EDB — 00Q

[00Q] A formula is well formed if it meets all the rules in the list in [00G]↺↻ and this additional rule: ”given a well-formed formula \(𝜙\) where the variable \(x\) is free, a formula of the form ”\(∀ x, 𝜙\)”, or ”\(∃ x, 𝜙\)” is a well-formed formula.”

We will say that a variable \(x\) is free in a well-formed formula if

• the formula is atomic and the variable \(x\) appears in it; or if

• the formula is of the form \(¬ 𝛼\) and the variable \(x\) is free in \(𝛼\); or even if

• the formula is of the form \(𝛼∧𝛽, 𝛼∨𝛽,𝛼⇒ 𝛽, 𝛼\iff 𝛽\) (or other logical connective introduced later) and the variable \(x\) is free in \(𝛼\) or \(𝛽\).

So in the formulas \((∀ x, 𝜙)\) or \((∃ x, 𝜙)\), the variable \(x\) is no longer free; we will say that ”the variable is quantified”.