As a former Objectivist, I understand the point being made.
That said, I no longer agree… I now believe that Ayn Rand made an axiom-level mistake. Existence is not Identity. To assume that Existence is Identity is to assume that all things have concrete properties, which exist and can therefore be discovered. This is demonstrably false; at the fundamental level of reality, there is Uncertainty. Quantum-level effects inherent in existence preclude the possibility of absolute knowledge of all things; there are parts of reality which are actually unknowable.
Moreover, we as humans do not have absolute knowledge of things. Our knowledge is limited, as is the information we’re able to gather about reality. We don’t have the ability to gather all relevant information to be certain of anything, nor the luxury to postpone decision-making while we gather that information. We need to make decisions sooner then that, and we need to make them in the face of the knowledge that our knowledge will always be imperfect.
Accordingly, I find that a better axiom would be “Existence is Probability”. I’m not a good enough philosopher to fully extrapolate the consequences of that… but I do think if Ayn Rand had started with a root-level acknowledgement of fallibility, it would’ve helped to avoid a lot of the problems she wound up falling into later on.
Existence is frequently defined in terms of identity. ‘exists(a)’ ≝ ‘∃x(a=x)’
To assume that Existence is Identity is to assume that all things have concrete properties, which exist and can therefore be discovered. This is demonstrably false; at the fundamental level of reality, there is Uncertainty.
Only if you’re an Objective Collapse theorist of some stripe. If you accept anything in the vicinity of Many Worlds or Hidden Variables, then nature is not ultimately so anthropocentric; all of its properties are determinate, though those properties may not be exactly what you expect from everyday life.
Quantum-level effects inherent in existence preclude the possibility of absolute knowledge of all things; there are parts of reality which are actually unknowable.
If “there are” such parts, then they exist. The mistake here is not to associate existence with identity, but to associate existence or identity with discoverability; lots of things are real and out there and objective but are physically impossible for us to interact with. You’re succumbing to a bit of Rand’s wordplay: She leaps back and forth between the words ‘identity’ and ‘identification’, as though these were closely related concepts. That’s what allows her to associate existence with consciousness—through mere wordplay.
Accordingly, I find that a better axiom would be “Existence is Probability”.
But that axiom isn’t true. I like my axioms to be true. Probability is in the head, unlike existent things like teacups and cacti.
Existence is frequently defined in terms of identity. ‘exists(a)’ ≝ ‘∃x(a=x)’
Isn’t that just kicking the can down the road? What does it mean for an x to ∃, “there is an x such that …”, there we go with the “is”, with the “be” with the “exist”.
RobbBB, in my experience, tends to give pseudo-precise answers like that. It seems like a domain confusion. You are asking about observable reality, he talks about mathematical definitions.
I’m not a frequent poster here, and I don’t expect my recommendations carry much weight. But I have been reading this site for a few years, and offline I deal with LWish topics and discussions pretty regularly, especially with the more philosophical stuff.
All that said, I think RobbBB is one of the best posters LW has. Like top 10. He stands out for clarity, seriousness, and charity.
Also, I think you shouldn’t do that thing where you undermine some other poster while avoiding directly addressing them or their argument.
All that said, I think RobbBB is one of the best posters LW has. Like top 10. He stands out for clarity, seriousness, and charity.
It certainly has not been my impression. I found my discussion with him about instrumentalism, here and on IRC, extremely unproductive. Seems like a pattern with other philosophical types here. Maybe they don’t teach philosophers to listen, I don’t know. For comparison, TheOtherDave manages to carry a thoughtful, polite and insightful discussion even when he disagrees. More regulars here could learn rational discourse from him.
Or maybe I’m falling prey to the Bright Dilettante trap and the experts in the subject matter just don’t have the patience to explain things in a friendly and understandable fashion. I’m not sure how to tell.
Also, I think you shouldn’t do that thing where you undermine some other poster while avoiding directly addressing them or their argument.
I take back the “pseudo-” part. His answers were precise, but from a wrong domain.
Seems like a pattern with other philosophical types here. Maybe they don’t teach philosophers to listen, I don’t know. For comparison, TheOtherDave manages to carry a thoughtful, polite and insightful discussion even when he disagrees. More regulars here could learn rational discourse from him.
Agree on both counts. I’ll second your advocacy of a TheOtherDave as a posting style role model. In particular he conveys the impression that he is far better than the average lesswrong participant at understanding what people are saying to him. (Rather than the all to common practice of pattern matching a few keywords to the nearest possible stupid thing that can be refuted.)
