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1988-06-28
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WFFs AND PROOFS
Introduction
WFFs AND PROOFS is a series of twenty-one games designed to help the
player become accustomed to symbol-handling and learn something about
mathematical logic, as well as to provide practice in abstract thinking
and, of course, to have some fun at the same time. The first few games
are relatively simple and can be played by both children and adults,
although children may require some help in understanding the instruc-
tions and basic concepts involved. The final games will provide a
challenge to even the most astute players, including experts in
mathematical logic.
There are 13 basic concepts gradually and repeatedly used throughout
the games: the definition of a WFF, the definition of a Proof, and 11
Rules of Inference. (Together, they comprise one formulation of the
system of logic called propositional calculus.)
The twenty-one games are played with 36 simulated dice cubes:
18 capital-letter cubes (C-cubes) with N, C, A, K, E and R.
18 small-letter cubes (s-cubes) with p, q, r, s, o and i.
The program randomly selects and displays on the screen a given number
of capital letters and small letters, without any graphic representa-
tion of them as cubes. However, in the instructions and prompts, those
letters will often be referred to as cubes, s-cubes, and C-cubes.
This program is a computerized and somewhat altered version of a board
game developed by Layman E. Allen. Persons desiring more detailed
background information and instruction on the basic concepts involved
in these games should obtain a copy of the instruction manual for the
original board game: "WFF 'N PROOF, The Game of Modern Logic" by
Layman E. Allen (Copyright 1962, Layman E. Allen). This computerized
version was developed using the 1969 edition of WFF 'N PROOF as a base,
but the computerized version contains several variations in procedures
and scoring. While the emphasis in the original project appears to
have been equally divided between learning and fun, the computerized
version attempts to put the emphasis on the latter. No conscious
effort was made to write these instructions, or those in the program
and its prompts, as a teaching tool for logic or at a level easily
understood by children.
The basic object of the games is to form a WFF which, in the study of
logic, represents the term "Well-Formed Formula". The games are based
on logic, with the small letters representing sentence variables and
the capital letters representing connectives, as will be discussed
later.
In order to make a WFF and play the first two games, however, it is not
necessary to apply specific meanings to the sentence variables nor to
learn the definitions of the connectives.
**********
PART I: GAMES 1 AND 2
DEFINITION OF A WFF
In the following definitions, "expression" refers to any given combina-
tion of one or more s- or C-cubes.
There are three and only three types of WFFs:
1. A single-letter expression that is p, q, r or s.
2. An expression that begins with an N and is followed by one WFF.
3. An expression that begins with C, A, K or E and is followed by
two WFFs. (These may be referred to as Cake-Wffs.)
An expression containing an i, o or R is *not* a WFF.
The following are examples of WFFs:
q (one of the specified single letters)
Nq (N plus one WFF: q)
Cpq (C plus two WFFs: p and q)
NCpq (N plus one WFF: Cpq)
NNCqq (N plus one WFF: NCpq)
CNpNq (C plus two WFFs: Np and Nq)
CqCNpNq (C plus two WFFs: q and CNpNq)
When any expression forms a WFF, it can be combined with other letters
to make a larger WFF. Thus, in the last example, two WFFs (Np and Nq)
are combined with C to make a single WFF (CNpNq), and that WFF then
becomes one of the two WFFs (the other being the first q) required
after the first C to form the complex Cake-WFF: CqCNpNq.
The following are *not* WFFs:
i (i is not included in the definition of a WFF)
o (o is not included in the definition of a WFF)
qN (a multi-letter WFF must begin with N, C, A, K or E)
Rq (R is not included in the definition of a WFF)
Cq (only one WFF follows the C; there must be two WFFs after C)
NCq (An N-WFF has one WFF after the N, but Cq is not a WFF, as seen
in the example above)
CAqrKs (Aqr is one of the two required WFFs after the C, but the K
only has one WFF after it and, hence, is not itself a WFF.)
For complex WFFs beginning with C, A, K or E, it is sometimes helpful
to use hyphens to separate the initial letter and each of the two
following WFFs. Thus, CqCNpNq may be written as C-q-CNpNq to show that
q and CNpNq are the two WFFs required after the C. Also, the separate
WFF, CNpNq, could be written as C-Np-Nq.
Additional Comments About WFFs
From the above discussion, it can be seen that, in order to form a WFF,
the letters i, o and R may be ignored and that any attempt to formulate
a WFF with those letters will be unsuccessful. (Those letters will
become useful in later games, but never become part of a WFF.)
Certain other conclusions can be drawn, which are mentioned below but
which are not necessary to understand the games and have little or no
relationship to the rules of logic. They are mentioned only because
they may facilitate the rapid formulation of a WFF from many available
cubes.
-- Once you have formed a basic WFF, you can add any number of Ns
in front of it and it will still be a WFF. If you add a C, A, K or E
in front of a basic WFF, you must add another WFF either immediately
after that letter or at the end of the basic WFF. For example:
If Nq is a WFF, so is NNNq.
If Cqq is a WFF, so is NNCqq.
