syntax

Syntax

First order syntax comprises followings

Constants symbols. Such as $B o b$ , $A l i ce$ , $M a t h e ma t i cs$ etc.
Variables symbols.
Functions symbols. Functions interpreted as Symbols taking input objects and returning output objects. They have arity (number of arguments). For example, $f a t h er O f (B o b)$ , etc. Function symbols follow snake case.
Predicate Symbols. Similar to functions but can be thought as specifying the property of given input. It defines relationships between/among objects. For example, $R i c h (B o b)$ defines $B o b i s R i c h$ . $Ow n s (B o b, C a r)$ etc. Predicate symbols follow CamelCase.
Quantifiers. Universal Quantifier $\forall$ and existential quantifier $\exists$ .
Logical connectives and negation.

Using these six components, FOL has

Terms
1. Constants such as $ma t h e m e t i cs$ , $a r i t hm e t i c$ , $d r a w in g$ etc.
2. Variables such as $x$ .
3. Functions of terms. Functions taking terms. $P (T_{1}, T_{2}, ..., T_{n})$ where $T_{i}$ are terms and $n$ is the arity of the function.
Formulas
1. Atomic formulas. $P (T_{1}, T_{2}, ..., T_{n})$ where $T_{i}$ are terms and $n$ is the arity of the formula.
2. Quantifiers, $\forall$ and $\exists$ .