1. ## Predicate Logic

Hello. I was wondering if anyone could explain this to me:

I want to prove the validity of this argument:

If you are bigger then someone and smaller then someone else, you are between these two. x is bigger then y and smaller than z. Thus x is between y and z.

I want to do it with these predicates:
(F(x,y)∧¬F(x,z)) F= x is bigger than y
(M(x,y,z)) M= x is between y and z

I cant see how this could work in a predicate tree or is there any other method? Hope I made myself understood.

2. Here is how I see 'betweeness':
$\displaystyle M(x,y,z) \equiv \left( {F(x,y) \wedge F(y,z)} \right) \vee \left( {F(z,y) \wedge F(y,x)} \right)$

3. It's not valid (and I'm even going to grant an unstated premise, viz. "For all x and y, we have x is bigger than y if and only if y is smaller than x").

Let 'A' be the universal quantifier.

given premise: Bxy & Sxz & ~y=z
unstated premise: Axy(Bxy <-> Syx)
conclusion: Txyz

This interpretation refutes:

universe = {0 1 2}
B = {<1 0> <2 1>}
S = {<0 1> <1 2>}
T = empty set
map x to 1
map y to 0
map z to 2

So at least one additional premise is needed:

additional premise: Axy((Bxy & Sxz -> Txyz)

But then the exercise is trivial.

4. Originally Posted by Plato
Here is how I see 'betweeness':
$\displaystyle M(x,y,z) \equiv \left( {F(x,y) \wedge F(y,z)} \right) \vee \left( {F(z,y) \wedge F(y,x)} \right)$
Sure, but the exercise did not include this. If it included it, then the exercise is trivial. And by not including it, the exercise is mistaken, since, without that definition (or axiom, whatever) of betweeness, the argument is invalid. (Not to mention that the exercise didn't stipiulate "x bigger than y iff y smaller than x".)

5. Originally Posted by MoeBlee
Sure, but the exercise did not include this. If it included it, then the exercise is trivial. And by not including it, the exercise is mistaken, since, without that definition (or axiom, whatever) of betweeness, the argument is invalid. (Not to mention that the exercise didn't stipiulate "x bigger than y iff y smaller than x".)
Do you know what axioms or otherwise are included in the question?
I have been around longer enough to know that posters only use a minimal amount of information.
Sometimes is out laziness, not knowing better, or not wanting to be caught posting questions.
I frankly think there is a lot more to this question than was posted.
In fact I think I know the textbook it came from.

6. Originally Posted by Plato
Do you know what axioms or otherwise are included in the question?
I have been around longer enough to know that posters only use a minimal amount of information.
Sometimes is out laziness, not knowing better, or not wanting to be caught posting questions.
I frankly think there is a lot more to this question than was posted.
In fact I think I know the textbook it came from.
Yes, of course, there may have been more given in the original question. I guess my point is that should be stated (e.g., one could say "given the ordinary definition of 'betweeness', etc.). Otherwise, a rigorous logician may point out that either not enough information has been given or, as in this case, the argument is actually invalid. I mean, the subject is logic and validity; it is not warranted to just assume that other premises have been given unless either it is mentioned generally that there are other natural premises taken as implicit or they are given explicitly.