Essay II
Be sure you have read, understood, and meticulously followed the new and improved instructions, since essays which fail to do so will be returned ungraded. If you have any question or puzzle, please don't hesitate to ask by email to berkich@gmail.com. Please note that this essay is due in class Thursday, 9/30, and worth 100 points.
You may recall my mentioning in class either Erdos or Polanyi, I couldn't remember which, saying "if you can't explain a theorem to a ten year old, you don't understand it yourself." I've since been unable to verify either ever said anything of the sort, although a book by Tom Parker, "Rules of Thumb", attributes the quote to IU mathematician G.S. Tahim. To be sure, there are lots of quotes from Erdos and Polanyi--all due respect to Tahim--well worth pondering. For example, Erdos tells us that,
“In a way, mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite. That's why mathematics is really my only interest.”
while Polanyi speaks to some of the dangers we currently face:
“The downfall of liberty which in every case followed the success of these attacks demonstrates in hard facts what we said before: that freedom of thought is rendered pointless and must disappear wherever reason and morality are deprived of their status as a force in their own right. When a judge in a court of law can no longer appeal to law and justice; when neither a witness, nor the newspapers, nor even a scientist reporting on his experiments can speak the truth as he knows it; when in public life there is no moral principle commanding respect; when the revelations of religion and of art are denied any substance; then there are no grounds left on which any individual may justly make a stand against the rulers of the day. Such is the simple logic of totalitarianism. A nihilistic regime will have to undertake the day-to-day direction of all activities which are otherwise guided by the intellectual and moral principles that nihilism declares empty and void. Principles must be replaced by the decrees of an all-embracing party line.”
But I digress.
The point the attribution of which I was mangling is nevertheless solid and bears emphasis: Take anything conceptually challenging--the distinction, say, between an argument being formally correct and an argument being factually correct. If you cannot explain it to a complete novice, you don't fully grasp the distinction yourself.
Now suppose you are chatting with a friend not in the class who asks, innocently enough, "What are these upside-down 'trees' you keep drawing?"
Such a simple question!
You, of course, answer by telling your friend, "I'm testing translations for validity."
Your friend, thoroughly confused by your glib answer, asks "So, what's validity?"
Answer your friend in this essay. In clear, readily comprehensible, and reasonably concise language, and using well-chosen examples for clarification, answer the question "what is validity?".
Examples are crucial in any explanation. First and foremost, use your own examples--i.e., not ones given in class--to explore the distinction between formal correctness and factual correctness. Second, your essay must contain,
- Four natural language arguments illustrating a sound argument, an argument which is valid but not sound having a false premise and a false conclusion, an argument which is valid but not sound having a false premise but a true conclusion, and an invalid argument having all true premises and a true conclusion;
- An explanation of formal correctness as demonstrated in the Method of Truth Tables which describes how the definition of validity via Truth Tables relates to the general definition of validity and gives two examples: the truth table for a PC-argument which is valid and the truth table for a PC-argument which is invalid; and, finally,
- An explanation of formal correctness as demonstrated in the Method of Analytic Tableaux which describes how the definition of validity via Analytic Tableaux relates to the general definition of validity and gives (using the very same example PC-arguments you used in explaining validity via the Method of Truth Tables above) the semantic tree for a PC-argument which is valid and the semantic tree for a PC-argument which is invalid.
That totals six novel example arguments: four natural language and two PC.
The truth tables and semantic trees you construct may of course be handwritten. They do not count towards the word-count on the essay. The only quotes you may use, which also do not count towards the essay word-count, are i) the general definition of validity, ii) the definition of PC-validity by the Method of Truth Tables, and iii) the definition of PC-validity by the Method of Analytic Tableaux.
One suggestion, if I may: Grab someone not in the class and ask them to read your essay. Find out where they stumble on your explanations and try to clarify for them, then go back and revise your essay accordingly.