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Suppose $A$ be an sentence letter, the text asks if $A\Rightarrow A$ is a wff of SL and a sentence of SL.

I understood that it is not a wff of SL because it is not contained in parenthesis. Yet I still do not get why it is not a sentence.

By definition of sentence, we should be able to assign truth value to $A \Rightarrow A$, and it is obviously true always, hence I think it is a sentence. However, the answer sheet says otherwise. Can you please explain why?

Thanks,

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When defining the syntax of a formal language, it is customary to require parentheses everywhere (almost) to keep things simple. Specifically, fully parenthesized expressions do not require precedence rules. (Do I take conjunctions before disjunctions? What about negation? And so on.)

I suppose that's what your text does. Then $A \Rightarrow A$ is not a wff because it breaks the syntax rules. If it breaks the syntax rules, it's not even considered when it comes to assigning "meaning." It's classified as nonsensical. Since it does not follow the rules, we have no way to parse it and assign a truth value to it.

What I suspect you are doing, is relaxing the syntax rules, admitting your expression among the wffs and then concluding that it is a sentence. Apparently, that's not what your book or instructor expects you to do.

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    Actually, my text allows setence not to follow syntax rules as long as its meaning is clear; for example, if A and B are a sentence letter, then $A \Rightarrow B$ is a sentence of SL, but it is not a wff because it is not contained in brackets (which was one of questions). But when they talk about $A\Rightarrow A$, it is neither a sentence nor a wff. – 2017-01-10
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    The common rule is that a sentence is a wff without free variables. Hence one can assign a truth value to it. In sentential logic there are no free variables, which makes all wffs sentences. Maybe you should check the definitions in your text. They appear to be somewhat non-standard. – 2017-01-10