Hazardous Material Beyond This Point. DO NOT ENTER Without a Hard Hat.
(HAHA! I CHANGED IT!)
TIME: NOVEMBER 1946
CHALLENGE: THE WORLD IS AT WAR OVER THE DEFINITION OF A SQUARE.
THE SOVIET CAMP (WHICH WILL FROM HERE ON BE REFERRED TO AS THE EASTERN "SQUARE") IS CERTAIN THAT ANYTHING CAN BE A SQUARE AS LONG AS IT IS FOR THE GOOD OF THE PEOPLE.
THE WEST STEADFASTLY SAYS THAT SOMETHING MUST HAVE FIVE EQUAL SIDES TO BE CONSIDERED A SQUARE.
YOU REPRESENT THE U.N. YOU MUST MEDIATE PEACE BETWEEN THESE TWO CONFLICTING POWERS OR THE WORLD AS WE KNOW IT WILL BE ANNIHILATED.
MAY GOD HELP US ALL.
There are many characteristics of good definitions that should be considered. If the U.N. creates an official definition of a square the East and the West may be able to put down their protractors and talk. A good definition is a definition that clearly describes and uniquely classifies an object or a group of objects. A good definition would be helpful in this case because the two groups of countries can't seem to agree!
Biconditional statements are statements that include a conditional and its converse. An example of this would be "If a figure has four equal side of equal length and four right angles then it is a square; if a figure is a square then it has four equal side of equal length and four right angles." If only the U.N. could come up with a definition like this that all parties would agree on! The rampant bloodshed must be stopped!
The definition that a square is a figure that has four equal side of equal length and four right angles can be broken up into two true conditionals, as follows: If a figure has four equal side of equal length and four right angles then it is a square. If a figure is a square then it has four equal side of equal length and four right angles.
AFTER MUCH DEBATE THE TWO SIDES HAVE DECIDED TO AGREE ON THE UNITED NATIONS' DEFINITION OF A SQUARE.
JUST AS THE DUST CLEARED, HOWEVER, THE "SQUARE" CHALLENGED THE ANGLE MEASURE POSTULATE THAT WAS CREATED BY THE UNITED NATIONS JUST EARLIER THIS YEAR.
FRANCE, ONE OF THE LARGEST CONTRIBUTORS TO THE POSTULATE, HAS DECLARED THAT THEY WILL BE SNOBBY TO AMERICANS UNTIL THE "SQUARE" RECOGNIZES THEIR POSTULATE.
WILL THE UNITED NATIONS SUCCEED NEXT TIME? TUNE IN NEXT WEEK, FOLKS....
(...just say "no".)
Socrates on Definitions
**WARNING: IF YOU HAVE NOT READ THE DIALOGUES OF PLATO OR DO NOT KNOW WHO MENO IS, GO NO FURTHER!**
(A Dialogue between Socrates and one of his pupils)
Tell me, oh great and knowledgeable one, why are definitions important?
Ah, but why are definitions important? In order to answer that question, should we not first examine the very principles, of what a definition is? Then define "definition" for me.
But how am I, who know so little, qualified to make such a statement.
What would you, yourself, consider a definition?
Well, I would consider a definition something that… that specifies an object or class of objects. Like a dog is a creature with fur and four legs that can be found in the streets, or in people's homes.
But by that definition, would not a cat, or even a rat be considered a dog?
Bearing this in mind, is there anything else you would add to your definition?
A dog is a creature with fur and four legs that can be found in the streets, or in people's homes. A dog also barks and is companionable with humans.
Is there anything else then that is important to bear in mind when making definitions? They should be specific, so that there can be no misunderstandings about what is meant by them. Call one of your servants over here.
Come here, boy.
Now listen carefully, boy. The definition of an isosceles triangle is a polygon with three sides. Understand?
It depends upon what a polygon is, Sir.
Why? Is the definition unclear?
Yes, Sir. I cannot understand what this thing of which you speak is, if you do not first tell me what a polygon is.
You are dismissed.
So therefore, should all definitions be made using only previously defined terms?
Yes, or commonly understood ones.
Very well then. So I could say that an arc is a line connecting at least two nodes, or points.
Of course not, "oh great one." You should know that it is necessary to define what lines or points are.
Since you are so wise, perhaps you could enlighten me. What are lines and points?
Er… well, it varies. In this sense they're… but sometimes they mean…
So can a definition also include terms that are purposefully undefined?
Such as point and line? Yes. But what would you consider, personally, a good way to express a definition?
Tell me. Is it true that if a polygon is a square, then it is a regular quadrilateral?*
Yes, I believe so.
Is it also true that if a polygon is a regular quadrilateral, then a polygon is a square?
Ingenious! Are you saying that a definition must be expressed as a biconditional?
Or a conjecture and its converse? But what is a biconditional?
A conjecture is an if a then b statement. Its converse is if b then a. And it makes perfect sense that a definition must be made up of a true biconditional.
So we have established that a good definition must specify an object or a class of objects, be specific, include only previously defined, commonly understood, or purposefully undefined terms, and expressed as a biconditional. Could it not be added that good definitions should not have too much or too little information in them?
Of course. And a definition is important because without them, confusion would ensue. If I were to explain a dog as I had the first time, then people would be unsure if a cat were a dog. And it's just as important with math, where if you don't have a good definition, you can end up with all sorts of differing formulas, particularly when working with proofs. Gee, thanks for teaching me why definitions are important, Socrates.
All in a day's work, my son.
I don't believe in regular quadrilaterals.
As, apparently, a staff member of the school newspaper, it has come to my attention that many of the definitions used in this school are faulty, broken, or otherwise in need of repair, and should be fixed or replaced immediately.
--That all definitions not complying with standards (see below) be immediately removed, be it for repair or replacement.
--That all existing definitions (those not removed for repair/replacement) should be checked.
--That all definitions used to replace faulty ones should be individually checked before being released into the school system
--That until all definitions in the system are effective, school should be closed.
--That definitions older than twelve years should be replaced, as they are often worn and abused by this time.
--It is suggested that any activity, be it during or after school, that requires precise, good definitions see the office immediately to apply for replacements for the old or imprecise definitions.
--Must specify an object or class of objects
--Must not be vague
--Should include only commonly understood, previously defined, or purposely undefined (as with “point” and “line”) words
--Must be able to be written as a biconditional
According to the math book, a “definition” is:
“A description that clearly and uniquely specifies an object or a class of objects.”
This can be written as a biconditional:
Something is a definition if and only if it is a description that clearly and uniquely specifies an object or a class of objects.
--Must not contain excess information (for example, adding things about horseshoes or oranges to the above definition)
--Must not have insufficient information (for example, not including “clearly and uniquely” in the above definition)
In conclusion, after discussion with both the principal and various English teachers, I have determined that it is vital for the education and safety of the students that the definitions be checked at once and replaced if neccesary. Please see the office for further details.
Escape the horror and Return to the Brink