I don’t know. Certainly there is some emphasis on charitable reading and steelmanning on this forum, but the results are mixed. Maybe it’s taught in psychology, nursing and other areas which require empathy.
This seems like something a rationalist course could profitably teach, especially if there are no alternative ways to learn it besides informal practice.
I’m a little unclear on what your criticism is. Is one of these right?
You’re being too precise, whereas I wanted to have an informal discussion in terms of our everyday intuitions. So definitions are counterproductive; a little unclarity in what we mean is actually helpful for this topic.
There are two kinds of existence, one that holds for Plato’s Realm Of Invisible Mathy Things and one that holds for The Physical World. Your definitions may be true of the Mathy Things, but they aren’t true of things like apples and bumblebees. So you’re committing a category error.
I wanted you to give me a really rich, interesting explanation of what ‘existence’ is, in more fundamental terms. But instead you just copy-pasted a bland uninformative Standard Mathematical Logician Answer from some old textbook. That makes me sad. Please be more interesting next time.
If your point was 1, I’ll want to hear more. If it was 3, then my apologies! If it was 2, then I’ll have to disagree until I hear some argument as to why I should believe in these invisible eternal number-like things that exist in their own unique number-like-thing-specific way. (And what it would mean to believe in them!)
Thank you, this framework helps. Definitely no to 1. Definitely yes to 2, with some corrections. Yes to some parts of 3.
Re 2. First, let me adopt bounded realism here, with physics (external reality or territory) + logic (human models of reality, or maps). Let me ignore the ultraviolet divergence of decompartmentalization (hence “bounded”), where Many Words, Tegmark IV and modal realism are considered “territory”. To this end, let me put the UV cutoff on logic at the Popper’s boundary: only experimentally falsifiable maps are worth considering. A map is “true” means that it is an accurate representation of the piece of territory it is intended to represent. I apologize in advance if I am inventing new terms for the standard philosophical concepts—feel free to point me to the standard terminology.
Again, “accurate map”, a.k.a. “true map” is a map that has been tested against the territory and found reliable enough to use as a guide for further travels, at least if one does not stray too far. Correspondingly, a piece of territory
is said to “exist” if it is described by an accurate map.
On the other hand, your “invisible mathy things” live in the world of maps. Some of them use the same term “true”, but in a different way: given a set of rules of how to form strings of symbols, true statements are well-formed finite strings. They also use the same term “exist”, but also in a different way: given a set of rules, every well-formed string is said to “exist”.
Now, I am not a mathematician, so this may not be entirely accurate, but the gist is that conflating “exist” as applied to the territory and “exist” as applied to maps is indeed a category error. When someone talks about existence of physical objects and you write out something containing the existential quantifier, you are talking about a different category: not reality, but a subset of maps related to mathematical logic.
I am not sure whether this answers your objection that
why I should believe in these invisible eternal number-like things that exist in their own unique number-like-thing-specific way. (And what it would mean to believe in them!)
but I hope it makes it clear why I find your replies unconvincing and generally not useful.
You’ve redefined ‘x exists’ to mean ‘x is described by a map that has been tested and so far has seemed reliable to us’, and ‘x is true’ correspondingly. One problem with this is that it’s historical: It commits us to saying ‘Newtonian physics used to be true, but these days it’s false (i.e., not completely reliable as a general theory)‘, and to saying ‘Phlogiston used to exist, but then it stopped existing because someone overturned phlogiston theory’. This is pretty strange.
Another problem is that it’s not clear what it takes to be ‘found reliable enough to use as a guide for further travels’. Surely there’s an important sense in which math is reliable in that sense, hence ‘true’ in the territory-ish sense you outlined above, not just in the map-ish sense. So perhaps we’ll need a more precise definition of territory-ish truth in order to clearly demonstrate why math isn’t in the territory, where the territory is defined by empirical adequacy.
I think your view, or one very close to yours, is actually a lot stronger (can be more easily defended, has broader implications) than your argument for it suggests. You can simply note that things like Abstract Numbers, being causally inert, couldn’t be responsible for the ‘unreasonable efficacy of mathematics’; so that efficacy can’t count as evidence for such Numbers. And nothing else is evidence for Numbers either. So we should conclude, on grounds of parsimony (perhaps fortified with anti-Tegmark’s-MUH arguments), that there are unlikely to be such Numbers. At that point, we can make the pragmatic, merely linguistic decision of saying that mathematicians are using ‘exists’ in a looser, more figurative sense.
Perhaps a few mathematicians are deluded into thinking that ‘exists’ means exactly the same thing in both contexts, but it is more charitable to interpret mathematics in general in the less ontologically committing way, because on the above arguments a platonistic mathematics would be little more than speculative theology. Basically, we end up with a formalist or fictionalist description of math, which I think is very plausible.
