Subject: HW #2: Negation of a disjunction
From: "William J. Rapaport" <rapaport@buffalo.edu>
Date: Thu, 17 Sep 2009 20:58:07 -0400 (EDT)

A student writes:
"It says in question 2 to use De Morgan's Law to find the negation of
the proposition "Either color is a scientific property or it is not a
property of material things.". You state later on that we can decide
which "or" to use. From what I have read, the exclusive or should be
used here, the problem I am finding though is that there is no De
Morgan's Law or any Law in Table 6 that will allow me to use the
exclusive or. Is it enough to write a truth table showing that it is
logically equivalent to ~p iff ~q, which I have found that it is, or
should I use the inclusive or, even though I know it is wrong, because
the Laws are made for it?"

Reply:
Good question! The "official" answer in the teacher's manual interprets
the "or" as **inclusive**, which makes the problem very simple. So,
feel free to do that.

Whether the "or" in the sentence should be inclusive or exclusive is a
matter of interpretation. Without further information, I don't think
it's obvious that "scientific properties" (whatever that is) can't be
"properties of material things" (whatever those are). So it's at least
open to interpretation that the "or" in this case could be inclusive.

However, if you want a challenge, try the **exclusive** interpretation.
To solve that problem, use the fact that (p+q) is logically equivalent
to ((pvq) ^ -(p^q)). (If you don't remember why that's the case, use
truth tables to convince yourself!) Then you can use DeMorgan's Law a
few times, as well as Double Negation, to get the answer.

Subject: HW #2: Negation of a disjunction -- follow up
From: "William J. Rapaport" <rapaport@buffalo.edu>
Date: Thu, 17 Sep 2009 21:07:37 -0400 (EDT)

I just re-read the previous question, which says, in part:
"...to use the exclusive or. Is it enough to write a truth table
showing that it is logically equivalent to ~p iff ~q, which I have
found that it is"

What is "it"? Which of the following are logically equivalent?

 (p  +  q)
-(p  +  q)
 (p <-> q)
(-p <-> -q)
-(p <-> q)

Use truth tables to find out. You may be surprised :-) 


Subject: HW #2: Negation of a disjunction -- follow up #2
From: "William J. Rapaport" <rapaport@buffalo.edu>
Date: Fri, 18 Sep 2009 07:20:18 -0400 (EDT)

A student writes:
"I just thought that "Either-or" meant either one or the other, not
both; I could be wrong though."

Reply:  
Some people use it that way, but others don't. So you can't rely on
that always being its meaning.

The student continues:
"Also, as a follow up, one of the propositions is 'it is not a property
of material things'. I'm wondering if the proposition can be
q='it is not a property of material things' or if it's proper to write
q='it is a property of material things' and then negate it later on."

Reply:
It's almost always better to let atomic proposition letters stand for
atomic propositions (there are a few exceptions, and we'll see one
pretty soon in lecture). So, in this case, let q = "it is a property of
material things" and then negate it to represent the second disjunct.