I’d say that such a definition still reused the term to be defined in the right part of the definition, wouldn’t you?
So your claim is that universal quantification, and identity and/or set membership, are all in effect just trivial linguistic obfuscations of existence?
Your definitions by necessity reduce to ‘exists(a) =(def) there is an x such that exists(x) and (x=a)’
The idea that existence is in some way a conceptual prerequisite for the particular quantifier is an interesting idea, and I could imagine good arguments being made for it. Certainly Graham Priest would agree with your above claim. But I don’t see any corresponding reason yet to think this about ‘exists(a)’ ≝ ‘a∈EXT(=)’.
It is trivial to show that if your universal or your existential quantifier’s domain (i.e. the possible values which x could take) were anything other than precisely those x’s for which exists(x) is true, the definition would be wrong
Why does that matter? It’s trivial to show that is the set of primary colors were a different set, then extensional definitions of the primary colors would fail. But this doesn’t undermine extensional definitions of primary colors.
Perhaps what you’re trying to get at is that we couldn’t construct the identity set, or the proper domain for our quantifiers, without prior knowledge that amounts to knowledge of which things exist? I.e. we couldn’t build an algorithm that actually gives us the right answers to ‘are a and b identical?’ or ‘is a an object in the domain of discourse?’ without first understanding on some level what sorts of things exist? Is that the idea? A definition then is unexplanatory (or ‘useless’) if the definiens cannot be constructed with perfect reliability without first grasping the definiendum.
The definition works only iff the domain of either the universally quantified or the existentially quantified version of the definition were precisely “the things that exist”, i.e. for which exists(x) returned true.
Yes… but, then, that’s true for every definition. Whatever definition of ‘bird’ we give will, ideally, return precisely the set of birds to us. It would be a problem if the two didn’t coincide, surely; so why is it equally a problem if the two do coincide? I can’t make an objection out of this, unless we go with something like the one in the previous paragraph.
If only. I disagree that it does (because of the above).
Well, it still does. You don’t use an ‘exists’ predicate in the logic. Your claim is philosophical or metasemantic; it’s not about what logical or nonlogical predicates we use in a system. Logicians have found a neat trick for reducing how many primitives they need to be expressively complete; you’re objecting in effect that their trick doesn’t help us understand the True Nature Of Being, but one suspects that this is orthogonal to the original idea, at least as many logicians see it.
Well, something ‘new’ to work with.
How ’bout identity?
‘Different part’ as in ‘not only the exact same concept which is to be explained’.
Are you saying that identity, existence, universal quantification, and particular quantification are all the exact same concept? If so, your concepts must be very multifaceted things!
However, the definition for exist we are discussing offers no such additional concept.
So you’re at a minimum saying that ‘∃’ and ‘exists’ are the same concept. Are you saying the same for ‘∀’, ‘¬’, ‘∈’, ‘=’, etc.?
So your claim is that universal quantification, and identity and/or set membership, are all in effect just trivial linguistic obfuscations of existence?
They are tools which in themselves can construct relationships that further describe that which is to be described. They just don’t in this case. Syntactic concatenation of operators doesn’t equal bountiful semantic content. Just like you can construct meaningless sentences even though those sentences still are composed of letters.
“a exists if there is some x which exists which is the exact same as a”, “If a and x are identical (the same actual thing) and x exists, we can conclude that a exists, since it is in fact x”.
These definitions can be used for most any property replacing “exists”. The particular usage of ‘∀’, ‘¬’, ‘∈’, ‘=’, “identity” or what have you in this case doesn’t add any content, or any concepts, if it’s just bloviating reducing to “if x exists, and x is a, then a exists”, or in short, if P(a) then P(a).
Whatever definition of ‘bird’ we give will, ideally, return precisely the set of birds to us.
Ideally. More leniently, “useful” would mean that given a definition, we would at least have some changed notion of whether at least one thing, or class of things, belongs in the set of birds or not. Even when someone just told you that a duck is a bird and nothing else, you would have learned something about birds. At least as an alien at least you could answer yes when pointed to a duck, if nothing else.
