The use of connectives can be iterated: that is, sentences containing connectives can themselves serve as components in yet longer sentences involving further connectives. Care needs to be taken with parentheses when this is done, as is the case when parentheses are used in arithmetic. Just as 2¸ (5´ 7) is not equal to (2¸ 5)´ 7, so the sentence:

(A & B) ÚC

is a very different sentence from:

A & (B ÚC)

Commentary on The Logic Book, pages 54-60:

Within Sentential Logic we can only represent connectives that are used truth-functionally.

So in symbolizing sentences care must be taken not to symbolize a non-truth-functional connective of English with a truth-functional connective of SL. Certain connectives in English can never be used truth functionally.

E.g.: the truth value of the sentence 'Mary left before Paul left' is not dependent only on the truth values of 'Mary left' and 'Paul left'. Suppose both the latter components are true; this tells us nothing of the order of their departures, hence it is consistent both with the truth and with the falsity of 'Mary left before Paul left'. This sentence must be symbolized in SL as a simple sentence.

Other connectives of English are sometimes used truth-functionally and sometimes not. 'If ... then. ...' is a notorious case discussed in the textbook.

Commentary on The Logic Book, pages 60-66:

What has to be added to the following phrase to convert it from nonsense to sense?

(1) The has three letters

The simplest addition is to place quotation marks as follows:

(2) 'The' has three letters

This example nicely illustrates the distinction between use and mention, and the troubles that arise if we confuse the two. In the nonsensical phrase (1) we are using 'The' inappropriately -- we really mean to be mentioning the word, which is what we do in the second phrase (2).

In discussing the French word 'rouge', we place the word in quotes; we are talking about it, hence we are mentioning it, not using it. If we are discussing the French language, then we shall be mentioning many of its terms, and it is called the object language of our study. The language we use to discuss the object language is called the metalanguage. In our study of SL, the language of sentential logic, SL is the object language; and since we happen to be using English to talk about SL, English is our metalanguage. If we were using Spanish to talk about SL, Spanish would be our metalanguage and SL our object language. If we were using English to talk about English, English would be both metalanguage and object language.

In Lesson 1, 'P' served as a sentence of SL to abbreviate 'Paul is happy'. Thus 'P' did service as a particular sentence of SL. But often, we want to talk about all sentences having a particular form (recall our enterprise of investigating logical form). Thus we might want to speak of all sentences comprised of two sentences connected by '&'. 'P & M' is but one example of such a sentence. In order to speak of all such sentences, it is convenient to employ metavariables. Script letters, such as 'P' and 'Q', are employed for this purpose. Thus we can say such things as:

for all sentences P, Q: P & Q is true if, and only if, P is true and Q is true.

By now, you should have an informal grip on the notion of a sentence of SL -- you should be able to recognize sentences of SL when you see them. But logicians seek formality; hence chapter two of the textbook concludes with a formal way of specifying what it takes to be a sentence of SL. This is the syntax of SL, and it consists of:

Vocabulary: the acceptable symbols of SL
Grammar: the rules that dictate when a string of SL symbols makes an SL sentence.

You need not worry too much at this stage if your grasp of this definition of SL sentencehood is somewhat hazy. The important thing at this point is to make sure that you have the ability to determine whether something is a sentence of SL. You should also be able to write out sentences of SL, and distinguish their main connectives, immediate sentential components, sentential components, and atomic components (see page 65 of the textbook). The ability to distinguish these is going to be extremely important. Check "Notes and Examples" for Lesson 2 after February 4^th to see if you got the main idea.

Note that you can determine whether or not an expression is a sentence of SL without asking any questions concerning the meaning of the expression. For example, you know that

((A & B) Ú (C É D))

is a sentence of SL despite the fact that we do not know the interpretations of the sentential letters. The notion of sentencehood is a purely syntactic or formal notion: it can be defined without concern for how the symbols of the language relate to the world. In the next lesson, we turn our attention to the semantics of SL: we shall look at the relation between the symbols of SL and the world.

Pages 59-60, nos. 1b,d

Pages 65-66, nos. 1b,d,f; 3b,d,f,h,j; 4b,d; 5b,d,f,h,j