HOW TO SOLVE A PROOF WITH MODUS PONENS AND MODUS TONENS
The way to solve truth functional deductive logic proofs is kind of awkward. There isn’t any set way to go about it. But that should not deter you, because there is always an easy way to solve any proof.
The best way to show you how to approach a proof is to do a sample problem and show you what to look for, what seems simplest in the long run, etc.
First, remember the two inference rules we are going to use (since at this point we only know of two of them for now).
Step 0. Now this is the sample problem we are going to try with only these two inference rules. We start with an already translated and lined problem with the conclusion at the end ( /~M ) .
Step 1. The first step is to put this proof into a standard proof table to actually begin to do the proof. If you do not remember how to do this, go back to symposium 7 for a more thorough explanation.
Step 2. This is the part of the proof that becomes difficult. There is really no set way to attack a proof, but there are ways of approaching a proof that make the solution a little easier. You may become familiar enough with proofs and the inference and replacement rules to see the patterns right off the bat and infer without a second thought and do it very precisely. But what if you are genuinely stumped and you don’t know in what direction to go?
POINT 1. Do every combination and inference possible and sort it all out later!
This is a very valuable trouble shooting tip. When in doubt, do everything so that you can select what works and what doesn’t. Keep in mind a proof does not have to be short, or solved in a single way. There are numerous ways to solve a single proof. It all depends on what argumentative style you have (what inference rules you favor most).
So let’s try every single combination and see what we come up with.
For me at least, I am most comfortable with Modus Ponens. So with that in mind, let’s go through the argument and find the tell tale signs of a modus ponens. Remember, a Modus ponens needs two things. 1) a conditional, and 2) the antecedent (first letter or compound statement) of the conditional.
Step 3. Now that we derived an inference, we need to go back to the beginning and look for any other instances of either Modus Ponens or Modus Tollens. Keep in mind that whatever you add to the proof can and will be included in the rest of you assumptions to solve the proof. Simply, you do not need to work with just lines 1-4. YOU USE ANY AND ALL LINES YOU HAVE IN BOTH YOUR ARGUMENT AND YOUR DERIVATIONS. It is best to exhaust all possibilities for any combinations in the argument (lines 1-4) before going into the proof for combinations. At this point for our proof, there are no such combinations exclusively in the argument (lines 1-4). But there is one instance using the derived S on line five.
Step 4. Now that we have a new derivation line, we again have to go to the beginning and see if any new combination can be revealed with our newest addition, ~R. And wouldn’t you know it… There is a possible combination.
So you successfully reached a conclusion for your argument with proven argumentative methods and inference rules in order to secure the conclusion of the above argument. Huzzah!!!
IF YOU HAVE ANY PROBLEMS (OR NOTICED THAT I SCREWED UP SOME WHERE) LET ME KNOW BECAUSE I’M HAPPY TO ANSWER ANY QUESTIONS OR COMMENTS.