Logic

In predicate logic can we have valid arguments if we make an existential claim in our premises and not in the conclusion? In other words can we simply rename the existential quantifer to a "particular" quantifer or something of the sort? Does this particular quantifer always have to carry existential import?

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If I understand your first question, the answer is no (unless the existential premise is superfluous). By an "existential claim," I take it you mean an existential generalization such as "There exists an x such that Fx," rather than a claim of the form "Fa," which implies an existential generalization. But you might wish to look into the rule of Existential Instantiation (or Existential Elimination in natural deduction systems); you'll find a brief summary of it here. I'm not sure I understand your second question. There are two ways of interpreting the universal and existential quantifiers: the objectual way and the substitutional way. I can't find a handy link to recommend, but if you search for discussions of those terms, you may find something relevant to your third question.

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