You see, we aren’t so different, you and I. Not once we bracket whether unexperienced cucumbers exist out there, anyway!
You’ve redefined ‘x exists’ to mean ‘x is described by a map that has been tested and so far has seemed reliable to us’, and ‘x is true’ correspondingly.
I disagree that this is a redefinition. You believe that elephants exists because you can go and see them, or talk to someone you trust who saw them, etc. You believe that live T-Rex (almost surely) does not exist because it went extinct some 60 odd million years ago. Both beliefs can be updated based on new information.
’Newtonian physics used to be true, but these days it’s false
That’s not at all what I am saying. Consider resisting your tendency to strawman. Newtonian physics is still true in its domain of applicability, it has never been true where it’s not been applicable, though people didn’t know this until 1905.
‘Phlogiston used to exist, but then it stopped existing because someone overturned phlogiston theory’
Again, a belief at the time was that it existed, a more accurate belief (map) superseded the old one and now we know that phlogiston never existed. Maps thought of as being reliable can be found wanting all the time, so the territory they describe is no longer believed to exist, not stopped existing. This is pretty uncontroversial, I would think. Science didn’t kill gnomes and fairies, and such. At least this is the experiment-bounded realist position, as far as I understand it.
You can simply note that things like Abstract Numbers, being causally inert, couldn’t be responsible for the ‘unreasonable efficacy of mathematics’; so that efficacy can’t count as evidence for such Numbers.
I can’t even parse that, sorry. Numbers don’t physically exist because they are ideas, and as such belong in the realm of logic, not physics. (Again, I’m wearing a realist hat here.) I don’t think parsimony is required here. It’s a postulate, not a conclusion.
Perhaps a few mathematicians are deluded into thinking that ‘exists’ means exactly the same thing in both contexts, but it is more charitable to interpret mathematics in general in the less ontologically committing way
Then I don’t understand why you reply to questions of physical existence with some mathematical expressions...
You see, we aren’t so different, you and I. Not once we bracket whether unexperienced cucumbers exist out there, anyway!
I disagree that this is a redefinition. You believe that elephants exists because you can go and see them, or talk to someone you trust who saw them, etc.
Sure, but ‘you believe in X because of Y’ does not as a rule let us conclude ‘X = Y’. I believe in elephants because of how they’ve causally impacted my experience, but I don’t believe that elephants are experiences of mine, or logical constructs out of my experiences and predictions. I believe elephants are animals.
Indeed, a large part of the reason I believe in elephants is that I think elephants would still exist even had you severed the causal links between me and them and I’d never learned about them. The territory doesn’t go away when you stop knowing about it, or even when you stop being able to ever know about it. If you shot an elephant in a rocket out of the observable universe, it wouldn’t stop existing, and I wouldn’t believe it had blinked out of existence or that questions regarding its existence were meaningless, once its future state ceased to be knowable to me.
Elephants don’t live in my map. But they also don’t live in my map-territory relation. Nor do they live in a function from observational data to hypotheses-that-help-us-build-rockets-and-iPhones-and-vaccines. They simply and purely live in the territory.
That’s not at all what I am saying. Consider resisting your tendency to strawman.
I’m not trying to strawman you, I’m suggesting a problem for how you stated you view so that you can reformulate your view in a way that I’ll better understand. I’m sorry if I wasn’t clear about that!
Newtonian physics is still true in its domain of applicability, it has never been true where it’s not been applicable, though people didn’t know this until 1905.
Right. But you said “‘accurate map’, a.k.a. ‘true map’ is a map that has been tested against the territory and found reliable enough to use as a guide for further travels”. My objection is that wide-applicability Newtonian Physics used to meet your criterion for truth (i.e., for a long time it passed all experimental tests and remained reliable for further research), but eventually stopped meeting it. Which suggests that it was true until it failed a test, or until it ceased to be a useful guide to further research; after that it became false. If you didn’t mean to suggest that, then I’m not sure I understand “map that has been tested against the territory and found reliable enough to use as a guide for further travels” anymore, which means I don’t know what you mean by “truth” and “accuracy” at this point.
Perhaps instead of defining “true” as “has been tested against the territory and found reliable enough to use as a guide for further travels”, what you meant to say was “has been tested against the territory and will always be found reliable enough to use as a guide for further travels”? That way various theories that had passed all tests at the time but are going to eventually fail them won’t count as ever having been ‘true’.
