculus Revisited: Boolean Algebra (PDF)
Regards,
Stephen
Chris Menzel
> SH: You have provided similar statements to McGee who is
> teaching an online class at MIT. He apparently considers it
> important to mention truth tables and tautologies as part of
> the background information leading into discussing FOPL.
Well, truth tables are not theoretically necessary, but they provide
students with a nice gentle introduction to the ideas of formal
languages and their interpretation, and to the idea of logical truth --
which, as noted countless times now, is in this context indicated by
the term "tautology".
> I have found several references to Monadic Predicate logic
> on the net which mention truth tables.
That's because there is, theoretically, a way to USE truth tables to
determine validity in general for monadic predicate logic. But note
what seems to be a sticking point for you and paul. Although there is a
way to USE truth tables for some purposes in predicate logic, truth
tables are not a part of the standard *semantics* of predicate logic,
even mondaic predicate logic, and in principle *cannot* provide such a
semantics in general, due to their finite character. In particular,
they cannot provide an adequate account of the meaning of the
quantifiers. You seem to be emphasizing the fact that they can be USED
in (some corners of) predicate logic. Paul seems to have picked up on
the fact that truth tables are not part of the standard SEMANTICS of
predicate logic, and hence, in particular, on the idea that "tautology"
-- which has essentially to do with the semantical methods of
propositional logic -- is an inherently propositional notion.
> When you teach (or taught) this logic class which goes into PL,
> did you provide the same background as McGee? Or did you
> perhaps omit it (full discussion of MPL) because you thought it
> might be confusing? What I'm trying to get at, is McGees
> decision to include the material I quoted a fairly standard decision,
> or would some qualified instructors choose to dismiss the material
> that McGee deems important enought to include in his lectures, re MPL?
I introduce predicate logic via monadic predicate logic. But I don't
talk about the TT method for deciding validity because, at that point,
I'm focusing on the meaning of the quantifiers, not validity, and I do
so by talking about simple interpretations. I do eventually introduce a
method of demonstrating invalidity that uses TTs, but only as a way
station en route to the destination of a predicate logic countermodel.
Chris Menzel
paul
<cme...@remove-this.tamu.edu> wrote:
>On Sun, 12 Dec 2004 02:30:55 -0500, paul <paul...@on-ramp.nl> said:
>> 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.
>
>Because it usefully picks out a certain class of logical truths, viz.,
>those that are true simply in virtue of their truth functional
>structure. If its meaning were broadened to include the logical truths
>of predicate logic, it would serve no purpose.
Thanks for your response, but I'm still unclear. Isn't (x)(Px v -Px)
(everything is either P or not-P) true in virtue of its truth
functional structure?
My hunch as to why most logicians limit the term "tautology" to
sentential logic is that when truth tables are applied to monadic
predicate statements based on some model with some domain of
individuals, one is not building a truth table right off the
quantified statement but (even if undeclared) one is instantiating the
quantified statement such that in the domain {a, b} (x)Px becomes Pa
and Pb. Once that's done, what's actually being analyzed is not a
quantified predicate statement but a statement that aside from the
quantification project could be reduced to a sentential statement.
Moreover, I think categorical predicate statements can be subjected
to truth table analysis (that's what I infer from Copi, "Symbolic
Logic," p.81) if (x)(Px -> Bx) in the domain {a, b} is instantiated as
(Pa -> Ba) & (Pb -> Bb)
and existential statements like Ex(Px & Bx) are instantiated as
(Pa & Ba) v (Pb & Bb).
Then we could build truth tables for them based on a model. But at
that point our analysis is operating *outside* quantified predicate
statements with statements that could as well be reduced to sentential
letters.
>We already have "logical
>truth" and "universal validity" (though the latter is rather less
>common; indeed, I can only recall seeing it in the LTF Gamut text --
>whose actual authors, BTW, are the frighteningly prolific Dutch logician
>Johan van Benthem and a couple of his colleagues).
Right, "L.T.F. Gamut" is by some means a collective pseudonym for
professors Benthem, Groenendijk, de Jongh, Stokhof, and Verkuyl.
Thanks for your input Chris.
- paul
Stephen Harris
>>original) that support what I said --
>>that the term "tautology" is not applied outside propositional logic"
>new) versus the one example of Partee. My question is why is "tautology"
> not normally used outside sentential logic? There must be a reason.
