Sentence vs proposition

We need some conventions on the terminology.

A sentence

is a meaningful group of words that express a statement, question, exclamation, request, command or suggestion.

A declarative sentence (stament, assertion) is a sentence stating a fact, like: "The rose is red".

In logic, there are declarative sentences and not e.g. questions or commands.

Unfortunately, we call the propositional calculus also sentential logic.

Thus, in propositional calculus we can replace a sentential variable $p_i$ with the declarative sentence: "The rose is red" and not with the question: "Which is the color of the rose?"

In propositional calculus, sentence and proposition are interchangeable, while in philosophical discourse, a proposition is usually an extra-linguistic entity: the content expressed by, the meaning of, the reference of a linguistc entity (a declarative sentence).

In predicate logic we have formulas with free variables (called open formulas), like: "$x$ is red".

The free variable acts as a pronoun; when we assert "it is red", the meaning of the assertion is disambiguated contextually: according to the context, "it" may stand for the book on the table, the car, the pen, and the truth value of the assertion may change accordingly.

In logic, we have to assign a denotation to the free variable $x$ (through e.g. a variable assigment fucntion) in order that the formula has a meaning (and a truth value).

In predicate logic, a formula without free variables (or closed formula), like "the rose in my hand is red" or "all the roses are red", is called a sentence.

Thus, in conclusion, either in propositional logic or in predicate one, a sentence is always meaningful, and it has always a definite truth value.

Propositional logic or sentential logic is the branch of logic that studies ways of joining or modifying entire propositions, statements or sentences to form more complicated ones. In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement.

Thus, a statement can be defined as a declarative sentence, or part of a sentence, that is capable of having a truth-value, such as being true or false. So, for example, the following are statements:

George W. Bush is the 43rd President of the United States. Paris is the capital of France.

The term proposition is sometimes used synonymously with statement. However, it is sometimes used to name something abstract that two different statements with the same meaning are both said to "express". In this usage, the two English sentences, "Callisto orbits Jupiter" and "Jupiter is orbitted by Callisto" would also be considered to express the same proposition. However, the nature or existence of propositions as abstract meanings is still a matter of philosophical controversy, thus the phrases "statement" and "proposition" are used interchangeably.

What is distinctive about propositional logic as opposed to other (typically more complicated) branches of logic is that propositional logic does not deal with logical relationships and properties that involve the parts of a statement smaller than the simple statements making it up. Therefore, propositional logic does not study those logical characteristics of the propositions below in virtue of which they constitute a valid argument:

George W. Bush is a president of the United States. George W. Bush is a son of a president of the United States. Therefore, there is someone who is both a president of the United States and a son of a president of the United States.

The recognition that the above argument is valid requires one to recognize that the subject in the first premise is the same as the subject in the second premise. However, in propositional logic, simple statements are considered as indivisible wholes, and those logical relationships and properties that involve parts of statements such as their subjects and predicates are not taken into consideration.

Adapated from http://www.iep.utm.edu/prop-log/