Numbers don’t physically exist because they are ideas, and as such belong in the realm of logic, not physics. (Again, I’m wearing a realist hat here.) I don’t think parsimony is required here. It’s a postulate, not a conclusion.
Postulates like ‘1 is nonphysical’, ‘2 is nonphysical’, etc. aren’t needed here; that would make our axiom set extraordinarily cluttered! The very idea that ‘ideas’ aren’t a part of the physical world is in no way obvious at the outset, much less axiomatic. There was a time when lightning seemed supernatural, a violation of the natural order; conceivably, we could have discovered that there isn’t really lightning (it’s some sort of illusion), but instead we discovered that it reduced to a physical process. Mental contents are like lightning. There may be another version of ‘idea’ or ‘thought’ or ‘abstraction’ that we can treat as a formalist symbol game or a useful fiction, but we still have to also either reduce or eliminate the natural-phenomenon-concept of abstract objects if we wish to advance the Great Reductionist Project.
It sounds like you want to eliminate them, and indeed stop even talking about them because they’re silly. I can get behind that, but only if we’re careful not to forget that not all mathematicians (etc.) agree on this point, and don’t equivocate between the two notions of ‘abstract’ (formal/fictive vs. spooky and metaphysical and Tegmarkish).
Then I don’t understand why you reply to questions of physical existence with some mathematical expressions...
Only because the apples are behaving like numbers whether you believe in numbers or not. You might not think our world does resemble the formalism in this respect, but that’s not obvious to everyone before we’ve talked the question over. A logic can be treated as a regimentation of natural language, or as an independent mathematical structure that happens to structurally resemble a lot of our informal reasoning and natural-language rules. Either way, information we get from logical analysis and deduction can tell us plenty about the physical world.
Re 2. First, let me adopt bounded realism here, with physics (external reality or territory) + logic (human models of reality, or maps). Let me ignore the ultraviolet divergence of decompartmentalization (hence “bounded”), where Many Words, Tegmark IV and modal realism are considered “territory”. To this end, let me put the UV cutoff on logic at the Popper’s boundary: only experimentally falsifiable maps are worth considering. A map is “true” means that it is an accurate representation of the piece of territory it is intended to represent. I apologize in advance if I am inventing new terms for the standard philosophical concepts—feel free to point me to the standard terminology.
I suspect you have, in fact, reinvented something. For reference, how does this “bounded realism” evaluate this statement:
On August 1st 2008 at midnight Greenwich time, a one-foot sphere of chocolate cake spontaneously formed in the center of the Sun; and then, in the natural course of events, this Boltzmann Cake almost instantly dissolved.
It makes no predictions; this is, in a sense, epiphenomenal cake—I know of no test we could perform that would distinguish between a world where this statement is false and one where it is true. Certainly tracking it provides us with no predictive power.
Yet is it somehow invalid? Is it gibberish? Can it be rejected a priori? Is there any sense in which it might be true? Is there any sense in which it might be false?
Sorry if I’m misinterpreting you here; I doubt this has much effect on your overall point.
How about this: Mathematicians have a conception of existence which is good enough for doing mathematics, but isn’t necessary correct. When you give a mathematical definition of existence, you are implicitly assuming a certain mathematical framework without justifying it. I think you would consider this criticism to be a variant of #2.
In particular, I also think about things mathematically, but when I do so, I don’t use first-order logic, but rather intuitionistic type theory. Can you give a definition for existence which would satisfy me?
I’m a mathematical fictionalist, so I’m happy to grant that there’s a good sense in which mathematical discourse isn’t strictly true, and doesn’t need to be.
Are you asking for a definition of an intuitionistic ‘exists’ predicate, or for the intuitionistic existential quantifier?
First, if you accept that mathematical constructs are fictional, why do you consider it valid to define a concept in terms of them? Second, I admit I wasn’t clear on this issue: The salient part of intuitionistic type theory isn’t intuitionism, but rather that it is a structural theory. This means that statements of the form “exists x, P(x)” are not well defined, but rather only statements of the form “exists x in A, P(x)” can be made.
I’m not saying it’s a very useful definition, just noting that it’s very standard. If we’re going to reject something it should be because we thought about it for a while and it still seemed wrong (and, ideally, we could understand why others think otherwise). We shouldn’t just reject it because it sounds weird and a Paradigmatically Wrong Writer is associated with it.
I agree with you that there’s something circular about this definition, if it’s meant to be explanatory. (Is it?) But I’m not sure that circularity is quite that easy to demonstrate. ∃ could be defined in terms of ∀, for instance, or in terms of set membership. Then we get:
‘exists(a)’ ≝ ‘¬∀x¬(a=x)’
or
‘exists(a)’ ≝ ‘a∈EXT(=)’
You could object that ∈ is similarly question-begging because it can be spoken as ‘is an element of’, but here we’re dealing we’re dealing with a more predicational ‘is’, one we could easily replace with a verb.
