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Q. If domains restrictions affect validity..?

Malac
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10/16/2012 9:48:33 PM
Posted: 4 years ago
OK. Another problem that's driving me crazy! It involves the restrictions of domains that I've noticed can affect the validity of an argument. Say for instance, an argument was valid in a domain containing only 2 entities. However, if that same argument was switched to a domain that contains only 3 entities, then it becomes invalid. An example of what I mean:

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A) Domain: Adam and Bob
For some x, x is playful. //(∃x)Px
Adam isn't playful. //~Pa
Therefore, Bob is playful. //Pb -----------------VALID ARGUMENT

B) Domain: Adam, Bob, AND Cassidy
For some x, x is playful. //(∃x)Px
Adam isn't playful. //~Pa
Therefore, Bob is playful. //Pb ----------------INVALID ARGUMENT

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As can be seen, beside the listed domains, both arguments in the examples shown are exactly the same.

By looking at the "expansions" of both quantified statements in EX: A and B, I can understand why each argument is either valid or invalid. *However,from logical texts I've read, a common method of creating a counter-example in predicate logic is to assume a model universe containing a small number of entities, and use just those entities as a way to lay out a now "complete list", in respect of the model, of every entity to assign to the logical structure of the argument's quantification. My main confusion is that because of the dependency of an argument's validity rests on the domain it's in, why is it acceptable to create a counter-example by using a model that may effect the original status of validity??

Anyone know why?