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I've read in one mathematics book that any definition is not a proposition. Why is it so? What definition of proposition makes the former into the latter?

EDIT: There is well-known analytic-syntactic distinction introduced by Kant. In it analytic propositions include definitions. If we say that definition is not a proposition then it implies a special case of analytical propositions are not propositions.

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A definition (in the mathematical context) is simply the granting of a new name. When we define pi to be the ratio of the circumference to the diameter of a circle, we are not making a claim of any kind; we're simply agreeing to use a given greek letter to substitute for a given notion.

A proposition, on the other hand, attempts to add new information (by predication, etc.)

EDIT:

Since the question was updated to mention Kant's analytic/synthetic distinction, I thought I'd update my answer as well.

Note that I said above "(in the mathematical context)"-- in mathematics, it is customary to draw a distinction between definitions and propositions. This distinction is not maintained the same way in Kantian (and post-Kantian) metaphysics; there, as you point out, a definition is a type of analytic proposition.

Certain words are "terms of art" in more than one domain-- it is important to take these in the appropriate context, and avoid slippages between them.

  • Your first sentence is correct. A proposition, on the other hand, is a statement that requires verification (or proof) by formal logic. – mdg Mar 25 '13 at 10:26
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I suppose it depends on how you define "definition", and although that sounds somewhat goofy, I say it will all seriousness. Typically, you use a definition as a sort of representation of an idea. In computer science, this would be like assigning a value to a variable, or a defining a function or method. In mathematics, it would be like establishing that the idea of the ratio of a circle's circumference to its diameter is pi, such that whenever you encounter the word or symbol pi you understand what it represents. In philosophy, definitions are used to capture the true "essence" of an idea or thing. In all these regards, the common factor is that a definition is merely a label for a previously known or understood concept. Linguistically, it is used as a tool for describing an otherwise tedious-to-explain concept in a single word. I could talk about pi in a sentence by only ever referring to "the ratio of any circle's circumference to its diameter", but after a while, talking about "the ratio of any circle's circumference to its diameter" could become rather burdensome, because "the ratio of any circle's circumference to its diameter" is somewhat of a long phrase. Definitions provide a way of capturing the essence of an idea into a single word to make our lives easier.

A proposition, on the other hand, is typically more than simply an equality relationship (var A = B) or label; it attempts to establish the validity the content as truth. In mathematics, it is known as a proof. In philosophy, it is known as an argument.

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I want to put forward a case for that definitions are propositions. One of my favorite ideas of all time is Quine's idea that there is no fundamental difference between a fact and a definition.

"Like other Analytic philosophers before him, Quine accepted the definition of "analytic" as "true in virtue of meaning alone". Unlike them, however, he concluded that ultimately the definition was circular. In other words, Quine accepted that analytic statements are those that are true by definition, then argued that the notion of truth by definition was unsatisfactory. " (http://en.wikipedia.org/wiki/Willard_Van_Orman_Quine#Rejection_of_the_analytic.E2.80.93synthetic_distinction)

The way I understand the theory, or my spin on it, is that ultimately all definitions have to be verified and falsified in the physical world just like facts do, since they are ultimately about the same thing; they are about our experiences of reality. When you define words with other words you are at the same time making claims about what is true (i.e. propositions). (side note: a made up a story can have definitions which are not about reality, but they are still about our experiences of reality since they are about a story which exist in a book, online or memory etc.) When you claim that e=mc2 you are both saying that mc2 is a useful definition of energy and that it is true that the function of energy can be determined by these measurements. A more straight forward example; when you say that the earth is round you are at the same time saying that round is a good definition of the shape of the earth and that it is true that the earth is round (it is actually no way near round since we have high mountains and deep valleys). When you attack the argument it can be about definition or fact, but if you change one you will have to change the other in order to stay coherent. If you say that 'almost round' is a better definition of the shape of the earth, then it is also true to you that the earth is 'almost round' and not round anymore. If you conclude that it is true that the earth is round then you are also saying that the shape of the earth is a useful definition of round.

An interesting consequence of this is that 1+1 is not necessarily 2. In order to claim that 1+1 is a useful definition of 2 you have to prove it in the physical world. We have a lot of proof since we have added up a lot of apples and bananas throughout history, but there are cases when it is not true. If you add 1 pile of sand to 1 pile of sand you don't get 2 piles of sand, you only get 1. So 1x+1x is not always 2x and so it is not always true that 1+1 is 2 and 1+1 is not always a useful definition of 2 in the same way that the shape of the earth is not always a useful definition of round (even though sometimes it is).

Summary: all definition are propositions because unless you can verify or falsify something which they are supposed to directly or indirectly be about they are not useful definitions because they ONLY try to describe something which does not exist i.e. something which does not have the slightest present or historical reference to reality. What use would 1+1=2 be if it had never been used to verify or falsify any real world events?

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Definitions are generally objective, lexical statements, while assertions are subjective and are often about events.

You can say that definitions are assertions if you reason that definitions are simply assertions that have been agreed upon.

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    I disagree with the first part of your answer since any definition is subjective in the sense that it's defined by some subject (human). In the meantime, propositions are extremely objective. I disagree with the second part too since there are propositions which are not definitions and are agreed upon. – Oleksandr Bondarenko Aug 18 '11 at 7:56
  • @Oleksandr If you're erring on the side of Subjectivism, then yes, however, since definitions are so fundamental to language and often very well agreed upon (nobody is going to assert that a door is a table with much weight) that we consider it objective fact. – digitxp Aug 18 '11 at 13:33
  • @Oleksandr With the second part, I said that definitions are a special kind of assertion, not that all propositions that are agreed upon are definitions. – digitxp Aug 18 '11 at 13:36
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Mathematics, not philosophy…

Since you read about the distinction between proposition and definition in a math book, the usual meaning(s) of "proposition" in philosophy doesn't really apply (and therefore the Kantian terminology is just misleading). In most linguistic/philosophical meanings of the term, all mathematical statements are propositions, including definitions. But this is not what the book means. What seems is alluded to in the book you read is the mathematical meaning of "proposition".

When doing mathematics, we can distinguish different kinds of statements, like axioms, definitions, theorems, lemmas, corollaries, conjectures, etc. The term "proposition" is usually reserved for theorems of no particular importance. Sometimes it is used to denote statements that are going to be proved. Finally, the term may be used as synonym for all proven statements. In either case, the main difference you might want to look into is that between definitions and theorems:

  • Definitions are precise descriptions of the meaning of a mathematical term. A definition supplies all and only those properties necessary to unambiguously fix the object described.
  • Theorems are proven mathematical statements, which may build on previous theorems and definitions. (In fact, it is common to state definitions describing the exact meaning of the terms used in the theorem before executing the proof.)

Theorems are proven mathematical statements. Definitions are also mathematical statements, but they are not proven, because there's nothing to prove.


You can read this entry on mathematical terminology to get a better understanding of the different kind of statements usually used in mathematical works. See also Some hints on mathematical style.

Some related discussions:

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