I suspect the above definitions look meaningful to those who have studied philosophy and mathematical logic because they have internalised the mathematical machinery behind ‘∃’. But a proper definition wouldn’t simply refer you to another symbol. Rather, you would describe the mathematics involved directly.
For example, you can define an operator that takes a possible world and a predicate, and tells you if there’s anything matching that predicate in the world, in the obvious way. In Newtonian possible worlds, the first argument would presumably be a set of particles and their positions, or something along those lines.
This would be the logical existence operator, ‘∃’. But, it’s not so useful since we don’t normally talk about existence in rigorously defined possible worlds, we just say something exists or it doesn’t — in the real world. So we invent plain “exists”, which doesn’t take a second argument, but tells you whether there’s anything that matches “in reality”. Which doesn’t really mean anything apart from:
Where P(w) is your probability distribution over possible worlds, which is itself in turn connected to your past observations, etc.
Anyway, the point is that the above is how “existence” is actually used (things become more likely to exist when you receive evidence more likely to be observed in worlds containing those things). So “existence” is simply a proposition/function of a predicate whose probability marginalises like that over your distribution over possible worlds, and never mind trying to define exactly when it’s true or false, since you don’t need to. Or something like that.
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’.)
As a former Objectivist, I understand the point being made.
That said, I no longer agree… I now believe that Ayn Rand made an axiom-level mistake. Existence is not Identity. To assume that Existence is Identity is to assume that all things have concrete properties, which exist and can therefore be discovered. This is demonstrably false; at the fundamental level of reality, there is Uncertainty. Quantum-level effects inherent in existence preclude the possibility of absolute knowledge of all things; there are parts of reality which are actually unknowable.
Moreover, we as humans do not have absolute knowledge of things. Our knowledge is limited, as is the information we’re able to gather about reality. We don’t have the ability to gather all relevant information to be certain of anything, nor the luxury to postpone decision-making while we gather that information. We need to make decisions sooner then that, and we need to make them in the face of the knowledge that our knowledge will always be imperfect.
Accordingly, I find that a better axiom would be “Existence is Probability”. I’m not a good enough philosopher to fully extrapolate the consequences of that… but I do think if Ayn Rand had started with a root-level acknowledgement of fallibility, it would’ve helped to avoid a lot of the problems she wound up falling into later on.
Also, welcome, new person!
Existence is frequently defined in terms of identity. ‘exists(a)’ ≝ ‘∃x(a=x)’
Only if you’re an Objective Collapse theorist of some stripe. If you accept anything in the vicinity of Many Worlds or Hidden Variables, then nature is not ultimately so anthropocentric; all of its properties are determinate, though those properties may not be exactly what you expect from everyday life.
If “there are” such parts, then they exist. The mistake here is not to associate existence with identity, but to associate existence or identity with discoverability; lots of things are real and out there and objective but are physically impossible for us to interact with. You’re succumbing to a bit of Rand’s wordplay: She leaps back and forth between the words ‘identity’ and ‘identification’, as though these were closely related concepts. That’s what allows her to associate existence with consciousness—through mere wordplay.
But that axiom isn’t true. I like my axioms to be true. Probability is in the head, unlike existent things like teacups and cacti.
Isn’t that just kicking the can down the road? What does it mean for an x to ∃, “there is an x such that …”, there we go with the “is”, with the “be” with the “exist”.
RobbBB, in my experience, tends to give pseudo-precise answers like that. It seems like a domain confusion. You are asking about observable reality, he talks about mathematical definitions.
I’m not a frequent poster here, and I don’t expect my recommendations carry much weight. But I have been reading this site for a few years, and offline I deal with LWish topics and discussions pretty regularly, especially with the more philosophical stuff.
All that said, I think RobbBB is one of the best posters LW has. Like top 10. He stands out for clarity, seriousness, and charity.
Also, I think you shouldn’t do that thing where you undermine some other poster while avoiding directly addressing them or their argument.
It certainly has not been my impression. I found my discussion with him about instrumentalism, here and on IRC, extremely unproductive. Seems like a pattern with other philosophical types here. Maybe they don’t teach philosophers to listen, I don’t know. For comparison, TheOtherDave manages to carry a thoughtful, polite and insightful discussion even when he disagrees. More regulars here could learn rational discourse from him.
