Can there be a different semantic for FOL formulas?
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From: William Elliot on 30 Jun 2010 02:49 On Tue, 29 Jun 2010, Nam Nguyen wrote: > Nam Nguyen wrote: >> Nam Nguyen wrote: >>> On and off, I've wondered if the entire FOL syntactical >>> proofs would be able to support different semantic >>> for "A" and "E": different than the typical quantifier >>> semantics "For all" and "There exists"? >>> >>> For example, if we let Ax[P(x)] and Ex[P(x)] to be specifiers >>> (instead of quantifiers) to mean "That x is P", "This x is P", >>> respectively, how far can we reason and still make sense? >>> That's violates syntactics, as bounded x is being interpeted as free x. >> The motivation for these questions is suppose there could be >> different semantics (for A and E), what would the role of >> G(PA) play as far as the _syntactical_ incompleteness of PA >> is concerned? > Nono. > 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? In addition, we can have Ax.~Px and Ax.Px which is contradiction. For example the model N with Pn <-> n even. ----
From: Nam Nguyen on 30 Jun 2010 03:18 William Elliot wrote: > On Tue, 29 Jun 2010, Nam Nguyen wrote: > >> Nam Nguyen wrote: >>> Nam Nguyen wrote: >>>> On and off, I've wondered if the entire FOL syntactical >>>> proofs would be able to support different semantic >>>> for "A" and "E": different than the typical quantifier >>>> semantics "For all" and "There exists"? >>>> >>>> For example, if we let Ax[P(x)] and Ex[P(x)] to be specifiers >>>> (instead of quantifiers) to mean "That x is P", "This x is P", >>>> respectively, how far can we reason and still make sense? >>>> > That's violates syntactics, as bounded x is being interpeted as free x. Huh? Isn't x bound to "That" or "This"? > >>> The motivation for these questions is suppose there could be >>> different semantics (for A and E), what would the role of >>> G(PA) play as far as the _syntactical_ incompleteness of PA >>> is concerned? >> > Nono. That's actually a question seeking for some reflection, not an assertion! > >> 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). > In addition, we can have Ax.~Px and Ax.Px > which is contradiction. For example the model N with Pn <-> n even. Could we still arrive at these conclusions, given the above semantics? (Right off the first glance, I don't see how).
From: William Elliot on 30 Jun 2010 03:43 On Wed, 30 Jun 2010, Nam Nguyen wrote: > William Elliot wrote: >>>> On and off, I've wondered if the entire FOL syntactical >>>> proofs would be able to support different semantic >>>> for "A" and "E": different than the typical quantifier >>>> semantics "For all" and "There exists"? >>>> >>>> For example, if we let Ax[P(x)] and Ex[P(x)] to be specifiers >>>> (instead of quantifiers) to mean "That x is P", "This x is P", >>>> respectively, how far can we reason and still make sense? >>>> >>>> Even if the "Specifier" semantics would fail, would there be >>>> others? >>> >>> The motivation for these questions is suppose there could be >>> different semantics (for A and E), what would the role of >>> G(PA) play as far as the _syntactical_ incompleteness of PA >>> is concerned? >>> >> None. > > Are you sure, given that in such cases there could no longer be natural > numbers? > Even if you had a non-standard model for PA, the syntactis would not change nor the theorems of the theory.
From: William Elliot on 30 Jun 2010 03:49 On Wed, 30 Jun 2010, Nam Nguyen wrote: >>>> Nam Nguyen wrote: >>>>> On and off, I've wondered if the entire FOL syntactical >>>>> proofs would be able to support different semantic >>>>> for "A" and "E": different than the typical quantifier >>>>> semantics "For all" and "There exists"? >>>>> >>>>> For example, if we let Ax[P(x)] and Ex[P(x)] to be specifiers >>>>> (instead of quantifiers) to mean "That x is P", "This x is P", >>>>> respectively, how far can we reason and still make sense? >>>>> >> That's violates syntactics, as bounded x is being interpeted as free x. > > Huh? Isn't x bound to "That" or "This"? > No. You are pointing to a specific element. If it was bound then that y, p(y) would mean the same as that x, p(x). That is false. For example, let 2x = 4 and 3y = 6. "That x is even integer" upon substitution of "bound" variables, "That y is even integer". >>> 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 Indeed it would as Ex is already inteperted as that for models. >> In addition, we can have Ax.~Px and Ax.Px >> which is contradiction. For example the model N with Pn <-> n even. > > Could we still arrive at these conclusions, given the above semantics? > (Right off the first glance, I don't see how). > Yes. There's at least one odd integer and there's at least one not odd integer. Thus that sematics violates the theorems of FOL or makes it inconsistent.
From: Frederick Williams on 30 Jun 2010 05:55
Nam Nguyen wrote: > > On and off, I've wondered if the entire FOL syntactical > proofs would be able to support different semantic > for "A" and "E": different than the typical quantifier > semantics "For all" and "There exists"? > > For example, if we let Ax[P(x)] and Ex[P(x)] to be specifiers > (instead of quantifiers) to mean "That x is P", "This x is P", > respectively, how far can we reason and still make sense? If you do that you are not so much giving different semantics for FOL as inventing a whole new language that just happens to use the same symbols as FOL and will therefore cause utter confusion. Note that you don't even have ~A~ = E and ~E~ = A. There _are_ semantics alternative to Tarski's, e.g. using complete Boolean algebras, and Hintikka's games. [Oh- I mean Tarski's "original" semantics, since he was one of the inventors of cylindric algebras, wasn't he?] -- I can't go on, I'll go on. |