Section 1: Sentence letters
In SL, capital letters are used to represent basic sentences. Considered only as a symbol of SL, the letter \(A\) could mean any sentence. So when translating from English into SL, it is important to provide a symbolization key . The key provides an English language sentence for each sentence letter used in the symbolization.
For example, consider this argument:
There is an apple on the desk. If there is an apple on the desk, then Jenny made it to class. .˙. Jenny made it to class.
This is obviously a valid argument in English. In symbolizing it, we want to preserve the structure of the argument that makes it valid. What happens if we replace each sentence with a letter? Our symbolization key would look like this:
A: There is an apple on the desk.
B: If there is an apple on the desk, then Jenny made it to class.
C: Jenny made it to class.
We would then symbolize the argument in this way:
\(A\)
\(B\)
.˙. \(C\)
There is no necessary connection between some sentence \(A\), which could be any sentence, and some other sentences \(B\) and \(C\), which could be any sentences. The structure of the argument has been completely lost in this translation.
The important thing about the argument is that the second premise is not merely any sentence, logically divorced from the other sentences in the argument. The second premise contains the first premise and the conclusion as parts . Our symbolization key for the argument only needs to include meanings for \(A\) and \(C\), and we can build the second premise from those pieces. So we symbolize the argument this way:
\(A\)
If \(A\), then \(C\).
.˙. \(C\)
This preserves the structure of the argument that makes it valid, but it still makes use of the English expression ‘If... then....’ Although we ultimately want to replace all of the English expressions with logical notation, this is a good start.
The sentences that can be symbolized with sentence letters are called atomic sentences , because they are the basic building blocks out of which more complex sentences can be built. Whatever logical structure a sentence might have is lost when it is translated as an atomic sentence. From the point of view of SL, the sentence is just a letter. It can be used to build more complex sentences, but it cannot be taken apart.
There are only twenty-six letters of the alphabet, but there is no logical limit to the number of atomic sentences. We can use the same letter to symbolize different atomic sentences by adding a subscript, a small number written after the letter. We could have a symbolization key that looks like this:
A
1
: The apple is under the armoire.
A
2
: Arguments in SL always contain atomic sentences.
A
3
: Adam Ant is taking an airplane from Anchorage to Albany. . . .
A
294
: Alliteration angers otherwise affable astronauts.
Keep in mind that each of these is a different sentence letter. When there are subscripts in the symbolization key, it is important to keep track of them.