Explaining “is a bird(x)” by referring to a set which by definition contains all things which are birds, without giving any further explanation or examples, and then saying that if x is in the set of all birds, it is a bird, doesn’t give us any information whatsoever about birds, and amounts to saying “well, if it’s a bird, and we postulate a set in which there would be all the birds, that bird would be in that set!”. Who woulda thunk?
Saying “there are chairs which exist” gives us more information about what exists means then the first two definitions we’re talking about.
Concerning the ‘exists(a)’ ≝ ‘a∈EXT(=)‘, I can’t comment because I have no idea what precisely is meant by that ‘extension’ of =. Is it supposed to be exactly restatable as equivalent the other two definitions? If so, naturally the same arguments apply. If not, can you give further information about this mysterious extension?
I think we share the same views, at least in spirit. I’m just not satisfied by your arguments for them.
First, your analogies weren’t relevantly similar to the original equations. Second, your previous arguments depended on somewhat mysterious notions of ‘concept containment’, similar to Kant’s original notion of analysis, that I suspect will lead us into trouble if we try to precisely define them. And third, your new argument seems to depend on a notion of these symbols as ‘purely syntactic’, devoid of semantics. But I find this if anything even less plausible than your prior objections. Perhaps there’s a sense in which ‘not not p’ gives us no important or useful information that wasn’t originally contained in ‘p’ (which I think is your basic intuition), but it has nothing to do with whether the symbol ‘not’ is ‘purely syntactic’; if words like ‘all’ and ‘some’ and ‘is’ aren’t bare syntax in English, then I see no reason for them to be so in more formalized languages.
Informally stated, a conclusion like ‘the standard way of defining existence in predicate calculus is kind of silly and uninformative’ is clearly true—its truth is far more certain than is the truth of the premises that have been used so far to argument for it. So perhaps we should leave it at that and return to the problem later from other angles, if we keep hitting a wall resulting from our lack of a general theory of ‘concept containment’ or ‘semantically trivial or null assertion’?
(I don’t claim to be able to identify all useless definitions as useless, just as I can’t label all sets which are in fact the empty set correctly. That is not necessary.)
I’m talking about the specific first two definitions you gave. Let me give it one more try.
foo(a) is a predicate, it evaluates to true or false (in binary logic). This is not new information (edit: if we go into the whole ordeal already knowing we set out to define the predicate foo(.)), so the letter sequence foo(a) itself doesn’t tell us anything new (e.g. foo(‘some identified element’)=true would).
You can gather everything for which foo(‘that thing’) is true in a set. This does not tell us anything new about the predicate. The set could be empty, it could have one element, it could be infinitely large.
We’re not constraining foo(.) in any way, we’re simply saying “we define a set containing all the things for which foo(thing) is true”.
Then we’re going through all the different elements of that set (which could be no elements, or infinitely many elements), and if we find an element which is the exact same as ‘a’, we conclude that foo(a) is true.
The ‘identity’ is not introducing any new specific information whatsoever about what foo(.) means. You can do the exact same with any predicate. If ‘a’ is ‘x’, then they are identical. You can replace any reference to ‘a’ with ‘x’ or vice versa.
Which variable name you use to refer to some element doesn’t tell us anything about the element, unless it’s a descriptive name. The letter ‘a’ doesn’t tell you anything about an element of a set, nor does ‘x’. And if ‘a’ = ‘x’, there is no difference. It’s the classical tautology: a=a. x=x. There is no ‘new information’ whatsoever about the predicate foo(.) there.
In fact, the definitions you gave can be exactly used for any predicate, any predicate at all! (… which takes one argument. The first two definitions, we’re still unclear on the third.) An alien could no more know you’re talking about ‘existence’ than about ‘contains strawberry seeds’, if not for how we named the predicate going in.