SH: I objected to your original statement because of "not applied" which
does not mean the same thing as "not normally". I did not dipute your
remarks about full predicate logic and valid. But used monadic predicate
logic as a counter example to "not applied".
I could have used Modal logic as another counter example; where the
term tautology is applied outside of propositional logic. Modal logic
is distinct from propositional logic, even when the adjective is
sentential modal logic.
Besides that Gamut wrote: "In predicate logic ... models M for the language
from which @ is taken are called universally valid formulas
(they are not normally called tautologies)
That statement means they are not normally called tautologies
"in predicate logic".
Your question transports the meaning above to something unsupported.
>>> paul wrote: "My question is why is "tautology"
>>> not normally used outside sentential logic? There must be a reason.
Outside of propositional logic does not mean the same thing as
"in predicate logic". There are several logics which are outside of or
distinct from propositional logic besides predicate logic. In one of
them, modal logic, I find the "tautology" term fairly common, which is
not as you state "not normally used outside sentential logic". Your
question, seems to me, should have been why isn't "tautology" normally
used in predicate logic, since "tautology" is more common in other logics.
Your statement/question leaves out the "in predicate logic" stipulation,
which is why your revision doesn't gain support from Gamut's quote.
Gamut doesn't say 'not normally found outside of propositional logic'.
I tried to keep the TT formatting aligned.
http://hume.ucdavis.edu/mattey/phi134/s5semant.htm
"A sentence (or formula) of a symbolic logic is logically true if and only
if it is true on all interpretations. For sentential logic, this amounts
to claiming that it has the value true on every line of its truth-table,
no matter what the value of its components. Thus, on Carnap's
interpretation,
every logically true formula of sentential logic expresses a proposition
which is necessary. This is part of what Bonevac asserts in his Thesis 1,
on page 231. [SH: The |_| with a _ at top is supposed to be a square box.]
'But what are we to say about the logical truth of modal formulas, i.e.,
formulas containing modal operators? We _ know that p --> p is a logical
truth of sentential logic, which will make |_|(p --> p) true on Carnap's
interpretation. But suppose that we want to know whether it is logically
true and therefore necessarily true. We do not have truth-tables for modal
formulas, so we cannot apply the definition of logical truth of sentential
logic to modal formulas. Something more is needed. For Carnap, it was an
adaptation of Leibniz's notion of a possible world.
'In my search for an explication [of logical truth] I was guided, on the
one hand, by Leibniz' view that a necessary truth is one which holds in
all possible worlds, and on the other hand, by Wittgenstein's view that
a logical truth or tautology is characterized by holding for all possible
distributions of truth values. Therefore the various forms of my definition
of logical truth are based either on the definition of logically possible
states or on the definition of sentences describing those states (state
descriptions). ("Autobiography, p. 63)'
Thus we can look at a possible world as a possible distribution of
truth-values, i.e., as a line on a truth table. A possible world, then,
matches sentence letters with truth values. A simple interpretation will
illustrate the transition from truth-tables to possible worlds. Let one
possible world w_1 match the single atomic formula p with the value true,
and another one w_2 match p with the value False. We determine the value
of a non-modal formula at a world by the values of its components at that
*same world*. Thus the value of p --> p at w_1 is true given that p is
true at the first world, and it is also true at w_2 given that the value
of p is false at the second world.
p p --> p
w1 True True
w2 False True
_
Now we can describe how to assign a value to |_|(p --> p). This formula
is true at a world just in case p --> p is true at all possible worlds.
The two worlds in our small interpretation both make the formula p --> p
_
true, so the modal formula |_|(p --> p) is true at both worlds, just as
p --> p is true at both worlds.
_
p p --> p |_|(p --> p)
w1 True True True
w2 False True True
The subtle difference between the evaluation of the modal and the
non-modal formula in the example is this: the non-modal formula is
evaluated on the basis of the value of its components in a single world,
while the modal formula is evaluated on the basis of the value of its
one component at two different worlds. In non-modal sentential logic,
one never uses information from another line on the truth-table to
determine the value of a formula on a given line.
