The Logic behind Logic

As a second year Geometry teacher, this year I have a second chance at explaining things better. It’s nice to have last year to look back on and have the benefit of knowing where I sucked it up big time in explaining. One of those things was in logic. Sure, I can explain simple truth tables but this year my class has the benefit of the one before them in terms of my explanations.

One thing that really stumped me in class last year was the conditional statement: p (hypothesis) implies q (conclusion), also known as if p then q. I had a very hard time explaining that a false hypothesis leads to a true statement, regardless of the truthfulness of the conclusion. I tried over and over to explain this but I think I was largely unsuccessful. This year has been much better and I was able to explain this confusion through the conditional statement’s logical equivalent: not p OR q. But also, it is also easier to understand when you understand that the “truth” of the statement is more like saying “from p, we can successfully prove q.” Maybe this doesn’t seem to make sense until you look at an example.

If trees grow from clouds, then the world is flat.

I can surely prove the world is flat if I start from a bogus position. So we would say that this statement is logically “true” because I can prove it. It’s easy to prove anything as long as the hypothesis is false.

How often are we convinced of conclusions that result from false hypotheses? I would say this happens really frequently because in most arguments (outside of the deeply philosophical kind that happen in classrooms), are formed out of unspoken hypotheses. For instance, it seems perfectly logical that Christians should only consider the Bible authoritative (no traditions). One of the many unspoken hypothesis is that Jesus abolished all Judaic religion, completing and perfecting the law of Moses. The problem is that this is a half-truth and actually irrelevant. Jesus did complete the law, but ancient Judaism had a form of worship that was not “the law.” Jews worshiped in a beautiful place, built in perfect unity with God’s instructions. Jews burned incense in perfect unity with God’s instructions. Jews crafted statues, sewed pictures, and completed rituals all in perfect unity with God’s instructions. The law was in place to keep the Jews from sinning but it did not entirely regulate their worship. Granted, in this example, my “unspoken hypothesis” is in some ways a red herring, since it is not relevant to the conclusion, but most people connect this hypothesis with the conclusion. This is just one example – sometimes I think we forget that our culture (secular and Christian), background, and the world around us (not necessarily the world in a secular sense) have formed us into the people we are.

How many other false hypotheses might I have?

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