Difference between If...then and Suppose...thenDate: 11/25/2002 at 22:02:35 From: Dan Subject: Proofs: difference between "if..then" and "Suppose..then" What is the difference between "If X, Then Y" and "Suppose X. Then Y."? Here, X and Y are given statements used in proving something. Can you explain this with an example? Thanks so much! Dan Date: 11/25/2002 at 23:51:09 From: Doctor Achilles Subject: Re: Proofs: difference between "if..then" and "Suppose..then" Hi Dan, Thanks for writing to Dr. Math. That is a very interesting and difficult question. In mathematical logic, there is a notion of a sentence. A sentence is one of three things: 1) A simple sentence, given by a letter. 2) The negation of a sentence. 3) The joining of two sentences by a connective (such as "and," "or," and "if...then") You can build very complicated sentences by applying (2) and (3) reiteratively. You can find a detailed explanation of sentences (and some other basic logic) at: A Crash Course in Symbolic Logic http://www.mathforum.com/dr.math/faq/symbolic_logic.html The sentence "If X, then Y" is just a single sentence. It is no more interesting logically than the sentence "P and Q" or "A or B" or "Not Z." It is simply a statement that X and Y have a certain truth- relationship to each other (specifically, that X cannot be true at the same time that Y is false). The problem with the English words "if...then" is that they have too many connotations associated with them. Informally, in everyday language, 'if' and 'then' can be used in dozens of ways. They are often used interchangeably in English with phrases such as "supposing ... conclude." So in English, you can argue that "if...then" is equivalent to "suppose...then," but not so in logic. This is okay for English, but in mathematical logic, we want to have a very strict definition of "if...then," that is to say, we want one and only one of the English uses to apply. In order to make sure that people don't get confused between the rather squishy and malleable English words "if...then," logicians use symbols. The symbol in logic for 'and' is ^. Thus "P and Q" is written (P^Q). The symbol in logic for 'or' is v. Thus "R or S" is written (RvS). The symbol in logic for 'not' is ~. Thus "Not A" is written ~A. The symbol in logic for 'if..then' is ->. Thus "If X, then Y" is written (X->Y). The use that has been chosen by logicians for the -> symbol is as follows: For any pair of sentences X and Y, the sentence (X->Y) means exactly and only "It is forbidden that X be true at the same time that Y is false". So "if X, then Y" is a single sentence. By contrast "Suppose X. Then Y." is actually two sentences. More than that, "Suppose X. Then Y." is a pair of commands. Commands are like orders; they tell you to do something (like "Sit." or "Do your homework."). Let's start by looking at the first command: "Suppose X." What is this telling us to do? Well, to suppose is basically to pretend for a minute that something is true. For instance, if I'm in a courtroom, I could say something like "Suppose the criminal is overweight." In doing this, I am asking the court to "play along" with me for a while by pretending that the criminal is overweight. This does not mean that I have proven that the criminal is overweight, merely that I think we can learn something by pretending for a while. Of course, everyone in the courtroom knows this, and sooner or later we will all have to stop supposing (pretending) and come back to reality. What about the second command: "Then Y."? I believe (and I hope you agree) that this command is implying something like "Then, conclude Y." Or, more explicitly "While you are supposing X, conclude Y." What this means is, I hope, fairly clear from the expanded command. We started with our first command by pretending that X is true. We can then do logical work on X (maybe we'll derive a contradiction from it, or combine it with things we know are true or other things we may also be pretending are true for the time being, etc.). Let's compare two problems you may see in a logic class: I) Suppose (A^B). Then, conclude A. II) ((A^B)->A) The first problem gives you a set of directions. It tells you two steps that it wants you to do. First, it wants you to suppose (A^B). [When I'm doing logic, I always use curly brackets {} to set off supposed or pretend sentences - see the crash course cited above for more.] { Starting a new supposition 1) (A^B) Assumption (supposition) Now that we have supposed (A^B), what is the next step? We have to conclude A. To do that we need the help of a rule of logic. This rule is called simplification (or, as I like to call it ^elimination). The rule says, if you have a sentence of the form (X^Y), then you are entitled to take from that either X or Y or both. Let's give it a shot: { Starting a new supposition 1) (A^B) Assumption (supposition) 2) A obtained from line 1 by simplification of the ^ [^elimination on 1] } Closing off the supposition That's it; we have done both parts of problem (I). Now let's turn to problem (II). Remember, it was: II) ((A^B)->A) This problem makes no sense by itself. It might as well have been: II') ((A^C)v(B->D)) It's just a sentence in symbolic logic. This is not a legitimate problem because we are not told to do anything. We can just as easily sit and look at the problem, cross it out, copy it exactly, hold it up to a mirror, or whatever we want. We need to know what we are to do with this sentence ((A^B)->A). Maybe the problem appears in a section with the heading "Copy the following symbolic logic sentences on another piece of paper." In that case, that is what we are to do. Maybe it appears in a section with the heading "Construct a truth table for the following symbolic logic sentences." Then we should do that. Maybe it appears in a section that says "Give an argument that the following sentences are tautologies." A 'tautology' is a sentence that is always true, no matter what. If we are told to do that, this is how we might proceed: The sentence ((A^B)->A) is a tautology because it is impossible for (A^B) to be true without both of the following conditions: (1) A is true and (2) B is true. Therefore, it is impossible for (A^B) to be true at the same time that A is false, and therefore the sentence satisfies the requirements of the -> symbol. [Notice that I didn't use the English words "if...then" at all in the argument.] So much for the difference between the sentence (X->Y) and the pair of commands: "Suppose X. Then, conclude Y." Despite the fact that they are very different, they are also closely related. It turns out that one way to prove that (X->Y) is a tautology is to follow the pair of commands "Suppose X. Then, conclude Y." To do this we need to go back to our problem (I) above: { Starting a new supposition 1) (A^B) Assumption (supposition) 2) A obtained from line 1 by simplification of the ^ [^elimination on 1] } Closing off the supposition And add a new rule to our repertoire. The rule is something I call: "->introduction". Here is how it goes. If you suppose x, and under that supposition conclude Y, then you are entitled to close off the supposition and state that (X->Y) is true. { Starting a new supposition 1) (A^B) Assumption (supposition) 2) A obtained from line 1 by simplification of the ^ [^elimination on 1] } Closing off the supposition 3) ((A^B)->A) ->introduction on lines 1-2 So the connection between the pair of commands "Suppose X. Then Y." and the sentence "If X, then Y." is that the pair of commands is a set of instructions for proving the sentence is true. However, do not confuse the set of commands with the product they produce. To do that is like trying to eat your mother's favorite cookie recipe for dessert. The set of commands is merely a set of instructions for making the sentence; it is not the sentence itself. The problem is that the set of commands is so simple that our intelligent minds see directly and immediately through the instructions to the conclusion. (And, unlike mother's cookies, the sentence tastes just as papery as the instructions.) I hope this helps. If you have other questions or you'd like to talk about this some more, please write back. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ |
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