If a definition is not meant to be explanatory, its usefulness in understanding that which is to be defined is limited.
Taking the two alternate formulations you offered, I can still hear the telltale “is” beating, from beneath the floor planks where you hid it:
‘exists(a)’ ≝ ‘¬∀x¬(a=x)’
The “∀” doesn’t refer to all e.g. logically constructible x, does it? Or to all computable x. For the definition to make sense, it needs to refer to all x that exist, otherwise we’d conclude that ‘exists(flying unicorns)’ is true. Still implicitly refers to that which is to be defined in its definition, rendering it circular.
‘exists(a)’ ≝ ‘a∈EXT(=)’
What is EXT(=)? Some set of all existing things? If so, would that definition do any work for us? Pointing at my chair and asking “does this chair exist”, you’d say “well, if it’s a member of the set of all existing things, it exists”. Why, because all things in the set share the “exist” predicate. But what does it mean for them to have the “exist” predicate in the first place? To be part of the set of all existing things, of course. Round and round …
Not much different from saying “if it exists, it exists”. Well, yes. Now what?
If a definition is not meant to be explanatory, its usefulness in understanding that which is to be defined is limited.
Exactly.
The “∀” doesn’t refer to all e.g. logically constructible x, does it? Or to all computable x. For the definition to make sense, it needs to refer to all x that exist, otherwise we’d conclude that ‘exists(flying unicorns)’ is true. Still implicitly refers to that which is to be defined in its definition, rendering it circular.
That’s one option for explaining the domain of ∀. Another is to simply say that that the domain is the universe, or that it’s everything, or that it’s unrestricted. All of those can be expressed without speaking in terms of existence.
If you have no idea what those ideas mean, but understand ‘exists’, then, sure, maybe you’ll need to demand that all those ideas be unpacked in terms of existence. But what of it? If you do understand those terms but not ‘exists’, then interdefining them can be cognitively significant for you. Broadly speaking, the function of a definition is to relate a term that isn’t understood to a term that is. If you already understand both terms, then the definition won’t be useful to you; but that isn’t a criticism of the definition, if other people might not understand both terms as well as you do. It’s just a biographical note about your own level of linguistic/conceptual expertise.
What is EXT(=)? Some set of all existing things?
It’s the extension of the identity predicate, a set of ordered pairs. Relational predicates of arity n can be treated as sets of n-tuples.
If so, would that definition do any work for us?
Do any work for who? What is it you want, exactly? If you’ve forgotten, the first thing I said to you was “I’m not saying it’s a very useful definition”. You don’t need to prove it’s circular in order to prove it’s useless, and if you did prove it’s circular (‘circular’ in what sense? is there any finite non-circular chain of definitions that define every term?) that very likely wouldn’t help demonstrate its uselessness. So what exactly are you trying to establish, and why?
Another is to simply say that that the domain is the universe, or that it’s everything, or that it’s unrestricted. All of those can be expressed without speaking in terms of existence.
Any domain which is not constrained to iterate/refer only to things which themselves exist would lead to wrong conclusions such as “flying unicorns exist”.
What is it you want, exactly?
To show that the definition you referred to, in all its variants, isn’t useful. I did not forget that you didn’t claim it was useful, just that it was common, but I also noticed you did not explicitly agree that it was not useful. If you do agree on that, there is no need to further dwell on useless rephrasings.
I agree that since the body of human knowledge is limited, any definition must eventually contain circles of some size. However, not all circles are created equal: To be useful, a definition must refer to some different part of your knowledge base, just because without introducing new information, there is nothing which could be useful.
“2 is defined as something with the property of being 2” isn’t useful because there is nothing new introduced. “That which exists, exists” isn’t useful for the same reason. Because all the definitions you referred to still contain “exist”, the additional information (“things in a set”) is superfluously added, the “exist” on the right part of the definition still isn’t unpacked. Hence, no additional information is introduced, and the definition useless, being equivalent to “2 is defined as 2″.
“Pain is when something which is in the set of ‘being able to experience pain’ experiences pain” just reduces to “pain is when pain”, which must be useless since it contains no additional concepts.
If the additional “identity” aspects etcetera helped any in explaining the concept of “exist”, then the definition would not need to refer again to just the same “exist” which the “identity” supposedly helped explain.
Any domain which is not constrained to iterate/refer only to things which themselves exist would lead to wrong conclusions such as “flying unicorns exist”.
