Can there be a different semantic for FOL formulas?
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From: Nam Nguyen on 1 Jul 2010 17:41 Ken Pledger wrote: > 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. :-) I (temporarily) agree here, in the sense it's going to complicate the search for new semantics may times more, if we don't leave ~, /\, \/ alone. What I'm really after is actually trying to explore the possibility of non-quantification semantics (such as relational semantics) that could be readily applicable to say, non-quantification fields such as biology or AI (if not more). But for the time being, I'd think any cohesive replacement would be a first-step win.
From: Nam Nguyen on 1 Jul 2010 17:50 Nam Nguyen wrote: > Ken Pledger wrote: >> 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. :-) > > I (temporarily) agree here, in the sense it's going to complicate > the search for new semantics may times more, if we don't leave ~, /\, > \/ alone. > > What I'm really after is actually trying to explore the possibility > of non-quantification semantics (such as relational semantics) that > could be readily applicable to say, non-quantification fields such > as biology or AI (if not more). > > But for the time being, I'd think any cohesive replacement would be a > first-step win. I also forgot to confess. I have one "darker" motive here: I don't have any kind "Weltanschauung" ("Worldview") of the absoluteness of the truths about the natural numbers and any cohesive alternative for the semantics of Ax, Ex (if not other logical symbols) would most likely support such motive.
From: Nam Nguyen on 1 Jul 2010 19:28 Nam Nguyen wrote: > > 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. For the case of single empty set theory, an axiom would be: ExAy[~(y e x)] /\ Axy[(Az[~(z e x)] /\ Az[~(z e y)]) -> (x=y)]
From: Nam Nguyen on 1 Jul 2010 23:39 Nam Nguyen wrote: > Nam Nguyen wrote: > >> 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. > > For the case of single empty set theory, an axiom would be: > > ExAy[~(y e x)] /\ Axy[(Az[~(z e x)] /\ Az[~(z e y)]) -> (x=y)] Yet another alternative that's non-quantification oriented is possible, I'd think. In this alternative, A and E would be used as a an "adjective", a modifier, to the semantics of the predicate symbol appearing after A or E. A now would mean "common" and E would mean "specific". For example, let H0 be an individual symbol, Mortal, Old, Male, European be 4 unary predicate symbols. Then let's consider the following formulas and the alternative semantics: 1) Ax[Mortal(x)] Being mortal is a "common" trait. 2) Ex[Mortal(x)] Being mortal is a "specific" trait. 3) Ex[Old(x)] Being old is a "specific" trait. 4) Ex[Male(x)] Being a male is a "specific" trait. 5) Ax[Mortal(x)] /\ Ex[European(x)] Being mortal is a "common" trait and being European is a "specific" trait. 6a) Ax[Mortal(x)] -> Ex[Mortal(x)] A "common" trait is a "specific" trait (in its own right). 6b) Ex[European(x)] -> Ax[European(x)] is _NOT_ allowed: A "specific" trait is not necessarily a "common" trait. 7a) ~Ax[Male(x)] <-> Ex[~Male(x)] Being a male isn't a "common" trait is equivalent to not-being a male is a "specific" trait. 7b) ~Ex[European(x)] <-> Ax[~European(x)] Being an European isn't a "specific" trait is equivalent to not-being an European is a "common" trait. 8) European(H0) H0 is an European. The nice thing about this alternate semantics is that _part_ of the canonical FOL "model" could still be retained. For instance, let U be set of the following human individuals: - h1: an old mortal European male, and is named H0 - h2: a mortal human being who isn't old. Then, w.r.t. U: - Ax[Mortal(x)] is true. - Ax[Old(x)] is false. - European(H0) is true. - Ax[European(x)] is neither true nor false. Well, that's just a draft version. Hopefully there aren't technical errors that couldn't overcome. But I guess time would tell.
From: Nam Nguyen on 2 Jul 2010 10:44
Nam Nguyen wrote: > Nam Nguyen wrote: >> Nam Nguyen wrote: >> >>> 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. >> >> For the case of single empty set theory, an axiom would be: >> >> ExAy[~(y e x)] /\ Axy[(Az[~(z e x)] /\ Az[~(z e y)]) -> (x=y)] > > Yet another alternative that's non-quantification oriented is > possible, I'd think. In this alternative, A and E would be used > as a an "adjective", a modifier, to the semantics of the predicate > symbol appearing after A or E. A now would mean "common" and E would > mean "specific". > > For example, let H0 be an individual symbol, Mortal, Old, Male, European > be 4 unary predicate symbols. Then let's consider the following formulas > and the alternative semantics: > > 1) Ax[Mortal(x)] Being mortal is a "common" trait. > > 2) Ex[Mortal(x)] Being mortal is a "specific" trait. > > 3) Ex[Old(x)] Being old is a "specific" trait. > > 4) Ex[Male(x)] Being a male is a "specific" trait. > > 5) Ax[Mortal(x)] /\ Ex[European(x)] > Being mortal is a "common" trait and being European is a "specific" > trait. > > 6a) Ax[Mortal(x)] -> Ex[Mortal(x)] A "common" trait is a "specific" > trait (in its own right). > > 6b) Ex[European(x)] -> Ax[European(x)] is _NOT_ allowed: A "specific" > trait is not necessarily a "common" trait. > > 7a) ~Ax[Male(x)] <-> Ex[~Male(x)] Being a male isn't a "common" trait > is equivalent to not-being a male is a "specific" trait. > > 7b) ~Ex[European(x)] <-> Ax[~European(x)] Being an European isn't > a "specific" trait is equivalent to not-being an European is a > "common" trait. > > 8) European(H0) H0 is an European. > > The nice thing about this alternate semantics is that _part_ of the > canonical FOL "model" could still be retained. For instance, let U > be set of the following human individuals: > > - h1: an old mortal European male, and is named H0 > - h2: a mortal human being who isn't old. > > Then, w.r.t. U: > > - Ax[Mortal(x)] is true. > - Ax[Old(x)] is false. > - European(H0) is true. > - Ax[European(x)] is neither true nor false. The phrase "neither true nor false" sounds familiar: it was one topic that was mentioned not long ago. We may have no choice to accept it but (1) = (pGC xor cGC) might be of the same status "neither true nor false" in the so-called The Standard Model of L(PA). > > Well, that's just a draft version. Hopefully there aren't technical > errors that couldn't overcome. But I guess time would tell. |