Allwein, G. Barker-Plummer, D. Barwise, Etchemendy, J., Liu, A. Languague, proof and...

(Parte 2 de 6)

Introduction

Essential instructions about homework exercises / 5 neither are the principles of logic. If your beliefs about a close friend logically imply that he would never spread rumors behind your back, but you flnd that he has, then your beliefs need revision. Logical consequence is central, not only to the sciences, but to virtually every aspect of everyday life.

One of our major concerns in this book is to examine this notion of logical consequence as it applies speciflcally to the language fol. But in so doing, we will also learn a great deal about the relation of logical consequence in natural languages. Our main concern will be to learn how to recognize when a speciflc claim follows logically from others, and conversely, when it does not. This is an extremely valuable skill, even if you never have occasion to use fol again after taking this course. Much of our lives are spent trying to convince other people of things, or being convinced of things by other people, whether the issue is in°ation and unemployment, the kind of car to buy, or how to spend the evening. The ability to distinguish good reasoning from bad will help you recognize when your own reasoning could be strengthened, or when that of others should be rejected, despite superflcial plausibility.

It is not always obvious when one claim is a logical consequence of others, but powerful methods have been developed to address this problem, at least for fol. In this book, we will explore methods of proof|how we can proof and counterexampleprove that one claim is a logical consequence of another|and also methods for showing that a claim is not a consequence of others. In addition to the language fol itself, these two methods, the method of proof and the method of counterexample, form the principal subject matter of this book.

Essential instructions about homework exercises

This book came packaged with software that you must have to use the book. In the software package, you will flnd a CD-ROM containing four computer applications|Tarski’s World, Fitch, Boole and Submit|and a manual that Tarski’s World, Fitch, Boole and Submitexplains how to use them. If you do not have the complete package, you will not be able to do many of the exercises or follow many of the examples used in the book. The CD-ROM also contains an electronic copy of the book, in case you prefer reading it on your computer. When you buy the package, you also get access to the Grade Grinder, an Internet grading service that can check the Grade Grinder whether your homework is correct.

About half of the exercises in the flrst two parts of the book will be completed using the software on the CD-ROM. These exercises typically require that you create a flle or flles using Tarski’s World, Fitch or Boole, and then submit these solution flles using the program Submit. When you do this, your solutions are not submitted directly to your instructor, but rather to our grad-

Essential instructions about homework exercises

6 / Introduction ing server, the Grade Grinder, which assesses your flles and sends a report to both you and your instructor. (If you are not using this book as a part of a formal class, you can have the reports sent just to you.)

Exercises in the book are numbered n:m, where n is the number of the chapter and m is the number of the exercise in that chapter. Exercises whose solutions consist of one or more flles that you are to submit to the Grade

Grinder are indicated with an arrow (ö ), so that you know the solutions areö vs. . to be sent ofi into the Internet ether. Exercises whose solutions are to be turned in (on paper) to your instructor are indicated with a pencil (. ). For example, Exercises 36 and 37 in Chapter 6 might look like this:

Use Tarski’s World to build a world in which the following sentences are all true: :: :

Turn in an informal proof that the following argument is logically valid: : : :

The arrow on Exercise 6.36 tells you that the world you create using

Tarski’s World is to be submitted electronically, and that there is nothing else to turn in. The pencil on Exercise 6.37 tells you that your solution should be turned in directly to your instructor, on paper.

Some exercises ask you to turn in something to your instructor in addition to submitting a flle electronically. These are indicated with both an arrow and a pencil (ö j. ). This is also used when the exercise may require a flle to be submitted, but may not, depending on the solution. For example, the next problem in Chapter 6 might ask:

Is the following argument valid? If so, use Fitch to construct a formal proof of its validity. If not, explain why it is invalid and turn in your explanation to your instructor.

Here, we can’t tel you deflnitely whether you’l be submitting a fle or turning something in without giving away an important part of the exercise, so we mark the exercise with both symbols.

