Re: [lojban] Re: A (rather long) discussion of {all}

[John E Clifford <clifford-j@sbcglobal.net>]

Correction and expansion

--- John E Clifford <clifford-j@sbcglobal.net> wrote:

> Trying again in a more straightforward way:
> 
> Singularist:
> 
> Domain D
> Masses M: all subsets of D with two or more members
> 
> Interpretation I
>  assigns to each name a member of  D u M
>  assigns to each predicate a set included in D u M
>  assigns to Y the relation between D u M and M that hold just in case the first relatum is
>  a member of or included in the second relatum. 
> 
> Assignment A assigns to each variable a member of D u M
> If A is an assignment, A(d/x) is an assignment just like A except for assigning d from 
> D u M to variable x in place of A(x)
> 
> R is a function from terms a to members of D u M, such that
> 	If a is a name, R(a) = I(a)
> 	If a is a variable, R(a) = A(a)
> 	If a = txF, R(a)  is some member d of D u M such that F is true for I and A(d/x)
> 
> A formula Pa is d-true for I and A iff  either I(a) e D and I(a) e I(P) or I(a) e M and for
>  every d e I(a) d e I(P).
> 
> A formula Pa is c-true for I and A iff I(a) e I(P).
> 
> A formula Pa is true for I and A iff it is either c-true or d-true
> 
> A formula ~F is true for I and A iff F is not true for I and A
> 
> A formula &FG is true for I and A iff both F and G are true for I and A
> 
> A formula ExF is true for I and A iff for some d e D u M, F is true for I and A(d/x)
> 
> 
> Pluralist:
> 
> Domain D
> 
> I is a relation whose first relatum is
	A name and whose second relatum is a member of d
	A predicate and whose second relatum is an n-place function over D into {0,1}
	Y and whose second relatum is an n+m-place function over D into {0,1}
Such that each name is related to at least one member of D, each predicate is related to exactly
one n-place function, for every n between 1 and the size of D, Y is related to exactly one
function fnm for each n, m between 1 and the size of D, such that I(Y)(nm) (d1â?¦dn e1â?¦em) = 1
iff  each di is identical to one of the es.

Since the array of functions for each predicate is unique as is the function for each number, we
can refer to the n-place function of a given predicate P as I(P)(n).

For convenience, we will abbreviate â??d1 â?¦ dn such that each di aIdiâ?? as I(a).  In the
sequence d1 â?¦ dn it is understood that 1) no two items are identical and 2) the order of the
items is not significant (the value of a function for d1â?¦dn in order is the same as the value
for any permutation of that order). In the definition of I(Y) above , the ds and the es are
separate sequences to which the above restrictions apply: a permutation of the whole conjoined
sequence that involved es and d being interdigitated would not necessarily yield the same value
and an e may be identical with a d, just not another e.

We say â??d1â?¦dn numbers nâ??

> 
> A is a relation between variable and members of D
> A(d1â?¦dn/x) is a relation just like A except that x is related to each of d1â?¦ dn rather than
> to
> the things it is related to by A
> 
> We use A(x) analogously to I(a)
> 
> In the same vein we can define 
> 	R(a) = I(a) if a is a name
> 	R(a) = A(a) if a is a variable
> 	R(a) is some d1â?¦dn such that F is true for I and A(d1â?¦dn/x) if a = txF
> 
> Pa is d-true for I and A iff for every d in R(a) I(P)(1)(d) = 1
> Pa is c-true for I and A iff R(a) numbers n and I(P)(n)(R(a)) = 1
> 
> F is true for I and A iff
> F = Pa and Pa is d-true for I and A or Pa is c-true for I and A.
> F = Yab and R(a) numbers n and R(b) numbers m and I(Y)(nm) (R(a)R(b)) =1
> F = ~G and G is not true for I and A
> F = &GH and both G and H are true for I and A
> F = ExG and, for some d1â?¦dn from D, G is true for I and A(d1â?¦dn/x)
> 
> 
> 
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>