What is actually a statement and what is truth?

Question

I feel like I am missing basic understanding of Mathematical logic. For instance, consider the statement

If Ram is a man then Ram has beard.

Firstly, can I attach a truth value to the above sentence? Or is it that, the truth values can be arbitrarily attached to any string of letters (which I call by the name statement)? That is, say, I form a set $S$ whose members are nothing but string of letters (or symbols of some sort). I then have a $f$ map from $S$ to set $\{T, F\}$ and then play with the logical operations. What I wish is that $f$ should not defy my common sense and the reality in which I am living. Is it as simple as that?

Coming back to the statement I made on Ram, what is making me agitated is that, as a whole it gives me the sense that, 'no no I cannot say that if he is a man, he should have a beard'. On the other, if Ram is man and Ram has beard, then $\implies$ says the above is true. Or is it that the statement is ambiguous? Or what actually is a 'statement' (and also 'truth'?). I require an explanation apart from simply saying that a 'statement is a declarative sentence with well defined truth values'. This worries me much - because they (some books) say that '$x$ is greater than $5$' is not a statement. They also say logical connectors are used between atomic statements to make compound statements and $\implies$ is a logical connecter. Then they say - '$\textbf{the statement}$ if $x$ is greater than 5, then $x-5=0$ is false'. I am very much confused about their way of explanations.

I have started worrying about my understanding after an age of 30 years. It might appear silly. I myself might feel this to be so silly after some light is shed upon this.

You can suggest even some good books/materials to build a proper understanding in mathematical logic.

Answer 1

they (some books) say that '$x$ is greater than $5$' is not a statement.

In these examples, what they mean is that the variable $x$ does not point to a fixed object (in this example, the context has not supplied $x$ with a fixed value), so formulae like '$x$ is greater than $5$' are merely propositional functions rather than propositions/statements.

(Recall that a function has a varying output that depends on its input. Do read the linked answer before continuing below.)

Then they say - '$\textbf{the statement}$ if $x$ is greater than 5, then $x-5=0$ is false'.

In these examples, they are writing 'if $x$ is greater than 5, then $x-5=0$' (which technically isn't a statement) as a shorthand for 'for each $x$, if $x$ is greater than 5, then $x-5=0$' (which is a statement). This is called implicit universal quantification.

If Ram is a man then Ram has beard.

Coming back to the statement I made on Ram, what is making me agitated is that, as a whole it gives me the sense that, 'no no I cannot say that if he is a man, he should have a beard'. On the other, if Ram is man and Ram has beard, then $\implies$ says the above is true. Or is it that the statement is ambiguous? Or what actually is a 'statement' (and also 'truth'?).

Does "Ram" refer to the particular person that you have described, who is distinct from other persons also called Ram? If so, then as you have pointed out, the above is a true statement.

On the other hand, if "Ram" is understood to be an arbitrary member of the collection of all persons named Ram, then we are reading 'If Ram is a man then Ram has beard' (which technically is not a statement) with an implicit universal quantification, that is, reading it as 'Every person named Ram who is a man has a beard'; in this case, the statement indeed needn't be true (and in our universe is likely false).