Why is "therefore" abbreviated as three dots forming a triangle?
J Asking
"therefore", which is abbreviated as three dots forming a triangle?
Skitt
> Can anyone explain the origin of the abbreviation for the word
> "therefore", which is abbreviated as three dots forming a triangle?
It is not an abbreviation; it is a mathematical symbol.
--
Skitt (in SF Bay Area) http://www.geocities.com/opus731/
I speak English well -- I learn it from a book!
-- Manuel (Fawlty Towers)
Mike Oliver
> J Asking wrote:
>> Can anyone explain the origin of the abbreviation for the word
>> "therefore", which is abbreviated as three dots forming a triangle?
>
> It is not an abbreviation; it is a mathematical symbol.
Which, of course, still leaves the question as to the origin of
that symbol. I don't know either.
I'm also curious how the custom started of writing a box (small hollow
square) at the end of a proof, replacing the older QED. At least I
know where that one came from: It stands for _quod_ego_dico_,
"because I say so".
R J Valentine
} J Asking wrote:
}
}> Can anyone explain the origin of the abbreviation for the word
}> "therefore", which is abbreviated as three dots forming a triangle?
}
} It is not an abbreviation; it is a mathematical symbol.
.
Yeah, like the old "Cogito . . sum".
--
R. J. Valentine <mailto:r...@smart.net>
Mark Brader
>>> "therefore", which is abbreviated as three dots forming a triangle?
>> It is not an abbreviation; it is a mathematical symbol.
> Which, of course, still leaves the question as to the origin of
> that symbol.
See Florian Cajori's "A History of Mathematical Notations". The
"therefore" sign was one of many invented by the Swiss mathematician
Johann Heinrich Rahn for his 1659 book "Teutsche Algebra", most of
which have long since disappeared.
Rahn used the sign either way up, indifferently, and so did the English
translation of the book that appeared in 1668. Some later authors used
only two raised dots, side by side. The idea of using the three-dot
sign one way up for "therefore" and the other way for "since", as we
know it now, dates from 1805.
> I'm also curious how the custom started of writing a box (small hollow
> square) at the end of a proof, replacing the older QED. ...
Now that one Cajori doesn't cover; I assume it was after his time.
The first time I personally saw that notation was about 30 years ago
in Donald Knuth's "The Art of Computer Programming".
--
Mark Brader | "I will be speaking today about work in progress,
Toronto | instead of completed research; this was not my
m...@vex.net | original intention when I chose the subject of
| this lecture, but the fact is I couldn't get my
| computer programs working in time." -- Donald Knuth
My text in this article is in the public domain.
Evan Kirshenbaum
> I'm also curious how the custom started of writing a box (small
> hollow square) at the end of a proof, replacing the older QED. At
> least I know where that one came from: It stands for
> _quod_ego_dico_, "because I say so".
I'm pretty sure that you're joking here, but just in case (and lest
anyone be misled) it comes from "quot erat demonstrandum", "that which
was to be proven".
--
Evan Kirshenbaum +------------------------------------
HP Laboratories |If only some crazy scientist
1501 Page Mill Road, 1U, MS 1141 |somewhere would develop a device
Palo Alto, CA 94304 |that would allow us to change the
|channel on our televisions......
kirsh...@hpl.hp.com | --"lazarus"
(650)857-7572
Mark Brader
> (...lest anyone be misled) it comes from "quot erat demonstrandum"...
I think Evan has misquodded that phrase.
--
Mark Brader, Toronto, m...@vex.net | "...but I could be wromg." --Rodney Boyd
John Ings
wrote:
>I'm also curious how the custom started of writing a box (small hollow
>square) at the end of a proof, replacing the older QED. At least I
>know where that one came from: It stands for _quod_ego_dico_,
>"because I say so".
quod erat demonstrandum
N.Mitchum
----
> Can anyone explain the origin of the abbreviation for the word
> "therefore", which is abbreviated as three dots forming a triangle?
It's a symbol, not an abbreviation. It's used in math and -- I
believe -- in logic. Where it came from, I can't tell you.