Or maybe I’m falling prey to the Bright Dilettante trap and the experts in the subject matter just don’t have the patience to explain things in a friendly and understandable fashion. I’m not sure how to tell.
I take back the “pseudo-” part. His answers were precise, but from a wrong domain.
Agree on both counts. I’ll second your advocacy of a TheOtherDave as a posting style role model. In particular he conveys the impression that he is far better than the average lesswrong participant at understanding what people are saying to him. (Rather than the all to common practice of pattern matching a few keywords to the nearest possible stupid thing that can be refuted.)
I can tell you from experience that ‘they’ don’t. Do you know who does teach this?
I don’t know. Certainly there is some emphasis on charitable reading and steelmanning on this forum, but the results are mixed. Maybe it’s taught in psychology, nursing and other areas which require empathy.
This seems like something a rationalist course could profitably teach, especially if there are no alternative ways to learn it besides informal practice.
I’m a little unclear on what your criticism is. Is one of these right?
You’re being too precise, whereas I wanted to have an informal discussion in terms of our everyday intuitions. So definitions are counterproductive; a little unclarity in what we mean is actually helpful for this topic.
There are two kinds of existence, one that holds for Plato’s Realm Of Invisible Mathy Things and one that holds for The Physical World. Your definitions may be true of the Mathy Things, but they aren’t true of things like apples and bumblebees. So you’re committing a category error.
I wanted you to give me a really rich, interesting explanation of what ‘existence’ is, in more fundamental terms. But instead you just copy-pasted a bland uninformative Standard Mathematical Logician Answer from some old textbook. That makes me sad. Please be more interesting next time.
If your point was 1, I’ll want to hear more. If it was 3, then my apologies! If it was 2, then I’ll have to disagree until I hear some argument as to why I should believe in these invisible eternal number-like things that exist in their own unique number-like-thing-specific way. (And what it would mean to believe in them!)
Thank you, this framework helps. Definitely no to 1. Definitely yes to 2, with some corrections. Yes to some parts of 3.
Re 2. First, let me adopt bounded realism here, with physics (external reality or territory) + logic (human models of reality, or maps). Let me ignore the ultraviolet divergence of decompartmentalization (hence “bounded”), where Many Words, Tegmark IV and modal realism are considered “territory”. To this end, let me put the UV cutoff on logic at the Popper’s boundary: only experimentally falsifiable maps are worth considering. A map is “true” means that it is an accurate representation of the piece of territory it is intended to represent. I apologize in advance if I am inventing new terms for the standard philosophical concepts—feel free to point me to the standard terminology.
Again, “accurate map”, a.k.a. “true map” is a map that has been tested against the territory and found reliable enough to use as a guide for further travels, at least if one does not stray too far. Correspondingly, a piece of territory is said to “exist” if it is described by an accurate map.
On the other hand, your “invisible mathy things” live in the world of maps. Some of them use the same term “true”, but in a different way: given a set of rules of how to form strings of symbols, true statements are well-formed finite strings. They also use the same term “exist”, but also in a different way: given a set of rules, every well-formed string is said to “exist”.
Now, I am not a mathematician, so this may not be entirely accurate, but the gist is that conflating “exist” as applied to the territory and “exist” as applied to maps is indeed a category error. When someone talks about existence of physical objects and you write out something containing the existential quantifier, you are talking about a different category: not reality, but a subset of maps related to mathematical logic.
I am not sure whether this answers your objection that
but I hope it makes it clear why I find your replies unconvincing and generally not useful.
You’ve redefined ‘x exists’ to mean ‘x is described by a map that has been tested and so far has seemed reliable to us’, and ‘x is true’ correspondingly. One problem with this is that it’s historical: It commits us to saying ‘Newtonian physics used to be true, but these days it’s false (i.e., not completely reliable as a general theory)‘, and to saying ‘Phlogiston used to exist, but then it stopped existing because someone overturned phlogiston theory’. This is pretty strange.
Another problem is that it’s not clear what it takes to be ‘found reliable enough to use as a guide for further travels’. Surely there’s an important sense in which math is reliable in that sense, hence ‘true’ in the territory-ish sense you outlined above, not just in the map-ish sense. So perhaps we’ll need a more precise definition of territory-ish truth in order to clearly demonstrate why math isn’t in the territory, where the territory is defined by empirical adequacy.
I think your view, or one very close to yours, is actually a lot stronger (can be more easily defended, has broader implications) than your argument for it suggests. You can simply note that things like Abstract Numbers, being causally inert, couldn’t be responsible for the ‘unreasonable efficacy of mathematics’; so that efficacy can’t count as evidence for such Numbers. And nothing else is evidence for Numbers either. So we should conclude, on grounds of parsimony (perhaps fortified with anti-Tegmark’s-MUH arguments), that there are unlikely to be such Numbers. At that point, we can make the pragmatic, merely linguistic decision of saying that mathematicians are using ‘exists’ in a looser, more figurative sense.
