Re: Logical Quantifiers Again

[William Hale]

In article
<a8593716-07c8-4858...@o9g2000yqa.googlegroups.com>,
MoeBlee <mode...@gmail.com> wrote:

> On Dec 24, 1:52 am, MoeBlee <modem...@gmail.com> wrote:
>
> CORRECTION:
>
> I wrote:
>
> > for example, first order model theory does not have with it one
> > particular universe
>
> Should be:
>
> for example, first order group theory does not have with it one
> particular universe
>
> MoeBlee

I don't know much about first order group theory. My understanding,
after a brief search on the internet, is that first order group theory
has a specific, but unspecified universe. That is, any universe that
meets the axioms of first order group theory is ok. Thus, a cyclic group
of order 5 could be an instance of that universe and a cyclic group of
order 37 could be another instance of that universe. Of course, first
order group theory is not about the specific instances: rather, it is
about a universe that satisfies the axioms of first order group theory.

In particular, I have the feeling that first order group theory cannot
treat quotient groups or subgroups or homomorphisms since there is just
one underlying universe. Is that true?

[Frederick Williams]

Aatu Koskensilta wrote:

> Yes, with the caveat that in general a formula doesn't express
> anything unless its non-logical symbols are given some interpretation.

How does one distinguish between logical symbols and non-logical
symbols? In an account of any particular theory one is told what the
logical symbols are, and they are usually chosen from a well-known
collection: 'and', 'for all', etc. But why _those_?

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[MoeBlee]

On Dec 24, 3:30 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:

> By group theory we usually mean the
> branch of mathematics studying groups, i.e. structures for which the
> group axioms hold. In the above you're talking about these axioms, not
> the branch of mathematics known as group theory.

Right. I too had mentioned that in one of my previous posts and I
addressed both senses - group theory as a general mathematical study,
as you just mentioned, and also the more narrow sense of first order
group theory.

MoeBlee

[MoeBlee]

On Dec 24, 10:28 am, William Hale <h...@tulane.edu> wrote:

> My understanding,
> after a brief search on the internet, is that first order group theory
> has a specific, but unspecified universe. That is, any universe that
> meets the axioms of first order group theory is ok.

Each interpretation of the language has a a specific universe for that
interpretation. But there are many different interpretations for the
language. And any interpretation that satisfies the axioms specifies a
specific group.

MoeBlee

[Nam Nguyen]

On 24/12/2011 10:31 AM, Frederick Williams wrote:
> Aatu Koskensilta wrote:
>
>> Yes, with the caveat that in general a formula doesn't express
>> anything unless its non-logical symbols are given some interpretation.
>
> How does one distinguish between logical symbols and non-logical
> symbols? In an account of any particular theory one is told what the
> logical symbols are, and they are usually chosen from a well-known
> collection: 'and', 'for all', etc. But why _those_?

Because the semantics of the logical symbols are invariant from
one axiomatization to the other. For instance, depending what
we'd mean by an empty set: with only one unique one or multiple
of them (urelement), the semantic of "xey" would be different,
but the semantic of "/\" (i.e. "and") would stay the same
for (xey /\ ~xey), in all axiomatization involving the binary
predicate symbol 'e'.


--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

[Nam Nguyen]

On 24/12/2011 2:03 AM, Aatu Koskensilta wrote:

> Yes, with the caveat that in general a formula doesn't express
> anything unless its non-logical symbols are given some interpretation.

And with that, a Pandora box has come into existence in the FOL
model-theoretical world; for, one can _cohesively interpret the epsilon_
_symbol 'e' _ in such a way that the following formula is meaningless
as a non-logical expression and would have no model:

Axy[(x=y) <-> Az[zex <-> zey]]

So much for the antique belief that every syntactically consistent
formal system would have a model.

[Aatu Koskensilta]

Frederick Williams <freddyw...@btinternet.com> writes:

> How does one distinguish between logical symbols and non-logical
> symbols? In an account of any particular theory one is told what the
> logical symbols are, and they are usually chosen from a well-known
> collection: 'and', 'for all', etc. But why _those_?

