Tarski’s World lets you represent simple, three-dimensional worlds inhabited by geometric blocks of various kinds and sizes, and test first-order sentences to see whether they are true or false in those worlds. We begin with instructions on how to start and stop Tarski’s World, and explain the basic layout of the screen.
- Getting Started
- The World Window
- The Keyboard and Sentence Windows
- The Evaluation Box
- Playing the game
The application is contained inside the folder called Tarski’s World Folder. Also in this folder is a folder called TW Exercise Files, in which you will find the exercise files referred to in the book.
Launching Tarski’s WorldTo run Tarski’s World, double-click on the application icon, which looks like an upside-down A floating next to a tetrahedron (pyramid). After a moment, the application will appear on your screen. Pull down each of the menus that appear in the menubar (File, Edit, Display, ...) to see the sorts of commands they contain.
The three main windowsThere are three main windows on the screen. The world window is the black window in the upper left. It contains a grid on which blocks are placed and, on the left, three square buttons showing a tetrahedron, a cube, and a dodecahedron (or soccer ball). Feel free to click on these buttons to see what happens.
The sentence window is the white window across the bottom of the screen. At first it contains only the numeral “1” inside. This is where sentences are entered and evaluated to see whether they are true or false in the world represented in the world window. Click once in the sentence window to activate it. Feel free to type something in the sentence window, say, “I'd rather be in Philadelphia.”
Finally, the keyboard window is the window to the right of the world window. This is the window we generally use to enter sentences of first-order logic. Click on the window to activate it. Feel free to play around by clicking on the buttons in the keyboard window.
There is one last thing to notice. On the right of the sentence window you will see the evaluation box. This is where Tarski’s World will display the results of its evaluations of sentences. We'll say more about this later.
Opening saved filesBoth worlds and sentence lists can be saved as files on your disk. Indeed, many prepackaged world and sentence files come with . To open a saved file, you use the Open... command on the File menu.
To open a file, pull down the File menu and choose Open.... If the world window was active when you did this, you will be presented with a list of any existing world files in the current folder. If the sentence window was active, you will be presented with a list of sentence files. You can get a list of the world files by clicking on the World files button at the bottom of the open dialog box. Similarly, you can get the list of sentence files by clicking on the Sentence files button.
You will have to navigate to the right folder to find the prepackaged files, which are in TW Exercise Files. Find this folder, select it, and then click Open, or simply double-click on the name. Feel free to open one of the files you see, say, Ackermann's World, but if you make any changes to the world, don't save them.
Starting new filesIf you want to start a new world or sentence file, choose New... from the File menu. You will then have to specify whether you want a new world or new sentence file.
Saving a fileTo save a list of sentences, choose Save Sentences or Save Sentences As... from the File menu. To save a world, choose Save World or Save World As... from the File menu.
If the file has never been saved before, a dialog box will appear giving you the option of naming the file you are about to create. If you were to hit the return key, or click the Save button, the file would be saved with the default name. You should type in some other name before hitting the return key or clicking Save. You should also make sure you are saving the file where you want it. Check the directory name at the top of the save dialog box. If you're not in the folder where you want to save the file, navigate to the right one by clicking on this name. If you are not familiar with navigating around the disks and folders on your computer, you should ask a fellow student for help.
Once a file has been saved, the name of the file appears in the title bar of the corresponding window. If you are working on a named file, the Save and Save As... commands behave differently. The first will save a new version of the file under the same name, and the old version will be gone. The second gives you a chance to create a new file, with a new name, and keeps the old file, with its name. For this reason, Save As... is the safer of the two options.
All files created by can be read by either the Macintosh or Windows version of the application.
Quitting (Exiting)Eventually you will want to leave Tarski’s World. To do this, choose Quit from the File menu. If you've made any unsaved changes to the files, will give you a chance to save them.
The World WindowTo put a block on the grid, simply click the button on the left that shows the type of block you want. Try this out. Also try holding down the Option key while you click on a block button. This opens up the block's “parameter window,” allowing you to specify other characteristics of the block. Click OK when you are satisfied with the characteristics.