----NM
Jitze Couperus
<kirsh...@hpl.hp.com> wrote:
>Mike Oliver <oli...@math.ucla.edu> writes:
>
>> I'm also curious how the custom started of writing a box (small
>> hollow square) at the end of a proof, replacing the older QED. At
>> least I know where that one came from: It stands for
>> _quod_ego_dico_, "because I say so".
>
>I'm pretty sure that you're joking here, but just in case (and lest
>anyone be misled) it comes from "quot erat demonstrandum", "that which
>was to be proven".
>
A quick google shows
"quot erat demonstrandum" 55
"quod erat demonstrandum" 6860
Jitze
Matti Lamprhey
>
> See Florian Cajori's "A History of Mathematical Notations". The
> "therefore" sign was one of many invented by the Swiss mathematician
> Johann Heinrich Rahn for his 1659 book "Teutsche Algebra", most of
> which have long since disappeared.
>
> Rahn used the sign either way up, indifferently, and so did the
> English translation of the book that appeared in 1668. Some later
> authors used only two raised dots, side by side. The idea of using
> the three-dot sign one way up for "therefore" and the other way for
> "since", as we know it now, dates from 1805.
Interesting. What does the book say about the origin of the '='? I
recently read a claim that it was first used by a Welshman, and Wales
isn't known for its mathematicians isn't it.
Matti
PS While I've got your attention, what were the usual "God's Wonderful
Railway"-type cod expansions of the other railway companies, LNER, LMS
and suchlike? I've searched the archives in vain.
Mark Brader
> Interesting. What does [Cajori's] book say about the origin of the
> '='? I recently read a claim that it was first used by a Welshman...
I could've told you that without checking Cajori that the = sign was
invented by Robert Recorde of England. He first used it in his 1557
book "The Whetstone of Witte". However, on checking his biography at
<http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Recorde.html>,
I see that he was indeed born in Wales. That page also explains the
multilingual pun in the book's title.
His = signs were much longer than ours, and varied in length from one
use to the next. The typographer would make them up out of small
pieces, and since the joints were visible they had an appearance like
"=====", which is very congenial to C programmers. :-) I believe the
actual page where he first defines the sign is reproduced on the Web
somewhere. And, of course, see also Michael Quinion's article at
<http://www.worldwidewords.org/articles/signs.htm>.
> PS While I've got your attention, what were the usual "God's Wonderful
> Railway"-type cod expansions of the other railway companies, LNER, LMS
> and suchlike? I've searched the archives in vain.
Well, geez, there was only one GWR, wasn't there, so who needs to create
expansions like that for the also-rans?
--
Mark Brader "[It] was the kind of town where they spell
Toronto trouble TRUBIL, and if you try to correct them,
m...@vex.net they kill you." -- Dead Men Don't Wear Plaid
Matti Lamprhey
> Matti Lamprhey:
> > Interesting. What does [Cajori's] book say about the origin of the
> > '='? I recently read a claim that it was first used by a
> > Welshman...
>
> I could've told you that without checking Cajori that the = sign was
> invented by Robert Recorde of England. He first used it in his 1557
> book "The Whetstone of Witte". However, on checking his biography at
>
<http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Recorde.html>,
> I see that he was indeed born in Wales. That page also explains the
> multilingual pun in the book's title.
Thanks!
> [...] And, of course, see also Michael Quinion's article at
> <http://www.worldwidewords.org/articles/signs.htm>.
>
>
> > PS While I've got your attention, what were the usual "God's
> > Wonderful Railway"-type cod expansions of the other railway
> > companies, LNER, LMS and suchlike? I've searched the archives in
> > vain.
>
> Well, geez, there was only one GWR, wasn't there, so who needs to
> create expansions like that for the also-rans?
My, my! A railway question that stumped Mark Brader! I'll take the
rest of the day off now.
Matti
PeterD
It won't be long before the world's democracies are replaced by
googlocracies.
A quick google shows
"No war in Iraq" 7906
"Nuke the bastards" 7907
sorry Iraqis, the people have decided.
--
Pd
rzed
"PeterD" <pd....@dsl.pipex.invalid> wrote in message
news:1fr1bcv.gh8qsn1fwjhdsN%pd....@dsl.pipex.invalid...
