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To say “Pegasus doesn’t exist” is to say “it is not the case that there is exactly one x which is a flying horse of Greek mythology”.

https://plato.stanford.edu/entries/nonexistent-objects/

  1. “Pegasus doesn’t exist”
  2. “It is not the case that Pegasus exists".

I think these two sentences are basically different. In (1), the verb is negated, but in (2) the whole clause is negated (it's like logical negation; such as not p or ~p).

Why do some philosopher including Russell paraphrase like this? They think "The king of France is bald" can be paraphrased as "it's the case that the king of France is bald", so they think it's false. I think "the king of France is bald" is truth-valueless.

So, my question is are "x isn't..." and "it's not the case that x is..." not the same?

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    Yes, they are "basically different", the whole point of paraphrase is to change the phrasing while expressing the same meaning. One can dispute that the meaning remains the same in Russell's paraphrase, and some do, but not on the grounds that a verb has been replaced by a clause or something else happened to the grammar.
    – Conifold
    Nov 23, 2023 at 9:24
  • @Conifold I'm a little confused. Russell paraphrased even "X exists" as "there is X", which is definitely different in meaning.
    – user68943
    Nov 23, 2023 at 9:29
  • @MauroALLEGRANZA I mean it's because if they are negated, "Pegasus doesn't exist" has no truthvalue because there isn't Pegasus, but "there isn't Pegasus" is true with the same reason.
    – user68943
    Nov 23, 2023 at 9:41
  • But "Pegasus" denotes nothing in "Pegasus doesn't exist"?
    – user68943
    Nov 23, 2023 at 9:50
  • Let us continue this discussion in chat.
    – user68943
    Nov 23, 2023 at 9:59

4 Answers 4

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In Fregean terms, a phrase has both a sense and a denotation. For a phrase like "the king of England", the sense is the concept of being king of England, and the denotation is the actual man who is the king of England. For a phrase like "the king of France", there is nothing that it denotes. For propositional sentences, the sentence denotes either True or False (or sometimes, arguably, nothing at all, like "the king of France").

You seem to be pointing out that the paraphrase in question does not have the same sense as the original sentence, which is true, but the goal of a paraphrase is not to duplicate the sense (which would be pointless); the goal of a paraphrase is to modify the sense in such a way that the denotation is the same in all cases. So the goal of Russel's paraphrase is to take a sentence where the denotation is unclear and to modify the sense in such a way that the new sentence can be argued to have the same denotation, but for the new sentence, the denotation is clearer.

Your complaint is basically that you do not agree with the argument that the new sentence has the same denotation as the original. This is a reasonable point of view, but to argue it, you need to argue for why you prefer a different denotation for the original sentence. It really isn't obvious what that denotation is, which is what spawned Russel's interest in the first place.

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Note that, "There is an x," can be reordered as, "An x is there," where "there" is an indexical for something like the world/reality/existence (as an atmosphere or environment in which things are said to exist). So we might even speak in terms of spatial quantification and say rather, "Somewhere is an x," and then we would paraphrase, "x doesn't exist," as, "Nowhere is an x" (not even in some metaphysical/eternal space, if we wish to countenance such a "place" or "realm").

But so is "is there" a predicate? Yes. Is it a property? If being-quantified-over is not a property, though, then it seems not. For then the somewhere-quantifier is not descriptive of an object "by itself," but is essentially relative (to "wherever").

Strongly recommended reading: Quantifiers and Quantification, sec. 4, in the Stanford Encyclopedia of Philosophy. I would also say to look over Kant's critique of the ontological argument, especially the part where he says that a hundred possible units of currency do not contain "less" than a hundred actual such units.

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In the philosophy of language, there is a notion known as deep structure which originates with Noam Chomsky. It is the idea that when one uses language for others, that one provides a shallow syntax which represents a method of communication that transfers a deeper meaning attached to the syntax. For instance:

Go!

Seems to be rather a simple imperative construction. But for starters, we can see that it hides the subject of the sentence so that:

(You) Go!

might be better. And yet, there may be more structure lying under the surface.

(You) Go! (away because I don't want you to be here.)

Thus, arguably, when philosophers reshape language, the goal is to find additional meaning that is hidden from the surface syntax to find what else is going on in someone's head or to phrase the construction to avoid problematic communication.

So, my question is are "x isn't..." and "it's not the case that x is..." not the same?

Not necessary. "It is not the case" makes explicit the truth-conditions of the statement. One might argue that the two statements are an example of language that is relevant to the deflationary theory of truth wherein a statement like "It is true that S" is seen as the same as "S". Yet, in linguistics, a good rule of thumb is that any difference in surface structure might express a subtle difference in deep structure. Perhaps to use the language "It is not the case that" is an act of performativity. For instance, in formal settings, one might hear a judge make statements like "I hereby declare..." where the purpose of this is not to communicate truth semantics, but to change social reality as in with a performative utterance.

Another possibility is the effects of adding additional surface syntax is to deal with contexts and possible worlds. There are extended philosophical theories like accessibility and discourse representation theory (SEP) that track existential quantification in multiple contexts for instance. Ideas like contexts, counterfactuals, segments of text, etc. might have additional surface syntax that indicates their presence.

So, the short of it is that Russell's expansion of the surface structure probably serves some purpose within the particulars of his theory. It is important to remember that tiny differences in surface syntax can have substantial differences in deep structure. (Consider the difference between 'strippers, Churchill, and Stalin' and 'strippers, Churchill and Stalin' (YT).) What those are in this particular case is a matter of interpreting Russell's theory fully.

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  • The "The king of France is bald" is what it is and Bertrand Russell's Theory of Descriptions cannot explain the logic of it. Dec 11, 2023 at 10:35
  • @Speakpigeon Fictionalism has swept away those ancient objections. It might help if you read some contemporary philosophy to understand the power of mathematical logic and modern analytic philosophy. plato.stanford.edu/entries/fictionalism
    – J D
    Dec 11, 2023 at 16:26
  • "Fictionalism has swept away" Academics don't even agree on whether it has any value. Dec 11, 2023 at 17:08
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Why do some philosopher including Russell paraphrase like this? They think "The king of France is bald" can be paraphrased as "it's the case that the king of France is bald", so they think it's false. I think "the king of France is bald" is truth-valueless.

And you are absolutely correct. If there is no king of France, the sentence "The king of France is bald" is obviously neither true nor false, and therefore not false, contrary to what Russell makes it out to be.

This is not Bertrand Russell who first discussed the sentence "The king of France is bald", it is well before Russell, Gottlob Frege, and Frege arrived at the same conclusion as you do, that the sentence is neither true nor false.

What no doubt spawned Russell's interest was that contrary to Frege, he realised that once your theory is that the implication α → β is the horseshoe ¬α ∨ β, as Russell, in that following Frege, did, then you cannot account for the fact that the sentence "The king of France is bald" is neither true nor false.

What to do? The problem for Russell was to interpret the sentence to make it either true or false. However, making it true would have looked silly, so he definitely had to make it false.

How to do that?

Oh, well, after postulating that the implication α → β is just the horseshoe ¬α ∨ β, Russell had to keep postulating a way through and say that the sentence really is "There is a king of France and he is bald". This is silly, too, because the sentence is what it is, and we already know what it is, and it is obviously not "There is a king of France and he is bald".

So, Russell's theory of descriptions cannot account even for such simple sentences as "The king of France is bald" when there is no king of France.

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