This example shows why Thesis 1 ("Every logical truth is necessary")
holds for the S5 semantics. Recall that a logical truth is one that
is true on every interpretation, and hence true at every possible world,
since an interpretation is equated to a possible world. A non-modal
logical truth is true on every line of a truth-table, and hence at any
possible world that can be constructed. A modal logical truth of S5
_
such as |_|(p --> p) is also true at every possible world, since by
virtue of being a logical truth it is true on every interpretation.
Hence the modal formula is necessary as well; in the present example,
_ _
|_| |_|(p --> p) is true at every possible world, and so on.
--------------------------------------------------------------
A proof system that applies rules of inference to an initial set of
premises can be thought of as a machine for cranking out theorems:
truths in the top, proven new truths out the bottom...
-----------------------------------------------------------
A tautology checker for Propositional Calculus
Here is a small piece of software that some of you might be interested in.
What the program does is to check whether a given propositional calculus
formula is a tautology or not.Click here for details.http://www.utexas.edu
/cola/depts/philosophy/faculty/asher/course/phl358/phl358_3.htm
-------------------------------------------------------------------
The set of all tautologies (PC) is the smallest system of modal logic
*every system closed under RPL contains all tautologies (by n=0)
-------------------------------------------------------------
2) Leibniz (on modern readings):
A model is not just a possible world, but a collection of possible worlds.
Hence a model is a collection of extensional models.
Regards,
Stephen
Stephen Harris
> On Sun, 12 Dec 2004 10:57:20 GMT, Stephen Harris said:
>
>> I have found several references to Monadic Predicate logic
>> on the net which mention truth tables.
>
> That's because there is, theoretically, a way to USE truth tables to
> determine validity in general for monadic predicate logic. But note
> what seems to be a sticking point for you and paul. Although there is a
> way to USE truth tables for some purposes in predicate logic, truth
> tables are not a part of the standard *semantics* of predicate logic,
> even mondaic predicate logic, and in principle *cannot* provide such a
> semantics in general, due to their finite character. In particular,
> they cannot provide an adequate account of the meaning of the
> quantifiers.
You seem to have reinforced Vann McGee's explanation:
"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: Thank you for your detailed response. I noticed in a thread
that Pat Hayes participates in that Sanskrit was a possible
formalizable natural language. But there was only an abstract available,
not a full paper. Perhaps the paper will be of some interest to you.
Knowledge Representation in Sanskrit and Artificial Intelligence by
Rick Briggs
http://www.aaai.org/Library/Magazine/Vol06/06-01/Papers/AIMag06-01-003.pdf
Regards,
Stephen
Chris Menzel
> ...
> term tautology is applied outside of propositional logic.
Hm, nothing I saw in what you quoted supports this claim. Did I miss a
quote where the validities of propositional modal logic are referred to
as "tautologies" by someone?
As you note, there is a limited use for truth tables -- of a sort -- in
the case of the propositional modal logic S5, but this is directly
analogous to their limited use in Monadic Predicate Logic (indeed, in a
certain sense, it's identical). But because the modal operators
function semantically as quantifiers over "possible worlds", a general
truth table style semantics for propositional modal logic is not
possible.
Chris Menzel
Ross A. Finlayson
Chris Menzel wrote:
Chris, no, they're just over "One World" instead, just like second order
logic is formalizable in first order logic.
Stephen, I really am impressed with the depth of your postings. Would you
briefly elaborate on your background? I researched you on the Internet
and would like to learn more about you.
If "tautology" is just modus ponens, modus tollens, synjunctive syllogism,
and identity, those are very useful for most true logical statements.
Thank you for posting these interesting arguments. That's argument in the
sense of rhetorical logic.
Ross Finlayson
Chris Menzel
> Chris Menzel wrote:
>
>> On Mon, 13 Dec 2004 00:44:52 GMT, Stephen Harris said:
>> > ...
>> > I could have used Modal logic as another counter example; where the
>> > term tautology is applied outside of propositional logic.
>>
>> Hm, nothing I saw in what you quoted supports this claim. Did I miss a
>> quote where the validities of propositional modal logic are referred to
>> as "tautologies" by someone?
>>
>> As you note, there is a limited use for truth tables -- of a sort -- in
>> the case of the propositional modal logic S5, but this is directly
>> analogous to their limited use in Monadic Predicate Logic (indeed, in a
>> certain sense, it's identical). But because the modal operators
>> function semantically as quantifiers over "possible worlds", a general
>> truth table style semantics for propositional modal logic is not
>> possible.