[...]
> It won't be long before the world's democracies are replaced by
> googlocracies.
>
> A quick google shows
> "No war in Iraq" 7906
> "Nuke the bastards" 7907
>
> sorry Iraqis, the people have decided.
Though, strictly speaking, the list of victims and the area involved expand
dramatically in the second case. I can't begin to calculate the cost of
*that* war.
--
rzed
John O'Flaherty
Last night, Jay Leno cracked that 'Iraq' is Arabic for 'Vietnam'.
--
john
Jacqui
> PeterD wrote
>> googlocracies.
>>
>> A quick google shows
>> "No war in Iraq" 7906
>> "Nuke the bastards" 7907
>>
>> sorry Iraqis, the people have decided.
>
> Though, strictly speaking, the list of victims and the area
> involved expand dramatically in the second case. I can't begin to
> calculate the cost of *that* war.
*Really* Smart missiles are able to read birth certificates?
Jac
R H Draney
news:3e5dcde6....@sd.znet.com:
> A quick google shows
>
> "quot erat demonstrandum" 55
> "quod erat demonstrandum" 6860
and:
"quod erat demonstratum" 192
...which one might expect from those with small Latin...one of the
hits on the first page also suggests "quite easily demonstrated"...I
think it's meant to be a joke....r
K. Edgcombe
>>
>> "quot erat demonstrandum" 55
>> "quod erat demonstrandum" 6860
>
This raises an interesting question as to why people Google for such things.
If there is a disputed usage or spelling in English, it may be interesting to
find out what, in fact, the majority of Web pages/newsgroup postings actually
say. It's a pretty crude way of trying to sample users of a language, but if
enough users of a language do in fact change over to a new spelling, then in
due course that spelling becomes correct, as we have all seen many times.
However, nothing that any number of modern speakers do can possibly change the
rules of Latin. It's dead, it's gone, we can't get at it. So unless the
Google refs actually turn out, improbably, to be verbatim quotes from the
period when Latin was still alive, I don't see that Google can help.
Evan's error (if it was he) is clearly a typo, anyway.
Katy
Evan Kirshenbaum
It was actually just a typo on my part. I know it's "quod".
--
Evan Kirshenbaum +------------------------------------
HP Laboratories |It's gotten to the point where the
1501 Page Mill Road, 1U, MS 1141 |only place you can get work done is
Palo Alto, CA 94304 |at home, because no one bugs you,
|and the best place to entertain
kirsh...@hpl.hp.com |yourself is at work, because the
(650)857-7572 |Internet connections are faster.
| Scott Adams
http://www.kirshenbaum.net/
Mark Brader
>>> PS While I've got your attention, what were the usual "God's
>>> Wonderful Railway"-type cod expansions of the other railway
>>> companies, LNER, LMS and suchlike? I've searched the archives in
>>> vain.
>> Well, geez, there was only one GWR, wasn't there, so who needs to
>> create expansions like that for the also-rans?
> My, my! A railway question that stumped Mark Brader! I'll take the
> rest of the day off now.
Stumped? I stand by my previous answer.
ObAUE: "cod expansion". The only time I've previously seen this use
of "cod" is in the phrase "cod Latin", as in my signature below.
--
Mark Brader, Toronto | "Domine, defende nos
m...@vex.net | Contra hos motores bos!" -- A. D. Godley
Mark Brader
> It won't be long before the world's democracies are replaced by
> googlocracies.
>
> A quick google shows
> "No war in Iraq" 7906
> "Nuke the bastards" 7907
>
> sorry Iraqis, the people have decided.
Nice joke!
Just for the record, the actual counts today, give or take the usual
heisengoogle fuzz, are:
"No war in Iraq" 4,860
"Nuke the bastards" 880
--
Mark Brader, Toronto | "Show that 17x17 = 289. Generalise this result."
m...@vex.net | -- Carl E. Linderholm
Matti Lamprhey
> >>> PS While I've got your attention, what were the usual "God's
> >>> Wonderful Railway"-type cod expansions of the other railway
> >>> companies, LNER, LMS and suchlike? I've searched the archives in
> >>> vain.
