On Translating ‘Some Individual Exists’ Fressellianly

An astute reader comments:

2. But can this presupposition be expressed (said) in this logic? Here is a little challenge for you Fressellians: translate 'Something exists' into standard logical notion. You will discover that it cannot be done. Briefly, if existence is instantiation, which property is it whose instantiation is the existence of something? Same problem with 'Nothing exists.' If existence is instantiation, which property is it whose non-instantiation is the nonexistence of anything? Similarly with 'Everthing exists' and 'Something does not exist.'

But couldn't we translate those expressions this way (assuming we have only two properties: a, b)?
1. "something exists" -> "there is an x that instantiates either a or b or ab"
2. "everything exists" -> "there is an x that instantiates a and there is a y that instantiates b and there is a z that instantiates ab"
3. "nothing exists" -> 1 is false
4. "something doesn't exist" -> 2 is false

I am afraid that doesn't work. We need focus only on on 'Some individual exists.' The reader's proposal could be put as follows. Given the properties F-ness and G-ness,

What 'Some individual exists' says is exactly what 'Either F-ness is instantiated or G-ness is instantiated' says.

I would insist however that they do not say the same thing, i.e., do not have the same meaning. The expression on the left says that some individual or other, nature unspecified, exists. The expression on the right, however, makes specific reference to the 'natures' F-ness and G-ness. Surely, 'Some individual exists' could be true even if there are are no individuals that are either Fs or Gs.

Note that it is not a matter of logic what properties there are. This is an extralogical question.

On the Frege-Russell treatment of existence, 'exist(s)' is a second-level predicate, a predicate of concepts, properties, propositional functions and cognate items. It is never an admissible predicate of individuals. Thus in this logic every affirmation of existence must say of some specified concept or property that it is instantiated, and every denial of existence must say of some specified concept or property that it fails of instantiation.

This approach runs into trouble when it comes to the perfectly meaningful and true 'Something exists' and 'Some individual exists.' For in these instances no concept or property can be specified whose instantiation is the existence of things or the existence of individuals. To head off an objection: self-identity won't work.

That there are individuals is a necessary presupposition of the Frege-Russell logic in that without it one cannot validly move from 'F-ness is instantiated' to 'Fs exist.' But it is a necessary presupposition that cannot be stated in the terms of the system. This fact, I believe, is one of the motivations for Wittgenstein's distinction between the sayable and the showable. What cannot be said, e.g., that there are individuals, is shown by the use of such individual variables as 'x.'

The paradox, I take it, is obvious. One cannot say that 'There are individuals' is inexpressible without saying 'There are individuals.' When Wittgenstein assures us that there is the Inexpressible, das Unaussprechliche, he leaves himself open to the retort: What is inexpressible? If he replies, 'That there are individuals,' then he is hoist by his own petard.

Surely it is true that there are individuals and therefore expressible, because just now expressed.

"The suicide of a thesis," says Peter Geach (Logic Matters, p. 265), "might be called Ludwig's self-mate . . . . " Here we may have an instance of it.

Posted

May 9, 2012

Existence, Frege, Russell, Wittgenstein

Bill Vallicella

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11 responses to “On Translating ‘Some Individual Exists’ Fressellianly”

Leo Carton Mollica

May 9, 2012

Dear Dr. Vallicella,
Could the reader’s suggestion possibly be emended by quantifying over predicates? Thus, we could write it as (∀F)~(∃x)(Fx), where “x” ranges over individuals. Does this solve the problem, or do you think this emended version still runs into difficulties?

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arash

May 10, 2012

Not sure If I understood this statement:
“Surely, ‘Some individual exists’ could be true even if there are are no individuals that are either Fs or Gs. ”
On the premiss that the smallest unit of *reality* is a state of affairs
and a state of affairs is by definition a compound of *logically* distinct objects, it follows that in *reality* no single object of our system can be given outside a certain combination.
What gives us the impression that objects can exist isolatedly
is a confusion between *logic* articulation and *real* articulation.
On this view, independent existence would hold between 2 atomic states of affairs not between their logical constituents.
The important epistemic consequence of this is that:
The statement “a, b, c, etc. exist” is equivalent to a metalinguistic list of *nouns* (i.e. linguistic units with their syntactic function)
but in order to get a, b, c, etc. as *logically* distinct units (i.e. corresponding to nouns with their syntactic function) I must get
ab, ac, bc, etc. (i.e. I must be shown how a, b, c, etc. synctactically operates in propositional-like contexts) beforehand, not vice versa.

