16.  AN EXPANDED RULE SET FOR DIRECT DERIVATIONS

In some ways I have been going against the direction taken by main textbooks.  Although I try to stay close to the way material is presented in these textbooks (especially in Patrick Hurley's widely used A Concise Introduction to Logic),  my own experience with student difficulties has led me to seriously reorder the way in which things appear.  For instance, I have material on symbolization in predicate and quantifier logic right away, and I begin with the equivalence rules (termed "replacement axioms in the Hurley text) rather than bringing them into play after we have already begn with simple derivations.  It is with the same intent that I have not yet brought in the inference rules used to work with expressions invovlving implication (those in which the arrow is the main connective). 

We could, in theory avoid such rules altogether.  Certainly the relationship expressed through the arrow (->) plays no role in the discussion of gates in electronics, even though it is seen as basic to formal logic otherwise.  That, however, probably goes too much against tradition and what will be expected of you, so at this point we are going to introduce three new rules for working with that arrow.

Two of these rules--which for the moment we will label "put" and "take"--involve what happens when we either meet a sufficient condition or fail to meet a necessary condition.  Let's say we have the idea that study is enough for Alice to pass.  We might express this S->P (or, if you want, Sa->Pa).  Once you tell me the truth of the matter is that Alice has studied, I should be confident in admitting that she will pass (either that, or  I was wrong  to say study would make this happen, even though I need to allow for the possibility that she might still pass even if she does not study).  As an argument form we have S->P,S |- P.  As a general pattern we could say that whenever we know the antecedent of a considtional is true we must accept that the consequent (here expressing a result of some kind) has to be true as well.
          P -> Q, P |- Q      MP  (this is short for modus ponens, which is Latin for "the putting pattern")

Now when you tell me something different, such as when you say Alice needs to study in order to pass, we are talking about study being a necessary condition and we are expressing it to the right of the arrow: P->S (or Pa->Sa).  Once you tell me that something necessary did not happen, I am able to admit that it has to be true that the result also did not happen.  We are taking away the result when we take away what was needed for it.  As a general pattern we now say that whever the consequent of a conditional is false we can be sure that the antecedent has to be false as well.
          P -> Q. ~Q |- ~P     MT (this is short for modus tollens, which is Latin for "the taking pattern")

What you need to remember above all is that, unlike our earlier use of operators for "and " as well as "or,"  the "if...then" or "only if" relationship expressed in a conditional is a one-way street.  We cannot "put" to the right (that is the formal fallacy of affirming the consequent) and we cannot "take" from the left (and that is the formal fallacy of denying the antecedent). 
       If Alice studies she will pass.  She is studyng.  Therefore she will pass.    Yes, this is the MP pattern.
       If Alice studies she will pass.  She has passed.  Therefore, she must have studied.    No, this is the mistake (fallacy) of affirming the consequent.  She may have been smart enough or the test easy enough that she could pass without study.
       Alice will pass only if she studies.  She is not studying.  Therefore, she will not pass.   Yes, this is the MT pattern.
       Alice will pass only if she studues.  She did not pass.  Therefore, she must not have studied.   No, this is the mistake (fallacy) of denying the antecedent.  She may still have studied but not passed because the test was just too hard.

Having these two new rules does make it somewhat easier to work with a few of the argument forms you have already had as exercises.
P -> (Q & R), P v S, ~S  |-  P & Q

1.  P -> (Q & R)
2.  P v S
3.  ~S                     /show P & Q
4.  P                      2,3,  DS
5.  Q & R               1,4,  MP
6.  Q                      5,  Simp
7.  P & Q                4,6,  Conj

P -> Q, Q -> R, S -> T, P v S, ~T  |-  R

1.  P -> Q
2.  Q -> R
3.  S-> T
4.  P v S
5.  ~T                              /show R
6.  ~S                             3,5,  MT
7.  P                               4,6,  DS
8.  Q                               1,7,  MP
9.  R                               2,8,  MP

Examining the last example you might ask reasonably enough why, if we already know that P imlies Q and Q implies R, we cannot jump to say that P implies R.  This, in fact, is exactly what we do with what is called a hypothetical syllogism (HS for short), and it will be the last new rule we will make use of.  It may not really be necessary (we can, after all, just repeat our MP steps), but it will prove helpful.

          P -> Q, Q -> R, |- P -> R    HS
EXERCISES  (ON YOUR OWN)

Symbolize the following valid arguments and provide derivations using the full package of equivalence and inference rules that we now have.

1.  If logic is fun and interesting, then it should be easy.  Logic turns out to be interesting even though it is not easy, so it must not be fun.
2.  Logic is easy if it is fun, and it is fun if it is interesting.  Alice will study unless logic is not easy, but since she is not studying it must not be interesting.
3.  Only if logic is not either easy or fun will it not be interesting.  Unless logic is easy Alice will not study.  Alice is studying, so this proves logic must be interesting.
4.  Alice will study only if logic is both easy and fun. Alice will pass only if she studies.  It follows that Alice will pass only if logic is easy.
5.  Every employee needs to work hard if there are to be some companies that are successful.  Some employees are not working hard.  That means no companies are successful.

Go to the answer key.