I'm having trouble understanding how to determine the logical form of a statement.
Here's my current understanding (I'm not sure if all of them are correct though):
p is a propositional variable means that p can be any proposition, so the Truth Value of p has 2 possibilities: T or F(but we don't know which one yet until we assign p to be a certain statement). I think it's kind of analogous to numerical variables, like let x be a natural number, then the "evenness" of x has 2 possibilities: even or odd (but we don't know which one yet until we assign a value to x, like 4 or 123...)
A logical form is an expression consist of the component statement variables and the logical connectives, and the truth table of logical form shows all the possible truth values of the form for all combinations of Truth Values of the statement variables. I like to think of the truth table for the logical form to be what all statements of that logical form mean logically (replacing the propositional variables with the respective component statements, of course).
If my understanding of propositional variable and logical form is correct, then here's what I don't understand: when determining the logical form of a statement, the general procedure is to first identify the component statements and give them names as variables, then translate the whole statement into an expression consist of the propositional variables and appropriate logical connectives. For example, say "the car is black and the bike is white". When identifying the component statement, the solution always says that denote p:= the car is black and q:= the bike is white. How can that work?
I thought what I should do is to completely remove the component statements and replace them with 2 propositional variables p and q, which results in "p and q", and then set up the truth table for that logical form and get to the logical form which is p∧q. If we denote p and q to be the component statements, then p and q are not propositional variables anymore, right? therefore, there will be no truth table for the initial statement (because "p and q" now is not a logical form, it's a statement so it can only have a single truth value). Then how can we determine what is the logical connective for the initial statement's logical form?
I guess I also want to ask is that can there be a truth table for a statement? I thought truth tables only exist for logical forms.*