predicate forms implications

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**Attempt:**

**(a)** Let be an interpetation, and let be an I-assignment.

Suppose that . We want . And we know that iff there is some with . But how do we show that we have such a d?

**(b)** Suppose , I have to show that . But I'm confused because assuming means there is some d in the domain with . What could I do? Are there better ways to show that the implication doesn't hold?

Re: predicate forms implications

Again, one should use the property that if x is not free in B, then iff .

For (a), suppose that v maps x to d, i.e., for some smaller v'. Then means , so and by the property above, .

For (b), suppose that , which means that for some d. However, v does not have to map x to this particular d, so there is no reason for . You should construct a counterexample.

Re: predicate forms implications

Re: predicate forms implications

Let the carrier of I be natural numbers and hold iff m is even. Let also v map a single variable x to 3. Then , but .