>
> >> Well, geez, there was only one GWR, wasn't there, so who needs to
> >> create expansions like that for the also-rans?
>
> > My, my! A railway question that stumped Mark Brader! I'll take the
> > rest of the day off now.
>
> Stumped? I stand by my previous answer.
Oh well, here's a rather substandard list:
http://homepages.tesco.net/~scotsnet/o.f.carter/fun/fun116.htm
>
> ObAUE: "cod expansion". The only time I've previously seen this use
> of "cod" is in the phrase "cod Latin", as in my signature below.
> --
> Mark Brader, Toronto | "Domine, defende nos
> m...@vex.net | Contra hos motores bos!" -- A. D. Godley
I think it fits quite well to describe an expansion which no-one assumes
is the true one.
Matti
Jerry Friedman
Been on bumper stickers for a few months.
--
Jerry Friedman
PeterD
> Peter D.:
> > It won't be long before the world's democracies are replaced by
> > googlocracies.
> >
> > A quick google shows
> > "No war in Iraq" 7906
> > "Nuke the bastards" 7907
> >
> > sorry Iraqis, the people have decided.
>
> Nice joke!
>
> Just for the record, the actual counts today, give or take the usual
> heisengoogle fuzz, are:
>
> "No war in Iraq" 4,860
> "Nuke the bastards" 880
Thanks for that. The problem is defining which phrases support which
position. "No war in Iraq" is reasonably obvious, but "nuke.." could
apply to any number of possible candidates. Especially Dubya supporters.
I've just been watching the news, and can't believe what a charade the
whole thing is. As if it has anything to do with dismantling missiles,
and nothing to do with personal power agendas of pissy little
politicians who are fairly sure they won't be on the receiving end of a
uranium enhanced bullet, more's the pity.
--
Pd
Joe Fineman
> I'm also curious how the custom started of writing a box (small
> hollow square) at the end of a proof, replacing the older QED.
It isn't always a hollow square; in fact, I think the use of a solid
black square or rectangle is older.
It is reputed to be the brainchild of Paul R. Halmos, a mathematician
at the University of Chicago. In fact, it is often called a halmos.
--
--- Joe Fineman j...@TheWorld.com
||: Fool me once -- shame on you. Fool me twice -- shame on me. :||
Simon R. Hughes
Google:
nuke washington: 113 000 results
nuke baghdad: 13 500 results
Hmmm.
--
Simon R. Hughes
"I often think there should exist a special typographical
sign for a smile -- some sort of concave mark, a supine
round bracket" -- Vladimir Nabokov, _Strong Opinions_.
Ben Zimmer
Mark Brader wrote:
>
> Matti Lamprhey and I (Mark Brader) wrote...
> >>> PS While I've got your attention, what were the usual "God's
> >>> Wonderful Railway"-type cod expansions of the other railway
> >>> companies, LNER, LMS and suchlike? I've searched the archives in
> >>> vain.
>
> of "cod" is in the phrase "cod Latin", as in my signature below.
Here's the relevant OED entry:
-------------
cod, n.5
2. A joke; a hoax, leg-pull; a parody, a 'take-off'. (See also E.D.D.
n.5) Also attrib. or quasi-adj., parodying, burlesque; 'mock'.
[...]
1952 GRANVILLE Dict. Theatr. Terms 46 Cod version, a burlesque of a
well-known play. 1959 Church Times 16 Jan. 4/4 The 'cod' Victorian
decorations tend to disguise the editor's underlying seriousness. 1959
Listener 29 Jan. 228/1 She obliged, initially in the delicious hiccup
polka, a cod of Old Vienna. Ibid. 228/2 Joyce Grenfell too, doing her
evergreen cod chorister. 1961 B. WELLS Day Earth caught Fire ii. 31
Pete picked up the empty tea mug and again used it as a cod mike.
'Alcoholics of the press, unite! 1962 Listener 5 July 36/1 The very
idiosyncratic cod cockney of the scenes. 1970 Guardian 11 May 8/2 The
cod version of 'Road to Mandalay'.
