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From: Frederick Williams on 30 Jun 2010 05:59 Nam Nguyen wrote: > A very close sibling of the current quantifier semantics of Ax and > Ex is to leave Ex semantically unchanged but alter Ax to mean > "For many x", instead "For all x". Would this be reasonable? There are _other_ quantifiers (as opposed to a reinterpretation of existing quantifiers thereby falsifying well-known principles as I indicated in my other post) such as "there exist uncountable many" and the like. -- I can't go on, I'll go on.
From: Ken Pledger on 30 Jun 2010 17:03 In article <ydCWn.2870$cO.588(a)newsfe09.iad>, Nam Nguyen <namducnguyen(a)shaw.ca> wrote: > William Elliot wrote: > > On Tue, 29 Jun 2010, Nam Nguyen wrote: > > .... > >> A very close sibling of the current quantifier semantics of Ax and > >> Ex is to leave Ex semantically unchanged but alter Ax to mean > >> "For many x", instead "For all x". Would this be reasonable? > >> > > No. What does many mean? How can you distinguish many x from > > some x, ie Ex? > > By syntactical inference: Ax -> Ex, but not Ex -> Ax. Semantically, > we could take "many" to mean "at least more than one x". (Though > I think the semantic of Ex might have to be altered, despite what > I previously claimed. But I'm not 100% sure yet if this Ex alteration > would be necessary in this case).... If your Ax meant semantically "for more than one x", then I think the equivalence Ex = ~A~x would force Ex to mean "for all x with at most one exception". That may not be what you're intuitively looking for. Still, your basic question is interesting. Ken Pledger.
From: Nam Nguyen on 1 Jul 2010 12:03 Ken Pledger wrote: > In article <ydCWn.2870$cO.588(a)newsfe09.iad>, > Nam Nguyen <namducnguyen(a)shaw.ca> wrote: > >> William Elliot wrote: >>> On Tue, 29 Jun 2010, Nam Nguyen wrote: >>> .... >>>> A very close sibling of the current quantifier semantics of Ax and >>>> Ex is to leave Ex semantically unchanged but alter Ax to mean >>>> "For many x", instead "For all x". Would this be reasonable? >>>> >>> No. What does many mean? How can you distinguish many x from >>> some x, ie Ex? >> By syntactical inference: Ax -> Ex, but not Ex -> Ax. Semantically, >> we could take "many" to mean "at least more than one x". (Though >> I think the semantic of Ex might have to be altered, despite what >> I previously claimed. But I'm not 100% sure yet if this Ex alteration >> would be necessary in this case).... > > > If your Ax meant semantically "for more than one x", then I think > the equivalence Ex = ~A~x would force Ex to mean "for all x with at > most one exception". That may not be what you're intuitively looking > for. I appreciate your feedback. It's indeed not so easy to find a pair of semantics that would rival the normal pair for Ax, Ex, whether or not it's in the realm of quantification, specification, or somewhere else. But while at it (seeking for different semantics for logical symbols), could we put ~ on the table as well, and in such case ~ might not necessarily mean "not" anymore (not 100% at least) and "for all x with at most one exception" would still cohesively pair with "for more than one x", whatever the semantically-altered ~ might now mean? After all, we shouldn't have Ex -> Ax, and "for all x with at most one exception" doesn't seem to imply "for more than one x", I'd think. Would you agree? > > Still, your basic question is interesting. Thanks for an encouragement. Though not being an alien from another galaxy, I'd like to how differently one might see a syntactical FOL "scripture". I'm just kidding of course on being from another place! On the other hand, since our FOL semantics are so quantification oriented, I just wonder if FOL reasoning would support descriptions of a portion of the physical universe with different kind of semantics?
From: Ken Pledger on 1 Jul 2010 17:15 In article <K%2Xn.13661$4B7.11308(a)newsfe16.iad>, Nam Nguyen <namducnguyen(a)shaw.ca> wrote: > .... > But while at it (seeking for different semantics for logical symbols), > could we put ~ on the table as well, and in such case ~ might not > necessarily mean "not" anymore (not 100% at least) and "for all x with at > most one exception" would still cohesively pair with "for more than one x", > whatever the semantically-altered ~ might now mean? > > After all, we shouldn't have Ex -> Ax, and "for all x with at most one > exception" doesn't seem to imply "for more than one x", I'd think. > Would you agree? .... A two-element model shows that. You might find it helpful to be as precise as possible about what you want. Tossing in ~ as an afterthought is fairly radical. :-) Ken Pledger.
From: Nam Nguyen on 1 Jul 2010 17:28
Nam Nguyen wrote: > Ken Pledger wrote: >> In article <ydCWn.2870$cO.588(a)newsfe09.iad>, >> Nam Nguyen <namducnguyen(a)shaw.ca> wrote: >> >>> William Elliot wrote: >>>> On Tue, 29 Jun 2010, Nam Nguyen wrote: >>>> .... >>>>> A very close sibling of the current quantifier semantics of Ax and >>>>> Ex is to leave Ex semantically unchanged but alter Ax to mean >>>>> "For many x", instead "For all x". Would this be reasonable? >>>>> >>>> No. What does many mean? How can you distinguish many x from >>>> some x, ie Ex? >>> By syntactical inference: Ax -> Ex, but not Ex -> Ax. Semantically, >>> we could take "many" to mean "at least more than one x". (Though >>> I think the semantic of Ex might have to be altered, despite what >>> I previously claimed. But I'm not 100% sure yet if this Ex alteration >>> would be necessary in this case).... >> >> >> If your Ax meant semantically "for more than one x", then I >> think the equivalence Ex = ~A~x would force Ex to mean "for all x >> with at most one exception". That may not be what you're intuitively >> looking for. > > I appreciate your feedback. It's indeed not so easy to find a pair of > semantics that would rival the normal pair for Ax, Ex, whether or not > it's in the realm of quantification, specification, or somewhere else. > > But while at it (seeking for different semantics for logical symbols), > could we put ~ on the table as well, and in such case ~ might not > necessarily mean "not" anymore (not 100% at least) and "for all x with at > most one exception" would still cohesively pair with "for more than one x", > whatever the semantically-altered ~ might now mean? > > After all, we shouldn't have Ex -> Ax, and "for all x with at most one > exception" doesn't seem to imply "for more than one x", I'd think. > Would you agree? Another alternative which I think would work is: - Leave the semantics of AxPx the same: "For all x, P(x)" - But ExPx would now mean: "For many x, P(x)", where "many" means "more than one". So AxPx -> ExPx still holds and we still can't infer AxPx from ExPx, and the equivalence "Ex = ~A~x " would still be meaningful. (It's still about quantification but it seems a good candidate for a _different_ semantics.) For example, the familiar formula in L(ZF): ExAy[~(y e x)] might now might mean urelements, instead of one single empty set. |