By the way, in giving instructions in the exercises, we will reserve the word \submit" for electronic submission, using the Submit program. We use \turnsubmitting vs. turning in exercises in" when you are to turn in the solution to your instructor.

When you create flles to be submitted to the Grade Grinder, it is important that you name them correctly. Sometimes we will tell you what to name the flles, but more often we expect you to follow a few standard conventions. Our naming conventions are simple. If you are creating a proof using Fitch, thennaming solution flles you should name the flle Proof n.m, where n:m is the number of the exercise. If you are creating a world or sentence flle in Tarski’s World, then you should call

Introduction

Essential instructions about homework exercises / 7 it either World n.m or Sentences n.m, where n:m is the number of the exercise. Finally, if you are creating a truth table using Boole, you should name it Table n.m. The key thing is to get the right exercise number in the name, since otherwise your solution wil be graded incorrectly. We’l remind you of these naming conventions a few times, but after that you’re on your own.

When an exercise asks you to construct a formal proof using Fitch, you will flnd a flle on your disk called Exercise n.m. This flle contains the proof set starting proofs up, so you should open it and construct your solution in this flle. This is a lot easier for you and also guarantees that the Grade Grinder will know which exercise you are solving. So make sure you always start with the packaged Exercise flle when you create your solution.

Exercises may also have from one to three stars (?, ??, ???), as a rough ? stars indication of the di–culty of the problem. For example, this would be an exercise that is a little more di–cult than average (and whose solution you turn in to your instructor):

Design a flrst-order language that allows you to express the following

Remember

1. The arrow (ö ) means that you submit your solution electronically.

2. The pencil (. ) means that you turn in your solution to your instructor.

3. The combination (ö j. ) means that your solution may be either a submitted flle or something to turn in, or possibly both.

5. Unless otherwise instructed, name your flles Proof n.m, World n.m, Sentences n.m, or Table n.m, where n:m is the number of the exercise.

6. When using Fitch to construct Proof n.m, start with the exercise flle Exercise n.m, which contains the problem setup.

Throughout the book, you will flnd a special kind of exercise that we call You try it exercises. These appear as part of the text rather than in You try it sections the exercise sections because they are particularly important. They either illustrate important points about logic that you will need to understand later or teach you some basic operations involving one of the computer programs

Essential instructions about homework exercises

8 / Introduction that came with your book. Because of this, you shouldn’t skip any of the You try it sections. Do these exercises as soon as you come to them, if you are in the vicinity of a computer. If you aren’t in the vicinity of a computer, come back and do them as soon as you are.

Here’s your flrst You try it exercise. Make sure you actually do it, right now if possible. It will teach you how to use Submit to send flles to the Grade Grinder, a skill you deflnitely want to learn. You will need to know your email address, your instructor’s name and email address, and your Book ID number before you can do the exercise. If you don’t know any of these, talk to your instructor flrst. Your computer must be connected to the internet to submit flles. If it’s not, use a public computer at your school or at a public library.

You try it

I 1. We’re going to step you through the process of submitting a flle to the

Grade Grinder. The flle is called World Submit Me 1. It is a Tarski’s World flle, but you won’t have to open it using Tarski’s World in order to submit it. We’l pretend that it is an exercise fle that you’ve created while doing your homework, and now you’re ready to submit it. More complete instructions on running Submit are contained in the instruction manual that came with the software.

I 2. Find the program Submit on the CD-ROM that came with your book.

Submit has a blue and yellow icon and appears inside a folder called Submit Folder. Once you’ve found it, double-click on the icon to launch the program.

I 3. After a moment, you will see the main Submit window, which has a rotating cube in the upper-left corner. The flrst thing you should do is flll in the requested information in the flve flelds. Enter your Book ID flrst, then your name and email address. You have to use your complete email address| for example, claire@cs.nevada-state.edu not just claire or claire@cs|since the Grade Grinder will need the full address to send its response back to you. Also, if you have more than one email address, you have to use the same one every time you submit flles, since your email address and Book ID together are how Grade Grinder will know that it is really you submitting flles. Finally, flll in your instructor’s name and complete email address. Be very careful to enter the correct and complete email addresses!

