Re: Blank nodes, "leaning", and the LEM

On Mar 23, 2011, at 6:39 AM, Gregg Reynolds wrote:

> I think the definition of leaning might be a little problematic so I could use a reality check.
> RDF Semantics defines (sort of) blank nodes as existentially quantified variables.  Ok.
> If I understand "leaning" correctly  (aside: something wrong with "normalization"?) it works something like the following.  Graph [1] below is not lean; graph [2] is the "lean" version of graph [1]:
> [1]  <ex:Pedro ex:owns _:x>,  <ex:Pedro ex:owns _:y>
> [2] <ex:Pedro ex:owns _:x>
> Informally, leaning is premised on the idea that both clauses in [1] "say the same thing", namely that Pedro owns something.  So one of the clauses can be removed.  (BTW, the example is inspired by the famous 'donkey sentence' "Every farmer who owns a donkey beats it".)

OK so far.
> I don't think this works logically.  

It does work logically.

> Treating the blank nodes as existentially quantified variables and translating into logic (using email markup from Z, where %E% means "There exists"), [3] is equivalent to [1]:
> [3]   %E% x, y @ Owns(Pedro, x) /\ Owns(Pedro y)
> Leaning substitutes x for y in [3].  (Alternatively: it substitutes z for both) and then removes the redundant clause.  But doesn't that violate the rules of substitution?

No, it doesn't. 

>  Both x and y are bound in [3]; substituting x for y or z for both changes the meaning of [3].

No, it doesn't. [3] is logically equivalent to 

%E% x @ Owns(Pedro, x)

> The problem I see is that even if the clauses in [1] individually mean "Pedro owns something", reading them as redundant means that conjunction must be vacuous.  

Which in this case it is, yes. A & A is redundant: it is logically equivalent to A.

> But I don't see how that can be right; logical /\ is meaningful.  So the clauses in [1] should be read "Pedro owns some x" and "Pedro owns some y"; substituting "some thing" for both changes the meaning.  It amounts to an unmotivated abstraction.

NO, it doesn't. YOu need to brush up on your logic. 
> To put it another way, the symbols 'x' and 'y' in [1] are not purely extensional.  Being distinct (bound) symbols, they have different intensions and are not interchangeable.

Exactly wrong. They are extensional, and they do not mean anything different, and they are interchangeable. 
> That's the syntactic view, but the semantic view is equally problematic.  The problem is that existentially quantified  sentences  characterize populations, not individuals; so existentially quantified variables do not denote individuals.

I think I see what you are intending to mean, but your way of phrasing it is misleading. 

>  Sentence [2] says that *some* individual in the population under consideration is owned by Pedro; it does not say that _:x denotes such an individual.  So [1] could characterize a population containing one thing that Pedro owns, or it could characterize a population containing two or more such individuals:  either _:x = _:y or _:x != _:y. We have no way of knowing which.  

True; and it does not matter, since the simple existential is true in both of them.

>   The Law of the Excluded Middle rears its ugly head.  


> If [1] were a *description* of a population we could survey it and decide which case holds; but I don't see how RDF can be treated as anything but a constructive language, meaning that [1] must be treated not as description but as prescription.  The problem then is that we don't know which population to construct.  Unless we outlaw the LEM, in which case  we construct a population with exactly two individuals owned by Pedro, and leaning is invalid.  At least I think that's how it would work.  My understanding is that outlawing the LEM would require choosing _:x != _:y in [3], but I'm not entirely confident that this is correct.  I guess it relies on the implicit assumption that  P != Q in P /\ Q.  Certainly one cannot *infer* P = Q from the conjunction.  Maybe the real problem is that data description languages and logical calculi are just different beasts.
> To put it yet another way, leaning seems to rely on the truth value of the clauses in [1] rather than their meanings.

Yes, exactly. That is what logic is all about, truth-values. 

>  But by that reasoning we could discard all true clauses except one in any conjunction.  Truth would be preserved,

No, it would not. 

> but meaning would be mangled.
> Leaning amounts to the selection of _:x = _:y over _:x != _:y.  I guess you could consider that one way of dealing with the LEM, but if I'm not mistaken it amounts to a redefinition of logical conjunction, which seems a little dubious.  Or maybe not; but it sure seems counter-intuitive.  It also seems incompatible with open-world semantics, which embraces "we don't know".
> Does that make sense?  Am I missing something?

What you are missing, basically, is how to think logically. It is much simpler than your way of thinking, but it requires a little mental discipline. Entailment is defined very simply: A entails B means that whenever A is true, B must also be true. IT refers only to *truth* of sentences. OK, consider the two sentences

A:  %E% x @ P(x)
B: %E% z, y @ P(z) & P(y)

Suppose A is true. Then there is something X such that P is true of X. Is the second sentence true under these circumstances? Yes, because that X can be the value for both z and y, and it makes both conjuncts true, and so the conjunction is true. Now suppose B is true: is A true? Obviously yes. Ergo, A and B each entail the other. Ergo, they are logically equivalent. 


> Thanks,
> Gregg

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Received on Wednesday, 23 March 2011 13:39:02 UTC