-------------
I also noticed the expression "cod-funk" not too long ago in a Guardian
review of a Peter Gabriel album:
http://www.guardian.co.uk/arts/reviews/story/0,11712,795028,00.html
"In 1986, he suffered a musical mid-life crisis, releasing So, an
album packed with ultra-commercial priapic cod-funk."
Googling turns up a few dozen hits on "cod-funk" suggesting that it's a
Briticism used to ridicule music that strives at funk but fails in an
unpleasant manner (prime culprits being Jamiroquai and various bands of
the mid-80s):
"that horrible, cod-funk guitar just screams 1985"
"the naff cod-funk of [REM's] 'Can't Get There From Here'"
"[Jay K's] inter-galactic cod-funk white boy disco..."
"the laboured cod-funk of Jamiroquai"
"the nineties saw [Prince] spiral into cod funk hell"
Jay K of Jamiroquai seems to attract quite a few "cod" epithets:
"coming from the slower and less cod P-funk side of his brain"
"increasingly bonkers cod funkster"
"cod Stevie Wonder squealing"
"cod-mystical lyrics"
"cod-West Country accent"
A few word-lists of British gay slang define "cod" as "vile" or "naff"
(=AmE "cheesy"):
http://www.chris-d.net/polari/
http://www.homeusers.prestel.co.uk/cello/Polari.htm
http://www.geocities.com/gaylanguage/AppendixB.htm
It strikes me that "ersatz" might be the best gloss for this new sense
of "cod", since it encompasses both phoniness and cheesiness.
--Ben
Michael J Hardy
> square) at the end of a proof, replacing the older QED. ...
I think that may have been invented by Paul Halmos.
Mike Hardy
Michael J Hardy
> > hollow square) at the end of a proof, replacing the older QED.
>
> It isn't always a hollow square; in fact, I think the use of a solid
> black square or rectangle is older.
>
> It is reputed to be the brainchild of Paul R. Halmos, a mathematician
> at the University of Chicago. In fact, it is often called a halmos.
I think it's mentioned in his autobiography too.
He is not _now_ at the U of Chicago, although it's not
inconceivable to me that that's where he was when he created
that symbol. He left the U of Chicago maybe 35 or 40 years
ago and went to the U of Michigan, then later to Indiana
University, and then to Santa Clara University, a Jesuit
institution in Santa Clara, California. I think he's not
entirely retired and is still there. -- Mike Hardy
Peter Morris
"Mark Brader" <m...@vex.net> wrote in message
news:gnl7a.228$S%.35859505@news.nnrp.ca...
> Matti Lamprhey:
> > Interesting. What does [Cajori's] book say about the origin of the
> > '='? I recently read a claim that it was first used by a Welshman...
>
> I could've told you that without checking Cajori that the = sign was
> invented by Robert Recorde of England. He first used it in his 1557
> book "The Whetstone of Witte". However, on checking his biography at
> <http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Recorde.html>,
> I see that he was indeed born in Wales. That page also explains the
> multilingual pun in the book's title.
It doesn't say WHY he chose that symbol for equals.
However, the symbol has always made sense to me. Just consider
the trio of symbols made of two lines, greater than, less than and equals.
If x<y then the small end of the lines is near x, so x is the smaller of
the two.
If x>y then the small end is near y so x is big and y is small. If x=y then
both
ends of the symbol are the same size, so x and y are the same size.
Did Robert Record use < and > ?
dcw
Peter Morris <no...@m.please> wrote:
>It doesn't say WHY he chose that symbol for equals.
I seem to remember someone saying something like "what could be
more equal than two parallel lines?" (I could hardly make it
vaguer than that. Sorry.)
David
Mark Brader
>> [<http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Recorde.html>]
>> doesn't say WHY [Recorde] chose that symbol for equals.
Huh? Yes it does:
# ... bicause noe 2 thynges can be moare equalle .
This is also quoted in the other page I cited in my posting,
<http://www.worldwidewords.org/articles/signs.htm> by Michael Quinion.
David Wood:
> I seem to remember someone saying something like "what could be
> more equal than two parallel lines?" (I could hardly make it
> vaguer than that. Sorry.)
