Basic symbolization

2. BASIC SYMBOLIZATION FOR COMPLETE STATEMENTS

What we start with are sometimes called atomic propositions--letters that express basic ideas that are the building blocks for everything else. Let's say we want to say that logic is easy. We would just have the letter E to represent this thought (and we completely disregard time, so that we could just as easily be talking about the past or the future and be representing the thoughts that logic was easy or that it will be easy). In the same way we could let F stand for "logic is fun" and S for Alice studies and P for Alice passes.

Note that because what the letters stand for depends on the code we opt to use (S could mean "Alice is a student" in one problem and "study is important" in another) they are often called constants, like the capital letters in an algebraic formula such as Ax² + Bx + C = y. We may also refer to them as terms when we are talking about parts of an expression. Later, when we talk about something called predicate or quantifier logic, we will use the term "variable" for the letters x and y and z.

What we really want to do, though, is express the possible relationships among these ideas. We do this by making use of a special set of symbols. We call them either operators or connectives, and they are the symbols that create, as it were, logical molecules from the logical atoms that are the individual letters.

Suppose we start with letters such as P or Q, which stand for complete positive statements. Here we will let P represent the statement "there is a picture on the screen," and we will let Q represent the statement "the audience is quiet."

We can negate the statement with the curl (or tilde): ~P (here this now reads "it's false that there is a picture on the screen" or "there is no picture on the screen").

We can say two different statements are both true with an ampersand: P & Q ("there is a picture on the screen and the audience is quiet").
    This sets up the relationship of conjunction.

We can say that at least one of two different statements is true with a wedge:   P v ~Q ("either there is a picture on the screen or the audience is not quiet")
    This sets up the relationship of disjunction (more specifically, what we call inclusive disjunction, which means that it is possible for both statements to be true).

We can set up a condition with the arrow: P -> Q ("if there is a picture on the screen then the audience is quiet" or "there is a picture on the screen only if the audience is quiet")
   This sets up the relationship of implication.
   red flag

This is the only relationship in which the order of the terms will matter. In the last two examples we could have had Q & P or ~Q v P without it changing what we call the truth value of the expression. When we express a condition it really does matter whether we say P -> Q or Q -> P, even though the English word order does not matter (if P then Q means the same as Q if P and P only if Q means the same as only if Q then P). We will see more about this in Chapter 3.

If we want to express the idea that the implication goes both ways we will use have two heads on the arrow: P <-> Q ("there is a picture on the screen if and only if the audience is quiet"
This sets up the relationship of equivalence or mutual implication. Here again the order of the terms does not matter.

We can create more complicated relationships by setting off our letters in groups with parentheses (which we refer to as punctuation and not as connectives). For instance, with P & (Q v R) we are saying that both P and the disjunction of Q and R are true. To build still more groupings inside a group we can either continue to use more parentheses or we can follow the practice from algebra of using brackets and braces: P -> (Q v (R & S)) or P -> [Q v (R & S)].

red flag

Once we have a more complex expression it becomes important to establish the main connective. In the example above it would be the arrow. The main connective is never what is inside a parenthesis, so if we have an expression such as ~(P v Q) the main connective is the curl (and we talk about the curl as expressing the negation of the disjunction--or, put another way, the scope of the curl is the wedge).

Keep in mind that quite different sentences might involve identical logical relationships since what gets lost in our symbolization is not only any sense of time (as when one thing has to happen before another) but the kind of contrast expressed through words such as "but" or "although." In the same way the word "because" used to explain why something happens can be expressed just through a conjunction. With that in mind let's look at some possible "translations" of different symbolic propositions using just the curl, ampersand, and wedge (in the next section we will be looking at various things we need to understand in order to work effectively in expressing conditional relationships).

E & F -- Logic is easy and it is fun. Logic is both easy and fun. Logic is easy because it is fun.
~E & F -- Although logic is not easy it is fun. Logic is not easy but it is fun.
E v ~F -- Logic is either easy or it is not fun. Unless logic is easy it will not be fun.
~(E & F) -- It is not the case that logic is both easy and fun. (This is equivalent to "Logic is either not easy or not fun.")
(E & F) v ~ I -- Unless logic is both easy and fun it will not be interesting. Logic is easy and fun unless it is not interesting.

EXERCISES

Symbolize each of the following. Use the following code for the individual ideas. Keep in mind that these can be separate sentences or just parts of a sentence, and also remember that we disregard time or sequence in our expressions.
          H: The students are happy.
          P: Alice is prepared.
          S: Alice is studying
          T: There is a test on Friday.

1. There is a test on Friday, but Alice is studying.
2. Unless Alice studies she will not be prepared, but she is studying.
3. The students are happy because there is not going to be a test on Friday.
4. There is a test on Friday, but Alice is not prepared although she did study.
5. It is the case either that there is no test on Friday and the students are happy or there is a test on Friday and the students are unhappy.
6. It is not true that Alice is studying because there is a test on Friday.

Check your work with the answer key.
Optional additional drill

The first red flag above is to alert you to the idea that the exact ordering of the terms in symbolization will matter only when we have the relationship of implication. The second red flag above is to alert you to the importance of always thinking in terms of a main connective when you look at an expression. Later on, when we work with derivations, I will keep stressing the importance of using what we call inference rules only with the main connective in an expression, and very shortly, when we learn to talk about an expression as true or false, you will see that this is decided by what happens with that main connective.