2.1.15: Summary
A first-order language consists of constant , function , and predicate symbols. Function and constant symbols take a specified number of arguments. In the language of arithmetic , e.g., we have a single constant symbol \(\Obj 0\) , one 1-place function symbol \(\prime\) , two 2-place function symbols \(+\) and \(\times\) , and one 2-place predicate symbol \(<\) . From variables and constant and function symbols we form the terms of a language. From the terms of a language together with its predicate symbol, as well as the identity symbol \(\eq[][]\) , we form the atomic formulas . And in turn from them, using the logical connectives \(\lnot\) , \(\lor\) , \(\land\) , \(\lif\) , \(\liff\) and the quantifiers \(\lforall{}{}\) and \(\lexists{}{}\) we form its formulas. Since we are careful to always include necessary parentheses in the process of forming terms and formulas, there is always exactly one way of reading a formula. This makes it possible to define things by induction on the structure of formulas.
Occurrences of variables in formulas are sometimes governed by a corresponding quantifier: if a variable occurs in the scope of a quantifier it is considered bound , otherwise free . These concepts all have inductive definitions, and we also inductively define the operation of substitution of a term for a variable in a formula. Formulas without free variable occurrences are called sentences .
The semantics for a first-order language is given by a structure for that language. It consists of a domain and elements of that domain are assigned to each constant symbol. Function symbols are interpreted by functions and relation symbols by relation on the domain. A function from the set of variables to the domain is a variable assignment . The relation of satisfaction relates structures, variable assignments and formulas; \(\Sat[,s]{M}{A}\) is defined by induction on the structure of \(A\) . \(\Sat[,s]{M}{A}\) only depends on the interpretation of the symbols actually occurring in \(A\) , and in particular does not depend on \(s\) if \(A\) contains no free variables. So if \(A\) is a sentence, \(\Sat{M}{A}\) if \(\Sat[,s]{M}{A}\) for any (or all) \(s\) .
The satisfaction relation is the basis for all semantic notions. A sentence is valid , \(\Sat{ {} }{A}\) , if it is satisfied in every structure. A sentence \(A\) is entailed by set of sentences \(\Gamma\) , \(\Gamma \Entails A\) , iff \(\Sat{M}{A}\) for all \(\Struct{M}\) which satisfy every sentence in \(\Gamma\) . A set \(\Gamma\) is satisfiable iff there is some structure that satisfies every sentence in \(\Gamma\) , otherwise unsatisfiable. These notions are interrelated, e.g., \(\Gamma \Entails A\) iff \(\Gamma \cup \{\lnot A\}\) is unsatisfiable.