Naming blocksTo name a block already on the grid, double-click on the block. This will open the Object Parameter window displaying the block's current shape, size, and names, if any. To give the block a name, simply choose the desired name by clicking in the box next to it. When you have chosen the block's name(s), click OK. If you know that you want to name a block (or change its size) when you add it to the world, you can save a bit of effort by holding down the Option key while you click the button to add the block. This immediately opens the Object Parameter window so that you can give the block a name or change its size.
In first-order logic, one object can have several names, but two objects cannot share the same name. Hence Tarski’s World lets you give a block more than one name, but once a name is used, that name cannot be assigned to another block.
Moving blocks
To move a block, position the cursor over the block and drag it to
the desired position. (That is, move the mouse's arrow over the
block, until the arrow turns into an 
. Then, with the button
depressed, move the mouse until the block is where you want it.) If
you move the block too close to the edge it will fall off.
To change a block's size or shape, follow the instructions for naming the block. You can alter the size or shape by clicking in the appropriate circles in the parameter window. Note that if two blocks are in immediately adjacent squares, then neither of them can be large. In that case, the Large option will be grey, and cannot be selected. When you've made your choices, click OK.
Deleting blocksTo delete a block, click on it. It will quiver with anticipation. Then press the backspace key on the (physical) keyboard. The block will jump off the edge. You can also drag the block off the edge of the grid and drop it.
Hiding labelsWhenever you name a block, Tarski’s World labels the block with its name. Of course, in the real world we only wear name tags at unpleasant social occasions. Like us, blocks in can have names without wearing labels. To hide the labels, simply choose Hide Labels from the Display menu. To redisplay the labels, choose Show Labels from the Display menu.
2-D viewLabels aren't the only things that can hide. Sometimes a small block can be obscured from view by another block in front of it. To get a bird's eye view of the world, choose 2-D View from the Display menu. To get back to the usual perspective, choose 3-D View from the Display menu.
Blocks can be moved, selected, and changed from the 2-D view in exactly the same way as the 3-D view. (You can even change to the 2-D view in the middle of playing the game; sometimes you will have to in order to pick an appropriate block, or to see what is referring to.)
Rotating WorldsTo rotate a world by 90 degrees in either direction, choose Rotate World Clockwise or Rotate World Counterclockwise from the Display menu. Such a rotation counts as a change to the world and will be saved when you save the world.
The Keyboard and Sentence WindowsThere are two ways to enter formulas into the sentence window, from the Keyboard window or from the physical keyboard. (From now on we will capitalize the word “Keyboard” whenever referring to the Keyboard window, and use lower case when referring to the computer's physical keyboard.) Most people find it easier to use the Keyboard than the keyboard. So we will begin by describing the use of the Keyboard.
Writing formulasTarski’s World makes writing first-order formulas quite painless. As you may have noticed while playing with the Keyboard, when you enter a predicate, like Tet or BackOf, the insertion point locates itself in the appropriate position for entering “arguments”--variables (u, v, w, x, y, z) or individual constants (a, b, c, d, e, f).
What this means is that a sentence like BackOf(a,b) can be entered into the sentence list with three mouse clicks in the Keyboard: first on the BackOf button, then on the a button, then on the b button. To enter the same thing from the physical keyboard would require 11 keystrokes.
Besides the various symbols used in the language, there are four more buttons in the Keyboard window, a Delete button, an Add Sentence button, a Verify button, and a Game button. The Add Sentence button is immediately below the Delete button (it looks like a curved arrow). The first allows you to delete unwanted symbols and spaces from the sentence window. It works just like the backspace key on the physical keyboard. The second allows you to add a new sentence to your sentence list after the sentence that contains the insertion point. The Verify and Game buttons do the same thing as the buttons in the evaluation box. We'll explain these buttons later.
In order to allow you to write more readable formulas, treats brackets
(“[ ]”) and braces (“{ }”) as completely equivalent to
parentheses. Thus, for example, you could write [Leftof(a,b)
Large(a)] and will read this as
(LeftOf(a,b)Large(a))} But you have to
type brackets and braces from the physical keyboard.
You can add comments to your sentences in a way that will be ignored by the program when it is checking to see if they are well formed or true. You do this by prefacing each line of text you want ignored by a semicolon (;). This will cause to ignore anything that follows on the same line.