You can probably replace foo(a) with exists(a) on your own …
That is why I reject the definition as wholly uninformative and useless. The most interesting part is that existing is described as a predicate at all, and that’s an (unexplained) assumption made before the fully generic and such useless definition is made.
Which of the above do you disagree with? (Regarding ‘concept containment’, I very much doubt we’d run into much trouble with that notion. An equivalent formulation to ‘concept containment’ when saying anything about a predicate would be ‘any information which is not equally applicable to all possible predicates’.)
So your claim is that universal quantification, and identity and/or set membership, are all in effect just trivial linguistic obfuscations of existence?
The idea that existence is in some way a conceptual prerequisite for the particular quantifier is an interesting idea, and I could imagine good arguments being made for it. Certainly Graham Priest would agree with your above claim. But I don’t see any corresponding reason yet to think this about ‘exists(a)’ ≝ ‘a∈EXT(=)’.
Why does that matter? It’s trivial to show that is the set of primary colors were a different set, then extensional definitions of the primary colors would fail. But this doesn’t undermine extensional definitions of primary colors.
Perhaps what you’re trying to get at is that we couldn’t construct the identity set, or the proper domain for our quantifiers, without prior knowledge that amounts to knowledge of which things exist? I.e. we couldn’t build an algorithm that actually gives us the right answers to ‘are a and b identical?’ or ‘is a an object in the domain of discourse?’ without first understanding on some level what sorts of things exist? Is that the idea? A definition then is unexplanatory (or ‘useless’) if the definiens cannot be constructed with perfect reliability without first grasping the definiendum.
Yes… but, then, that’s true for every definition. Whatever definition of ‘bird’ we give will, ideally, return precisely the set of birds to us. It would be a problem if the two didn’t coincide, surely; so why is it equally a problem if the two do coincide? I can’t make an objection out of this, unless we go with something like the one in the previous paragraph.
Well, it still does. You don’t use an ‘exists’ predicate in the logic. Your claim is philosophical or metasemantic; it’s not about what logical or nonlogical predicates we use in a system. Logicians have found a neat trick for reducing how many primitives they need to be expressively complete; you’re objecting in effect that their trick doesn’t help us understand the True Nature Of Being, but one suspects that this is orthogonal to the original idea, at least as many logicians see it.
How ’bout identity?
Are you saying that identity, existence, universal quantification, and particular quantification are all the exact same concept? If so, your concepts must be very multifaceted things!
So you’re at a minimum saying that ‘∃’ and ‘exists’ are the same concept. Are you saying the same for ‘∀’, ‘¬’, ‘∈’, ‘=’, etc.?
This paper might interest you; it also discusses translatability into alien languages with different ways e.g. of quantifying: Being, existence, and ontological commitment.
They are tools which in themselves can construct relationships that further describe that which is to be described. They just don’t in this case. Syntactic concatenation of operators doesn’t equal bountiful semantic content. Just like you can construct meaningless sentences even though those sentences still are composed of letters.
“a exists if there is some x which exists which is the exact same as a”, “If a and x are identical (the same actual thing) and x exists, we can conclude that a exists, since it is in fact x”.
These definitions can be used for most any property replacing “exists”. The particular usage of ‘∀’, ‘¬’, ‘∈’, ‘=’, “identity” or what have you in this case doesn’t add any content, or any concepts, if it’s just bloviating reducing to “if x exists, and x is a, then a exists”, or in short, if P(a) then P(a).
Ideally. More leniently, “useful” would mean that given a definition, we would at least have some changed notion of whether at least one thing, or class of things, belongs in the set of birds or not. Even when someone just told you that a duck is a bird and nothing else, you would have learned something about birds. At least as an alien at least you could answer yes when pointed to a duck, if nothing else.
Explaining “is a bird(x)” by referring to a set which by definition contains all things which are birds, without giving any further explanation or examples, and then saying that if x is in the set of all birds, it is a bird, doesn’t give us any information whatsoever about birds, and amounts to saying “well, if it’s a bird, and we postulate a set in which there would be all the birds, that bird would be in that set!”. Who woulda thunk?
