Let's get back to the problem of demonstrating argument validity. You know how to construct derivations which demonstrate the validity of valid sentence logic arguments. Now that you have a basic understanding of quantified sentences and what they mean, you are ready to extend the system of sentence logic derivations to deal with quantified sentences.
Let's start with a short review of the fundamental concepts of natural deduction: To say that an argument is valid is to say that in every possible case in which the premises are true, the conclusion is true also. The natural deduction technique works by applying truth preserving rules. That is, we use rules which, when applied to one or two sentences, license us to draw certain conclusions. The rules are constructed so that in any case in which the first sentence or sentences are true, the conclusion drawn is guaranteed to be true also. Certain rules apply, not to sentences, but to subderivations. In the case of these rules, a conclusion which they license is guaranteed to be true if all the sentences reiterated into the subderivation are true.
A derivation begins with no premises or one or more premises. It may include subderivations, and any subderivation may itself include a subderivation. A new sentence, or conclusion, may be added to a derivation if one of the rules of inference licenses us to draw the conclusion from previous premises, assumptions, conclusions, or subderivations. Because these rules are truth preserving, if the original premises are true in a case, the first conclusion drawn will be true in that case also. And if this first conclusion is true, then so will the next. And so on. Thus, altogether, in - any case in which the premises are all true, the final conclusion will be true.
The only further thing you need to remember to be able to write sentence logic derivations are the rules themselves. If you are feeling rusty, please refresh your memory by glancing at the inside front cover, and review chapters 5 and 7 of Volume I, if you need to.
Now we are ready to extend our system of natural deduction for sentence logic to the quantified sentences of predicate logic. Everything you have already learned will still apply without change. Indeed, the only fundamental conceptual change is that we now must think in terms of an expanded idea of what constitutes a case. For sentence logic derivations, truth preserving rules guarantee that if the premises are true for an assignment of truth values to sentence letters, then conclusions drawn will be true for the same assignment. In predicate logic we use the same overall idea, except that for a "case" we use the more general idea of an interpretation instead of an assignment of truth values to sentence letters. Now we must say that if the premises are true in an interpretation, the conclusions drawn will be true in the same interpretation.
Since interpretations include assignment of truth values to any sentence letters that might occur in a sentence, everything from sentence logic applies as before. But our thinking for quantified sentences now has to extend to include the idea of interpretations as representations of the case in which quantified sentences have a truth value.
You will remember each of our new rules more easily if you understand why they work. You should understand why they are truth preserving by thinking in terms of interpretations. That is, you should try to understand why, if the premises are true in a given interpretation, the conclusion licensed by the rule will inevitably also be true in that interpretation.
Predicate logic adds two new connectives to sentence logic: the universal and existential quantifiers. So we will have four new rules, an introduction and elimination rule for each quantifier. Two of these rules are easy and two are hard. Yes, you guessed it! I'm going to introduce the easy rules first.