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Introduction of the concept of a first-order language.
We now understand what a vocabulary is, and we've learned about models, the semantic entities corresponding to vocabularies. It's time now to turn to the notion of first-order formula e. First-order formulae are derived from first-order languages. And a first-order language defines how we can use a vocabulary to form complex, sentence-like entities. Starting from a vocabulary, we then build the first-order language over that vocabulary out of the following ingredients:
The Ingredients.
All of the symbols in the vocabulary. We call these symbols the non-logical symbols of the language.
A countably infinite collection of variables , and so on.
The Boolean connectives (negation),
(implication),
(disjunction), and
(conjunction).
The quantifiers (the universal quantifier) and
(the existential quantifier).
The round brackets ) and (. (These are essentially punctuation marks; they are used to group symbols.)
Items 2-5 are common to all first-order languages: the only aspect that distinguishes first-order languages from one another is the choice of non-logical symbols (that is, of vocabulary).
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