Chapter 8 Use and mention

Consider these two sentences:

• ‣
  ​

  Justin Trudeau is the Prime Minister.

• ‣
  ​

  The expression ‘Justin Trudeau’ is composed of two uppercase letters and eleven lowercase letters

When we want to talk about the Prime Minister, we use his name. When we want to talk about the Prime Minister’s name, we mention that name, which we do by putting it in quotation marks.

There is a general point here. When we want to talk about things in the world, we just use words. When we want to talk about words, we typically have to mention those words. We need to indicate that we are mentioning them, rather than using them. To do this, some convention is needed. We can put them in quotation marks, or display them centrally in the page (say). So this sentence:

• ‣
  ​

  ‘Justin Trudeau’ is the Prime Minister.

says that some expression is the Prime Minister. That’s false. The man is the Prime Minister; his name isn’t. Conversely, this sentence:

• ‣
  ​

  Justin Trudeau is composed of two uppercase letters and eleven lowercase letters.

also says something false: Justin Trudeau is a man, made of flesh rather than letters. One final example:

• ‣
  ​

  “ ‘Justin Trudeau’ ” is the name of ‘Justin Trudeau’.

On the left-hand-side, here, we have the name of a name. On the right hand side, we have a name. Perhaps this kind of sentence only occurs in logic textbooks, but it is true nonetheless.

Those are just general rules for quotation, and you should observe them carefully in all your work! To be clear, the quotation-marks here do not indicate reported speech. They indicate that you are moving from talking about an object, to talking about a name of that object.

These general quotation conventions are very important for us. After all, we are describing a formal language here, TFL, and so we must often mention expressions from TFL.

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 .

For the most part, the object language in this chapter has been the formal language that we have been developing: TFL. The metalanguage is English. Not conversational English exactly, but English supplemented with some additional vocabulary to help us get along.

Now, we have used uppercase letters as sentence letters of TFL:

These are sentences of the object language (TFL). They are not sentences of English. So we must not say, for example:

• ‣
  ​

  $D$ is a sentence letter of TFL.

Obviously, we are trying to come out with an English sentence that says something about the object language (TFL), but ‘$D$’ is a sentence of TFL, and not part of English. So the preceding is gibberish, just like:

• ‣
  ​

  Schnee ist weiß is a German sentence.

What we surely meant to say, in this case, is:

• ‣
  ​

  ‘Schnee ist weiß’ is a German sentence.

Equally, what we meant to say above is just:

• ‣
  ​

  ‘$D$’ is a sentence letter of TFL.

The general point is that, whenever we want to talk in English about some specific expression of TFL, we need to indicate that we are mentioning the expression, rather than using it. We can either deploy quotation marks, or we can adopt some similar convention, such as placing it centrally in the page.

However, we do not just want to talk about specific expressions of TFL. We also want to be able to talk about any arbitrary sentence of TFL. Indeed, we had to do this in section 6.2, when we presented the inductive definition of a sentence of TFL. We used uppercase script letters to do this, namely:

These symbols do not belong to TFL. Rather, they are part of our (augmented) metalanguage that we use to talk about any expression of TFL. To explain why we need them, recall the second clause of the recursive definition of a sentence of TFL:

1. 2.
   ​

   If $𝒜$ is a sentence, then $\mathrm{\neg }𝒜$ is a sentence.

This talks about arbitrary sentences. If we had instead offered:

1. 2${}^{\prime }$.
   ​

   If ‘$A$’ is a sentence, then ‘$\mathrm{\neg }A$’ is a sentence.

this would not have allowed us to determine whether ‘$\mathrm{\neg }B$’ is a sentence. To emphasize:

‘$𝒜$’ is a symbol (called a metavariable ) in augmented English, which we use to talk about expressions of TFL. ‘$A$’ is a particular sentence letter of TFL.

But this last example raises a further complication, concerning quotation conventions. We did not include any quotation marks in the second clause of our inductive definition. Should we have done so?

The problem is that the expression on the right-hand-side of this rule, i.e., ‘$\mathrm{\neg }𝒜$’, is not a sentence of English, since it contains ‘$\mathrm{\neg }$’. So we might try to write:

1. 2${}^{\prime \prime }$.
   ​

   If $𝒜$ is a sentence, then ‘$\mathrm{\neg }𝒜$’ is a sentence.

But this is no good: ‘$\mathrm{\neg }𝒜$’ is not a TFL sentence, since ‘$𝒜$’ is a symbol of (augmented) English rather than a symbol of TFL.

What we really want to say is something like this:

1. 2${}^{\prime \prime \prime }$.
   ​

   If $𝒜$ is a sentence, then the result of concatenating the symbol ‘$\mathrm{\neg }$’ with the sentence $𝒜$ is a sentence.

This is impeccable, but rather long-winded. But we can avoid long-windedness by creating our own conventions. We can perfectly well stipulate that an expression like ‘$\mathrm{\neg }𝒜$’ should simply be read directly in terms of rules for concatenation. So, officially, the metalanguage expression ‘$\mathrm{\neg }𝒜$’ simply abbreviates:

the result of concatenating the symbol ‘$\mathrm{\neg }$’ with the sentence $𝒜$

and similarly, for expressions like ‘$(𝒜\wedge ℬ)$’, ‘$(𝒜\vee ℬ)$’, etc.

One of our main purposes for using TFL is to study arguments, and that will be our concern in part III. In English, the premises of an argument are often expressed by individual sentences, and the conclusion by a further sentence. Since we can symbolize English sentences, we can symbolize English arguments using TFL.

Or rather, we can use TFL to symbolize each of the sentences used in an English argument. However, TFL itself has no way to flag some of them as the premises and another as the conclusion of an argument. (Contrast this with natural English, which uses words like ‘so’, ‘therefore’, etc., to mark that a sentence is the conclusion of an argument.)

So, we need another bit of notation. Suppose we want to symbolize the premises of an argument with ${𝒜}_1$, …, ${𝒜}_n$ and the conclusion with $𝒞$. Then we will write:

The role of the symbol ‘$\therefore$’ is simply to indicate which sentences are the premises and which is the conclusion.

Strictly, the symbol ‘$\therefore$’ will not be a part of the object language, but of the metalanguage. As such, one might think that we would need to put quote-marks around the TFL-sentences which flank it. That is a sensible thought, but adding these quote-marks would make things harder to read. Moreover—and as above—recall that we are stipulating some new conventions. So, we can simply stipulate that these quote-marks are unnecessary. That is, we can simply write:

without any quotation marks, to indicate an argument whose premises are (symbolized by) ‘$A$’ and ‘$A\to B$’ and whose conclusion is (symbolized by) ‘$B$’.