2.1.1. A connective

We are interested in logical forms as a way of stating general laws of entailment. Let us begin by looking at cases of entailment that seem to involve the word and. Here is an example:

In attempting to understand any fact, it is useful to collect related facts. One way to search for related facts about entailment is to look for cases involving sentences similar in grammatical form to those above. If we follow this route, we run into entailments like this:

And we will eventually hit upon a general pattern like this:

Although we will move on to more general patterns, any pattern that abstracts from particular words makes the label formal logic appropriate.

If we look a little farther afield, we also find examples like

which follows the pattern

This pattern can be seen to operate also in examples of the first group if we paraphrase them, transforming

for instance, into

In applying a pattern by first paraphrasing, we treat a sentence as having a form that is hidden by its surface appearance. Much of our analysis of logical form will involve this sort of transition.

Assuming we are willing to apply the second pattern by way of a paraphrase, we have a fairly general law of inference in which the word and plays a key role. If we look at what this role entails, we see that and marks a particular sort of compound sentence formed of component sentences, one that we will label a conjunction. So the word and is a sign for an operation that forms conjunctions. We will call an operation that forms compound sentences out of component sentences a connective, and we will refer to the connective we are considering here as conjunction, marking it with the sign ∧ (one of whose names is logical and). (The use of the term conjunction for both the operation of conjoining and the compound that results may seem confusing, but it follows a pattern that is used fairly often in English—as when the word distribution is used for both the act of distributing and its result.) It will often be convenient to employ a further related term and refer to the components of a conjunction as its conjuncts.

Using these ideas, we can express our analysis of That bear is large and edgy as

and we can express our principle of entailment as

This symbolic notation can save space but it is often convenient to use English to mark conjunction. When we do this, we will use the construction both … and … and write it (as done here) using a special type. So the principle above could be stated as

(The reason for using the particle both in addition to and will be discussed later.)

At this point, we have reached a stage like that reached by a physicist who recognizes pressure, temperature, and volume as physical quantities and has formulated a law relating them but who does not know why the law holds. That is, we have a generalization about entailment that we can apply in special cases, but we cannot say why this generalization is true. What is it about conjunction that makes this sort of entailment work?

We can find an answer by again scaring up some more facts. Notice, for example, that the entailment that got us started is matched by a second.

Moreover, we can see not only that the sentence That bear is large and edgy entails each of the two sentences That bear is large and That bear is edgy but also that it is entailed in turn by the two taken together.

If we abbreviate the longer sentence by B and the two shorter sentences as L and N, respectively, we have collected the following facts:

And checking other cases of conjunction would show us that these are instances of three general laws.

Or, using English to express the forms,

So far, all we have done is to accumulate more general laws, but it is often easier to understand a larger number of facts because a pattern can begin to emerge.