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### 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.

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/
```
Associated Topics:
High School Logic

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