Formal definition of proposition

Question

Let us consider the following definitions of proposition

1) Proposition is a well-defines statement

2) Proposition is a declarative statement which has to be either true or false but not both

There is a lot of confusion if we did not include the context for such statement.

For some time, keep natural language statements aside and consider the following

$$1 + 1 = 2$$

Since context is immaterial, we can decide that it is not a proposition(true for natural numbers and false for binary numbers). So, even in mathematics, context in which we are declaring statements is important.

Now coming to natural language, no real world natural language statement can be a proposition without including context.

Hence the two definitions presented above does not formally define what proposition is;

Now my doubt is: What is the formal definition for proposition?

Answer 1

The term proposition has a broad use in philosophy : from Aristotle since modern times.

For the present discussion, we can agree on two different interpretations; either :

they are the bearers of truth-value, i.e. linguistic entities that are said to be either true or false and nothing else,

or :

they are the meanings of declarative sentences, i.e. non-linguistic entities related to linguistic expressions.

According to Logical positivists, propositions are "statements" that are truth-bearers i.e. that are either true or false and nothing else.

This view is the most similar to that adopted by mathematical logic :

Propositions in modern formal logic are parts of a formal language. A formal language begins with different types of symbols. These types can include variables, operators, function symbols, predicate (or relation) symbols, quantifiers, and propositional constants.

Symbols are concatenated together according to rules in order to construct strings to which truth-values will be assigned. The rules specify how the operators, function and predicate symbols, and quantifiers are to be concatenated with other strings.

A proposition is then a string with a specific form. The form that a proposition takes depends on the type of logic.