Not parallel lines, but *gemowe* (i.e. twin), lines.
--
Mark Brader, Toronto | "Mark is probably right about something,
m...@vex.net | but I forget what" -- Rayan Zachariassen
Joe Fineman
> > > I'm also curious how the custom started of writing a box (small
> > > hollow square) at the end of a proof, replacing the older QED.
> >
> > It isn't always a hollow square; in fact, I think the use of a solid
> > black square or rectangle is older.
> >
> > It is reputed to be the brainchild of Paul R. Halmos, a mathematician
> > at the University of Chicago. In fact, it is often called a halmos.
>
> I think it's mentioned in his autobiography too.
>
> He is not _now_ at the U of Chicago, although it's not
> inconceivable to me that that's where he was when he created that
> symbol.
Probably not, in fact. I failed to locate him with Google, so I took
Chicago out of the only book by him that I have -- _Naive Set Theory_,
which does *not* use the end-of-proof symbol.
--
--- Joe Fineman j...@TheWorld.com
||: To do good is virtuous, and to wish good to be done is :||
||: amiable, but to wish to do good is as vain as it is vain. :||
frank green
R H Draney wrote:
> "quite easily demonstrated"
ROFL. I must remember that one.
Per Rønne
> J Asking wrote:
>
> > Can anyone explain the origin of the abbreviation for the word
> > "therefore", which is abbreviated as three dots forming a triangle?
>
> It is not an abbreviation; it is a mathematical symbol.
I thought the proper mathematical symbol for that were:
=> like x > 3 => x > 2.
The other way around:
<= like x > 3 <= x > 4
And logical equivalence:
<=> like x - 5 = 0 <=> x = 5
--
Per Erik Rønne
Mark Brader
> I thought the proper mathematical symbol for [therefore] were:
>
> => like x > 3 => x > 2.
No, that's the mathmatical symbol for "implies". "x > 3, therefore
x > 2" might appear as two connected statements in a proof (and would
commonly use the "therefore" sign, not the word). "x > 3 => x > 2"
is a single statement relating the inequalities x > 3 and x > 2.
> The other way around: <= like x > 3 <= x > 4
> And logical equivalence: <=> like x - 5 = 0 <=> x = 5
These have meanings analogous to the => case.
--
Mark Brader | "You wake me up early in the morning to tell me
Toronto | I am right? Please wait until I am wrong."
m...@vex.net | -- John von Neumann, on being phoned at 10 am
Per Rønne
> Per Rønne:
> > I thought the proper mathematical symbol for [therefore] were:
> >
> > => like x > 3 => x > 2.
>
> No, that's the mathmatical symbol for "implies".
What's the difference?
--
Per Erik Rønne
Mark Brader
> > > I thought the proper mathematical symbol for [therefore] were:
> > > => like x > 3 => x > 2.
Mark Brader:
> > No, that's the mathmatical symbol for "implies".
Per Rønne:
> What's the difference?
I explained that in the part of the posting you edited out.
--
Mark Brader, Toronto "Logic is logic. That's all I say."
m...@vex.net -- Oliver Wendell Holmes
Evan Kirshenbaum
1 > 2 => 4 is prime
is a true statement, but wouldn't be if "=>" were read as "therefore".
"A (logically) implies B" means that either A (the antecedent) is
false or B (the consequent) is true.
--
Evan Kirshenbaum +------------------------------------
HP Laboratories |Its like grasping the difference
1501 Page Mill Road, 1U, MS 1141 |between what one usually considers
Palo Alto, CA 94304 |a 'difficult' problem, and what
|*is* a difficult problem. The day
kirsh...@hpl.hp.com |one understands *why* counting all
(650)857-7572 |the molecules in the Universe isn't
|difficult...there's the leap.
http://www.kirshenbaum.net/ | Tina Marie Holmboe
Martin Ambuhl
> p...@ronne.invalid (Per Rønne) writes:
>
>
>>Mark Brader <m...@vex.net> wrote:
>>
>>
>>>Per Rønne:
>>>
>>>>I thought the proper mathematical symbol for [therefore] were:
>>>>
>>>>=> like x > 3 => x > 2.
