I'm reading a book about discrete math written by Kenneth H.Rose and in it he states that in mathematical logic,

A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both [1].

Isn't it redundant to use the word "declarative" here? My reasoning is that any statement in English that is either true or false must syntactically be declarative. I'm feeling anxious though, because I think I'm missing out on some meaning and don't want to misinterpret this statement.


[1] K. H. Rosen, "The foundations: Logic and Proofs" in Discrete Mathematics and It's Applications, 7th Ed. : Mcgraw Hill

  • The book in question is a textbook. If you are reading it in the context of a class, you may be better off directing such questions to the instructor.
    – jsw29
    Apr 9 at 20:45

There are four different types of sentence in English. These are:

Declarative - making a statement normally

Interrogative - asking a question

Exclamatory - making a statement but with emotional involvement

Imperative - making a request or giving a command

The same website as above says that there are two ways to form an exclamatory statement: by making it exclamatory in function and by using the exclamatory form. The first makes a statement but with an element of excitement; in writing this is indicated by ending it with an exclamation mark. The second begins with 'what' or 'how', is not a question (in writing does not end with a question mark and in speech does not end on a rising tone) but does have a change in the normal word order.

Exclamatory sentences make statements and can, therefore, be untrue, even if the speaker or writer conveys emotional involvement. For instance "I enjoyed my trip to Jupiter!" or "How helpful the stegosaurus was when we went back in time!"

All this means that the word 'declarative' is not redundant because mathematical propositions are not exclamatory statements, they are only ever declarative.

  • So just for clarification, a sentence like "I enjoyed my trip to Jupiter!" can't be used as a mathematical proposition?
    – lmn32
    Apr 8 at 17:17
  • @lmn32 That is correct, if you accept Rose's definition. More realistically neither "Two plus two equals five in some integer base!" nor "How steep the curve of the graph of the equation y = x¹⁰⁰ is!" can be mathematical propositions because they are exclamatory sentences. The first is obviously untrue and the second is obviously true but they are both exclamatory which Rose says is not a characteristic of a mathematical proposition.
    – BoldBen
    Apr 8 at 22:38
  • The definition that was quoted in the question was a definition of a proposition, not specifically of a mathematical proposition. Also, even though the exclamation 'I enjoyed my trip to Jupiter!' is not a itself a proposition, it can be analysed in a way that shows it to incorporate the proposition 'I enjoyed my trip to Jupiter'.
    – jsw29
    Apr 9 at 20:41
  • @jsw29 The OP states that the book they are quoting deals specifically with mathematical logic and actually asks whether the word 'declarative' is superfluous in the definition of a mathematical proposition. I was pointing out that it is not superfluous, if you use an exclamatory sentence then, according to Rose, you are not creating a mathematical proposition. You may or may not agree with Rose but the question was about Rose's definition of a mathematical proposition and whether there was redundancy in the use of the word 'declarative'. Nothing more.
    – BoldBen
    Apr 10 at 22:46
  • Of course, declarative was not superfluous. I was trying to clarify the confusion about the use of mathematical in this context: mathematical logic is a set tools, broadly mathematical in nature, for analysing propositions in general, not specifically mathematical ones. The comment that exclamations may involve propositions even though they are not themselves propositions was not a criticism of the answer, but an additional explanation. In any event, what the OP is asking belongs to the first week of an introductory logic class, and not to this site.
    – jsw29
    Apr 11 at 15:50

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