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mattghg

May 10, 2012

I guess this takes me outside of ‘Fresselian logic’, but surely ‘something exists’ can be expressed in a second-order logic by saying that some property is instantiated by some individual?
∃P ∃x P(x)
And surely we need to be able to ask other questions relating to propositions that can only be expressed in a second-order logic, like ‘why are most professors over 50 years old?’

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peterlupu

May 10, 2012

Bill,
How about the following in classical logic:
1. Something exists: (Ex) (x=a), for an arbitrary a;
2. Everything exists; (x)(Ey)(x=y);
3. Nothing exists: (x)~(x=x);
4. Something does not exists; (Ex)~(x=x).
As for Free logic, there is an existence predicate which would carry the translations.

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Bill Vallicella

May 10, 2012

Leo,
What your formula says is that no property is instantiated. But if ‘Nothing exists’ is true, then there are no properties.

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Bill Vallicella

May 10, 2012

Matt,
Your formula entails that something exists. But ‘Something exists’ does not entail your formula. For all logic can tell us, there could be individuals but no properties, or individuals and second-level properties, but no first-level properties. Your ‘P’ is a first-level property variable.

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Bill Vallicella

May 10, 2012

Aresh says, “On the premiss that the smallest unit of *reality* is a state of affairs . . . .”
Where does this premise come from? The smallest units of reality in Frege-Russell logic are the individuals that satisfy first-level predicates.
The monadic state of affairs *a’s being F* decomposes into the individual a and the property F-ness, which are ‘smaller’ units of reality than the state of affairs.

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Bill Vallicella

May 10, 2012

Hi Peter,
How goes it?
>>1. Something exists: (Ex) (x=a), for an arbitrary a;<< That doesn't say that something exists; it says that a exists. 'a' here is not a variable but a placeholder for a name such as 'Socrates.' If a exists, then something exists, but if something exists it doesn't follow that a exists. >>2. Everything exists; (x)(Ey)(x=y);<< Your formula does not say that everything exists; it says that everything is identical to something or other. >> 3. Nothing exists: (x)~(x=x);
4. Something does not exists; (Ex)~(x=x).<< The first does not say that nothing exists; it says that nothing is self-identical. The possibility -- if it is a possibility -- that nothing exists is not the possibility that everything is self-diverse. The second does not say that something does not exist; it says that something is self-diverse. Vulcan does not exist; ergo, something does not exist; but that is not to say that something, namely Vulcan, is self-diverse. Existence and self-identity are not the same property.

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arash

May 10, 2012

You are right but I was still trying to understand the way you are using the Frege-Russell logic to interpret Wittgenstein’s stand, and to me what W. has in mind corresponds more to what I reported in my previous comment
“On the premiss that the smallest unit of *reality* is a state of affairs . . . .”
is how I interpret what you recalled in a previous post: “the world is the totality of facts (Tatsachen) not of things (Dinge)”

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peterlupu

May 11, 2012

Bill,
“Existence and self-identity are not the same property.”
Well, this assumes that existence is a property and more specifically a property of individuals (first-order property). But setting that aside, the suggestion I have made above (following Hintikka) is that identity can be used to express existence. So if something is identical to a, then something exists (and so on for the rest).
“…but if something exists it doesn’t follow that a exists.”
Here you use ‘a’ as a name for a particular object. In this sense, you are right. But (as you say) a here is used more like a place holder. Hence, it is neither a name nor a variable. Since I have not used ‘a’ as a name, the above objection does not apply to the suggestion I made.
The central issue as I see it is whether individual-existence can be fully expressed only in terms that appeal to the individual whose existence is asserted and nothing else or not. If the later, then individual-existence, if it can be expressed at all, must be expressed by appealing to something beyond the individual. Hence, Fresselians think of existence as a second-order concept.

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David Brightly

May 11, 2012

Bill, my suggestion is that we call the concept at the root of the Porphyrean tree ‘Object’. Then I think we can translate ‘Something exists’ into ∃x.Object(x), ie, there is at least one object. More here.

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On Translating ‘Some Individual Exists’ Fressellianly

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11 responses to “On Translating ‘Some Individual Exists’ Fressellianly”

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