Creating a list of sentencesTo create a whole list of sentences, you first enter one sentence, and then choose Add Sentence After from the Edit menu. You are given a new, numbered line, and can then enter a new sentence. If you hit the Return key, this will not start a new sentence, but will simply break your existing sentence into two lines. Use Add Sentence After!
Instead of choosing Add Sentence After from the Edit menu, you can do this from the Keyboard window by clicking the Add Sentence button (the roundish arrow) or you can do it directly from the keyboard in two ways. You can type Shift-Return (that is, type Return while holding the shift key down) or use the keyboard equivalent shown in the menu (Command-A).
To insert a new sentence in your list before the current sentence, choose Add Sentence Before from the Edit menu.
Moving from sentence to sentence
You will often need to move from sentence to sentence within a list of
sentences. (The reason is that the evaluation box applies only to one
sentence at a time, the one in which the insertion point is present.)
You can move the insertion point with the up and down arrow keys (
, 
) on the keyboard
or by clicking on the sentence of interest with the mouse. The left and right
arrow keys (
, 
) on the keyboard also move the
insertion point, but only within a single sentence.
If you hold down the Option key, the up arrow takes you to the first sentence of the list, the down arrow takes you the last sentence of the list, and the left and right arrows take you to the beginning and the end of the current sentence.
Deleting sentencesTo delete a whole sentence and renumber the sentences that remain, choose Delete Sentence from the Edit menu. First make sure the insertion point is somewhere in the sentence you want to delete.
Note that you cannot highlight parts of two different sentences and then delete them. If you want to delete a sentence boundary, you must use the command Delete Sentence from the Edit menu.
Typing symbols from the keyboardSentences can be entered into the sentence window by typing them on the physical keyboard. When typing predicates in the blocks language, you must be sure to spell them correctly and to capitalize the first letter (since otherwise they will be interpreted as names, not predicates). You also have to insert your own punctuation: parentheses after the predicate, and commas to separate multiple “arguments” (as in Between(a,x,z)). To get the logical symbols use the keyboard equivalents shown below
| Symbol | Key | Symbol | Key |
 | ~ |  | # |
| & |  | | |
| $ |  | % |
 | @ |  | / |
 | _ |  | \ |
 | ^ |
These are the same keyboard equivalents used in Fitch and Boole.
Either the sentence window or the Keyboard window must be “active” before typing on the physical keyboard will have any effect. If you type and nothing shows up, that's because the world window is currently the active window. To activate another window, just click in it somewhere.
Cutting, copying, and pastingIf you want to change the order of the sentences in a list, or copy a sentence from one file to another, use the cut, copy, and paste functions.
If you highlight a string of symbols and then choose Cut or Copy from the Edit menu, the string of symbols is stored on the computer's clipboard. The difference between the two commands is that Cut deletes the highlighted symbols from their present position, while Copy leaves them in place. You can't see the contents of the clipboard, but the symbols will be there until you cut or copy something else to the clipboard.
Once something is on the clipboard, it can be pasted anywhere you want it. Just put the insertion point at the desired place and choose Paste from the Edit menu. A copy of the string of symbols on the clipboard will be inserted. You can paste several copies at several different points, if you want to.
You can copy sentences out of Tarski’s World and paste them into Fitch or Boole, and vice versa.
PrintingTo print your sentences or world, choose the appropriate Print... command from the File menu. If your computer is not connected to a printer, this probably won't work.
If your sentences print with incorrect symbols, quit , find the font suitcase labeled “Tarski” on your CD-ROM, drag the font suitcase onto your (closed) System Folder, and then re-launch . This will only be necessary if you have unusual fonts in your System Folder that violate Apple's font numbering conventions.
The Evaluation BoxThe evaluation box appears on the right of the sentence window. It is what ties the sentence and world windows together.
Verifying syntax and truthAs you will learn, only some strings of symbols are grammatically correct, or well formed, as we say in logic. These expressions are usually called well-formed formulas, or wffs. And only some of these are appropriate for making genuine claims about the world. These are called sentences. Sentences are wffs with no free variables. You will learn about these concepts in the text.
