But before we do, let's look at the main reason I have used PLN as a teaching language for symbolic logic. This is the fact that the truth value of the last letter of a string expresses the value of the string itself. This makes it particularly easy to test for the validity of propositional arguments (those using P's and Q's but not predicate expressions or quantifiers) by using a computer, and (provided you are working with Windows and Microsoft Explorer) you are invited to download and use the program "TTlines" in the files on our group in YahooGroups to try this out for yourself.
The most important point right now is to keep in mind what we mean when we say that an argument form is valid. It means first that when all the premises are true the conclusion cannot be false. It also means that if we express the argument form as a conditional (what happens automatically when you use the program "TTlines") it will be a tautology (the final column will always be "true").
This concept is the key to testing for validity using a truth table. The main difficulty is that such a table can become quite long and tedious (although not for the computer), so it helps to have a pencil-and-paper technique that can shorten the process. We do have such a technique in what I call the reverse method: we start by seeing whether we can have a false conclusion at the same time that we have all true premises. If it is not possible to find such a "bad line" by working with forced choices then the argument is valid. Click on to see this method explained in my SLIPS pages.
Shortly we will be learning to work with a more efficient technique that will allow us to deal with most argument forms, but the key concept is the same: with a valid argument it will be inconsistent to have all true premises and a false conclusion.
In the next assignment you are to see if you can put together the last two things we have learned to do for propositional arguments: test for validity and provide a derivation for the arguments that are valid. (The examples below will also involve symbolization, but in the assignment we will deliberately use wffs that involve relationships somewhat more complicated than those we find in ordinary English sentences. )
Use the following code for these examples.
P: Logic is easy.
Q: The tests are hard.
R: The students are ready.
S: The instructor is happy.
Symbolize and test for validity. If the argument is not valid, give the values of the variables that produce a bad line (a line in the truth table in which all the premises are true but the conclusion is false). If the argument is valid, provide a direct or an indirect derivation.
Example 1.
If logic is not easy but the students are
ready then the instructor is happy. Logic is not easy unless the
tests are not hard. The tests are not hard. Therefore, the instructor
is happy.
PNRASC, PNQNO, QN |- S
(testing
for validity using the reverse method:
P Q R S || PNRASC | PNQNO | QN | S
0 0 0 0 0 1 0 0 0 1 0 1 0 1 1
0 1 0 )
invalid: there is a bad line when all the
variables are false.
Example 2.
Logic will not be easy unless the tests are
not hard. The students are ready only if logic is easy. The
instructor is happy only if the students are ready. The instructor
is happy. Therefore, the tests are not hard.
PNQNO, RPC, SRC, S |- QN
(testing for validity using the reverse method:
P Q R S || PNQNO | RPC | SRC | S | QN
1 1 1 1 1 0 1 0 x 1 1 1 1 1 1
1 1 0
working backwards, we find that we must have all our variables true because
of the last four strings, but this makes it impossible to have the first
string true, so we know the argument form is valid)
1. PNQNO Ass
2. RPC
Ass
3. SRC
Ass
4. S
Ass
5. R
3,4 C elim
6. P
2,5 C elim
7. PNN
6 DN
8. QN
1,7 O elim