------------------------------------------------------------------------ SUBJECT: POSITION PAPER 3: COMMENTS (LONG DOCUMENT!) ------------------------------------------------------------------------ Statistically, you did very much better on PP3 than on the two previous ones, which means you're learning something! The class average was B, and the grades ranged between 16 and 30 points (out of 30): C- to A. I'll return them on Wednesday. But... 1. TERMINOLOGICAL PROBLEMS ------------------------------------------------------------------------ There are still some serious misunderstandings about the nature of how to analyze or evaluate arguments, and about the language for doing that. For instance, some of you said things like "Premise 1 is either false or invalid" or "Premise 2 is valid but false". These make no sense. So let me try once more... A premise is a sentence: Sentences can only be true or false; they cannot be valid or invalid. (And even if you thought that you could call a sentence "valid", I would have expected that you thought that that meant that it was true, so it would make no sense to say that it was "valid but false".) Instead of repeating the definitions of these terms, let me try a new approach: Consider a set of sentences. Sentences, as I've said, are either true or false. As you may recall from the beginning of the semester, a sentence is true if and only if ("iff") it corresponds to reality, i.e., iff it correctly describes some part of the world. (And it's false otherwise.) You can't always or easily tell whether a sentence is true, so we can relax this a bit and say that sentences can be such that either you agree with them or you don't. (And, of course, you always have to say *why* you do or don't agree.) But sometimes sentences are related to each other in various ways. Sometimes, for instance, they contain "matching parts" (there are some examples later on) or are about the same topics. Another way is for some sentences to be reasons for believing another sentence. In that case, the sentences that are reasons are called "premises", the sentence that is the one they are reasons for is called a "conclusion", and the set of the premises and conclusion is called an "argument". The conclusion of an argument can, of course, be true or false, i.e., you can agree with it or not. But here's the tricky part: Besides being "absolutely" or "independently" true or false (agreeable or disagreeable), a conclusion can also be RELATIVELY TRUE: More precisely: a conclusion can be TRUE *RELATIVE TO* THE TRUTH OF ITS PREMISES. What this means is that you can have a situation in which a sentence is, let's say, *absolutely* or *independently* false (or you disagree with it), but it could also be true *relative* to some premises. How could that be? Easy: If the world is such that, whenever it makes the premises true, then it also makes the conclusion true, we say that the conclusion is true relative to the premises. But note that this is a conditional statement: "*Whenever* the world makes the premises true, then...". But sometimes the world might *not* make the premises true. And then we can't say anything about the truth of the conclusion. To link this to the terminology we've been using all semester: When a conclusion is true relative to its premises, then the argument is said to be valid. There is, however, one other point that bears repeating: What's wrong with the following claim (that some of you made!)?: "I agree with all of the premises, and I believe that all of the arguments are valid, but I disagree with the conclusion." What's wrong is that that's inconsistent! Valid arguments are "truth-preserving": The premises lead from truths to truths. If you start out with true premises, then a valid argument won't lead you to a false conclusion. So, if you believe all premises of a valid argument, but *don't* believe the conclusion, then you have to rethink things. One of those premises must be false, or one of the arguments must be invalid. And if you rethink it, and *don't* find the place where you disagree, then you *must* accept the conclusion! That's logic :-) 2. DETERMINING WHETHER AN ARGUMENT IS VALID ------------------------------------------------------------------------ It's one thing to say that you think that an argument is valid. It's another to say *why* you think so. Just saying that it seems logical isn't enough. There are several ways to convince your reader that an argument is valid. I'll list a few, and then apply some of them to PP3. First, you can try to convince your reader that if the world had made the premises true, then the world would have to have made the conclusion true, i.e., that the conclusion would have to be true if the premises were true. If you think the argument is *invalid*, then you have a slightly easier task: Find a situation in which the premises *are* true but in which the conclusion is *false*. Second, you could show that the argument follows a generally accepted "rule of inference", like "modus ponens": If P, then Q. P. Therefore, Q. or a rule that generalizes this: For anything, x, if x has property P, then x has property Q. Some specific thing, c, has property P. Therefore, c has property Q. or a set-theoretic version: All things that are in class A are also in class B. This thing, c, is in class A. Therefore, c is in class B. Third, you can reason with the premises. Let me illustrate this with PP3: 1. A physical object can compute iff it can do what a UTM can do. 2. A computer is (by definition) any physical device that can compute. 