The basic idea is that they are logical in the sense that they "mean
the same" in any structure, on any interpretation. Giving a principled
mathematical (or philosophical) account of this notion is a classical
problem -- on which old Alfred T. himself had a few things to say. You
will find illuminating discussion (and helpful references for further
study) in Sol Feferman's recent paper /Which Quantifiers are Logical?/:

http://math.stanford.edu/~feferman/papers/WhichQsLogical(text).pdf

and in the _Standford Encyclopedia of Philosophy_ article on logical
constants:

http://plato.stanford.edu/entries/logical-constants/

Quantifier here means generalized quantifier in the sense of Lindström,
so the usual propositional connectives in particular count as
quantifiers.

--
Aatu Koskensilta (aatu.kos...@uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen."
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

[Nam Nguyen]

On 24/12/2011 11:57 AM, Nam Nguyen wrote:
> On 24/12/2011 2:03 AM, Aatu Koskensilta wrote:
>
>> Yes, with the caveat that in general a formula doesn't express
>> anything unless its non-logical symbols are given some interpretation.
>
> And with that, a Pandora box has come into existence in the FOL
> model-theoretical world; for, one can _cohesively interpret the epsilon_
> _symbol 'e' _ in such a way that the following formula is meaningless
> as a non-logical expression and would have no model:
>
> Axy[(x=y) <-> Az[zex <-> zey]]

Where "meaningless" here could be defined as being well-formed but
having no model.

[Uergil]

In article
<1788d530-ff86-4342...@n6g2000vbg.googlegroups.com>,
Tony Orlow <bony...@gmail.com> wrote:

> Well, yes, as are 'A' and 'E'. The fact that "Ax f(x)" is equal to
> "~Ex ~f(x)" and that "Ex f(x)" is equal to "~Ax ~f(x)" is generally
> accepted, and essentially the same as Morgan's Law relating "and" and
> "or".

If you are referring to Augustus de Morgan, and his "laws" that say
"Not (A and B)" is equivalent to "(not A) or (not B)" and so on, those
are "de Morgan's laws"
--
"Ignorance is preferable to error, and he is less
remote from the- truth who believes nothing than
he who believes what is wrong.
Thomas Jefferson

[Aatu Koskensilta]

Nam Nguyen <namduc...@shaw.ca> writes:

> So much for the antique belief that every syntactically consistent
> formal system would have a model.

I'm afraid I'm simply unable to follow your line of thought here. A
formula such as:

Axy[(x=y) <-> Az[zex <-> zey]]

does not express anything until it is explained what relation the symbol
e should be taken to stand for and what the quantifiers range over. (And
saying e is in some sense meaningless is not much of an explanation.)
But whether a formula (or a bunch of formulas) is consistent,
contradictory, whether it logically entails some other formula, and so
on, does not depend on any such explanation.

[MoeBlee]

On Dec 24, 11:31 am, Frederick Williams

<freddywilli...@btinternet.com> wrote:

> How does one distinguish between logical symbols and non-logical
> symbols?

The logical symbols are the quantifiers and sentential connectives.
The non-logical symbols are the predicate and function symbols.
However, often, the 1-place predicate symbol '=' is an exception and
is regarded as a logical symbol.

> In an account of any particular theory one is told what the
> logical symbols are, and they are usually chosen from a well-known
> collection: 'and', 'for all', etc.  But why _those_?

(1) The logical symbols are common to all languages, but the non-
logical symbols are different for different languages. (2) It is
convenient in certain formlations to distinguish between the logical
symbols and non-logical symbols.

MoeBlee

[MoeBlee]

On Dec 24, 1:54 pm, MoeBlee <modem...@gmail.com> wrote:

> (1) The logical symbols are common to all languages, but the non-
> logical symbols are different for different languages. (2) It is
> convenient in certain formlations to distinguish between the logical
> symbols and non-logical symbols.

Reading more posts, it occurred to me that (2) requires elaboration:

The logical symbols are invariant in interpretation, while the non-
logical symbols have different interpretations per different
structures for the language. (One exception is in Enderton's method
where a structure assigns a universe to the universal quantifier. So
Enderton's method is that a structure is one function as opposed to a
pair <U F> where U is the universe and F is the interpretation
function for the non-logical symbols.)

