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Disorderly Deduction

Date: 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/ 
Associated Topics:
High School Logic

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