Introduction

Essential instructions about homework exercises / 9

to enter it by hand each time. You can do this by choosing Save As

J4. If you are working on your own computer, you might want to save the information you’ve just entered on your hard disk so that you won’t have from the File menu. This will save all the information except the Book ID in a flle called Submit User Data. Later, you can launch Submit by doubleclicking on this flle, and the information will already be entered when the program starts up.

J5. We’re now ready to specify the flle to submit. Click on the button Choose

Files To Submit in the lower-left corner. This opens a window showing two flle lists. The list on the left shows flles on your computer|currently, the ones inside the Submit Folder|while the one on the right (which is currently empty) will list flles you want to submit. We need to locate the flle World Submit Me 1 on the left and copy it over to the right.

The flle World Submit Me 1 is located in the Tarski’s World exercise flles folder. To flnd this folder you will have to navigate among folders until it appears in the flle list on the left. Start by clicking once on the Submit Folder button above the left-hand list. A menu will appear and you can then move up to higher folders by choosing their names (the higher folders appear lower on this menu). Move to the next folder up from the Submit Folder, which should be called LPL Software. When you choose this folder, the list of flles will change. On the new list, flnd the folder Tarski’s World Folder and double-click on its name to see the contents of the folder. The list will again change and you should now be able to see the folder TW Exercise Files. Double-click on this folder and the flle list will show the contents of this folder. Toward the bottom of the list (you will have to scroll down the list by clicking on the scroll buttons), you will flnd World Submit Me 1. Double-click on this flle and its name will move to the list on the right.

J6. When you have successfully gotten the flle World Submit Me 1 on the righthand list, click the Done button underneath the list. This should bring you back to the original Submit window, only now the flle you want to submit appears in the list of flles. (Macintosh users can get to this point quickly by dragging the flles they want to submit onto the Submit icon in the Finder. This will launch Submit and put those flles in the submission list. If you drag a folder of flles, it will put all the flles in the folder onto the list.)

J7. When you have the correct flle on the submission list, click on the Submit Files button under this list. Submit will ask you to conflrm that you want to submit World Submit Me 1, and whether you want to send the

Essential instructions about homework exercises

10 / Introduction results just to you or also to your instructor. In this case, select Just Me. When you are submitting flnished homework exercises, you should select Instructor Too. Once you’ve chosen who the results should go to, click the Proceed button and your submission will be sent. (With real homework, you can always do a trial submission to see if you got the answers right, asking that the results be sent just to you. When you are satisfled with your solutions, submit the flles again, asking that the results be sent to the instructor too. But don’t forget the second submission!)

I 8. In a moment, you will get a dialog box that will tell you if your submission has been successful. If so, it will give you a \receipt" message that you can save, if you like. If you do not get this receipt, then your submission has not gone through and you will have to try again.

I 9. A few minutes after the Grade Grinder receives your flle, you should get an email message saying that it has been received. If this were a real homework exercise, it would also tell you if the Grade Grinder found any errors in your homework solutions. You won’t get an email report if you put in the wrong, or a misspelled, email address. If you don’t get a report, try submitting again with the right address.

I 10. When you are done, choose Quit from the File menu. Congratulations on submitting your flrst flle.

Congratulations

Here’s an important thing for you to know: when you submit flles to the

Grade Grinder, Submit sends a copy of the flles. The original flles are stillwhat gets sent on the disk where you originally saved them. If you saved them on a public computer, it is best not to leave them lying around. Put them on a °oppy disk that you can take with you, and delete any copies from the public computer’s hard disk.

To the instructor

Students, you may skip this section. It is a personal note from us, the authors, to instructors planning to use this package in their logic courses.

(Parte 2 de 6)