Section 4: Sentences of SL
The sentence ‘Apples are red, or berries are blue’ is a sentence of English, and the sentence ‘(\(A\)∨\(B\))’ is a sentence of SL. Although we can identify sentences of English when we encounter them, we do not have a formal definition of ‘sentence of English’. In SL, it is possible to formally define what counts as a sentence. This is one respect in which a formal language like SL is more precise than a natural language like English.
It is important to distinguish between the logical language SL, which we are developing, and the language that we use to talk about SL. When we talk about a language, the language that we are talking about is called the object language . The language that we use to talk about the object language is called the metalanguage .
The object language in this chapter is SL. The metalanguage is English— not conversational English, but English supplemented with some logical and mathematical vocabulary. The sentence ‘(\(A\)∨\(B\))’ is a sentence in the object language, because it uses only symbols of SL. The word ‘sentence’ is not itself part of SL, however, so the sentence ‘This expression is a sentence of SL’ is not a sentence of SL. It is a sentence in the metalanguage, a sentence that we use to talk about SL.
In this section, we will give a formal definition for ‘sentence of SL.’ The definition itself will be given in mathematical English, the metalanguage.
|
sentence letters with subscripts, as needed |
\(A\),\(B\),\(C\),...,\(Z\) \(A\) 1 ,\(B\) 1 ,\(Z\) 1 ,\(A\) 2 ,\(A\) 25 ,\(J\) 375 ,... |
| connectives | ¬,&,∨,→,↔ |
| parentheses | ( , ) |
We define an expression of sl as any string of symbols of SL. Take any of the symbols of SL and write them down, in any order, and you have an expression.
Well-formed formulae
Since any sequence of symbols is an expression, many expressions of SL will be gobbledegook. A meaningful expression is called a well-formed formula . It is common to use the acronym wff ; the plural is wffs.
Obviously, individual sentence letters like \(A\) and \(G\) 13 will be wffs. We can form further wffs out of these by using the various connectives. Using negation, we can get ¬\(A\) and ¬\(G\) 13 . Using conjunction, we can get \(A\)&\(G\) 13 , \(G\) 13 &\(A\), \(A\)&\(A\), and \(G\) 13 &\(G\) 13 . We could also apply negation repeatedly to get wffs like ¬¬\(A\) or apply negation along with conjunction to get wffs like ¬(\(A\)&\(G\) 13 ) and ¬(\(G\) 13 &¬\(G\) 13 ). The possible combinations are endless, even starting with just these two sentence letters, and there are infinitely many sentence letters. So there is no point in trying to list all the wffs.
Instead, we will describe the process by which wffs can be constructed. Consider negation: Given any wff \(\mathcal{A}\) of SL, ¬\(\mathcal{A}\) is a wff of SL. It is important here that \(\mathcal{A}\) is not the sentence letter \(\mathcal{A}\). Rather, it is a variable that stands in for any wff at all. Notice that this variable \(\mathcal{A}\) is not a symbol of SL, so ¬\(\mathcal{A}\) is not an expression of SL. Instead, it is an expression of the metalanguage that allows us to talk about infinitely many expressions of SL: all of the expressions that start with the negation symbol. Because \(\mathcal{A}\) is part of the metalanguage, it is called a metavariable .
We can say similar things for each of the other connectives. For instance, if \(\mathcal{A}\) and \(\mathcal{B}\) are wffs of SL, then (\(\mathcal{A}\) & \(\mathcal{B}\)) is a wff of SL. Providing clauses like this for all of the connectives, we arrive at the following formal definition for a well-formed formula of SL:
1. Every atomic sentence is a wff.
2. If \(\mathcal{A}\) is a wff, then ¬\(\mathcal{A}\) is a wff of SL.
3. If \(\mathcal{A}\) and \(\mathcal{B}\) are wffs, then (\(\mathcal{A}\) & \(\mathcal{B}\)) is a wff.
4. If \(\mathcal{A}\) and \(\mathcal{B}\) are wffs, then (\(\mathcal{A}\)∨\(\mathcal{B}\)) is a wff.
5. If \(\mathcal{A}\) and \(\mathcal{B}\) are wffs, then (\(\mathcal{A}\)→\(\mathcal{B}\)) is a wff.
6. If \(\mathcal{A}\) and \(\mathcal{B}\) are wffs, then (\(\mathcal{A}\)↔\(\mathcal{B}\)) is a wff.
7. All and only wffs of SL can be generated by applications of these rules.
Notice that we cannot immediately apply this definition to see whether an arbitrary expression is a wff. Suppose we want to know whether or not ¬¬¬D is a wff of SL.
Looking at the second clause of the definition, we know that ¬¬¬\(D\) is a wff if ¬¬\(D\) is a wff. So now we need to ask whether or not ¬¬\(D\) is a wff. Again looking at the second clause of the definition, ¬¬\(D\) is a wff if ¬\(D\) is. Again, ¬\(D\) is a wff if \(D\) is a wff. Now \(D\) is a sentence letter, an atomic sentence of SL, so we know that \(D\) is a wff by the first clause of the definition. So for a compound formula like¬¬¬\(D\), we must apply the definition repeatedly. Eventually we arrive at the atomic sentences from which the wff is built up.
Definitions like this are called recursive . Recursive definitions begin with some specifiable base elements and define ways to indefinitely compound the base elements. Just as the recursive definition allows complex sentences to be built up from simple parts, you can use it to decompose sentences into their simpler parts. To determine whether or not something meets the definition, you may have to refer back to the definition many times.