To see if what you have written in the sentence window is a sentence, and if so, whether it is true in the world currently displayed, click on the Verify button (either in the Keyboard window or in the evaluation box). Alternatively, you can hit the Enter key on the physical keyboard. If you want to check a whole list of sentences, choose Verify All Sentences from the Edit menu. Alternatively, type Option-Enter or Command-F.
When you verify a sentence, the results are displayed in two places.
In the evaluation box, checkmarks will appear showing whether the
current formula is a well-formed sentence of first-order logic,
whether it can be evaluated in the current world, and finally whether
it is true in the current world. (Well-formed sentences will not be
evaluable in a world if they either contain predicates that
does not understand or contain names that are not assigned to any
block in the world.) These results are also displayed in the margin to
the left of the sentence: “
” indicates that the formula is not
well-formed or not a sentence; “
” indicates that the formula is a
sentence of first-order logic, but not evaluable in the current world;
and “T” or “F” indicates that the sentence is true or false
in the world.
When you stake out a claim about a world with a complex sentence, you
are committed not only to the truth of that sentence, but also to
claims about its component sentences. For example, if you are
committed to the truth of a conjunction AB (read “A and B”) then you are also committed
both to the truth of A and to the truth of B.
Similarly, if you are committed to the truth of the negation A (read “not A”), then you are committed
to the falsity of A.
This simple observation allows us to play a game that reduces complex commitments to more basic commitments. The latter claims are generally easier to evaluate. The rules of the game are part of what you will learn in the body of this book. Here, we will explain the kinds of moves you will make in playing the game.
To play the game, you need a guess about the truth value of the current sentence in the current world. This guess is your initial commitment. The game is of most value when this commitment is wrong, even though you won't be able to win in this case.
To start the game, click the Game button in the evaluation box or in the Keyboard window. Tarski’s World will begin by asking you to indicate your initial commitment. At this point, how the game proceeds depends on both the form of the sentence and your current commitment. A summary of the rules can be found in Table 9.1 in Chapter 9 of the textbook.
Picking blocks and sentencesAs you see from the game rules, at certain points you will be asked to pick one sentence from a list of sentences. You do this by clicking on the desired sentence and then clicking OK.
At other points in the game, you will be asked to pick a block satisfying some formula. You do this by moving the cursor over the desired block and selecting it. Then click OK. Tarski’s World assigns a name to the chosen block, for example n1, and labels it.
Backing up and giving upnever makes a mistake in playing the game. It will win if it is possible for it to win, that is, if your initial commitment was wrong. However, you may make a mistake, and so lose a game you could have won. All it takes is some bad choices along the way. will take advantage of you. It will not tell you that you made a bad move until it has won, when it will inform you that you could have won. What this means is that there are two ways for you to lose: if you were wrong in your initial assessment, or if you make a faulty choice in the play of the game. To put this more positively, if you win a game against the computer, then you can be quite sure that your initial assessment of the sentence, as well as all subsequent choices, were correct.
To make up for the edge the computer has, allows you to retract any choices you have made, no matter how far into the game you've gone. So if you think your initial assessment was correct but that you've made a bad choice along the way, you can always retract some moves by clicking on the Back button. If your initial assessment really was correct, you should, by using this feature, eventually be able to win. If you can't, your initial commitment was wrong.
If, halfway through the play of the game, you realize that your assessment was wrong and understand why, you can stop the game by clicking the Quit button or closing the game window. This ends the game, but does not exit from Tarski’s World.
When to play the gameIn general, you won't want to play the game with every sentence. The game is most illuminating when you have incorrectly assessed a sentence's truth value, but are not sure why your assessment is wrong. When this happens, you should always play the game without changing your commitment. Tarski’s World will win, but in the course of winning, it will usually make clear to you exactly why your assessment was wrong. That's the real value of the game.
You might wonder what happens when you play the game with a correct assessment. In this case, if you play your cards right, you are guaranteed to win. But does not simply give up. At those points in the game when it needs to make choices, it will make them more or less randomly, hoping that you will blunder somewhere along the line. If you do, it will seize the opportunity and win the game. But, as we have noted, you can always renege by backing up.