MoeBlee

[MoeBlee]

Orlow, I SHOULD (whether I do or not) make this my last post to you on
this subject. You continue to make claims and arguments that are mired
in your lack of understanding of the basics of first order languages,
especially your virtually complete unfamiliarity with basic semantics
for first order languages. Thus, each of my explanations, in a vacuum
of your own familiarity with the subject, results in you returning
with more confusions on the subject.

On Dec 24, 11:00 am, Tony Orlow <bonyto...@gmail.com> wrote:
> On Dec 24, 2:52 am, MoeBlee <modem...@gmail.com> wrote:

> all sorts of talk about models,
> structures, variable assignments, or whatever, that don't have
> anything to do with the original question.

They have EVERYTHING to do with the question. I've SAID what they have
to do with the question.

> > My remarks don't depend on a premise that all knowledge comes from
> > textbooks. I'll repeat what I said:
>
> Moe: "What textbook in mathematical logic are you using to understand
> these
> basics?"

I did not mean for that to imply that knowledge comes only from
textbooks. So perhaps the question would have been better worded:

Is there a textbook you're using to understand these basics? If so,
what textbook is it?

> > You are making certain claims about first order logic and a proposed
> > change to it. As I said, if you tell me a specific textbook that you
> > refer to for your understanding of first order logic, then I may be
> > able to refer you to the appropriate passages in that textbook that
> > explain the crucial particulars you're overlooking.
>
> It's a little presumptuous to assume that I am overlooking details in
> some text, when what I am doing is running past other minds an
> alternative formulation of quantification that has occurred to me
> while considering how basic propositional calculus is extended to
> "ordered" logic.

No, you're making claims about how you can "run past" an aspect of
first order logic but you don't understand really what is actually
involved in that claim, since your arguments don't take into account a
number of the crucial details about the subject of first order logic.

> I have yet to see any objection that doesn't rely on
> assuming that x is some particular object, rather than the variable
> that it was obviously meant to represent.

For about the thrid time: There is the symbol 'x'. It is a variable.
Then, per a structure for the language and per an assignment for the
variables, the symbol 'x' is mapped to some member of the universe for
the structure.

> > As far as I can tell, you are claiming that you are able to dispense
> > with the universal quantifier as a primitive by instead adopting
> > primitive symbols 'U' and 'e'. But you have not shown how that would
> > work to express the actual semantics of universal quantification.
>
> > It's as if you are claiming that you know how to dispense with one of
> > the parts in the Fiat automobile engine and replace it with something
> > else you've devised. But when you start to describe how you'd do that,
> > it is clear that you don't know how a Fiat engine works.
>
> That's a nice analogy, I suppose, but completely off point. If you can
> give some kind of example where it doesn't work, that would be
> helpful, but I haven't seen any such exception yet.

I already explained what is lacking in your claim. You don't
understand it, because you don't have a grasp of the basics of this
subject.

To be clear: I do not claim that there cannot be a formulation of
first order logic in which the universal quantifier is replaced with
'U' and 'e' while still preserving the ability to express, with an
alternative method of semantics, that something holds for all members
of the universe. Rather, my point is that you have not yourself shown
how do accomplish such a thing. Most plainly, you have not specfied an
actual and coherent alternative semantical method that you've shown to
work. And I've given you some of the considerations that you would
need to address in doing that. Unfortunately, my remarks are going
right into the ditch, since you don't know the basics of syntax and
semantics of first order languages.

> Is
> there a particular question you have about what I am suggesting?

I've already mentioned the key problems with your claim that I find
most salient. Morevover, again, you have not given a coherent and
rigorous alternative semantics that accomplishes what you claim.
You're handwaving while you don't understand the crucial and basic
considerations you're waving right over.

> > But, in brief, a structure for a language L is a function that assigns
> > to the universal quantifier a non-empty set (this set we call 'the
> > universe'), assigns to each n-place predicate symbol of L an n-place
> > relation on the universe (where n=0, the assignment is to a truth
> > value, and where n=1, the assignment is to a subset of the universe),
> > and assigns to each n-place function symbol of L an n-place function
> > on the universe (where n=0, i.e., the symbol is a constant, the
> > assignment is to a member of universe).