>>>
>>>No, that's the mathmatical symbol for "implies".
>>
>>What's the difference?
>
>
> 1 > 2 => 4 is prime
>
> is a true statement, but wouldn't be if "=>" were read as "therefore".
> "A (logically) implies B" means that either A (the antecedent) is
> false or B (the consequent) is true.
As a small expansion:
Implication is a proposition
p => q ; meaning (~p || q)
But "therefore" is a term used in an argument, in
which assertions are made to yield a conclusion, as in
1) p ; 'p' is true
2) p => q ; 'p => q' is true
3) q ; therefore 'q' is true
(Modus potens)
or
1) ~q ; 'q' is false
2) p => q ; 'p => q' is true
3) ~p ; therefore 'p' is false
(Modus tolens)
We move on to the 2nd 5 minutes of a course in logic soon.
Per Rønne
> 1 > 2 => 4 is prime
>
> is a true statement, but wouldn't be if "=>" were read as "therefore".
> "A (logically) implies B" means that either A (the antecedent) is
> false or B (the consequent) is true.
OK. Not (1>2) or (4 is prime).
--
Per Erik Rønne
Phil Carmody
> Mark Brader <m...@vex.net> wrote:
>
>> Per Rønne:
>> > I thought the proper mathematical symbol for [therefore] were:
>> >
>>
>> No, that's the mathmatical symbol for "implies".
>
> What's the difference?
x > 3 => x > 2
x>3
.
. . x>2
Is an application of a law of inference, in this case Modus Ponens.
The 'therefore' symbol is more akin to the '|-' symbol. It's a statement
of the validity of the derivation of the formula that follows it.
For example, in most useful logics you'll have an equivalence between
((a & (a->b)) -> b)
and
{a, a->b} |- b
so the distinction between |- and -> is blurred to those who
don't know the details, because where you've got one form it's
trivial to convert it into the other form.
Phil
Phil
Phil Carmody
> p...@ronne.invalid (Per Rønne) writes:
>
>> Mark Brader <m...@vex.net> wrote:
>>
>> > Per Rønne:
>> > > I thought the proper mathematical symbol for [therefore] were:
>> > >
>> > > => like x > 3 => x > 2.
>> >
>> > No, that's the mathmatical symbol for "implies".
>>
>> What's the difference?
>
> 1 > 2 => 4 is prime
>
> is a true statement, but wouldn't be if "=>" were read as "therefore".
You've assumed a fixed model.
If I had a model where "1>2" was true and where "1>2 => 4 is prime",
then yes, I could say "therefore 4 is prime" is true.
Phil
Phil Carmody
>> I'm also curious how the custom started of writing a box (small hollow
>
> Now that one Cajori doesn't cover; I assume it was after his time.
> The first time I personally saw that notation was about 30 years ago
> in Donald Knuth's "The Art of Computer Programming".
When I first saw it (yup, Knuth, though about 20 years ago) I assumed it
was a pun and created the folk-etymology of the square being a 'quad' -
short for 'quod ...'
Phil
Evan Kirshenbaum
Even in a model in which "1>2" was true, I don't see how the statement
"1>2 therefore 4 is prime" would be a true statement, unless the model
also proved "1>2 implies 4 is prime".
--
Evan Kirshenbaum +------------------------------------
HP Laboratories |If I may digress momentarily from
1501 Page Mill Road, 1U, MS 1141 |the mainstream of this evening's
Palo Alto, CA 94304 |symposium, I'd like to sing a song
|which is completely pointless.
kirsh...@hpl.hp.com | Tom Lehrer
(650)857-7572
Aaron J. Dinkin
[someone wrote:]
>>> A quick google shows
>>>
>>> "quot erat demonstrandum" 55
>>> "quod erat demonstrandum" 6860
>>
>
> This raises an interesting question as to why people Google for such
> things.
Two reasons, I think: (1) to find out how frequent a particular error is
(in this case, approximately 1% of cases on the Web); and (2) in case
you're not sure which of two possibilities is the correct version, to find
out which is right.
-Aaron J. Dinkin
Dr. Whom