Disorderly DeductionDate: 09/23/2011 at 00:30:19 From: Crystal Subject: law of syllogism I know what the law of syllogism is, and honestly I've written to Dr. Math about it a lot and I'm starting to understand it better (thank you!). But here is a problem that I don't get because it doesn't follow the "if p->q and q->r then p->r" law of syllogism form. If the sum of the angles of a polygon is 720 degrees, then it has six sides. If the polygon is a hexagon, then the sum of the angles is 720 degrees. My teacher said that the conclusion would be If a polygon is a hexagon, then it has 6 sides. That doesn't follow (to use an analogy introduced to me by a math doctor recently) the "pipes" of the law of syllogism. I just keep getting more of these weird questions from my teacher, but when I look online for help, the only syllogisms I find are ones I already know how to do. Date: 09/23/2011 at 09:53:03 From: Doctor Peterson Subject: Re: law of syllogism Hi, Crystal. Let's translate this into symbols. First, what are the simple statements here? If the sum of the angles of a polygon is 720 degrees, then it has six sides. If the polygon is a hexagon, then the sum of the angles is 720 degrees. I see the following, just taking them in order as the come: p = the sum of the angles of a polygon is 720 degrees q = it [the polygon] has six sides r = the polygon is a hexagon Using those symbols, we have if p then q if r then p Or, in fully symbolic logic terms, p->q r->p Clearly we don't have EXACTLY "p->q, q->r," but that's just because we didn't define p, q, and r in the most helpful order. Since you have absorbed my comparison to plumbing, let's see whether we can arrange these "pipes" so they line up correctly: r->p p->q Can you see how this becomes this? r->q The key here is that the rules are not all about the particular letters you use, but about relationships. (A plumber doesn't get confused if he pulls out two pipes from his truck in the wrong order!) If you have two conditional statements such that the conclusion of one is the condition of the other, then the law of syllogism says you can connect them together to make one new conditional statement. Translating our conclusion, r->q, back into words using our definitions, we have if r, then q If the polygon is a hexagon, then it [the polygon] has six sides. Or, smoothing out the readability a bit, If a polygon is a hexagon, then it has six sides. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 09/23/2011 at 21:00:41 From: Crystal Subject: law of syllogism Okay, so this is another question on the law of syllogism! I understand the hexagon question now (thank you!); but is it because these statements ... p->q r->p ... are basically biconditionals that you can rewrite them like this? r->p p->q I'm just trying to understand it more, and my teacher isn't doing a good job explaining it to me. But I've emailed about this question before and it's getting a lot easier for me to understand! Date: 09/23/2011 at 22:18:34 From: Doctor Peterson Subject: Re: law of syllogism Hi, Crystal. No, nothing here is a biconditional. In fact, for this particular example, everything would have been true if the conditionals were replaced with biconditionals -- which is why this example is not a good example to illustrate the issues. The conclusion would still be valid if we replaced everything with different statements -- even total nonsense: If a borogove is mimsy, then mome raths outgribe. If toves are slithy, then a borogove is mimsy. We can conclude that If toves are slithy, then mome raths outgribe. That's because if toves are slithy, then a borogove is mimsy, and since a borogove is mimsy, mome raths outgribe. Do you see why? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 09/25/2011 at 13:37:30 From: Crystal Subject: law of syllogism So in your nonsense example, you can draw that conclusion because some of the p's and q's "cross out"? and because the statements are assumed true, you can just write the remaining hypothesis and conclusion as an if-then statement? I just want to make this clear. I've emailed a lot to Dr. Math and it's helped a LOT!!! Date: 09/25/2011 at 20:56:49 From: Doctor Peterson Subject: Re: law of syllogism Hi, Crystal. Yes, you can think of it as canceling p's or q's in the sense that it doesn't matter what the letters are, only whether they match up. But they have to match up in the right way: one a conclusion; the other a condition. In the form "p->q, q->r implies p->r," we have p->q q->r ------- p---->r In my example, "p->q, r->p" becomes this: p->q r->p ------- r---->q Here, the p's connect the two conditional statements. The order is different, but the relationship is the same. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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