PLEASE do you understand what I wrote above or not?

> > to show that one can dispense with the universal quantifier by
> > instead using the symbols 'U' and 'e' requires specifying a specific
> > semantics for those symbols and such that that semantics fulfills the
> > same semantical properties for the universal quantifier.
>
> > But you have not done that. Moreover, in my earlier posts I mentioned
> > certain points that you would have to overcome in order to to supply a
> > successful semantics as just described.
>
> I rather think I have done that sufficiently. In ZF there is no
> explanation of what 'e' *means*.

You're completely missing the point and working in a fog as you're
conflating differnent things. But it's hopeless if you continue to
refuse to learn how the semantics for first order languages actually
work.

'e' is a 2-place predicate symbol. It has no interpretation onto
itself. It is given an interpretation with a structure for the
language. The interpretation for 'e' per a given structure for the
language is some 2 place relation on the universe for the structure.

But, more fundamentally, whether we ourselves mention any structure
for the language, there is still the GENERAL definition of 'structure
for a language' so that at least we know what a structure for the
language is.

Now, you are using 'U' and 'e' as special symbols to capture a FIXED
notion of universal quantification. That is a different situation from
'e' ordinarily as a 2-place relation symbol that does not itself have
a FIXED interpretation (in the sense that even though we generally
intend 'e' to stand for membership, our semantics itself does not
REQUIRE that interpretation).

So your situation is more analogous to when we take '=' as having a
FIXED interpretation, where we require that for ANY interpretation,
'=' must map to the equality relation on the universe

> When I said U is the collection of
> all conceivable objects, then xeU is vacuously true, as soon as x is
> conceived.

That is handwaving waffle.

What you need to do is give a rigorous specification of your
alternative semantics. But you can't do that, because you don't have
the tools to do it, because you refuse to learn the basics of this
subject.

> > Please refer to a textbook in basic mathematical logic, such as, I
> > would suggest, Enderton's 'A Mathematical Introduction To Logic'.
>
> Again, that is not an objection, but an instruction.

Correct. But I've given you specific objections. Your responses to
those objections are confusions. That is why I recommend that you find
out how the Fiat engine actually runs before you start claiming that
it doesn't need a fuel system.

> If you have any
> actual logical objection to the replacement of 'A' with 'e' and 'U',
> please do state it succinctly.

I HAVE.

> > The interpretation is given by a structure for the language. If the
> > language has constants, then the structure (which is a certain kind of
> > function), among other things, assigns to each constant a member of
> > the universe for the structure.
>
> If I had included a statement, "xeX", by itself, then 'x' may be taken
> as some sort of constant, but I did not.

You raised a point about variables and constants. I merely answered
it.

> I used 'x' as a variable in
> the condition for an implication, as per the basic logical 2-place
> operator "->".

I don't know what you refer to in ordinary first order logic with "x
as a variable in condition for implication" where 'x' is an individual
variable. You seem to have built up your own very personal hodge podge
of notions about first order logic, based on various bits and pieces
of you've read here and there. That is hurting you terribly. You need
to get a good book and read a systematic account of the syntax and
semantics for first order languages.

> > The next step is to give, in addition to a structure, an assignment
> > for the variables, which is a function that assigns to each variable a
> > member of the universe for the structure.
>
> So, if I say "AneN (n+1)eN", then I have to specify which natural
> number n refers to?

No, I said no such thing.

What I said is that IF you give a structure for the language and an
assignment for the variables, and if 'n' is a variable, then n is
assigned to some member of the universe. I did not say that is
required just to utter the formula you mentioned. Structures and
assignments for the variables are used for the DEFINTIONS of
'satisfied' and 'true'; but I did not claim that they are needed
merely to utter formulas.

PLEASE, the notions of structure for a language and assignment for the
variables are basic notions in beginning mathematical logic. I don't
know what end you think you achieve by bickering about the basic
definitions when you don't even know or understand the definitions and
how they are part of the approach to semantics for first order
languages.

> No, incorrect. The statement is true given *any*
> natural number n. I am not required to assign a value to n in order to
> assert this axiomatic statement.

