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The problem I am facing is why we can’t treat a definite description as a statement about exactly one object having certain properties.

For example the statement: “The author of Evangeline is Henry Wadsworth” is translated into predicate logic format, according to my textbook, as:

(∃x){Ax ∧ Hx ∧ (∀y)[Ay→y=x]} (1)

Why can't I consider the statement as asserting the existence of precisely one object possessing the property of being Henry Wadsworth and also being the author of Evangeline? If I can do so, then the representation would be:

(∃x){Ax ∧ Hx ∧ (∀y)[(Ay ∧ Hy)→y=x]} (2)

In a nutshell, why can’t I use statement (2) as a representation of the definite description “The author of Evangeline is Henry Wadsworth”?

The format used in statement (2) is the format used for representing statements about exactly one object.

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    Because a definite description asserts existence of precisely one object that answers the description, not just existence of precisely one object that answers the description and is Henry Wadsworth. Your preferred form weakens the assertion. By the way, your formula does not match your text, it should be (∃x){Ax ∧ Hx ∧ (∀y)[(Ay ∧ Hy)→y=x]}.
    – Conifold
    Mar 19 at 0:20

2 Answers 2

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To expand a little on Conifold's comment...

(∃x){Ax ∧ Hx ∧ (∀y)[Ay → y=x]}

This states, there is at least one thing that is the author of Evangeline and is named Henry Wadsworth, and there is at most one thing that is the author of Evangeline. Hence it states that there is exactly one author of Evangeline, but there may be many people named Henry Wadsworth.

(∃x){Ax ∧ Hx ∧ (∀y)[(Ay ∧ Hy) → y=x]}

This states that there is at least one thing that is the author of Evangeline and is named Henry Wadsworth, and there is also at most one thing that is the author of Evangline and is named Henry Wadsworth. Hence, exactly one such person.

So, which you use depends on whether you intend to express that the name Henry Wadsworth is unique. In real life, names are usually not unique. In the standard way formal logic is done, names are usually assumed to have a unique referent, since this avoids ambiguity. In fact, it would be more common to represent Henry Wadsworth as a name rather than a predicate. In which case the second formula would come out as:

Ah ∧ (∀y)[Ay → y=h]

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    Isn't there a difference between "... is named Henry Wadsworth" and "... is Henry Wadsworth"? The latter assumes that "Henry Wadsworth" is a symbol for a unique individual (even though the name may not be unique, the one who is an author is well known); it's more like "The author ... is that guy."
    – Barmar
    Mar 19 at 14:57
  • Yes, I would agree that the natural way of understanding, "The author of Evangeline is Henry Wadsworth" is to treat the name as uniquely identifying 'that guy'. At least in most contexts, where that guy is well known. Natural language is messy though, and in practice the pragmatics of language will always enter into the understanding.
    – Bumble
    Mar 19 at 21:37
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Because

There is a unique author of Evangeline, and this author is Henry Wadsworth

and

There is a unique author of Evangeline who is Henri Wadsworth

are two different assertions. The first implies the second, but the second does not imply the first.

Your textbook says that "The author is..." is equivalent to the more explicit "There is a unique author, and this author is...".

Whether this equivalence is true is debatable, and a matter of definitions rather than a universal truth*, but what is clear is that "There is a unique author" is not equivalent to "There is a unique Henri Wadsworth author".

*See "The present king of France is bald".

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