If I’m not misunderstanding you, you’re advocating a view like Graham Priest does here, that our quantifiers should range over anything we can meaningfully talk about (if not wider?) until we restrict them further. I’m inclined to agree. We both dissent from the orthodox definition I posted above, then. You’ll need to dig up a Quinean if you want to hear counter-arguments.
I also noticed you did not explicitly agree that it was not useful.
Well, I’m sure it’s been useful to someone at some point. It lets logicians get away without appealing to an ‘exists’ predicate. Logicians are generally much more attached to ‘is identical to’ than to ‘exists’. Again, you’ll have to explain exactly what kind of use you want out of the ideal Definition of Existence so I can evaluate whether the above ones I tossed about are useful with respect to that goal. What are some examples of new insights or practical goals you were hoping or expecting to achieve by defining ‘exists’?
To be useful, a definition must refer to some different part of your knowledge base
Could you say more about what you mean by ‘different parts of your knowledge base’? Is there a heuristic for deciding when things are parts of the same knowledge base?
“2 is defined as something with the property of being 2” isn’t useful because there is nothing new introduced.
Is “2 is defined as SS∅” useful? Or “2 is defined as {{},{{}}}”? Or “2 is defined as 1+1”? Are there any useful definitions of 2?
Because all the definitions you referred to still contain “exist”
What do you mean by “contain”? They didn’t make reference to existence twice. You noted we could reverse the definitions or build a chain, but that’s true of any definitions. (If they weren’t dreadfully boring, we’d probably not call them definitions.)
Do you mean that they presupposed an understanding of existence, i.e., if you didn’t first understand existence then you couldn’t understand my definitions? Or do you mean that concepts are combinatorial, and the concepts I appealed to all have as components the concept ‘existence’?
“Pain is when something which is in the set of ‘being able to experience pain’ experiences pain” just reduces to “pain is when pain”, which must be useless since it contains no additional concepts.
Your definitions are circular in the strong sense that they’re of the form ‘… a … = … a …’. But interesting and useful identities and equalities can re-use the term on both sides. Generally they then reduce to predications. For instance, “pain occurs when something experiences pain” is a pretty hideous attempt at a definition, but it doesn’t reduce to “pain is when pain” (which isn’t even a sentence); it reduces to “pain is an experience”. That’s potentially useful, but it would’ve been more useful if we hadn’t dressed it up as though it were an analysis.
All of this seems a bit beside the point, though. None of the definitions I cited re-used the same term, whereas all the examples you made up to criticize them do re-use the same term on both sides of the definition. If your goal is to draw an analogy that problematizes certain practices in mathematical logic, you should include at least some problem cases that look like the formulas I first posted.
What do you mean by “contain”? They didn’t make reference to existence twice (...) None of the definitions I cited re-used the same term
That’s probably our main point of contention, since I’d argue that they do. Not evident when doing shallow parsing on a very superficial level, but plainly there nonetheless.
Say I gave you this definition: “2 is defined as (the following in ROT13) ‘gur ahzrevp inyhr bs gur jbeq gjb’”, with the ROT13 part (for your convenience) spelling “the numeric value of the word two”. I’d say that such a definition still reused the term to be defined in the right part of the definition, wouldn’t you?
Your definitions by necessity reduce to ‘exists(a) =(def) there is an x such that exists(x) and (x=a)’
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:
Say the domain set contained only {blue, green}, so x could only match to blue or green. Then exists(a) would only return true for blue and green. Not enough!
Say the set allowed for x to match to anything which is conceivable, such as a flying spaghetti monster (or whatever). Then exists(flying spaghetti monster) would evaluate to ‘true’, since there would be such an x. Too much!
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.
Hiding in rather plain sight, don’t you think? Even the ROT13 offered more obscurity.
It lets logicians get away without appealing to an ‘exists’ predicate.
If only. I disagree that it does (because of the above).
What are some examples of new insights or practical goals you were hoping or expecting to achieve by defining ‘exists’?
Well, something ‘new’ to work with. Where ‘we’ could go from there would probably depend on the concepts the definition relates ‘exists’ to. As with so much else, no practical goal other than the usual mental onanism. In our particular exchange, mostly showing that the definition you gave cannot be useful.
Could you say more about what you mean by ‘different parts of your knowledge base’? Is there a heuristic for deciding when things are parts of the same knowledge base?
