Relational Logic, as defined in Lesson 6, allows us to axiomatize worlds with varying numbers of objects. The main restriction is that the worlds must be finite (since we have only finitely many constants to refer to these objects).
Often, we want to describe worlds with infinitely many objects. For example, it would be nice to axiomatize arithmetic over the integers or to talk about sequences of objects of varying lengths. Unfortunately, this is not possible due to the finiteness restriction of Relational Logic.
One way to get infinitely many terms is to allow our vocabulary to have infinitely many object constants. While there is nothing wrong with this in principle, it makes the job of axiomatizing things effectively impossible, as we would have to write out infinitely many sentences in many cases.
In this lesson, we explore an alternative to Relational Logic, called Herbrand Logic, in which we can name infinitely many objects with a finite vocabulary. The trick is to expand our language to include not just object constants but also complex terms that can be built from object constants in infinitely many ways. By constructing terms in this way, we can get infinitely many names for objects; and, because our vocabulary is still finite, we can finitely axiomatize some things in a way that would not be possible with infinitely many object constants.
In this lesson, we proceed through the same stages as in the introduction to Relational Logic. We start with syntax and semantics. We then discuss evaluation and satisfaction. We look at some examples. And we conclude with a discussion of some of the properties of Herbrand Logic.