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

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

First round of corrections, thanks to xorxes.

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

> I send this along for corrections and questions before using it (in its revised form) to answer
> Maxim's questions.
> 
> Singular v. Plural Semantics
> 
> Language:
> 
> Variables:
> Names: 
> Predicates:  
Relation: Y
> Sentential connectives: ~, & (others by usual definitions)
> Quantifiers: E
> Descriptor: t
> 
> Terms: a variable is a term, a name is a term,  if F is a formula containing free variable
>  x, then txF is a term.
> Formula:  A predicate followed by a term is a formula, A followed by two terms is a 
> 	formula, a formula preceded by ~ is a formula, two formulas preceded by & is a
>  formula, a formula preceded by a variable preceded by E is a formula
> A formula contains a free variable x just in case there is an occurrence of x in that formula
> which is not in any subformula which begins Ex nor in a term which begins tx
> 
> A sentence is a formula which contains no free variables. 
> 
> A singularist model:
> 
> Domain D: a non-empty set
> Masses M:  Power D â?? 0. the set of all non-empty subsets of D
> Concepts: 
> 
> Interpretation: a function, I that assigns to:  
Each concept an object from M, with at least one concept for each singleton in M
> Each name a concept
> Each predicate a function from concepts into {0, 1}
> I(Y) is the function from pairs of concepts such that I(A)(c1,c2) = 1 iff  I(c1) is included in
> I(c2)
> 
> A is an assignment iff A is a function from variables to concepts
> A(c/x) is an assignment just like A except that it assigns the concept c to variable x instead
> of A(x).
> 
If a is a term, R(a) = I(a) if a is a name, R(a) = A(a) if a is a variable,  R(a) is a concept c
such that F is true for I and A(c/x), if a = txF
> 
> i is an individual just in case i is in M and is a subset of each of its subsets (is identical
> with each of its subsets, has only one member, i is a singleton).
> 
> Where P is a predicate and a a term, Pa is d-true for I and A iff  for every individual i
> included in I(R(a)) and for every concept c s.t. I(c) = i, I(P)(c) = 1
> 
> Where P is a predicate and a a term, Pa is c-true for I and A iff  I(P)(R(a)) = 1
> 
> A Pluralist model
> 
> Domain: Some things
> Concepts
> 
C is a relation between concepts and items in D, such that for every d in D, there is at least
once c such that c is related by C only to d [We designate a selected such concept C/d, for each
d]
> 
> 
> An interpretation I is a function which assigns 
> To each name a concept
> To each predicate a function from concepts into {0,1}
To Y the function from pairs of concepts into {0,1} such that I(A)(c1,c2)) = 1 iff 
for every thing d such that c1Cd holds, c2Cd holds
> 
> An assignment A is a function from variable to concepts
> A(c/x) is an assignment just like A except for assigning c to x in place of A(x).
> 
For term a, R(a) = I(a) if a is a name, R(a) = A(a) if a is a variable, is a concept c such
that F is true for C,I and A(c/x) if a = txF
> 
Pa is d-true for C,I and A iff  for every d such that R(a)Cd, I(P)(C/d) = 1
Pa is c-true for C,I and A iff  I(P)(R(a)) = 1
> 
> In either case,
> 
> A formula F is true for [C,]I and A 
> 
> If it is Pa, for some predicate P and some term a and either Pa is d-true for [C,]I and A or Pa
> > is c-true for [C,]I and A
> 
> If it is Yab and I(Y)(R(a) R(b)) =1
> 
> If it is ~S for some formula S and S is not true for [C,]I and A
> 
> It is &GH for some formulae G and H and both G and H are true for [C,] I and A
> 
> It is ExG for some variable x and some formula G and for some concept c, G is true for [C,] I
and > A(c/x)
> 
> Otherwise not.
> 
> 
> 
> 
> 
> 
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> 
>