First-order languages: syntax and two more tree rules
Syntactically, a first-order language is like a propositional language in that it is the set of all sentences which can be constructed from some class of ‘atomic’ components using a specified set of logical operations. However, there are two important differences: first, the set of connectives is fixed, the same for all first-order languages; and second, the atomic sentences of a first-order language are now not sentence letters, single and indivisible, but themselves constructed from a specified vocabulary of logical and extralogical items. The extralogical items are themselves sub-divided into a ‘descriptive’, or referential, part and a structural part. These categories of vocabulary item are specified as follows (the boldface capital letter L refers to an arbitrary first-order language):
(i) L’s logical vocabulary contains the same connectives ∧, ∨, ¬ and → as the standard propositional language of Chapter 3 and in addition the two quantifiers and ∃.