Determining
whether an argument is valid is easy to mechanically test for in some cases.
For example, there is the argument: “If the apple I am holding is red then I am
happy. The apple I am holding is red. Therefore, I am happy.” This is an easy
one to test for since there are only two possibilities for whether or not the
sentence “the apple I am holding is red” is true and two possibilities for
whether or not the sentence “I am happy” is true. Thus, there are only 4 ways
the “world” could be. Either my apple is red and I am happy, my apple is not
red and I am happy, my apple is red and I am not happy, or my apple is not red
and I am not happy. Now, on to the argument. In the argument, we are assuming
the conditional “if the apple I am holding is red then I am happy’ and that “the
apple I am holding is red”. Our second assumption limits us to two possibilities:
either the apple I am holding is red and I am happy, or the apple I am holding
is red and I am not happy. Furthermore, the second possibility makes one of our
assumptions false (“If the apple I am holding is red then I am happy”). Thus,
we are stuck with the world in which I am happy and my apple is red. This
obviously makes the conclusion of the argument “I am happy” true. By modeling
all of the possible worlds, we were able to show that the argument is valid.
But some other arguments are harder to prove to be valid; in fact, some are
infinitely harder. What about: “every child is innocent. Adam is a child.
Therefore, Adam is innocent.” Intuitively, this argument is a good one. Yet, it
is impossible to model all of the different ways the world could be. This is
because, unlike in my first argument, the thing that holds a truth-value (true
or false) in this argument is a concrete individual attached to a relation. For
example, the sentence “Eric is a child” would be true or false. With this in
mind, there could be any number of children and any subset of them could be
innocent. This would make it very hard to show that the argument is valid,
since in order to be valid, we would have to show that the conclusion is true
in every single world in which our premisses are true, of which there are
infinitely many. What would be easier would be to prove that an argument is
bad, since this would only require a description of a world in which the
argument is invalid. And, if it is invalid in one world, it is invalid in
general.
As a young child, I despised tomatoes. Unfortunately for me, however, my entire family was very fond of them. To my parent’s dismay, I would always decline when they offered me tomatoes in a sandwich or salad. It was hard for them to believe that tomatoes would render such a positive taste in their mouths, but not in a mouth engendered by them. Thus, they made it their goal to get me to like tomatoes; they were sure that my dislike for them was a fad that would soon wear off. They resorted to putting pureed tomatoes in my salads. But I outsmarted them: I would proceed to “disinfect” each individual piece of lettuce, each individual piece of cabbage, and each individual piece of spinach, with the precision of a neurosurgeon until there was no trace of tomato left on my salad. My hatred for tomatoes slowly became an integral part of my identity. I made friends and enemies over my opinions on tomatoes. During recess, I would sit on the bench and talk about tomatoes with my friends. Soon
First of all you should know that I liked this post, but (and not a "but the truth is" or a "but unfortunately" or a bad but or anything) it sounds like Mrs. Boca or Mr. Buck decided to analyze valid and invalid arguments. I could see you going all the way to the point where you have quantified everything and made formulas and all that. I can't imagine all the issues you would run into with trying to quantify strictly qualitative data like this, but I would be interested to see someone like you or a math teacher try it.
ReplyDeleteInteresting post! I like how it connects to your previous post. I had to make some diagrams as I read to follow along. I’m curious about what inspired you to write about the validity of arguments. Was there a time in your life that prompted you to think more deeply about it? Are you interested in the theory?
ReplyDeleteThis is an unconventional, yet interesting post. This also goes off your previous post. In my head, the validity of arguments seems like a simple concept, but now that you explain it in words, there's much more to consider than just regular old logic. You need to consider subsets, universes/worlds, and the conditionals of an argument. I assume the examples you provided are really basic so I can't imagine what you'd need to do for longer, more in depth arguments. It's too much for my little and fragile head...
ReplyDeletelol @adam's little, fragile, snow-filled, bean head.
ReplyDeleteNice job! I like how you over-analyze something so seemingly small and unimportant like happiness based on apple color. Like Camden said above, it seems like something a math teacher would do. I'd call it "Formulaic". I've tried this technique before in writing but I always sound like I'm rambling so I applaud you for your success. I'm curious though, where did you come up with this topic? It seems like something you've done extensive research on before.