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

Could you expand the definitions with some examples or brief descriptions?


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:


What's a variable?


Names:


What's a name?


Predicates:
Relation: Y


What's the difference between a predicate and a relation?


Sentential connectives: ~, & (others by usual definitions)
Quantifiers: E


Putting quantifiers up here will lead to a limited
version/understanding of my position. A quantifier is just a certain
type of relation. Given an identity "the students", a quantifier is
(roughly) "['the students'] is [students] of number [zo'e]".


Descriptor: t


What's a descriptor?



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.


What's a free variable?


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


So a predicate is an abstraction, while a formula is an instance of
this? "Runs" would be a predicate, and "Alice runs" (or "runs(alice)")
would be a formula? What's a relation?


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


I don't understand what you mean here.



A sentence is a formula which contains no free variables.

A singularist model:

Domain D: a non-empty set


What is a set?


Masses M:  Power D – 0. the set of all non-empty subsets of D


A mass is a set of all non-empty subsets of D? No. A mass is a certain
type of identity.


Concepts:

Interpretation: a function, I that assigns to:
Each concept an object from M, with at least one concept for each singleton in M


Object from M? What is an object? Singleton?


Each name a concept


Each name is (probably) not a concept. A name refers to an identity.
While an identity may be a special case of a concept, I avoid this
position because it fails to explain the sharp distinction between
instances and abstractions (identities and concepts; Alice and human),
and my urge to treat a perfect clone of X as Y (instead of thinking
them both X until they differentiate).


Each predicate a function from concepts into {0, 1}


Relation, predicate, function, formula. How are these different?


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


What is an assignment?

What is A (regardless of being an assignment or not)?

So variables are identities, and concepts are abstractions? What else,
if not a concept, would a certain variable be 'functioned' to?


A(c/x) is an assignment just like A except that it assigns the concept c to variable x instead
of A(x).


Example?



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


What is R?



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).



If I understand you correctly, my clarification is that a mass is an
identity. This opposes the pluralist view in that the pluralist mass
is not an identity.


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



I don't understand what "d-true" means.


Where P is a predicate and a a term, Pa is c-true for I and A iff  I(P)(R(a)) = 1



Nor "c-true"


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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