Notes:First order language

Source: Books:A Course on Mathematical Logic - Shashi Mohan Srivastava

Definition

A first order language, $L$, consists of two types of symbols:

1. Logical symbols
   • A sequence of variables, $x_1,x_2,\ldots$ (this is the alphabetical order of the variables)
   • logical connectives, $\neg$ (negation), $\vee$ (disjunction - posh way of saying "or"),
   • a logical quantifier $\exists$ (existential qualifier) and
   • the equality symbol, $=$
2. Non-logical symbols (which vary from theory to theory)
   • A set of constant symbols, $\{c_i\vert i\in I\}$,
   • for each positive integer, $n\in\mathbb{N}_{\ge 1}$ a set of $n$-ary function symbols, $\{f_j\vert j\in J_n\}$
   • for each positive integer, $n\in\mathbb{N}_{\ge 1}$ a set of $n$-ary relation symbols, $\{p_k\vert k\in K_n\}$

Terminology

• Expression - any finite sequence of symbols of a language.