Perhaps a few mathematicians are deluded into thinking that ‘exists’ means exactly the same thing in both contexts, but it is more charitable to interpret mathematics in general in the less ontologically committing way, because on the above arguments a platonistic mathematics would be little more than speculative theology. Basically, we end up with a formalist or fictionalist description of math, which I think is very plausible.
You see, we aren’t so different, you and I. Not once we bracket whether unexperienced cucumbers exist out there, anyway!
I disagree that this is a redefinition. You believe that elephants exists because you can go and see them, or talk to someone you trust who saw them, etc. You believe that live T-Rex (almost surely) does not exist because it went extinct some 60 odd million years ago. Both beliefs can be updated based on new information.
That’s not at all what I am saying. Consider resisting your tendency to strawman. Newtonian physics is still true in its domain of applicability, it has never been true where it’s not been applicable, though people didn’t know this until 1905.
Again, a belief at the time was that it existed, a more accurate belief (map) superseded the old one and now we know that phlogiston never existed. Maps thought of as being reliable can be found wanting all the time, so the territory they describe is no longer believed to exist, not stopped existing. This is pretty uncontroversial, I would think. Science didn’t kill gnomes and fairies, and such. At least this is the experiment-bounded realist position, as far as I understand it.
I can’t even parse that, sorry. Numbers don’t physically exist because they are ideas, and as such belong in the realm of logic, not physics. (Again, I’m wearing a realist hat here.) I don’t think parsimony is required here. It’s a postulate, not a conclusion.
Then I don’t understand why you reply to questions of physical existence with some mathematical expressions...
I’m not nearly as optimistic.
Sure, but ‘you believe in X because of Y’ does not as a rule let us conclude ‘X = Y’. I believe in elephants because of how they’ve causally impacted my experience, but I don’t believe that elephants are experiences of mine, or logical constructs out of my experiences and predictions. I believe elephants are animals.
Indeed, a large part of the reason I believe in elephants is that I think elephants would still exist even had you severed the causal links between me and them and I’d never learned about them. The territory doesn’t go away when you stop knowing about it, or even when you stop being able to ever know about it. If you shot an elephant in a rocket out of the observable universe, it wouldn’t stop existing, and I wouldn’t believe it had blinked out of existence or that questions regarding its existence were meaningless, once its future state ceased to be knowable to me.
Elephants don’t live in my map. But they also don’t live in my map-territory relation. Nor do they live in a function from observational data to hypotheses-that-help-us-build-rockets-and-iPhones-and-vaccines. They simply and purely live in the territory.
I’m not trying to strawman you, I’m suggesting a problem for how you stated you view so that you can reformulate your view in a way that I’ll better understand. I’m sorry if I wasn’t clear about that!
Right. But you said “‘accurate map’, a.k.a. ‘true map’ is a map that has been tested against the territory and found reliable enough to use as a guide for further travels”. My objection is that wide-applicability Newtonian Physics used to meet your criterion for truth (i.e., for a long time it passed all experimental tests and remained reliable for further research), but eventually stopped meeting it. Which suggests that it was true until it failed a test, or until it ceased to be a useful guide to further research; after that it became false. If you didn’t mean to suggest that, then I’m not sure I understand “map that has been tested against the territory and found reliable enough to use as a guide for further travels” anymore, which means I don’t know what you mean by “truth” and “accuracy” at this point.
Perhaps instead of defining “true” as “has been tested against the territory and found reliable enough to use as a guide for further travels”, what you meant to say was “has been tested against the territory and will always be found reliable enough to use as a guide for further travels”? That way various theories that had passed all tests at the time but are going to eventually fail them won’t count as ever having been ‘true’.
Postulates like ‘1 is nonphysical’, ‘2 is nonphysical’, etc. aren’t needed here; that would make our axiom set extraordinarily cluttered! The very idea that ‘ideas’ aren’t a part of the physical world is in no way obvious at the outset, much less axiomatic. There was a time when lightning seemed supernatural, a violation of the natural order; conceivably, we could have discovered that there isn’t really lightning (it’s some sort of illusion), but instead we discovered that it reduced to a physical process. Mental contents are like lightning. There may be another version of ‘idea’ or ‘thought’ or ‘abstraction’ that we can treat as a formalist symbol game or a useful fiction, but we still have to also either reduce or eliminate the natural-phenomenon-concept of abstract objects if we wish to advance the Great Reductionist Project.
