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

[John E Clifford <lojban-out@lojban.org>]

--- Maxim Katcharov <maxim.katcharov@gmail.com> wrote:

> 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?

An expression that stands in the place of a name but may have a different referent on each
occasion (hence the name of it).  The useful ones are bound by quantifiers.  (Lojban variables
are, for example, {da, de, di})

> > Names:
> 
> What's a name?


An expression with a fixed referent.
 
> > Predicates:
> > Relation: Y
> 
> What's the difference between a predicate and a relation?

Predicates are one-placed *take one argument to make a formula), relations are more (in this case,
two).

> > 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]".

We are modeling Lojban, whose quantifiers are first of all over variables, in just this position. 
I have not incorporated into this simplified version any of the derivative uses of quantifiers
(including enumeration), because they have no special properties (that I know of yet) connected
with the issue of singular versus plural.  What do you have in mind?
 
> > Descriptor: t
> 
> What's a descriptor?

Converts a formula into a name (cf. {le, lo} and the like).

> >
> > 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?

One not in the scope of a quantitifier on it.
 
> > 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?

Well, ""abstraction" isn't quite right, though it is incoplete without its subject.  "runs(alice)"
is a sentence.  A relation is like a predicate but has more arguments.
 
> > 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.

x is free in Fx but not in ExFx.
 
> >
> > A sentence is a formula which contains no free variables.
> >
> > A singularist model:
> >
> > Domain D: a non-empty set
> 
> What is a set?

Well, I suppose I had Cantorean (usual set theoretical) sets in mind, but nothing hangs on that. 
L-sets would do as well or we could just have a definite bunch (in the none-technical sense) of
things.
 
> > 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.

This is the set of masses, each mass is a non-empty set of things in D.  What does "type of
identity" mean? 

> > 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?

An object is, in this case, just something in D.  A singleton is a set with exactly one member.
 
> > Each name a concept
> 
> Each name is (probably) not a concept. A name refers to an identity.

Note, this sentence is part of what an interpretation does: assigns to each name a concept.

> 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).

I don't see what this is all about.  A concept here is just another abstract entity in the
metalanguage of the given language.  I suppose its name may have some useful associations but none
that need interfere here. For example it has nothing to do with the differences between things (in
D) and properties -- what predicates mean.  I don't understand where clones come in.
 
> > Each predicate a function from concepts into {0, 1}
> 
> Relation, predicate, function, formula. How are these different?

Do you know any of the language of set theory or mathematics?  These are pretty rudimentry.  But
if need be I can back up a bit more.  A predicate is an expression which needs one term to make a
formula, a relation is an expression which needs two terms to make a formula, a function is a
mapping from one set of things into another set, a formula a predicate or relation filled out with
appropriate number of terms or some compound of one or more such by connectives and quantifiers.

> > 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?

A function which gives each variable a meaning for the nonce.

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

I don't understand this question.
 
> So variables are identities, and concepts are abstractions? What else,
> if not a concept, would a certain variable be 'functioned' to?

I don't understand your terminology here.  Variables are expression in the language; I gather that
identities are not. I am not sure whether concepts are abstraction -- I am probably inclined to
think of them as thoughts, but nothing hans on what they are, only on the role they play. xorxes
has suggested another approach -- which I found harder to adapt to what I understood to be your
position -- in which variables were "functioned" to things (in D).
 
> > A(c/x) is an assignment just like A except that it assigns the concept c to variable x instead
> > of A(x).
> 
> Example?

A simple assignment Q assigns Charlie to every variable; now for some reason we change and for the
particular variable y17, we assign George.  This new function is Q(George/y17).
 
> >
> > 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?

It is a function from terms to their referents. 
> >
> > 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.

Again, I don't underand your use of "identity".  The last sentence is just a definition saying
that we are going to use the word "individual" to refer to sets (Masses) that have 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 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.

Distributively
 
> > 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"

Collectively (i.e., non-distributively)
 
> > 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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