If It Doesn't Follow a Form, Don't Draw a Conclusion
Date: 09/22/2011 at 08:32:57 From: Crystal Subject: law of syllogism and law of detachment Sometimes with the law of syllogism, the p's and q's don't line up with the formula. What are some examples where there is no conclusion to the laws of syllogism? What about for the laws of detachment? I've tried books and Internet sites and found nothing to help me with my problem. I can't even find any examples online.
Date: 09/22/2011 at 09:02:01 From: Doctor Peterson Subject: Re: law of syllogism and law of detachment Hi, Crystal. I'm not sure I understand. These laws always DO have a conclusion, and they are always correct. It sounds like you are asking about something very different: situations in which it may LOOK like these laws apply, but they really don't. The answer there would depend on what YOU think is close enough to the real thing as to confuse you. Is this an assignment you were given? How is it actually worded? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 09/22/2011 at 17:00:53 From: Crystal Subject: Law of Syllogism and Law of Detachment I know what the law of syllogism is, and what the law of detachment is. I know how to draw conclusions from syllogisms like this: If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle. It would be: If a quadrilateral is a square, then it is a rectangle. And for the law of detachment, when it is in this form, I know how to solve it: p->q is a true statement if p is true then q is true These are easy because they follow the form. However, there are some examples that I am not sure about. I get really confused and have no idea how to proceed when the statements are mixed up -- when they are not in this form: p->q and q->r are true then p->q is a true statement For example, Conditional: If a road is icy, then driving conditions are hazardous. Statement: Driving conditions are hazardous. Since this is with the law of detachment, wouldn't you conclude "it is icy outside"? My teacher said that because there are two conclusions, you can't make up another conclusion. I don't get it. Here's another, which deals with the law of syllogism: Conditional 1: If you spend money on it, then it is a business. Conditional 2: If you spend money on it, then it is fun. My teacher said that this has no conclusion, either. I asked about it, but she didn't really explain it to me well. And my math book also doesn't help me at all. Why does this one have no conclusion? I just don't understand these different forms. I've labeled these examples as p and q and stuff, but no matter how much I try to figure it out, I can't understand why they would have no conclusion. It makes absolutely no sense!
Date: 09/22/2011 at 23:20:35 From: Doctor Peterson Subject: Re: Law of Syllogism and Law of Detachment Hi, Crystal. As Crystal wrote to Dr. Math On 09/22/2011 at 17:00:53 (Eastern Time), > For example, > > Conditional: If a road is icy, then driving conditions are hazardous. > Statement: Driving conditions are hazardous. > > Since this is with the law of detachment, wouldn't you conclude "it is > icy outside"? My teacher said that because there are two conclusions, > you can't make up another conclusion. I don't get it. I suppose what your teacher meant is that the second statement affirms the conclusion of the first statement rather than its condition. The law of detachment has the form p->q, and p, therefore q That is, if you know that q happens whenever p happens, and you also know that p did happen, then q must happen. The example about icy roads and driving conditions is NOT like that; it has the form p->q, and q You can't apply this law; in fact, there is no law that you can apply, so you have no way to make a conclusion. Look at the details of the example, which illustrates the issue nicely. (Not all examples are even true, much less showing why the logic makes sense.) The conditional statement says that if the road is icy, then driving becomes hazardous. That makes sense. What it does NOT say is that ice is the ONLY thing that can make driving hazardous! There are other reasons to be careful when you drive -- flooding, sun glare, bars having just let out, or whatever. So if someone tells you, "Look out, the driving is hazardous right now," you can't know what the cause is. It might be hazardous for any number of reasons. You CAN'T conclude with any confidence that the road is icy. Does that make sense? > Here's another, which deals with the law of syllogism: > > Conditional 1: If you spend money on it, then it is a business. > Conditional 2: If you spend money on it, then it is fun. > > My teacher said that this has no conclusion, either. I asked about it, > but she didn't really explain it to me well. And my math book also > doesn't help me at all. Why does this one have no conclusion? The law of syllogism says that ... if p->q and q->r, ... then you can conclude that ... p->r. It's like a pipeline of reasoning: if p is true, then q is true, and therefore so is r. The conclusion of one statement has to be the condition of the next, so the reasoning flows in the right direction. The example about how you spend money has the form p->q, and p->r The pipes don't connect the right way to be able to say that q->r or r->q This is a less clear example; here's a situation of the same form that may be easier to follow: If it rains, I will be wet. If it rains, the road will be slippery. Can I conclude that if I am wet, the road will be slippery? No, maybe I'm wet because I just took a shower. Can I conclude that if the road is slippery, I am wet? No, maybe there was an oil spill on the road. So we can't make any definite conclusion; we don't know enough to say more than we have been told. > I just don't understand these different forms. I've labeled these > examples as p and q and stuff, but no matter how much I try to figure > it out, I can't understand why they would have no conclusion. It makes > absolutely no sense! The idea here is that if you can't put the argument in some form from which you can make a conclusion, then you can't make a conclusion. There are other forms you don't know yet, so you can't always be sure that no conclusion can possibly be drawn; but when statements do not conform to any of the forms you know, that is all you can say. YOU can't draw a conclusion, simply because none of the rules you know applies. Put another way: when given logical statements that do not follow the forms you know, conclude "No conclusion is possible"! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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