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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.

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The Pointer
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    I'd say that a proposition is the same thing as a statement or an assertion, and may be true or false. But I'm just an ignorant layman; let's see what the philosophers and logicians say. – bof Dec 24 '16 at 03:31
  • @bof Myself even more so. – The Pointer Dec 24 '16 at 03:32

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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.

Reese Johnston
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  • Excellent answer! You have my gratitude. – The Pointer Dec 24 '16 at 03:39
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    @ThePointer: I'd argue that Solow is simply wrong, by his own definition and in his own context, because if you're still trying to prove a statement then you haven't proven it yet and so can't claim it as a proposition (in the context of a mathematical writing). By the time you write it down as a proposition you would have thought that you had proven it already, and so it is never a statement that you are trying to prove. – user21820 Dec 24 '16 at 08:23
  • @user21820 What do you think is the correct definition of "proposition"? – The Pointer Dec 24 '16 at 08:27
  • @ThePointer: Reese has more or less given you a possible reason for the mismatching definitions you've seen for "proposition". I was just saying that Solow's definition as stated is not even compatible with those. And it has nothing to do with the hypothesis issue, where you derive conclusions under the assumption of the truth of hypotheses. Makes sense? – user21820 Dec 24 '16 at 08:57
  • @user21820 I understand, but I'm curious of what the correct definition is then? Or, If Solow's definition is logically inconsistent, even in the context of Reese's post, is there another word we should be using instead of "proposition"? If you have any opinions, I'd be better having heard them. – The Pointer Dec 24 '16 at 09:03
  • For context, Reese was correct when he said that Solow's use of "proposition" is in the context of "theorem", "lemma", and "corollary". – The Pointer Dec 24 '16 at 09:06
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    @ThePointer: Okay there are two issues that I can see here. Firstly, it is just a (small?) error on Solow's part in his definition of "proposition". I can easily believe that he only uses that word in writing for a statement that he believes he has proven, in line with current conventions though contrary to his definition. Secondly, I personally never use "proposition" to refer to statements that I claim are true, but rather I use "theorem", which is perfectly consistent with the definition in logic, or "lemma" for a stepping-stone theorem. But this is just my personal preference. =) – user21820 Dec 24 '16 at 09:16
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    @ThePointer: But anyway I'm just nitpicking; there is no need for you to worry about definitions of words used to described conventions in mathematical writings. You just need to understand the definitions of mathematical objects, unless you're actually interested in analyzing mathematics linguistically or philosophically or something like that. =) – user21820 Dec 24 '16 at 09:20
  • @user21820 Interesting. Thank you for sharing your opinion. – The Pointer Dec 24 '16 at 09:21
  • @ThePointer: It reminds me of the use of "if" in definitions when it actually means "iff". It irks me in the same way so I always use "iff". I guess the basic point is that expressions conventionally used in mathematics do not always agree with the usage in logic. – user21820 Dec 24 '16 at 09:30
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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.

Nitin Uniyal
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