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’.)
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’.)