ADDITIONAL MATERIAL: TRUTH VALUES AND TRUTH TABLES


Symbolic logic ordinarily works with just two values, true and false (or on and off or 1 and 0). The only statements that can be symbolized, then, are those which can properly be described as true or false, regardless of whether we know which they are (what we mean by saying they are "truth functional"). A sentence such as "Jane studies" is either true or false (assuming there is someone named Jane that we can be talking about in whatever context--real or fictitious--we have in mind). A sentence such as "Jane might be studying" is not exactly in the same category since the corresponding negative sentence "Jane might not be studying" is not properly a contradiction--both could be true. Also, a sentence such as "Jane ought to study" can be both true or false depending on the angle from which her situation could be looked at (it could be true from the angle that she needs to study to pass her test and false from the angle that by staying home to study she could lose her job). For that reason we exclude probability statements and most judgments about decisions from what we can symbolize in a strict deductive system.  We are saying they are not truth functional.

We begin with the idea that we have very basic ideas--statements--that could be expressed as short sentences. Sometimes we refer to these as atomic sentences and think of other expressions as built out of them just as molecules are built out of atoms. A code expresses the smallest possible set of these, which means we usually express them in present tense and without a word such as "not." Every other statement then will be composed of some combination of these atomic sentences, and whether a given statement is true or false depends on whether its basic parts are true or false.

There are only a handful of useful logical relationships, which we indicate through our operators or connectives.   Suppose we start with a code that reads as follows:

E: Logic is easy
F: Logic is fun

Technically, a negated letter is a compound expression, but what we are especially interested in are compound propositions expressing

conjunction:  E & F (logic is easy and fun)
disjunction:  E v ~F (either logic is easy or it is not fun)
keep in mind that this is the sense of "or" we have when we say you could have either cream or sugar with your coffee; saying one is true does not make the other false, although for the expression to be true knowing one is false will make the other true  (we call this the inclusive sense); we have no separate symbol for exclusive disjunction, and ordinarily we symbolize phrases with "or" or "unless" as examples of inclusive disjunction
implication:  E->F (if logic is easy then it is fun) and F->E (logic is easy if it is fun)  -- we call this a conditional (note that the English word order is not the key)
equivalence:   E<->F (logic is easy if and only if it is fun--and notice that there are four words used to express this, not just "only if") -- we call this a biconditional


A basic truth table for any two letters works this way:

P
 Q
 ~P
 ~Q
 P & Q
 P v Q
 P -> Q
 P <-> Q
T
 T
 F
 F
 T
 T
 T
 T
T
 F
 F
 T
 F
 T
 F
 F
F
 T
 T
 F
 F
 T
 T
 F
F
 F
 T
 T
 F
 F
 T
 T

Note that a conjunction is true only when both parts are true, a disjunction is false only when both parts are false, a conditional is false only when the lefthand part (the antecedent) is true and the righthand part (the consequent) is false, a biconditional is false only when both parts are different.  You should memorize this basic chart.

With a truth table we can show all the possible ways each of the sentences above could work out to be true or false.

E
 F
 E & F
 E v ~F
 E->F
 F->E
 E<->F
T
 T
 T
 T
 T
 T
 T
T
 F
 F
 T
 F
 T
 F
F
 T
 F
 F
 T
 F
 F
F
 F
 F
 T
 T
 T
 T

One thing to note right off is that the expressions E v ~F and F -> E have identical truth tables; this means they are equivalent, so that (as we will see) one could be substituted for the other.

Some terms to know: 

We use the same basic rules as expressions become longer and more complicated.  For instance, if we have [E v (F & ~E)] -> ~F  we would evaluate it from the inside out.  Let's say we are given that E is false but F is true.     (To avoid confusion, let's use 1 and 0 in place of T and F and color the steps)

E  F  ||  [E v (F & ~E)] -> ~F
0  1        0 1 1  1 1 0   0  0 1

first we find that (F & ~E) is true, then that the disjunction in brackets is true, and this makes the entire expression false

The original  sentence might express this thought: "If it is true that logic is easy unless it is both fun and not easy then logic is not fun."  If we start with the idea that logic really is not easy but really is fun, then the entire statement is false.   (For practice, see what would happen when we know that logic really is both easy and fun, when it is easy but not fun, and when it is neither easy nor fun.)

AN INTERACTIVE DRILL ON WORKING WITH TRUTH VALUES

strongly recommended!
LEARNING TO WORK WITH TRUTH TABLES

A PRACTICE EXERCISE ON SYMBOLIZATION AND TRUTH VALUES
This is an optional exercise.  I will send you a link to the answer key after I receive it.