>>
>> Chris Menzel
>
> Chris, no, they're just over "One World" instead, just like second order
> logic is formalizable in first order logic.
Second-order logic is not "formalizable" in first-order logic in any
meaningful sense.
I really don't understand why you prefer spouting out clueless nonsense
over actually learning something about this subject matter.
Ross A. Finlayson
Chris Menzel wrote:
Perhaps that is a contradictory assumption on your part. I would certainly
think so as I post words that are strongly framed.
I might not learn as rapidly as I did, say, as a teen, almost twenty years
ago.
I might learn more rapidly.
Why don't you re-view your contribution to this thread and think about what
you've been covering.
I hope that then you would see better, and perhaps understand some more,
about this mathematical logic.
So don't I.
Ross
Chris Menzel
>> > Chris, no, they're just over "One World" instead, just like second order
>> > logic is formalizable in first order logic.
>>
>> Second-order logic is not "formalizable" in first-order logic in any
>> meaningful sense.
>>
>> I really don't understand why you prefer spouting out clueless nonsense
>> over actually learning something about this subject matter.
>
> Perhaps that is a contradictory assumption on your part. I would certainly
> think so as I post words that are strongly framed.
I don't know what that has to do with anything, but let me put it less
stridently. Given that you seem to be aware of the fact that you are
"just learning," I don't understand what possesses you to speak as if
you have already mastered the subject matter. The rational thing would
seem to be to make sure that you really know what, say, the words
"second-order logic" mean before you make an unqualifiedly false
assertion that simply shows that you don't have any coherent idea of
what they mean. I mean, wouldn't it make better sense, say, to post
something like, "There seem to be some interesting connections between
modal logic and first- and second-order predicate logic. [In fact, this
is true.] Can anyone elaborate on this, or point me to some relevant
resources so I can study them?"
I am probably naive about the extent to which people desire to be
rational these days...
Chris Menzel
Owen
logic is formalizable in first order logic."
I agree with you Ross.
Higher order logic of monadic forms is decidable.
AFAx(Fx) is reduced to AF(Fa & Fb) which is reduced to:
{}a & {}b & {a}a & {a}b & {b}a & {b}b & {a,b}a & {a,b}b.
...and so on for for any order.
Higher order monadic (modal) logic with identity is also reducible to
propositional logic, in this way.
"Ross A. Finlayson" <r...@tiki-lounge.com> wrote in message
news:41BD7ADB...@tiki-lounge.com...
Stephen Harris
SH: Yes. Also I have become quite wary of using the word general.
> Chris Menzel
>SH wrote:
>> I could have used Modal logic as another counter example; where the
>> term tautology is applied outside of propositional logic.
Because Ed Zalta acknowledged your sage advice, I regarded him as a
good source. I didn't say that there was no relationship to propositional
logic when I mentioned sentential modal logic. I said that it was distinct
from propositional logic and that the term "tautological" comes up normally.
He mentions two approaches, the first uses infinity, and appears to have
three columns in the truth table which is enought to distinguish it IMO. The
second was finite (Enderton) and actually seemed closer to predicate logic,
but involves a different definition of tautology, meaning the term is
applied
normally outside of propositional logic with a different definition also.
http://mally.stanford.edu/notes.pdf by Ed Zalta (some of pages 18-23)
"The problem we face first is that we want to distinguish the tautologies
in some way from the rest of the formulas that are valid. We can't just
use the notion `true at every world in every model', for that is just
the notion of validity. So to prove that the tautologies, as a class,
are valid, we have to distinguish them in some way from the other valid
formulas. The basic idea we want to capture is that the tautologies have
the same form as tautologies in propositional logic. ...
[SH: This establishes their relatedness, IMO, now onto differences.]
...
Remark: These examples and exercises show that our definition of a
tautology allows us to prove that certain formulas are tautologies.
The definition is reasonably simple and serves us well in subsequent
work. But, for arbitrary @, there is no mechanical way of finding
arguments such as the one in the above Example that establish that @
is a tautology if indeed it is. Moreover, you can't mechanically use
the definition to show, for a given tautology, that indeed it is a
tautology, since you can't check every assignment f . Even if we start
with a language based on the set Omega = {p_1}, it would take a very
long time to even specify a basic assignment of the quasi-atomic
subformulas (since, as we have seen, Omega* will be an infinite set).
