As it stands, the grammar you've specified is an ambiguous grammar. i.e. something like

a AND b OR c

could have two parse trees, depending on how you parenthesize.

The role of Shunting-yard algorithm is to remove this ambiguity. If you specify operator precedence, it gives you a unique parenthesization for your expression. So yes, while it does give you an AST for your expression, it's not really meant for parsing.

Why I say this is because if you have a proper parse tree, then every node would have an associated nonterminal (i.e. <logical>, <comparison>, etc). Shunting-yard, in essence, treats each expression to be of the same kind.

So if you were to use it, it would recognize this grammar:

<expr> ::= <expr> "AND" <expr>
        |  <expr> "OR" <expr>
        |  "NOT" <expr>
        |  <expr> "==" <expr>
        |  <expr> "!=" <expr>
        |  <expr> "<=" <expr>
        |  <expr> ">=" <expr>
        |  <expr> "<" <expr>
        |  <expr> ">" <expr>
        |  "(" <expr> ")"    
        |  <constant>
        |  <addition>
        |  <subtraction>
        |  <multiplication>
        |  <division>

But the thing is, you don't make a distinction between expression that evaluate to a numeric value vs expressions which evaluate to a boolean value in your productions. Yet, you're using different nonterminals for logical expressions and other expressions.

That is to say, your grammar – in its current form – accepts ( 1 + 2 ) AND @x, and ( 1 == 2) >= 3 as valid.

So you might as well define your grammar to be the above grammar. And then you can incorporate it into Shunting-yard algorithm as suggested by amon in comments - by assigning precedence to the logical and relational operators.

You could use C++ Operator Precedence as a reference when assigning precedence to your logical and relational operators.

As it stands, the grammar you've specified is an ambiguous grammar. i.e. something like

a AND b OR c

could have two parse trees, depending on how you parenthesize.

The role of Shunting-yard algorithm is to remove this ambiguity. If you specify operator precedence, it gives you a unique parenthesization for your expression. So yes, while it does give you an AST for your expression, it's not really meant for parsing.

Why I say this is because if you have a proper parse tree, then every node would have an associated nonterminal (i.e. <logical>, <comparison>, etc). Whereas Shunting-yard, in essence, treats each expression to be of the same kind.

So if you were to use it, it would recognize this grammar:

<expr> ::= <expr> "AND" <expr>
        |  <expr> "OR" <expr>
        |  "NOT" <expr>
        |  <expr> "==" <expr>
        |  <expr> "!=" <expr>
        |  <expr> "<=" <expr>
        |  <expr> ">=" <expr>
        |  <expr> "<" <expr>
        |  <expr> ">" <expr>
        |  "(" <expr> ")"    
        |  <constant>
        |  <addition>
        |  <subtraction>
        |  <multiplication>
        |  <division>

But the thing is, you don't make a distinction between expression that evaluate to a numeric value vs expressions which evaluate to a boolean value in your productions. Yet, you're using different nonterminals for logical expressions and other expressions.

That is to say, your grammar – in its current form – accepts ( 1 + 2 ) AND @x, and ( 1 == 2) >= 3 as valid.

So you might as well define your grammar to be the above grammar. And then you can incorporate it into Shunting-yard algorithm as suggested by amon in comments - by assigning precedence to the logical and relational operators.

You could use C++ Operator Precedence as a reference when assigning precedence to your logical and relational operators.