How to decide if an expression is a formula in mathematical logic?

Question

I have a midterm exam today but still can't get a grasp on how to decide if given expressions are formulae or not. The three tasks are: $$1. \quad\forall y \forall x \forall yPxy $$ $$2. \quad\forall y(P \to \forall yQxy))$$ $$3. \quad\forall x(Qx)(Px \to Lx)$$

I understand the rules that atomic formulae are formulae and if A and B are formulae then $A \land B, A \lor B, \lnot A, A \to B$ are formulae and if x is a variable then $(\forall xA)$ and $(\exists xA)$ are formulae too and also if $t_1$ and $t_2$ are terms then $t_1 = t_2$ is a formula too.

Based on these rules how to I decide that given three mathematical expressions are formulae or not?

Answer 1

You have to follow the syntactical specification of the language.

Your examples are about first-order logic and not propositional calculus.

Thus, you have to consider atomic formulas, like :

$Pxy, t_1=t_2$, etc.

If $P$ is a binary predicate symbol, then $Pxy$ is an atomic formula, and thus a formula, and also $∀x∀yPxy$ is.

What about $∀y∀x∀yPxy$ ? It depends on the details of the syntactical rules...

If the rule for quantifiers is :

if $\varphi$ is a formula and $x$ is a variable, then $\forall x\varphi$ and $\exists x\varphi$ are formulas,

then the expression above is a correctly written formula (well-formed formula).

Regarding $∀y(P→∀yQxy)$, it depends if $P$ is a $0$-ary predicate symbol.

The third formula is not well written; thus, it is not a formula.