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

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

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: A
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
Each name a concept
Each predicate a function from concepts into {0, 1}
I(A) 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 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).

Where P is a predicate and a a term, Pa is d-true for I and A iff  for every individual i included
in 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 (C/d)


And interpretation I is a function which assigns 
To each name a concept
To each predicate a function from concepts into {0,1}
To A the function from pairs of concepts into {0,1} such that I(A)(R(a)R(b)) = 1 iff 
for every thing d such that R(a)Cd holds, R(b)Cd 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 I and A (c/x) if a = txF

Pa is d-true for I and A iff  for every d such that R(a)Cd, I(P)(C/d) = 1
Pa is c-true for I and A iff  I(P)(R(a)) = 1

In either case,

A formula F is true for I and A 

If it is Pa, for some predicate P and some term a and either Pa is d-true for I and A or Pa is
c-true for I and A

 If it is Aab and I(A)(R(a) R(b)) =1

If it is ~S for some formula S and S is not true for I and A

It is &GH for some formulae G and H and both G and H are true for I and A

It is ExG for some variable x and some formula G and for some concept c, G is true for 
	I and A(c/x)

Otherwise not.