Which of the following is a statement/proposition? 1. We ought to have a good balance between work and relaxation. 2. Not all mathematical...

The terms "statement" and "proposition" are technical terms used in philosophical analysis, but the terms are used differently in formal logic and in informal logic or argument theory. 

In formal logic, a proposition is something that has a truth value. The same proposition can, however, be expressed in multiple ways or multiple different statements. Thus "some cats are black" and "black coats are found on some felines" would express the same logical proposition with the...

The terms "statement" and "proposition" are technical terms used in philosophical analysis, but the terms are used differently in formal logic and in informal logic or argument theory. 

In formal logic, a proposition is something that has a truth value. The same proposition can, however, be expressed in multiple ways or multiple different statements. Thus "some cats are black" and "black coats are found on some felines" would express the same logical proposition with the same truth value, a value dependent on the empirical evidence of there being black cats. The propositional content of a sentence is its truth claim. Your second sentence, "Not all mathematical statements can be reduced to set theory," expresses a proposition. Because an "ought" statement expresses a moral imperative, it is of a different category than a formal logical proposition in that it is not making a clearly verifiable claim.

While some philosophers consider that there are moral truths, and thus that "ought" statements contain propositions in the sense of having truth values, many philosophers argue that to assign a truth value to an "ought" statement is to commit the "naturalistic fallacy."

In informal logic or argumentation, a proposition is a claim to be argued. Under this definition, both of your sentences would constitute propositions. Both sentences are declarative statements.