Notes:First order language

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

Definition

A first order language, [ilmath]L[/ilmath], consists of two types of symbols:

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

Terminology

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