It sounds like you want to eliminate them, and indeed stop even talking about them because they’re silly. I can get behind that, but only if we’re careful not to forget that not all mathematicians (etc.) agree on this point, and don’t equivocate between the two notions of ‘abstract’ (formal/fictive vs. spooky and metaphysical and Tegmarkish).
Only because the apples are behaving like numbers whether you believe in numbers or not. You might not think our world does resemble the formalism in this respect, but that’s not obvious to everyone before we’ve talked the question over. A logic can be treated as a regimentation of natural language, or as an independent mathematical structure that happens to structurally resemble a lot of our informal reasoning and natural-language rules. Either way, information we get from logical analysis and deduction can tell us plenty about the physical world.
I suspect you have, in fact, reinvented something. For reference, how does this “bounded realism” evaluate this statement:
It makes no predictions; this is, in a sense, epiphenomenal cake—I know of no test we could perform that would distinguish between a world where this statement is false and one where it is true. Certainly tracking it provides us with no predictive power.
Yet is it somehow invalid? Is it gibberish? Can it be rejected a priori? Is there any sense in which it might be true? Is there any sense in which it might be false?
Sorry if I’m misinterpreting you here; I doubt this has much effect on your overall point.
How about this: Mathematicians have a conception of existence which is good enough for doing mathematics, but isn’t necessary correct. When you give a mathematical definition of existence, you are implicitly assuming a certain mathematical framework without justifying it. I think you would consider this criticism to be a variant of #2.
In particular, I also think about things mathematically, but when I do so, I don’t use first-order logic, but rather intuitionistic type theory. Can you give a definition for existence which would satisfy me?
I’m a mathematical fictionalist, so I’m happy to grant that there’s a good sense in which mathematical discourse isn’t strictly true, and doesn’t need to be.
Are you asking for a definition of an intuitionistic ‘exists’ predicate, or for the intuitionistic existential quantifier?
(Note: I added a link in my previous comment)
First, if you accept that mathematical constructs are fictional, why do you consider it valid to define a concept in terms of them? Second, I admit I wasn’t clear on this issue: The salient part of intuitionistic type theory isn’t intuitionism, but rather that it is a structural theory. This means that statements of the form “exists x, P(x)” are not well defined, but rather only statements of the form “exists x in A, P(x)” can be made.
I should probably let Rob answer for himself, but he did say that existence is frequently defined in terms of identity, not by identity.
I’m not saying it’s a very useful definition, just noting that it’s very standard. If we’re going to reject something it should be because we thought about it for a while and it still seemed wrong (and, ideally, we could understand why others think otherwise). We shouldn’t just reject it because it sounds weird and a Paradigmatically Wrong Writer is associated with it.
I agree with you that there’s something circular about this definition, if it’s meant to be explanatory. (Is it?) But I’m not sure that circularity is quite that easy to demonstrate. ∃ could be defined in terms of ∀, for instance, or in terms of set membership. Then we get:
‘exists(a)’ ≝ ‘¬∀x¬(a=x)’
or
‘exists(a)’ ≝ ‘a∈EXT(=)’
You could object that ∈ is similarly question-begging because it can be spoken as ‘is an element of’, but here we’re dealing we’re dealing with a more predicational ‘is’, one we could easily replace with a verb.
I suspect the above definitions look meaningful to those who have studied philosophy and mathematical logic because they have internalised the mathematical machinery behind ‘∃’. But a proper definition wouldn’t simply refer you to another symbol. Rather, you would describe the mathematics involved directly.
For example, you can define an operator that takes a possible world and a predicate, and tells you if there’s anything matching that predicate in the world, in the obvious way. In Newtonian possible worlds, the first argument would presumably be a set of particles and their positions, or something along those lines.
This would be the logical existence operator, ‘∃’. But, it’s not so useful since we don’t normally talk about existence in rigorously defined possible worlds, we just say something exists or it doesn’t — in the real world. So we invent plain “exists”, which doesn’t take a second argument, but tells you whether there’s anything that matches “in reality”. Which doesn’t really mean anything apart from:
or in a more suggestive format
Where
P(w)is your probability distribution over possible worlds, which is itself in turn connected to your past observations, etc.Anyway, the point is that the above is how “existence” is actually used (things become more likely to exist when you receive evidence more likely to be observed in worlds containing those things). So “existence” is simply a proposition/function of a predicate whose probability marginalises like that over your distribution over possible worlds, and never mind trying to define exactly when it’s true or false, since you don’t need to. Or something like that.
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.