Chapter 2
The Syntax of Propositional Logic (PL)
We will now introduce a formal language which shall be called "PL" for
"propositional logic." PL is called a formal language to
distinguish it from natural languages such as German, English, and Japanese.
Natural languages are learned at our parents' knees. Formal languages are artifacts,
learned from texts like this one. A formal language can be constructed for a variety
of purposes; our purpose in constructing PL is to capture an essential part of the logical
form of our language, for the purpose of classifying arguments as good or bad arguments
(and for some related purposes, to be explained in a bit).
Consider the following argument:
| If Ignat passes the class, Ignat is eligible for graduation. Ignat passes the class. Therefore, Ignat is eligible for graduation. |
What is the structure of this argument? Note first that the first premise is composed of
smaller sentences, and that those smaller sentences appear as independent sentences in the
second premise and in the conclusion. There really are just two small sentences in
the entire argument, but they are put together in a compound sentence in the first
premise. If we use capital letters to stand for the two sentences in the argument:
| A | Ignat passes the class. |
| B | Ignat is eligibe for graduation. |
We can rewrite the argument in a way which makes its form more apparent:
| If A then B A Therefore B |
We call sentences like A and B simple sentences and the sentence "If A then B" a complex sentence, for the obvious reason that it is made up of simple sentences.
The syntax of a language is just a specification of the rules for putting sentences together.
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