Definitions of 'proper noun' describe it as a noun designating a particular being or thing.

But that is not enough, because in "I am riding my bike.", 'my bike' is a particular bike.

So the definitions add that a proper noun does not take a limiting modifier (such as 'my', 'this', 'a', 'an', ...).

Does anyone know of a satisfactory definition of 'proper noun' that does not resort to the limiting- modifier exclusion?
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The modifier is not part of the noun. A proper noun is limited to the proper name of any individual person, place or thing. It is considered a noun phrase in itself (Eiffel Tower). Whether it is preceded by a determiner (the Eiffel Tower-- the real one-- or my Eiffel Tower-- the toy one) is irrelevant.

There is no reason not to accept the limiting-modifier exclusion any more than you would not accept the fact that a proper noun is not a verb. 'Bike' is not a proper noun.
I don't think the use of limiting adjectives IS irrelevant.

Webster's Third New International Dictionary defines 'proper noun' : " a noun that designates a particular being or thing, does not take a limiting modifier, and is usu. capitalized in English."

The American Heritage Dictionary: "a noun designating by name a being or thing without a limiting modifier."

Perhaps the dictionaries are saying that a proper noun sufficiently specifies a particular being or thing so that a limiting adjective is not needed to identify the being or thing intended. Saying 'my', or 'this' or 'some', or 'a' Eiffel Tower is redundant or confusing because there is only one. (In your example 'my (toy) Eiffel Tower', there are several toy Eiffel Towers, so here it is a common noun.)

The remarks in the definitions of 'proper noun' about limiting adjectives may just be usage comments and not essential to the definitions. But, as worded, the remarks seem to be integral to the definitions.

I still have two questions:

1. Is there a better definition of 'proper noun', one which does not refer to limiting adjectives and which better communicates the specifying action of a proper noun?

2. The prohibition against using limiting adjectives with proper nouns seems to need some qualifying: her (France's) Eiffel Tower. And what about 'the'? The Eiffel Tower, The United States, ... .
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What if we approach it in a different way.

If I say 'show me your bike' to anyone who says 'I am riding my bike', I will be shown a different object each time.

If I say 'show me the Eiffel Tower' to anyone who says 'I am looking at the Eiffel Tower', I will always be shown the same object.

If I say 'show me Lydia' to anyone who says 'I am looking at Lydia', I will be shown a multitude of Lydias ('Lydiae'?).

'Lydia' is a non-exclusive proper noun. In everyday life, therefore, confusions do arise:

"I'm going out with Lydia tonight."
"Lydia? MrP won't like that. He's had his eye on Lydia for some time."
"Oh, I don't mean MrP's Lydia. I mean MrQ's..."

In practice, 'Lydia' is sufficient for most situations. But ?every non-exclusive proper name is implicitly part of a longer proper-name-string, which directs us exactly, e.g.

'Lydia Ostermeyer, 512 Rue Paul Cézanne, Roissy-en-Brie 67055, France.'

As in this conversation between two artificial intelligences:

"I'm going out tonight with Lydia Ostermeyer, 512 Rue Paul Cézanne, Roissy-en-Brie 67055, France."
"I'm glad you were specific. Otherwise I might have thought you meant MrQ's Lydia, who lives at..."

Hello, everyone,. This is a tough one ... it seems to me few questions were more disputed than this one during 20th century. I'm not competent for this discussion at all, but I'd like to put my some word in here.

[1] In Montague Grammar, proper names, definite descriptions, indefinite descriptions and all the other quantified terms are considered as expressions which belong to the same syntactic category T (term) (MM's a noun phrase).
(Proper names) G.W.Bush, Lydia Ostermeyer1(with an index, MrP's), Lydia Ostermeyer2(MrQ's), etc.
(Definite Descriptions) the president of America, the woman who lives at ~, my (toy) Eiffel Tower, etc.
(Other quantified terms) an American, a French, several students, every man, one cat, etc.

[2] The most simple answer to such a question as is, I think, Kripke's following thought.

(1) Proper nouns are considered as ... that is, it refers to one and the same object, in every possible world (situation).
(2) Definite descriptions are considered as ... that is, it's reference may be different in different situations.
(3) And we can tell one from the another by the following test:

# X (in this world) could be not X (in some another world).

(4) If some expression is a definite description, then we can put it in X:

#The president of America (in this world, i.e. G.W.Bush) could be not the president of America (in some another possible world).

(5) If some expression is a proper name, then we cannot put it in X ... it's meaningless, or at least not as acceptable as the sentence in (4).

#G.W.Bush could be not G.W.Bush (in another world).

(Actually I'm not so sure. This is what I understand from some other's handbook....)
(A correction notice)

Hello, there is a indecisive part in my explanation and it makes me uneasy. I wrote in (5)
[ , or at least not as acceptable as the sentence in (4).] , but I'd like to retract this.

We should consider such sentences as meaningless:

*Aristotle could be not Aristotle.
*Beethoven chould be not Beethoven.
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Hello guys

OED says about "proper noun" in the entry of "proper" as below. I think it is very close to Mr P's opinion. Unlike Webster and Heritage it doesn't mention anything about its relation to determiners. Personally I like the definition by Mill which is quoted last below.

proper [3-b] (grammatical term)
"Proper" is applied to a name or noun which is used to designate a particular individual object (e.g. a person, a tame animal, a star, planet, country, town, river, house, ship, etc.). A proper name is written with an initial capital letter. The same proper name may be borne by many persons in different families or generations, or by several places in different countries or localities; but it does not connote any qualities common to and distinctive of the persons or things which it denotes. A proper name may however receive a connotation from the qualities of an individual so named, and be used as a common noun, as a Hercules, a Cæsar (Kaiser, Czar), a Calvary, an atlas.

[1440] Charles, proper name, Carolus. [1551] (Wilson) In this proposition Cato is the noun proper, which belongs to one man only. [1690] (Locke) If we had reason to mention particular horses, as often as particular men, we should have proper names for the one, as familiar as for the other; and Bucephalus would be a word as much in use, as Alexander. [1720] (Waterland) Supposing Jehovah to be merely a proper name. [1843] (Mill) Proper names have strictly no meaning: they are mere marks for individual objects.

By the way, I would like to ask you why some proper names are modified by "the" in English. For example, why do you "the Thames", not just "Thames"? Is there any other Thames than the river?

Good morning, paco. I won't refute OED's definition. But....Mill's definition,

# Proper names have strictly no meaning: they are mere marks for individual objects.

Many philosophers were not content with such a vague explanation....Emotion: smile

I found an interesting site: maybe it helps as regard to your last question...?

Can we say:

"A proper noun denotes an object whose address does not change with a change of speaker or situation."

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