Assignment 2: Question 5 and 6

Having trouble with these questions - I don't know what I'm doing...

This is kind of what I am thinking for question 6 and was hoping that someone could comment on it to tell me if I'm on the right track.

The formula says that for all x, it is not the case that P(x,x). And I assume that we have to come up with a meaning for P? So then, a model where the formula is true is if P means 'greater than' and a model where the formula is false is if P is equality.

Is that right? Is that the idea to give P different meanings? Or is it x that we are supposed to assign meaning to?

Thanks

I do not think we need to have a meaning for P - I do not think it matters what P means. I used the model in question 7 as a "template".
I *think* (although, I could be wrong) what they are asking for, is for us to give two models - an alphabet of values for the model, A^m and a set of pairs of values, P^m - for which the sentence will be true, and another model for which the sentence will be false.
I have the same meaning of the formula as you - For all x, it is not the case there P(x, x)

I am less sure about question 5, but I think we have to do something similar. Build up one model (alphabet and pair values) which will work for both formula's to be true.

Hope this helps (and to everyone else, if I am on the wrong track, feel free to let me know! )

I kind of see what you're getting at. I guess I don't understand how the P^M values are created. For example, if you look at question 7 they give you a model and in the P^M of that model there are pairs but there are three variables in the formula. How do you interpret P^M?

Like InSaNiE, I also used Question 7 as a model for constructing models for question 5, 6 and 8 (sorry about the pun )
but I'm also not sure if that is what they actually want from us.

What I did was create a very very simple alphabet (A^M) and then slowly build up P^M in such a way as to satisfy the formulas.

I interpret P^M as concrete instances of P. So I read the formula and try to understnd it generally (as you would do normally) and then apply P^M to the formula to see if it satifies the formula.

So, for example if the formula has forall (Vx) then you need to verify P^M for each value in A^M but with exists (Ex) then you only need to find one case.

I hope I made sense.

Hmm, I'm still not quite getting it.

So for question 7 are you supposed to take the values for PM and then plug them into the sentence? For example, do you start with a,a and then see if the sentence equates to true for that?

rezrovs Wrote:
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> Hmm, I'm still not quite getting it.
>
> So for question 7 are you supposed to take the
> values for PM and then plug them into the
> sentence? For example, do you start with a,a and
> then see if the sentence equates to true for that?

Pretty much yes. Plug the values in P^M into the sentence determine if the sentence is true. For the model to satisfy the sentence each value in P^M needs to make the sentence true.

I do not have a clue what to do. :-(
Maybe by end of today, hopefully.

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The sun is always shining, but it is far away.

Question 7 has a model. But there is no function set? Is A^M a concrete set of functions of the model?

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The sun is always shining, but it is far away.

kiolb Wrote:
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> Question 7 has a model. But there is no function
> set? Is AM a concrete set of functions of the
> model?

If I understood it correctly, then yes, A^M is a concrete instance of the function set F (nullary functions in this instance). You don't need an actual function set since A^M defines it.

Thanks

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The sun is always shining, but it is far away.