Can there be a different semantic for FOL formulas?
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From: Nam Nguyen on 30 Jun 2010 00:22 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?
From: Nam Nguyen on 30 Jun 2010 00:29 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? > > 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?
From: Nam Nguyen on 30 Jun 2010 01:48 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? >> >> 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? 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?
From: William Elliot on 30 Jun 2010 02:37 On Tue, 29 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? >> >> 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.
From: Nam Nguyen on 30 Jun 2010 02:48
William Elliot wrote: > On Tue, 29 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? >>> >>> 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? |