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

["Maxim Katcharov" <maxim.katcharov@gmail.com>]

On 7/14/06, John E Clifford <clifford-j@sbcglobal.net> wrote:
> --- 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.

An expression is anything that tells us what is being related?

> > > 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).

Relation: Alice [throws] the ball
Predicate: Alice [throws the ball]

yes? Or does a predicate ignore the last bit ("Alice [throws]"), and
is therefore just like a relation, but only different in terms of
number?

> > > 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?

Language is a representation of thought. I think that there is a
distinction between describing the consequences and describing what
actually occurs. Grant that "26 students" is actually a certain
identity, perhaps fitting "X of type 'student' of number 26". We can
say that "26 students" is a set of 26, if this fits our purposes. The
fact that it is not anything like a set in the mind is irrelevant.
This describes a consequence - because of X, we can look at it as Y.

I assert that there must be a one-to-one relation between things. I
can't imagine two things being related otherwise (perhaps you can?).
At one end, you have something, and at the other end, you have
something else. Would each of these students be simply 'lifted' up
into the consciousness? If so, then how would the mind determine what
to 'lift'? If there's nothing there besides the students, then how
does it know to lift those 26? Is it connected to each of them? The
response was no (it's a mass, after all). I've already elaborated on
how this is incorrect regardless (a human cannot conceive of that many
identities [...]).

Now, while we /can/ view this in the pluralist sense, but the conflict
regards which of the two ways of treating the language most
approximates thought. I say that my version does - thought treats a
"mass" as in identity, which then allows you to expand {lu'o} using
{gunma}. This isn't just a "way of seeing it" (something based on
consequences), it's what actually happens. The pluralist view is a way
of seeing it.


So, regarding quantifiers. My point is that I treat them differently -
they're a shortcut for describing a relationship between "the
students" and a number. Your treatment of them seems to be that
they're something quite separate. I don't know the exact explanation
you offer that connects them to how we think. I agree that "the
students" being of number 26 means that we can treat it as 26
identities being related to something, even though these identities
are yet-undefined, but this is a description of a consequence.

> > > 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.

What does being in the scope of a quantifier mean?

> > > 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.

Why is there a free variable x?

So a free variable is just something with {zo'e} in its "of number..." slot.

> > >
> > > 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.

So a domain, D, is just a "meta"-set (i.e. it doesn't matter what kind
of set, we just need a way to talk about several things)? Domain seems
a strange thing to call it.

> > > 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.

I don't understand. An example? A mass is different from any sort of
set. A mass is an identity, especially one that implies that there are
other identities (doesn't matter what they are) as parts of it
(relationship "[the mass] has parts [those identities]").

> What does "type of identity" mean?

What I call a concept, or an abstraction. Human, joy, etc. We see a
one-thing, our referent, we form its identity in our mind, which could
be of type "bear", or "group", or "mass".

> > > 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.

I don't think I understand what is occurring here. In D, there are
masses? I understand that D is a set of some sort, but which masses
does it have in it?

> > > 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.

What is a concept? Is it a predicate?

> > 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.

A concept to me means something along the lines of an abstraction. A
human, time, etc. When you say "Alice is a human", 'human' is a
concept, or something that you mentally "grasp", or can recognize.

What I said was probably not relevant, as I think I misread.

As for clones, just a hypothetical situation, where if a 'named' cup
was cloned, I would not have them named the same, while something like
a concept would still be there - they'd both be cups.

> > > 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,

In what sense is it a mapping? That each element in the set fulfills
the relationship that both sets are part of?

Do you mean a mapping of "Fred" to Fred? Perhaps an example?

> 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.

if not an assignment, what would A be? If it's not a function from
variables to concepts, what do you call it? and what do you call it
before you decide that it is or isn't a function from variables to
concepts?

> > 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.

An identity is some one thing. A cat, a group, a mass (by my version),
a single thing in that set of 26 that is wearing a hat.

> 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).

Er, if not a concept (I think that what you call a concept I call an
identity), then what would be assigned to the variable?

> > >
> > > 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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