The connective that you look to first in decomposing a sentence is called the main logical operator of that sentence. For example: The main logical operator of ¬(\(E\) ∨ (\(F\) → \(G\))) is negation, ¬. The main logical operator of (¬\(E\) ∨(\(F\) → \(G\))) is disjunction, ∨.
Sentences
Recall that a sentence is a meaningful expression that can be true or false. Since the meaningful expressions of SL are the wffs and since every wff of SL is either true or false, the definition for a sentence of SL is the same as the definition for a wff. Not every formal language will have this nice feature. In the language QL, which is developed later in the book, there are wffs which are not sentences.
The recursive structure of sentences in SL will be important when we consider the circumstances under which a particular sentence would be true or false. The sentence ¬¬¬\(D\) is true if and only if the sentence ¬¬\(D\) is false, and so on through the structure of the sentence until we arrive at the atomic components: ¬¬¬\(D\) is true if and only if the atomic sentence \(D\) is false. We will return to this point in the next chapter.
Notational conventions
A wff like (\(Q\)&\(R\)) must be surrounded by parentheses, because we might apply the definition again to use this as part of a more complicated sentence. If we negate (\(Q\)&\(R\)), we get¬(\(Q\)&\(R\)). If we just had \(Q\)&\(R\) without the parentheses and put a negation in front of it, we would have ¬\(Q\)&\(R\). It is most natural to read this as meaning the same thing as (¬\(Q\)&\(R\)), something very different than ¬(\(Q\)&\(R\)). The sentence ¬(\(Q\)&\(R\)) means that it is not the case that both \(Q\) and \(R\) are true; \(Q\) might be false or \(R\) might be false, but the sentence does not tell us which. The sentence (¬\(Q\)&\(R\)) means specifically that \(Q\) is false and that \(R\) is true. As such, parentheses are crucial to the meaning of the sentence.
So, strictly speaking, \(Q\)&\(R\) without parentheses is not a sentence of SL. When using SL, however, we will often be able to relax the precise definition so as to make things easier for ourselves. We will do this in several ways.
First, we understand that \(Q\)&\(R\) means the same thing as (\(Q\)&\(R\)). As a matter of convention, we can leave off parentheses that occur around the entire sentence .
Second, it can sometimes be confusing to look at long sentences with many, nested pairs of parentheses. We adopt the convention of using square brackets ‘[’ and ‘]’ in place of parenthesis. There is no logical difference between (\(P\)∨\(Q\)) and [\(P\)∨\(Q\)], for example. The unwieldy sentence (((\(H\) → \(I\))∨(\(I\) → \(H\)))&(\(J\)∨\(K\))) could be written in this way: (\(H\) → \(I\))∨(\(I\) → \(H\))&(\(J\)∨\(K\))
Third, we will sometimes want to translate the conjunction of three or more sentences. For the sentence ‘Alice, Bob, and Candice all went to the party’, suppose we let \(A\) mean ‘Alice went’, \(B\) mean ‘Bob went’, and \(C\) mean ‘Candice went.’ The definition only allows us to form a conjunction out of two sentences, so we can translate it as (\(A\)&\(B\))&\(C\) or as \(A\)&(\(B\) & \(C\)). There is no reason to distinguish between these, since the two translations are logically equivalent.
There is no logical difference between the first, in which (\(A\)&\(B\)) is conjoined with \(C\), and the second, in which \(A\) is conjoined with (\(B\)& \(C\)). So we might as well just write \(A\)&\(B\) & \(C\). As a matter of convention, we can leave out parentheses when we conjoin three or more sentences.
Fourth, a similar situation arises with multiple disjunctions. ‘Either Alice, Bob, or Candice went to the party’ can be translated as (\(A\)∨\(B\))∨\(C\) or as \(A\)∨(\(B\)∨\(C\)). Since these two translations are logically equivalent, we may write \(A\)∨\(B\)∨\(C\).
These latter two conventions only apply to multiple conjunctions or multiple disjunctions. If a series of connectives includes both disjunctions and conjunctions, then the parentheses are essential; as with (\(A\)&\(B\))∨\(C\) and \(A\)&(\(B\)∨\(C\)). The parentheses are also required if there is a series of conditionals or biconditionals; as with (\(A\) → \(B\)) → \(C\) and \(A\) ↔ (\(B\) ↔ \(C\)).
We have adopted these four rules as notational conventions , not as changes to the definition of a sentence. Strictly speaking, \(A\)∨\(B\)∨\(C\) is still not a sentence. Instead, it is a kind of shorthand. We write it for the sake of convenience, but we really mean the sentence (\(A\)∨(\(B\)∨\(C\))).
If we had given a different definition for a wff, then these could count as wffs. We might have written rule 3 in this way: “If \(\mathcal{A}\), \(\mathcal{B}\), ... \(\mathcal{Z}\) are wffs, then (\(\mathcal{A}\) & \(\mathcal{B}\) & ... & \(\mathcal{Z}\)), is a wff.” This would make it easier to translate some English sentences, but would have the cost of making our formal language more complicated. We would have to keep the complex definition in mind when we develop truth tables and a proof system. We want a logical language that is expressively simple and allows us to translate easily from English, but we also want a formally simple language. Adopting notational conventions is a compromise between these two desires.