What is a proposition? Conflicting definitions.

Question

My textbook, How to Read and Do Proofs by Daniel Solow, defines a proposition as, "... a true statement of interest that you are trying to prove".

Other people seem to define a proposition as being possibly true or false, but not both: Difference between a proposition and an assertion, https://www.math.ku.edu/~jhart/Math_Logic_Student.pdf.

It seems logical to me that a proposition is something that can be either true or false, but not both. After all, the entire reason we strive to discover proofs is to definitively demonstrate that something is true; it would be paradoxical to assume beforehand that it is true!

However, my understanding is that we always attempt to prove a proposition by first assuming that the hypothesis is true; from this point, we work towards proving that the conclusion is true.

Is Solow's definition of proposition false? When he says that we assume that the proposition is true, is he actually referring to the hypothesis? Which definition of proposition is correct and why?

Thank you.

Answer 1

The two definitions are in different contexts. Solow's definition of "proposition" is in the same context as words like "theorem", "lemma", and "corollary"; these are terms used when writing a proof in mathematical English. In that context, a proposition must be true, for the same reason that a corollary must be true - you're trying to prove it! Note, however, that this means that a sentence can't be a proposition until you've proven it - until then, it's just a conjecture.

The other context is in formal logic, where a "proposition" is a statement like $P \wedge Q$ (or at least, an English sentence that can be translated into formal logic). In that context, a proposition is indeed a statement that can be true or false, but not both. If you're trying to do something about formalizing natural language, this is the context you're using.

To take an analogy: A "ring" in everyday life is a circular piece of jewelry worn on a finger; a "ring" in abstract algebra is a mathematical structure obeying certain axioms. Though these definitions clearly conflict, neither definition is "wrong" - they just apply in different contexts.

Answer 2

It seems logical to me that a proposition is something that can be either true or false, but not both.

Yes, you are correct with the logic that "A proposition is a statement which is either true or false but not both."

However, my understanding is that we always attempt to prove a proposition by first assuming that the hypothesis is true; from this point, we work towards proving that the conclusion is true.

I think you are referring here to the set of propositions which collectively form an argument.An argument is a set of $n$ propositions in which all except the $nth$ statement are called premises or hypothesis and the $n^{th}$ statement is called conclusion. An argument is said to be valid if the conjunction of all the premises implies the conclusion i.e. you prove the $n^{th}$ proposition by assuming all the $(n-1)$ propositions to be true irrelevant of the fact that how absurd they actually are in the real world. For example:

            *All Cats are birds.
             Birds fly in the sky.
     Therefore, all cats fly in the sky.*

The above argument is valid irrespective of the absurd nature of its premises.