So the definition of `tautology' per se doesn't offer a mechanical
procedure to discover, for a given @, whether or not @ is a tautology,
since strictly speaking, you would have to check an infinite number of
assignments (none of which you can even specify completely).
[SH: This is the basics of modal logic and it keeps referring to tautology.]
** But we know from work in propositional logic that the truth table
method gives us a mechanical procedure by which we can discover
whether or not a given @ is a tautology. Have we lost anything in the
move to modal logic? Actually, we haven't, for there is a way to
construct such a decision procedure that tests for tautologyhood. **
[SH: This does not appear to be exactly the same as TT in prop. logic.]
Such a procedure will be described in the Digression that follows
(disinterested readers, or readers who don't wish to interrupt the train of
development of the concepts, may skip directly ahead to (14) ).
[SH: skipping most of the Digression which starts on page 20.]Digression:
... "Consequently, if for every i, each member f of F_i is such that
f (@) = T (i.e., if the value T appears in every row of the final
column of the truth table), then @ is a tautology. This is our
mechanical procedure for checking whether @ is a tautology. The
reader should check that T does appear in every row under the
column headed by @ in the above example.
Of course, this intuitive description of a decision procedure depends
on our having a precise way to delineate of the truth-functionally
relevant subformulas of @, and on a proof that whenever f and f' agree
on the relevant quasi-atomic formulas in @, then they agree on @. The
latter shall be an exercise." ...
[SH: This method seems different to me than the construction of the
propositional logic TT. There seems more to it and more column(s).]
Alternative Section2: Tautologies are Valid (following Enderton)
In some developments of propositional logic (Enderton's, for example),
the notion of tautology is: @ is a tautology iff @ is true in all the
extensions of basic assignments of its atomic subformulas. The difference
here is that instead of being defined for all the atomic formulas in the
language, basic assignments f* are defined relative to arbitrary sets of
atomic formulas.
The basic assignments for a given formula @ will be functions that assign
truth values to every member of the set of atomic subformulas in @. An
extended assignment f is then defined relative to a basic assignment f*,
and extends f* to all the formulas that can be constructed out of the set
of atomic formulas over which f* is defined. So, for a given formula @,
f extends a given basic assignment f* by being defined on all the formulas
that can be constructed out of the set of atomic subformulas in @. Such fs
will therefore be defined on all of the subformulas in @, including @
itself. The definition of a tautology, then, is: @ is a tautology iff for
every basic assignment f* (of the atomic subformulas in @), the extended
assignment f (based on f*) assigns @ the value T .
One advantage of doing things this way is that for any given formula
@, there will be only a finite number of basic assignments, since there
will always be a finite number of atomic subformulas in @. Whenever
there are n atomic subformulas of @, there will be 2^n basic assignment
functions. Thus, our decision procedure for determining whether an
arbitrary @ is a tautology will simply be: check all the basic
assignments f* to see whether f assigns @ the value T .
In this section, we redevelop the definitions of the previous section
for those readers who prefer Enderton's definition of tautology. The
twist is that we have to define basic assignments relative to a given
set of quasi-atomic formulas. So for any given @, the basic assignments
f* will be defined on the set of quasi-atomic subformulas in @. Then we
extend those basic assignments to total assignments defined on all the
formulas constructible from such sets of quasi-atomics (these will there-
fore be defined for the subformulas of @ and @ itself). To accomplish all
of this, we need to define the notions of subformula, quasi-atomic formula,
and basic truth assignment to a set of quasi-atomic formulas, and then,
finally, extended assignment, before we can define the notion of a
tautology. Readers who are not familiar with Enderton's method, or who
have little interest in seeing how the method is adapted to our modal
setting, should simply skip ahead to section3. ....
Remark: Not only does our definition allow us to prove that a given
formula @ is a tautology, it gives us a decision procedure for
determining, for an arbitrary @, whether or not @ is a tautology. The
set of quasi-atomic subformulas of @ (Omega*_@) is finite. Suppose it
has n members. Then we have only to check 2^n basic assignments f* and
determine, in each case, whether f assigns @ the value T . So our modal
logic has not lost any of the special status that propositional logic has
with regard to the tautologies. **Indeed, there is a simple way to show that
tautologies in our modal language correspond with tautologies in
propositional language."**
[SH: This second method seems closer to propositional logic.]