Saying “there are chairs which exist” gives us more information about what exists means then the first two definitions we’re talking about.
Concerning the ‘exists(a)’ ≝ ‘a∈EXT(=)‘, I can’t comment because I have no idea what precisely is meant by that ‘extension’ of =. Is it supposed to be exactly restatable as equivalent the other two definitions? If so, naturally the same arguments apply. If not, can you give further information about this mysterious extension?
I think we share the same views, at least in spirit. I’m just not satisfied by your arguments for them.
First, your analogies weren’t relevantly similar to the original equations. Second, your previous arguments depended on somewhat mysterious notions of ‘concept containment’, similar to Kant’s original notion of analysis, that I suspect will lead us into trouble if we try to precisely define them. And third, your new argument seems to depend on a notion of these symbols as ‘purely syntactic’, devoid of semantics. But I find this if anything even less plausible than your prior objections. Perhaps there’s a sense in which ‘not not p’ gives us no important or useful information that wasn’t originally contained in ‘p’ (which I think is your basic intuition), but it has nothing to do with whether the symbol ‘not’ is ‘purely syntactic’; if words like ‘all’ and ‘some’ and ‘is’ aren’t bare syntax in English, then I see no reason for them to be so in more formalized languages.
Informally stated, a conclusion like ‘the standard way of defining existence in predicate calculus is kind of silly and uninformative’ is clearly true—its truth is far more certain than is the truth of the premises that have been used so far to argument for it. So perhaps we should leave it at that and return to the problem later from other angles, if we keep hitting a wall resulting from our lack of a general theory of ‘concept containment’ or ‘semantically trivial or null assertion’?
(I don’t claim to be able to identify all useless definitions as useless, just as I can’t label all sets which are in fact the empty set correctly. That is not necessary.)
I’m talking about the specific first two definitions you gave. Let me give it one more try.
foo(a) is a predicate, it evaluates to true or false (in binary logic). This is not new information (edit: if we go into the whole ordeal already knowing we set out to define the predicate foo(.)), so the letter sequence foo(a) itself doesn’t tell us anything new (e.g. foo(‘some identified element’)=true would).
You can gather everything for which foo(‘that thing’) is true in a set. This does not tell us anything new about the predicate. The set could be empty, it could have one element, it could be infinitely large.
We’re not constraining foo(.) in any way, we’re simply saying “we define a set containing all the things for which foo(thing) is true”.
Then we’re going through all the different elements of that set (which could be no elements, or infinitely many elements), and if we find an element which is the exact same as ‘a’, we conclude that foo(a) is true.
The ‘identity’ is not introducing any new specific information whatsoever about what foo(.) means. You can do the exact same with any predicate. If ‘a’ is ‘x’, then they are identical. You can replace any reference to ‘a’ with ‘x’ or vice versa.
Which variable name you use to refer to some element doesn’t tell us anything about the element, unless it’s a descriptive name. The letter ‘a’ doesn’t tell you anything about an element of a set, nor does ‘x’. And if ‘a’ = ‘x’, there is no difference. It’s the classical tautology: a=a. x=x. There is no ‘new information’ whatsoever about the predicate foo(.) there.
In fact, the definitions you gave can be exactly used for any predicate, any predicate at all! (… which takes one argument. The first two definitions, we’re still unclear on the third.) An alien could no more know you’re talking about ‘existence’ than about ‘contains strawberry seeds’, if not for how we named the predicate going in.
You can probably replace foo(a) with exists(a) on your own …
That is why I reject the definition as wholly uninformative and useless. The most interesting part is that existing is described as a predicate at all, and that’s an (unexplained) assumption made before the fully generic and such useless definition is made.
Which of the above do you disagree with? (Regarding ‘concept containment’, I very much doubt we’d run into much trouble with that notion. An equivalent formulation to ‘concept containment’ when saying anything about a predicate would be ‘any information which is not equally applicable to all possible predicates’.)