Due to the complexity of the sentences of quantification theory it is often difficult to see quite what they mean. One of the best ways to understand them is to consider one or more interpretations of a given sentence. Each symbolization key specified a particular interpretation. And, in certain cases, we could provide a specification of a "world". In many cases we could see that two sentences did not say the same thing by way of constructing a world in which the one was true but the other was false. Consider the following two sentences:

Let us look at a very small world:

For it to be true that:

there must be at least at least one object x such that if it is an F then it is a G. There is one such object, in fact in this world there are two. Consider the object a:

This statement is true so there is an object, namely a such that it is if an F then it is a G. b is also an object such that:

is also true. But in this world there is no object which is both F and G. So the claim that:

is not true. This world exhibits the fact that these claims are not logically equivalent.

Next consider the following two sentences:

Note that the first sentence is universally quantified while the second sentence is a conjunction.

Notice that for each of the three objects in our universe it is true that if it is either F or G then it is H. b is not H, but that does not matter since it is neither F nor G. Now let us look at the second sentence. Since it is a conjunction it will be true just in case both the left conjunct and the right conjunct are true. The left is true, that is, every object which is F is also H. And the right is true since every object which is G is also H. So both are sentences come out true in this world. Can we infer that they are logically equivalent on the basis of our examination of this world? The answer is no. We are considering only one world, only one interpretation. As it happens these are logically equivalent but we have not proven this claim.

In some cases we will not wish to, or will not be able to, provide a picture of the world in question. For example if I wish to speak of all natural numbers I am hardly in a position to write down an infinite set of statements. Here what we might to is to specify our symbolization key. Consider the

Let us look at the following two sentences:

If you consider both of these for a moment you should see that they are true in this interpretation. Now consider this sentence.

This looks rather like the first sentence above, but it is not the same. It says something holds of everything. So it should hold of 0.

This says that there is a number such that 0 is a successor of that number. But in this domain there is no such number. So the statement is false in this domain. You must be vary careful about the order of quantifiers and the variables which they bind.

We saw in sentential logic that certain sentences were tautologies. Those sentences came out true in every interpretation. There are sentences in quanitification theory which have a similar feature. They can be counted on to come out true in every interpretation in a non-empty domain, that is, a domain in which there is at least one object. We shall call such sentences quantificational logical truths. And argument validity is the same in quantification theory as it was before. That is, an argument is quantificationally valid just in case there is no interpretation in a non-empty domain in which all the premises are true while the conclusion is false. We will not, due to the complexities of quantification theory, undertake to provide a detailed account of quantificational logical truth. Suffice it to say that such accounts can be provided and are provided in more advanced texts.

That portion of quantification theory which involves only one-place predicates is similar to sentential logic in one respect. In sentential logic we saw that truth tables provided us with a finite, mechanical means of determining whether a given sentence was or was not a tautology. It can be show, though we shall not do so, that a quantificational sentence which uses only n one-place predicates is a quantificational logical truth if and only if it is true in every interpretation in a domain with 2ⁿ objects. Notice that this also gives us a means of determining whether a given sentence is a contradiction. If it is, its negation will be a logical truth. Given this we could in principle "write down" all those worlds and determine in a finite amount of time whether or not the sentence we were considering was a logical truth or a contradiction. However the same is not true in full quantification theory. There are there sentences, for example, which are false in every finite domain but true in some infinite domains.

We can charactize a system as decidable if and only if there is a procedure whereby we can in a finite number of steps determine that a sentence is a logical truth if it is and that it is not a logical truth if it is not. Sentential logic was decidable in this sense. Full quantification theory is not decidable in this sense. We can construct a device that will generate logical truths. But at any particular finite point we cannot say that a sentence that has not yet been generated is not a logical truth. For all we know it may be generated at some later point. We did not "need" derivations in sentential logic in that in principle truth tables provided us with a means of answering any question which we had. But since quantification theory is not decidable derivations or some equivalent of them are essential.