3. The human brain is a physical object that can do what a UTM can do. 4. Therefore, the human brain is a computer. This is valid (but if you argued that it was invalid and gave a counterexample to show that 1,2,3 could be true while 4 was false, I gave you full credit). Here's why it's valid: 1 says: A physical object, x, can compute iff x can do what a UTM can do 2 says: A thing, x, is a computer iff x is a physical device that can compute. We can logically combine these to get: x is a computer iff x is a physical object that can do what a UTM can do. (If A is true iff B is true, and B is true iff C is true, then A is true iff C is true--just eliminate the "middleman".) (Call this "intermediate conclusion" premise 2.5.) (Some of you pointed out that this requires a missing premise to the effect that "device" = "object". If you believe that equality, then the argument so far is valid. If you don't believe it, then the argument may still be valid, but will be unsound. For the sake of this example, let's assume that "device" = "object".) 3 says: The human brain is a physical object that can do what a UTM can do. Combining these last two sentences, we get: The human brain is a computer. But that's sentence 4. We've just shown that it's true relative to the truth of the premises, so the argument 1,2,3/.'.4 is valid. Now consider the argument 1,2,5/.'.6 (By the way, some of you missed this argument altogether. But I pointed it out in the grading scheme!) 5 says: A UTM can execute ("do") MS Word. We can combine this with 2.5 to get: If x is a computer, then x can execute MS Word. That's sentence 6. So, we've just shown why 1,2,5/.'.6 is valid. Finally, consider 4,6/.'.7: We can combine 4 and 6 to get: The human brain can execute MS Word. That's why 4,6/.'.7 is valid. 3. OTHER MISUNDERSTANDINGS ------------------------------------------------------------------------ There were some other misunderstandings--having to do with substantive concepts--that I'd like to clear up. First, some of you are not clear about the relationship between TMs and UTMs. Although all UTMs are TMs, most TMs are not UTMs. A TM computes exactly one algorithm; it is a hardwired computer capable of doing only one thing. A UTM is a stored-program, general-purpose computer. If you give it a suitable program, it can execute that program, so it can do what any other TM can do (if given that TM's program). Your laptop is a UTM. Well, more precisely, your laptop is a physical implementation of a UTM. You can program it, or load programs into it, that will allow it to do any computable task. If you run out of memory, you can buy some more. (Unfortunately, if you run out of time, you can't buy any more.) And contrary to what many of you think, your laptop *does* have a "tape"; it's called "random access memory" (RAM), and it's somewhat more flexible than a TM "tape", but it plays the same role as the tape. (Actually, the combination of the memory in the hard drive plus RAM corresponds to the tape.) Second, several of you confused "do not" with "cannot". Just because you have a computer that *does not* run MS Word doesn't mean that your computer *cannot* run it. As counterexamples to the statement that "any computer can execute MSWord", some of you gave: calculators iPhones Linux machines Macs There are problems with all of these. First, what kind of calculator? If it's a non-programmable one, then it's not a UTM and we wouldn't consider it a "computer" for the sake of this argument. If it's programmable, and if it's TM-equivalent--i.e., can, in principle, compute anything a TM can compute--then, given enough memory, it could run MS Word. Ditto for iPhones (and I think some "smart phones" do run a version of it). Maybe there's no version of MSWord for Linux machines, but that doesn't mean there couldn't be. For one thing, the operating system (Linux) is irrelevant; all that counts is the CPU: If MS Word's algorithm could be compiled into the Linux machine's machine language, then the Linux machine could run MSWord. And--some of you will be surprised to learn--Macs *do* run MSWord! I use it frequently on my Mac. (And the Mac OS X operating system, just like Linux, is a version of the Unix operating system, so that's not the stumbling block for Linux.) Moreover, I was able to run *the very same MS Word program* on my old PowerPC Mac and on my new Intel Mac, so even the machine language is somewhat irrelevant, as long as there's a way to compile the algorithm into it. (Some of you had other reasons for thinking that some computers couldn't run MS Word, having to do with whether UTMs can halt or not. That's a different issue.) 4. INTERESTING OBSERVATIONS ------------------------------------------------------------------------ On a positive note, some of you had some very interesting observations that I would like to record: * Unlike a TM tape, human memory is subject to forgetting. This may mean that humans can't compute everything that a UTM can. But if humans couldn't forget, they might not be able to compute as much either. (For a literary discussion of this, see Jorge Luis Borges's short story, Funes the Memorious; http://evans-experientialism.freewebspace.com/borges.htm) * Humans, in any case, are not limited to their memory. They can use scrap paper to help them keep track of complicated computations. For a philosophical discussion of this, see: Clark, Andy, & Chalmers, David J. (1998), "The Extended Mind", Analysis 58: 10-23. http://consc.net/papers/extended.html (and for more on this topic, see: http://www.cse.buffalo.edu/~rapaport/575/F08/situatedcog.html) * Some of you pointed out that a human-brain implementation of MS Word might not be very efficient, but that's not what the argument claimed. Nor did the argument claim that it would be *reasonable" to "run" MSWord "on" a human brain :-) It only claimed that it would be possible. A very, very long time ago, if I wanted to write a long document, I might have hired a human scribe to "process" my "words", and he (or she) might have used some of MS Word's algorithms to do that :-) * Some of you distinguished between a human brain doing *what* a UTM can do vs. doing it *how* the UTM does it. This, of course, is what distinguishes between input-output specifications and algorithms. In the argument, I did have in mind that the human brain could actually execute MS Word's algorithms, not merely that it could process words (we know it can do that!).