games

Logic is a game (uh huh, sure...)

Well, think about games for a minute. There is not really one common characteristic that defines a game as such (a point emphasized by the Austrian philosopher Ludwig Wittgenstein). Rather, there are "family traits" so that we might imagine overlapping groups of things we definitely consider to be games. Things like football and chess and solitaire and paintball.

So let's narrow what we mean to games in which a score is important. There are going to be rules for scoring and winning, and we expect something about these rules.

They should be consistent, so that it cannot happen that by looking at one section of a rulebook we find that Jack wins but by looking at another section we find that Jill wins.
They should be complete, so that we do not come up with a situation in which we cannot find a rule that tells us what to do.
They should allow the score to be decidable, meaning that we should not have a situation in which we couldn't tell whether Jack wins or Jill wins.

Arithmetic (or number theory, to be more fancy) is an example of a game in which we have two out of three of these characteristics. We have consistency and we have decidability, but, as someone named Gödel showed us, number theory is not and cannot be complete. By this we mean that there are more things true than we could ever prove to be true.

Symbolic logic is an example of a game in which, up to a point, we do have all three characteristics (we can come up with some relationships about which we could say we cannot decide whether they would be true or not).

Now the reason I ask you to be thinking about symbolic logic as a game is this:

We are working with techniques for mapping one system on to another. This means we have to have ways of translating one kind of organization into another. This sounds exotic, but whenever we read something out loud we are mapping a system of visual symbols on to a system of sounds. Whenever we do this by reading into a tape recorder, we have a machine mapping the sounds on to a system of electrical impulses that in turn maps these sounds on a piece of magnetic tape.

Here is an example of the mapping we work with. We have the sentence "Some students read every book" that can be symbolized as Ex(Sx & Ay(By -> Rxy)). What we have done, if you think through this with me without running from the screen in terror, is rethink the ideas so that the sentence has become "there is an unknown someone such that this someone is a student and for every thing else, if that thing is a book then that someone is reading that something." We will learn how to do this close to the end of the course, so do not panic. Instead just think how clever you will be if you stick with this. And think how much fun it is going to be to amaze people with your skill. Right...

Well, we do not stop here. We might go on to think about what would happen if we also said that anyone who reads every book is a genius. Would it also have to be true that we have at least one student who is a genius? (The answer, by the way, is "yes.") Now, can we also "prove" that this has to be so? Here is where we work with techniques for showing that when we know certain things already, other things follow.

Symbolic logic, then, has three components to it:

symbolization (as in the example above)
testing for validity (showing whether or not something does have to be "true")
derivations (showing why this has to happen)

We are going to go through all this step by step. Your task is to learn the game, but like any game requiring a measure of skill (poker and basketball as well as chess), practice is very important. And it may even be fun as well.

To get a better idea of the kind of game we are playing,

I am going to ask you to work with a pretty simple game that in several ways is like what we do in symbolic logic. We are going to use four letters from the alphabet:

a (which comes early and is a vowel)
b (which comes early but is not a vowel)
d (which comes late and is not a vowel)
e (which comes late and is a vowel)

In this game we have what we call well-formed formulas (wff's) that must obey these rules:

a late letter may not be followed by a different late letter (I could have dd or ee but not de or ed)
a wff must have at least one letter but cannot have more than six letters

In order for something to happen, we are going to start off with two wff's that are given as axioms: abd and ead

Here are the rules for changing one wff into another, but it is understood that they cannot be applied in any way that would violate the above conditions for a wff :

Rule 1: Any two letters may be reversed (abd could become bad)
Rule 2: An initial letter may be removed (abd could become bd)
Rule 3: Any wff can be attached to the end of any other wff (abd could become abdbad)

Now this is the game: we want to see what wffs can be derived from the two axioms above. We need to show every step of the derivation. Here is an example. Given abd we want to derive bead.

1. abd        axiom
2. adb        1 rule 1
3. ead        axiom
4. adbead 2,3 rule 3
5. dbead    4 rule 2
6. bead      5 rule 2

How many other English words are wff's that could also be derived with these axioms and these rules?
Are there any English words that are wff's that could not be derived from these axioms and these rules?

What I am asking you, if you think about it, is whether the game I've proposed can be described as having the property of completeness.

I'm encouraging all of you to make sure you sign on to "symboliclogic"in eGroups (see the icon and the link on the syllabus page) and submit your answers. You'll know you have mastered the game when you can provide a derivation for a word that another student has suggested cannot be derived. Make this your first practice exercise.