Chapter 4 First steps to symbolization

Consider this argument:

• ​

  It is raining outside.

• ​

  If it is raining outside, then Jenny is miserable.

• ∴
  ​

  Jenny is miserable.

and another argument:

• ​

  Jenny is an anarcho-syndicalist.

• ​

  If Jenny is an anarcho-syndicalist, then Dipan is an avid reader of Tolstoy.

• ∴
  ​

  Dipan is an avid reader of Tolstoy.

Both arguments are valid, and there is a straightforward sense in which we can say that they share a common structure. We might express the structure thus:

• ​

  $A$

• ​

  If $A$, then $C$

• ∴
  ​

  $C$

This looks like an excellent argument structure. Indeed, surely any argument with this structure will be valid. And this is not the only good argument structure. Consider an argument like:

• ​

  Jenny is either happy or sad.

• ​

  Jenny is not happy.

• ∴
  ​

  Jenny is sad.

Again, this is a valid argument. The structure here is something like:

• ​

  $A$ or $B$

• ​

  not-$A$

• ∴
  ​

  $B$

A superb structure! Here is another example:

• ​

  It’s not the case that Jim both studied hard and acted in lots of plays.

• ​

  Jim studied hard.

• ∴
  ​

  Jim did not act in lots of plays.

This valid argument has a structure which we might represent thus:

• ​

  not-($A$ and $B$)

• ​

  $A$

• ∴
  ​

  not-$B$

These examples illustrate an important idea, which we might describe as validity in virtue of form. The validity of the arguments just considered has nothing very much to do with the meanings of English expressions like ‘Jenny is miserable’, ‘Dipan is an avid reader of Tolstoy’, or ‘Jim acted in lots of plays’. If it has to do with meanings at all, it is with the meanings of phrases like ‘and’, ‘or’, ‘not,’ and ‘if …, then …’.

In parts II to IV, we are going to develop a formal language which allows us to symbolize many arguments in such a way as to show that they are valid in virtue of their form. That language will be truth-functional logic, or TFL.

In section 2.3, we first introduced the notion of formal validity, and contrasted it with other kinds of validity related to what kinds of counterexamples we consider. It bears repeating that there are plenty of arguments that are valid, but not for reasons relating to their form. Take an example:

• ​

  Juanita is a vixen.

• ∴
  ​

  Juanita is a fox.

It is impossible for the premise to be true and the conclusion false, since ‘vixen’ just means ‘female fox’. So the argument is (conceptually) valid. The validity is not explained by the form of the argument. To see this, we can give an invalid argument with the same form, e.g.:

• ​

  Juanita is a vixen.

• ∴
  ​

  Juanita is a cathedral.

Equally, consider the argument:

• ​

  The sculpture is green all over.

• ∴
  ​

  The sculpture is not red all over.

Again, it seems there can be no case where the premise is true and the conclusion false, for nothing can be both green all over and red all over. So the argument is valid, but here is an invalid argument with the same form:

• ​

  The sculpture is green all over.

• ∴
  ​

  The sculpture is not shiny all over.

This argument is invalid, since it is possible to be green all over and shiny all over. (One might paint the sculpture with an elegant shiny green varnish.) Plausibly, the validity of the first argument is keyed to the way that colours (or colour-words) interact, but, whether or not that is right, it is not simply the form of the argument alone that makes it valid.

The important moral can be stated as follows. At best, TFL will help us to understand arguments that are valid due to their form.

We started isolating the form of an argument, in section 4.1, by replacing subsentences of sentences with individual letters. Thus in the first example of this section, ‘it is raining outside’ is a subsentence of ‘If it is raining outside, then Jenny is miserable’, and we replaced this subsentence with ‘$A$’.

Our artificial language, TFL, pursues this idea absolutely ruthlessly. We start with some sentence letters. These will be the basic building blocks out of which more complex sentences are built. We will use single uppercase letters as sentence letters of TFL. There are only twenty-six letters of the alphabet, but there is no limit to the number of sentence letters that we might want to consider. By adding subscripts to letters, we obtain new sentence letters. So, here are five different sentence letters of TFL:

We will use sentence letters to represent, or symbolize, certain English sentences. To do this, we provide a symbolization key , such as the following:

$A$:

It is raining outside

$C$:

Jenny is miserable

In doing this, we are not fixing this symbolization once and for all. We are just saying that, for the time being, we will think of the sentence letter of TFL, ‘$A$’, as symbolizing the English sentence ‘It is raining outside’, and the sentence letter of TFL, ‘$C$’, as symbolizing the English sentence ‘Jenny is miserable’. Later, when we are dealing with different sentences or different arguments, we can provide a new symbolization key; as it might be:

$A$:

Jenny is an anarcho-syndicalist

$C$:

Dipan is an avid reader of Tolstoy

It is important to understand that whatever structure an English sentence might have is lost when it is symbolized by a sentence letter of TFL. From the point of view of TFL, a sentence letter is just a letter. It can be used to build more complex sentences, but it cannot be taken apart.