A definition must establish some relation of any kind to some other concept, or predicate. ‘Different part’ as in ‘not only the exact same concept which is to be explained’.
You are right with “pain is an experience” offering a connection to some other concept, and thus being potentially useful. However, the definition for exist we are discussing offers no such additional concept. You can define a set of things which share an attribute for anything (that set could be empty), that’s no new information regarding the thingie in question (unless you start listing examples), it does not constrain the concept space in any way.
FWIW, if someone said “pain: pain is an experience”, that would be quite a poor definition, but as you correctly pointed out, at least we would’ve learned something new.
A good litmus test may be “if you were tasked with explaining your concepts to some strange alien, could it potentially glean anything from your definition”? Pain is an experience: yes (new information). Exists(x): you can define a set for all x’s for which exists(x) is true: no (alien looks at you uncomprehendingly).
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’.)
If a definition is not meant to be explanatory, its usefulness in understanding that which is to be defined is limited.
Taking the two alternate formulations you offered, I can still hear the telltale “is” beating, from beneath the floor planks where you hid it:
The “∀” doesn’t refer to all e.g. logically constructible x, does it? Or to all computable x. For the definition to make sense, it needs to refer to all x that exist, otherwise we’d conclude that ‘exists(flying unicorns)’ is true. Still implicitly refers to that which is to be defined in its definition, rendering it circular.
What is EXT(=)? Some set of all existing things? If so, would that definition do any work for us? Pointing at my chair and asking “does this chair exist”, you’d say “well, if it’s a member of the set of all existing things, it exists”. Why, because all things in the set share the “exist” predicate. But what does it mean for them to have the “exist” predicate in the first place? To be part of the set of all existing things, of course. Round and round …
Not much different from saying “if it exists, it exists”. Well, yes. Now what?
Exactly.
That’s one option for explaining the domain of ∀. Another is to simply say that that the domain is the universe, or that it’s everything, or that it’s unrestricted. All of those can be expressed without speaking in terms of existence.
If you have no idea what those ideas mean, but understand ‘exists’, then, sure, maybe you’ll need to demand that all those ideas be unpacked in terms of existence. But what of it? If you do understand those terms but not ‘exists’, then interdefining them can be cognitively significant for you. Broadly speaking, the function of a definition is to relate a term that isn’t understood to a term that is. If you already understand both terms, then the definition won’t be useful to you; but that isn’t a criticism of the definition, if other people might not understand both terms as well as you do. It’s just a biographical note about your own level of linguistic/conceptual expertise.
It’s the extension of the identity predicate, a set of ordered pairs. Relational predicates of arity n can be treated as sets of n-tuples.
Do any work for who? What is it you want, exactly? If you’ve forgotten, the first thing I said to you was “I’m not saying it’s a very useful definition”. You don’t need to prove it’s circular in order to prove it’s useless, and if you did prove it’s circular (‘circular’ in what sense? is there any finite non-circular chain of definitions that define every term?) that very likely wouldn’t help demonstrate its uselessness. So what exactly are you trying to establish, and why?
Any domain which is not constrained to iterate/refer only to things which themselves exist would lead to wrong conclusions such as “flying unicorns exist”.
To show that the definition you referred to, in all its variants, isn’t useful. I did not forget that you didn’t claim it was useful, just that it was common, but I also noticed you did not explicitly agree that it was not useful. If you do agree on that, there is no need to further dwell on useless rephrasings.
I agree that since the body of human knowledge is limited, any definition must eventually contain circles of some size. However, not all circles are created equal: To be useful, a definition must refer to some different part of your knowledge base, just because without introducing new information, there is nothing which could be useful.
“2 is defined as something with the property of being 2” isn’t useful because there is nothing new introduced. “That which exists, exists” isn’t useful for the same reason. Because all the definitions you referred to still contain “exist”, the additional information (“things in a set”) is superfluously added, the “exist” on the right part of the definition still isn’t unpacked. Hence, no additional information is introduced, and the definition useless, being equivalent to “2 is defined as 2″.
“Pain is when something which is in the set of ‘being able to experience pain’ experiences pain” just reduces to “pain is when pain”, which must be useless since it contains no additional concepts.
If the additional “identity” aspects etcetera helped any in explaining the concept of “exist”, then the definition would not need to refer again to just the same “exist” which the “identity” supposedly helped explain.
