User:Gert

Quantification

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For the following exercises, assume the existence of a model with the universe of discourse below:

U = { cat1,cat2,cat3,dog1,dog2,dog3 }

Now do the following exercises about the interpretation function I of the model.

Exercise 1 For I(cat¹) and I(lilly), specify one plausible value each so that the formula cat¹(lilly) becomes true in M.

Check your answer

I(cat¹) = {<cat1>,<cat2>,<cat3>}

For I(lilly), each of the following 3 values will now make cat¹(lilly) true in M:
a. I(lilly) = cat1 , or
b. I(lilly) = cat2 , or
c. I(lilly) = cat3 .

We prove this for case (b):

1. [[cat¹(lilly)]]^M = 1 iff <[[lilly]]^M> ∈ I(cat¹)
2. iff <I(lilly)> ∈ I(cat¹)
3. iff <cat2> ∈ {<cat1>,<cat2>,<cat3>}

Hence: since (3) is the case, so are (2) and the righthand side of (1). Finally, since the righthand side of (1) is the case, the lefthand side is the case as well. Hence [[cat¹(lilly)]]^M is true in model M. That is what we intended to prove!