8.  TESTING FOR EQUIVALENCE WITH TRUTH TABLES

So far you have learned to symbolize statements and decide their truth value.  Now it is time to do something with this knowledge.

Earlier you were told that we can use symbolic logic to see whether any two expressions are logically equivalent, meaning that their truth values are identical in all situations.  This, remember, is not at all the same as saying that they suggest completely identical thoughts otherwise.  red flag For instance, when we say that logic is fun if it is easy and logic has to be fun in order to be easy (both of which can be symbolized as E->F)  we are shifting our focus between the thought that being easy is enough to make logic fun and the thought that it is a requirement that logic be fun before we can say it is easy.  Logically, however, we are being told the same thing, just as when we shift between sayng Jack is Jill's brother and Jill is Jack's sister. 

What we are now interested in is what other expressions would be logically equivalent to E->F.  Let's look at these four:  F->E  (logic is easy if it is fun), ~E v F  (unless logic is not easy it will be fun),  E v ~F  (unless logic is easy it will not be fun), and ~F-> ~E (if logic is not fun then it will not be easy).  We will match up their truth tables as well as the truth table for E->F side by side.
E
 F
 E->F
 F-> E
 ~E v F
 E v ~F
 ~F -> ~E
1
 1
 1
 1
 1
 1
 1
1
 0
 0
 1
 0
 1
 0
0
 1
 1
 0
 1
 0
 1
0
 0
 1
 1
 1
 1
 1

What we find is that ~E v F as well as ~F -> ~E  have the same values in all positions as E ->F, but F->E  and E v ~F do not (even though they have the same values as each other).  We then say that they are logically equivalent.  red flag What this means in practice is that we could always replace one with the other and not have changed the information provided. 

SOME TERMS WE USE

When any two expressions are true at least once at the same time (they have a 1 on the same line in a truth table), we call them consistent.  When they do not we call them inconsistent. 

When any two expressions are always true or false at the same time, we call them equivalent. 

When any two expressions are such that when one is true the other must be false (they have opposite values on every line of a truth table), we call them contradictory.  An example would be P and ~P.  All contradictory expressions are inconsistent, but not all inconsistent expressions are contradictory (imagine two statements that both are false, as when we might say that Jack comes in first in the race and Jill comes in first in the race but in fact neither does).

When an expression is true no matter what (every line is 1), we call it a tautology.  An example would be P v ~P. 


EXERCISES (ON YOUR OWN)
Use truth tables to decide the equivalence of each of the following pairs of expressions.  Remember that with three variables your truth tables will have to have eight lines.  To keep things simple, indicate the values only for the main connectives.

1.  P -> (Q v R)  and  (P v Q) -> R
2.  P -> (Q v R)  and  (P & Q) -> R
3.  P -> (Q & R)  and  (P & Q) -> R
4.  P -> (Q & R)  and  (P v Q) -> R
5.  P -> (Q v R)  and  (~P v Q) v R
6.  P -> (Q & R)  and  (~P v Q) & (~P v R)
7.  P -> (Q -> R)  and  (P & Q) -> R

Go to the answer key.