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Fun With Logic

Posted By David Nilsen On September 11, 2012 @ 7:00 am In Logic & Rhetoric,Philosophy | 2 Comments

This post could also be titled Boring The Audience To Death, or perhaps How To Lose Readers In 15 Seconds Or Less.  So, here is my brief apologetic for the worthiness of this post.  First, this is kind of a book review.  Second, logic is extremely important, yet too often neglected.  Whether you are interested in doing Apologetics, or merely being a better reader and interpreter of the Bible, logic is key.  People forget that logic is a discipline that needs to be studied and practiced, not taken for granted.  Finally, I think this is fun, and so I must subject everyone to my reverie.

In Oxford’s Logic: A Very Short Introduction [1], Graham Priest skillfully (and briefly) presents the basics of logical reasoning.  Logic is a lot like math, and once you have the rules down like mathematical formulas it is much easier to take those basic rules and apply them to more complicated problems.  Priest doesn’t just present the answers and move on, though.  All good philosophers enjoy problems and questions as much, if not more, than the answers.  So, at the end of each chapter, Priest will mention in passing, “Oh, by the way, here is a potential problem that totally shatters everything we’ve been discussing.  Got that?  All right, then, moving on.”  At the end of the chapter on conditionals (that is, “if-then” statements), Priest presents such a dilemma.  I would like to attempt a solution.

He starts with a pretty straightforward form of argument, one we use everyday.  The first premise (P1) is If a, then b.  P2 is If b, then c.  The conclusion is Therefore, if a, then c.  All these letters making your eyes glaze over yet?  Here’s a typical example.

P1: If you go to Rome, you will be in Italy.
P2: If you go to Italy, you will be in Europe.
C: Therefore, if you go to Rome, you will be in Europe.

So far so good.  But Priest, devious philosopher that he is, proposes an example that uses the exact same formula, where the premises are true, but the conclusion is clearly false.

P1: If Smith dies before the election, Jones will win.
P2: If Jones wins the election, Smith will retire and take her pension.
C: Therefore, if Smith dies before the election, she will retire and take her pension.

What are we to make of this?  See if you can work it out, and then scroll down and we’ll compare notes.

I believe the issue here is one of equivocation.  In P1, when we say “Jones will win the election” what we mean is something like “Jones will win by default.”  In other words, we are smuggling into the premise the unspoken idea that victory in this situation is of a specific kind.  The kind of victory in question, then, changed in P2.  After all, the only reason that P2 is true is because we assume that Smith has not in fact died, but will graciously accept defeat instead.  We can breathe easy again, knowing that the basic rules of logic remain unharmed.

Priest hasn’t finished trying to undermine our confidence in everyday rules of inference, however.  Another seemingly indisputable form of argument goes like this.  P1:  If a, then b.  Conclusion:  Therefore, if a and c, then b.  Let’s use another example to get at what all the letters are saying.

P1:  If a number is higher than 10, then it is higher than 5.
C:  Therefore, if a number is higher than 10 and lower than 100, then it is higher than 5.

Now for Priest’s counter-example.

P1: If Smith jumps from a very tall building, he will die.
C:  Therefore, if Smith jumps from a very tall building and wears a parachute, he will die.

Oopse!  Once again the simplist rules of logic seem to be in question.  What can we say?  Once again, work it out on your own, then meet me at the next paragraph.

As before, I think equivocation is the culprit here.  In P1, when we say that Smith will die when he jumps from the building, we are making the unspoken assumption that he is not a mutant man-bird hybrid, he is not wearing a jetpack, etc., etc., and that he is not wearing a parachute.  So we could actually rewrite P1 as If Smith jumps from a very tall building without some device to soften his landing, he will die.  But as soon as we do so, P1 and the first half of the Conclusion are now in contradiction (we are in fact saying, “If a and not a, then b”, which is clearly false in any case, regardless of what a and b stand for).  Once again, our confidence in the everyday rules of inference is restored.

If anyone sees that I have made a mistake here, please repond in the comments below.  I should also add that it is still possible that there are other counter-examples to these two rules of logic that do not involve equivocating.  If you can think of any, please make sure to comment.  Everyone else, go pick up this book (or something similar) and start having your own fun with logic!


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[1] Logic: A Very Short Introduction: http://www.amazon.com/dp/0192893203/?tag=evangeoutpos-20+very+short+intro

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