SH: This is my evidence that "tautology" is normally used in Modal logic,
which is outside of sentential logic, which disuptes paul's contention.
paul wrote:
>>original) ... "that support what I said --
>>that the term "tautology" is not applied outside propositional logic"
which paul revised to:
>new) "My question is why is "tautology"
> not normally used outside sentential logic? There must be a reason."
Chris wrote:
> Hm, nothing I saw in what you quoted supports this claim. Did I miss a
> quote where the validities of propositional modal logic are referred to
> as "tautologies" by someone?
Ed Zalta wrote in introducing the basics of modal logic (91 pages!):
"Basic Concepts in Modal Logic" by Edward N. Zalta
"The problem we face first is that we want to distinguish the tautologies
in some way from the rest of the formulas that are valid. We can't just
use the notion `true at every world in every model', for that is just
the notion of validity. So to prove that the tautologies, as a class,
are valid, we have to distinguish them in some way from the other valid
formulas. The basic idea we want to capture is that the tautologies have
the same form as tautologies in propositional logic. ... Ed also wrote:
"I am also indebted to Chris Menzel, Kees van Deemter, Nathan Tawil,
Greg O'Hair, and Peter Apostoli. Finally, I am indebted to the Center
for the Study of Language and Information, which has provided me with
offce space and and various other kinds of support over the past years."
[SH: I didn't think I had to quote this again since you had already read
it.]
SH wrote: Did Ed falsely claim you reviewed his draft of this paper?! :-)
Sometimes I leave the living room to get something from the bathroom
and by the time I get to the bedroom, I've forgotten my mission.
Really, I'm just kidding around with this. It was over 9 years ago, besides
maybe he ignored your red ink! In reading Ed's description, Modal TT
may not be exactly general, but requires some finesse. I had a little
trouble
with the paper. If propositional logic TT are quite mechanical and general,
and Modal TT are less so, then that is a distinction between them, which
did not prevent the usage of tautology several times in Zalta's paper.
Best regards,
Stephen
paul
>On Mon, 13 Dec 2004 00:44:52 GMT, Stephen Harris said:
>> ...
>> I could have used Modal logic as another counter example; where the
>> term tautology is applied outside of propositional logic.
>
>Hm, nothing I saw in what you quoted supports this claim. Did I miss a
>quote where the validities of propositional modal logic are referred to
>as "tautologies" by someone?
Stephen seems to see support in any data for his notion that the term
"tautology" is wide ranging. Even after you explained that it is not
http://groups-beta.google.com/group/sci.logic/msg/66d18bd9ec10c304
he responded as if you supported his view and then he goes on
promoting it, frenetically posting hundreds of lines of material.
BTW, I was hoping you might be able to answer my questions to you
http://groups-beta.google.com/group/sci.logic/msg/094ca5c91edc4be8
which may have been lost in Stephen's voluminous postings.
Thanks for your input.
- paul
Stephen Harris
>
>> On Mon, 13 Dec 2004 00:44:52 GMT, Stephen Harris said:
>> > ...
>> > I could have used Modal logic as another counter example; where the
>> > term tautology is applied outside of propositional logic.
>>
>> Hm, nothing I saw in what you quoted supports this claim. Did I miss a
>> quote where the validities of propositional modal logic are referred to
>> as "tautologies" by someone?
>>
>
>
> Stephen, I really am impressed with the depth of your postings. Would you
> briefly elaborate on your background? I researched you on the Internet
> and would like to learn more about you.
>
I have a CIS degree. I'm a computer tech. About 20 years ago I became
interested in AI after reading Hofstadter's "Godel, Escher and Bach".
What I'm actually quite good at is research. So when Chris made his
"did I miss a quote" remark, I looked for a way to answer it that would
include some humor. So I found this paper by Zalta that mentioned
Chris in the thank yous, which also connected Modal logic to Truth
tables and propositional logic, answering that point, and mentioned
the word "tautology" several times. It was recreational.
And I've read a lot of papers in different areas but am no expert in
any area, I've read mostly about AI. So in all my reading I've come
up with five key concepts: Chance, Causality, Counterfactuals,
Indistinguishability, and Self-reference. My good results are due
to my one strong talent, research. Thanks for your compliment.
Regards,
Stephen