Symbolic logic is really about tautologies

in that when we say an argument form is valid we mean that, written as a single wff, its value would always be true.

Let's compare a couple of arguments.

Logic is easy if it is fun, and it is fun, so it must be easy. F -> E, F |- E
Logic is easy only if it is fun, and it is fun, so it must be easy. E -> F, F |- E

in both cases we'll start off by listing the variables alphabetically
 
E F F->E F E
true true true true true
true false true false true
false true false true false
false false true false false
E F E->F F E
true true true true true
true false false false true
false true true true false
false false true false false

In the second pattern we see something that should not happen when an argument is deductively valid.  By definition, a valid argument rules out the possibiity of true premises and a false conclusion.  We have a "bad line" that shows this can happen, and this is enough to tell us the pattern is invalid.  (Imagine a story in which logic is very hard but it's fun anyway--like this course.)

Now let's compare the truth tables when we rewrite both arguments as single expressions with implication signs as the main connectives (as though we now stress the hypothetical quality of the premises).  Also, instead of "true" and "false" we will use the numbers 1 and 0.
 
 
((F->E)    &   F)  ->  E  ((E->F)    &   F)  ->   E
     1   1    1   1   1      1   1   1   1    1
     1   0    0   1   1      0   0   0   1   1
     0   0    1   1   0      1   1   1   0   0
     1   0    0   1   0      1   0   0   1   0

In the lefthand wff we have the value of the conditional as true all the way down, but that is not so in the righthand wff.

We could show the same thing with all valid argument forms.  All of them are tautologies.

This fact allows us a very mechanical way of establishing validity for both propositional and monadic predicate logic by means of truth tables.  (Later we will see why there is no mechanical test for predicate logic when we move into two-place or dyadic predicates.)
 

Even argument forms with quantifiers should be seen the same way.   (This is extra material.)

Again, let's assume that some students are ambitious and anyone ambitious works hard.  Thinking about it, we realize that in a world of even just two individuals we would have some students who work hard, so this would hold true for a world of any size.  Imagine that Alberto is a student, then we know that he works hard, and this means there is someone who is both a student and works hard.

But let's vary the story and say that all students are ambitious, and some ambitious individuals do work hard.  Now we recognize that we could have Alberto be someone who is not a student even though he is ambitious.  He might work hard, but there might not be any student who keeps up with him so that it would be imaginable that no student works hard.

(Ex(Sx&Ax)&Ax(Ax->Wx))->Ex(Sx&Wx)        (Ax(Sx->Ax)&Ex(Ax&Wx))->Ex(Sx&Wx)

Here we could set up an interpretation that presents individuals a and b and considers all possibilities,but since this would be like having a truth table with six variables (and the number of lines doubles for each variable)  it is not a particularly efficient method.

Instead we could show that in the righthand form there would be at least one exception so that it is not a tautology, but any intended exception in the lefthand form would lead to a contradiction.

Aa  Ab  Sa  Sb  Wa  Wb  (Ex(Sx&Ax)&Ax(Ax->Wx))->Ex(Sx&Wx)
  1    1       1    0     0     1              1       1  x                  0   0
For this not to be a tautology, we have to have one instance of the implication being false, and this could only happen if the antecedent is true while the consequent is false.  To have the consequent false we'll say Alberto is a student but not working hard (while someone else not a student--Barbara, for instance-- might be working hard).  However, now we make it impossible to say that all students work hard.

Aa  Ab  Sa   Sb  Wa  Wb  (Ax(Sx->Ax)&Ex(Ax&Wx))->Ex(Sx&Wx)
  1    1    0     1       1    1        1                1  1                 0    0
Alberto satisfies the condition that there is someone who is ambitious and working hard, but we have the possibility that no student is both ambitious and working hard.  This shows us that the argument form is not a tautology.