Re: basic logic help

A Aitken wrote:
I learned basic logic in an (intro to) math course.

We learned that p -> q in logic does not have to mean that p is
related to q, and that we should treat p and q as entirely different

We also learned that something like "n^2 + 1 is not a perfect squar" is
not a proposition. It is a predicate (or propositional function), and
can be denoted by P(n). Therefore, "n^2 + 1 is not a perfect square"
or P(n) can not be used as p/q listed above.

Personally, I'd rather avoid 'proposition', 'predicate', and
'propostional function' here and instead say that you've given examples
of open formulas (since they have at least one free variable) as
opposed to sentences (which have no free variables).

Of well formed formulas (which we'll just call 'formulas') there are
two kinds: Open formulas, which have at least one free variable, and
sentences, which have no free variables (a sentence may have variables,
but all variables in a sentence are bound by a quantifier or other
variable binding operator). So all sentences are formulas, but not all
formulas are sentences.

For P(n) to be used as p/q in p->q we have to bind "n" somehow, by
either specifying a value for n, or by quantifcation of P(n).

No, we can use open formulas, but we have to keep in mind that there
are differences between open formulas and sentences.

If we do
neither, then P(n) can not be considered a proposition, and can't be
used as p/q in p->q.

If we leave 'n' free (unbound) then, yes, P(n) is not a sentence. But
we can still use P(n).

Valid implications:

"cows are big" implies P(3)
"there exists an integer such that P(n) is true" implies "the sky is

Those aren't valid, but they are sentences. (Except the second one
should be "There exists and integer n such that (P) is true." So 'n' is
bound, thus the formula is a sentence.)

Invalid implications:

"cows are big" implies P(n)
"sky is blue" implies "n^2 is larger than 4"

Those are invalid, but not because they're open formulas. But, yes,
they're different from your first pair, since the first was a pair of
sentences and the above is a pair of open formulas.

[I am not saying what I've written is correct, I'm just saying that
this is what I got from the course].

Everything is fine, until I run into mathematical statements like:

If "n is an element of positive integers", then "n^2 is not a perfect

This statement (if I ignore the finer details of logic lessons) makes
perfect sense to me, but I can't fit it into the mold of my logic

(It makes sense, but it is false. But that's not the point.) The point
is that often writers and lecturers leave out the quantifiers, which we
take tacitly to be included. So the sentence that is really being
asserted would be, "For all n, if n is a positive integer then n^2 is
not a perfect square."

"n is an element of positive integers" is a proposition

No, it's not. It's an open formula.

"n^2 is not a perfect square" is not a proposition (it's a predicate).

Right, it too is an open formula.

So the sentence version is:

For all n, if n is an integer then n^2 is not a perfect square.

Therefore the implication is nonsense? Applying my background in
logic, it is nonsense.

When rendered as a sentence it is false but not nonsense. And it is not
nonsense as an open formula, while it is still not satisfied under the
usual interpretation of the words in it.

But using normal mathematical sense, the statement makes perfect sense.
Somehow, the hypothesis holds information that helps us determine the
truth value of the conclusion (predicate). The p->q in this case is
something where p is intertwined with the q, and the whole thing makes

Right. It makes sense as an open formula. And it also makes sense when
you quantify it to make it a sentence.

Yet I can not decompose it using my logic background. I want to be
able to break this p->q apart using the rules I've learned, without
needing to apply "common sense"

I guess one way around this is to restate the whole thing as "For every
n element of positive integers, n^2 is not a perfect square." This is
fine, because the universal quantifcation of the predicate makes
perfect sense.

The open formula makes sense too; but making it a sentence by
quantifying it gives it another aspect of sense. You'll learn about the
differences as you study logic more.

But is it possible to convert the p->q into this form using logic
rules? I sure can't do it, because p->q in the above case doesn't make
sense under the rules I use.

Yes, when reading math texts, you have to keep in mind that the
quantifiers are often left out.

Can someone help me get some traction on my difficulties? Any help
will be much appreciated.

I highly recommend the book 'Logic: Techniques Of Formal Reasoning' by
Kalish, Montague, and Mar. That book does a great job of explaining
these kinds of issues. If you study that book thoroughly, then you'll
be in great shape to deal with these kinds of things.