If Cqq is a WFF, so are CCqqp and CsCqq and NNNCrCqq.
-- Any WFF without an N will have one more s-cube than C-cubes
(because it will be either a single s-cube or will require two
WFFs after the C-Cube).
-- Any WFF with only one N in it will have an equal number of s-
cubes and C-cubes (because N is a C-cube).
-- Any WFF with two or more Ns in it will have fewer s-cubes than
C-cubes and, for each N over one, there will be one more C-cube
than s-cubes (because N is a C-cube and requires only one WFF
after it).
Examples: Cqq has one more s-cube than C-cubes.
NCqq has an equal number of s- and C-cubes.
NNCqq has two more C-cubes than s-cubes.
*****
Any procedural matters not covered in these instructions will be
explained by prompts displayed by the program while playing the game.
(For example, these instructions do not specify how to indicate that
you have finished forming a WFF, but the prompts during play of the
game will explain what to do.)
*****
GAME 1: SHAKE-A-WFF
Object: Form the longest possible WFF from as many as eight s- and C-
cubes.
Playing: You will start with 3 cubes (1 C-cube and 2 s-cubes) selected
at random. Use them to form the longest possible WFF
according to the rules for forming the three types of WFFs
explained above.
Your proposed WFF will be evaluated and you will be advised
whether or not it is a WFF. You will also be shown one of
the longest possible WFFs available from the cubes provided
to you.
If you have correctly formed a WFF and if it has the same
number of letters as the computer's example, you will be
given another cube and presented with another random combina-
tion from which you must form another longest-possible WFF.
If you indicate that no WFF can be formed from the available
cubes and one could have been formed, or if your WFF is not
as long as the computer's example, then one cube will be
removed and you will be presented with another random
combination from the remaining cubes.
If you have no cubes left, you lose.
When you have 8 cubes, if you manage to form one of the
longest possible WFFs from the cubes presented, you win.
NOTE: Before reading any further in these instructions, you should
play Game 1 until you become adept at forming WFFs. You should not
proceed to the next game until you have won a game of Shake-A-WFF.
*****
GAME 2: COUNT-A-WFF
Object: Within a given level of difficulty, form a correct WFF of any
length from the cubes provided within a given amount of time
to achieve points.
Levels Cubes Seconds
------ ----- -------
1 10 30
2 15 45
3 20 60
4 25 75
5 30 90
6 36 108
Playing: You select the level of play. You will then be provided a
random selection of cubes from which you must form a WFF of
any length within the time allowed for the level you chose.
The number of cubes that will be presented and the time
allowed for each level are shown above. (Level 6 is the
easiest; level 1 the most difficult.)
For each cube in a correct WFF, you receive 1 point. For
each cube in an incorrect WFF, you lose 1 point. If you have
not completed a WFF when time runs out, or if you declare
that no WFF can be made from the available letters and one
can be made, you lose 3 points.
If you achieve 50 points at any level of play, you win. If
your score sinks to -50, you will be given an opportunity to
return to the Shake-A-WFF game or to quit.
Color: If you have a color monitor, you may specify the Color
Variation for this game. It requires that red cubes take
precedence over green cubes. In this case, your WFF either
must have all red cubes or all red cubes must precede any
green cubes.
If this variation is selected, you will receive an additional
5 points when you follow the color precedence, and failure to
follow the order of color precedence will result in a loss of
5 points. In either case, you will still win or lose the
appropriate number of points for forming a correct or
incorrect WFF.
If you form a correct WFF with all green cubes and there are
red cubes available from which a WFF could have been formed,
you will win points for the correct WFF but will lose 5
points for failing to follow color precedence.
If you fail to complete a WFF before time runs out and there
are red cubes available from which a WFF could have been
formed, you will lose 5 points.
For example, suppose that, among all the cubes provided, all
but the following are o, i or R:
Red = p, C Green = q, s, N
Without regard for color, you could form several 4-cube WFFs,
such as NCpq or CNqs. But, giving red precedence, you should
form only p, Cpq, Cqp, Cps, Csp, CpNq or CpNs. If you used
NCpq, you would receive 4 points for the correct WFF but
would lose 5 points for failing to follow color precedence,
for a net loss of 1 point. On the other hand, if you used
Cpq, you would receive 3 points for the correct WFF and 5
points for following color precedence, for a total win of 8
points. (CpNq or CpNs would win 9 points.)
Cube Selection: In Game 1, you simply typed your proposed WFF at the
indicated point on the screen. In this game, to select one of the
available cubes for your proposed WFF, you must use the arrow keys to
move the cursor to the cube you want and press Return. When you press
Return, the cube at the cursor's location will be moved from among the
group of available cubes to the next available space on the line where
your proposed WFF is to be formed. Be careful: if you make a mistake,
you cannot replace the selected letter and select a different one.
Once selected by pressing Return, the cube becomes a permanent part of
your WFF and its position cannot be changed.
NOTE: Before reading any further in these instructions, you should
play Game 2 to sharpen your skills in forming WFFs. You should not
proceed to the next game until you achieve 50 points playing Count-A-
WFF.
**********
