Law

A law is a universally quantified equality of terms.

For example, there is the law of commutativity: $\forall a,b$ $a\cdot b=b\cdot a$ Notice there is the universal quantifier $\forall$ and an equality of terms.

Statements formed using the existential quantifier $\exists$ are not laws for example there is the field axiom which states “All elements in $(\mathcal{S},\cdot )$ except $0$ must have an inverse.” This is formally written as: $\forall x,x\ne 0⟹\exists \bar{x}$ There are many reasons why this is not a law:

• There is an inequality not an equality of terms
• it is an implication rather than an equality of terms
• it is not strictly universally quantified, there is an existential quantifier