If I’m not misunderstanding you, you’re advocating a view like Graham Priest does here, that our quantifiers should range over anything we can meaningfully talk about (if not wider?) until we restrict them further. I’m inclined to agree. We both dissent from the orthodox definition I posted above, then. You’ll need to dig up a Quinean if you want to hear counter-arguments.
Well, I’m sure it’s been useful to someone at some point. It lets logicians get away without appealing to an ‘exists’ predicate. Logicians are generally much more attached to ‘is identical to’ than to ‘exists’. Again, you’ll have to explain exactly what kind of use you want out of the ideal Definition of Existence so I can evaluate whether the above ones I tossed about are useful with respect to that goal. What are some examples of new insights or practical goals you were hoping or expecting to achieve by defining ‘exists’?
Could you say more about what you mean by ‘different parts of your knowledge base’? Is there a heuristic for deciding when things are parts of the same knowledge base?
Is “2 is defined as SS∅” useful? Or “2 is defined as {{},{{}}}”? Or “2 is defined as 1+1”? Are there any useful definitions of 2?
What do you mean by “contain”? They didn’t make reference to existence twice. You noted we could reverse the definitions or build a chain, but that’s true of any definitions. (If they weren’t dreadfully boring, we’d probably not call them definitions.)
Do you mean that they presupposed an understanding of existence, i.e., if you didn’t first understand existence then you couldn’t understand my definitions? Or do you mean that concepts are combinatorial, and the concepts I appealed to all have as components the concept ‘existence’?
Your definitions are circular in the strong sense that they’re of the form ‘… a … = … a …’. But interesting and useful identities and equalities can re-use the term on both sides. Generally they then reduce to predications. For instance, “pain occurs when something experiences pain” is a pretty hideous attempt at a definition, but it doesn’t reduce to “pain is when pain” (which isn’t even a sentence); it reduces to “pain is an experience”. That’s potentially useful, but it would’ve been more useful if we hadn’t dressed it up as though it were an analysis.
All of this seems a bit beside the point, though. None of the definitions I cited re-used the same term, whereas all the examples you made up to criticize them do re-use the same term on both sides of the definition. If your goal is to draw an analogy that problematizes certain practices in mathematical logic, you should include at least some problem cases that look like the formulas I first posted.
That’s probably our main point of contention, since I’d argue that they do. Not evident when doing shallow parsing on a very superficial level, but plainly there nonetheless.
Say I gave you this definition: “2 is defined as (the following in ROT13) ‘gur ahzrevp inyhr bs gur jbeq gjb’”, with the ROT13 part (for your convenience) spelling “the numeric value of the word two”. I’d say that such a definition still reused the term to be defined in the right part of the definition, wouldn’t you?
Your definitions by necessity reduce to ‘exists(a) =(def) there is an x such that exists(x) and (x=a)’
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:
Say the domain set contained only {blue, green}, so x could only match to blue or green. Then exists(a) would only return true for blue and green. Not enough!
Say the set allowed for x to match to anything which is conceivable, such as a flying spaghetti monster (or whatever). Then exists(flying spaghetti monster) would evaluate to ‘true’, since there would be such an x. Too much!
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.
Hiding in rather plain sight, don’t you think? Even the ROT13 offered more obscurity.
If only. I disagree that it does (because of the above).
Well, something ‘new’ to work with. Where ‘we’ could go from there would probably depend on the concepts the definition relates ‘exists’ to. As with so much else, no practical goal other than the usual mental onanism. In our particular exchange, mostly showing that the definition you gave cannot be useful.
A definition must establish some relation of any kind to some other concept, or predicate. ‘Different part’ as in ‘not only the exact same concept which is to be explained’.
You are right with “pain is an experience” offering a connection to some other concept, and thus being potentially useful. However, the definition for exist we are discussing offers no such additional concept. You can define a set of things which share an attribute for anything (that set could be empty), that’s no new information regarding the thingie in question (unless you start listing examples), it does not constrain the concept space in any way.
FWIW, if someone said “pain: pain is an experience”, that would be quite a poor definition, but as you correctly pointed out, at least we would’ve learned something new.
A good litmus test may be “if you were tasked with explaining your concepts to some strange alien, could it potentially glean anything from your definition”? Pain is an experience: yes (new information). Exists(x): you can define a set for all x’s for which exists(x) is true: no (alien looks at you uncomprehendingly).
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’.)
I’ve had this circular discussion with RobbBB for a couple of hours. Maybe you will have better luck.