PLEASE just study the basics of this subject. Again, I did not require
you to assign a value to the variable 'n'. Rather, you would find out,
were you to actually read a book, that the semantics for universal
quantification precede through a notion of assignments for the
variables.

> > No, for a given first order language and for a given structure for
> > that language, that structure has a universe specified by the
> > structure. And there is no restriction on what the members of the
> > universe may be. Only that the universe must be non-empty. Then with
> > an assignment for the variables per that structure, each variable is
> > assigned to some member of that universe.
>
> Okay. And a flying pink elephant is a member of U. Would you like to
> discuss this in the context of flying pink elephants?

You're not making any coherent point.

I said that there is no restriction on what the members of the
universe may be. That means that for any object, there is some (of
course, many) universe(s) that that object is a member of. I'm not
claiming that given the properties 'flying', 'pink' and 'elephant'
there is some object that has all three of those properties.

The point is that for any language and any non-empty set (no matter
its particular elements), said non-empty set is the universe for some
structure for said language.

> > I take it
> > that 'f' is a 1-place predicate symbol there.
>
> Yes, it is, with a value of "true" or "false".

That makes no sense. 0-place predicate symbols are assigned a value
'true' or 'false' per an interpretation. 1-place predicate symbols are
assigned a subset of the universe for the structure. Then fx is
evaluated as satisfied or not satisfied by an assignment for the
variables according to whether the object assigned to 'x' is a member
of the subset assigned to 'f'.

> > Note that it is NOT the case that in general Ax fx <-> fx.
>
> Does "fx" mean "f(x)", where f is a statement regarding variable x? If
> so, then indeed, Ax f(x) <-> f(x), unless you are asserting that f(x)
> can be true and false at the same time?

No, you're plain, flat out incorrect. Ask ANYBODY who knows about this
subject. Better yet, get a good book and read to understand for
yourself.

An, to be clear, "Ax fx <-> fx" is not "Ax(fx <-> fx)" by the
ordinary convention that the scope of a quantifier is the least
formula to the right of the quantifier.

It is NOT the case that in general we have |- Ax fx <-> fx.

That is, it is NOT the case that in general we have "for all x, fx"
iff "fx".

> > (However we do have the meta-theorem:
>
> > |- Ax fx iff |- fx
>
> > But that is much different from |- Ax fx <-> fx, which, as I've said,
> > does not in general hold.)
>
> I guess you've said that. What textbook are you quoting?

Now you're being a smart-ass again.

Just get a texbook, such as Enderton, read the material
systematically, and you will see exactly how to prove what I wrote
above.

MoeBlee

[MoeBlee]

On Dec 24, 10:13 am, Tony Orlow <bonyto...@gmail.com> wrote:
> On Dec 24, 1:39 am, Dan Christensen <Dan_Christen...@sympatico.ca>

> I don't understand
> why you think that 'x' is some particular object in this context.

You would if you would just read a textbook.

Fx is a an open formula. It has no semantical value except through a
structure that gives interpretation to 'F' and an assigment for the
variables that assigns 'x' to some member of the universe for the
structure. Fx is satisfied per the structure and assignment for the
variables iff the object assigned to 'x' is a member of the set
assigned to 'F'.

AxFx is a sentence. It has no semantical value excpet through a
structure that gives an interpretation to 'F'. And AxFx is true iff Fx
is satisfied for ALL assignments for the variables.

MoeBlee

[Nam Nguyen]

On 24/12/2011 12:50 PM, Aatu Koskensilta wrote:
> Nam Nguyen<namduc...@shaw.ca> writes:
>
>> So much for the antique belief that every syntactically consistent
>> formal system would have a model.
>
> I'm afraid I'm simply unable to follow your line of thought here. A
> formula such as:
>
> Axy[(x=y)<-> Az[zex<-> zey]]
>
> does not express anything until it is explained what relation the symbol
> e should be taken to stand for and what the quantifiers range over. (And
> saying e is in some sense meaningless is not much of an explanation.)

Right. I've not explained it yet. But I'm going to below.

> But whether a formula (or a bunch of formulas) is consistent,
> contradictory, whether it logically entails some other formula, and so
> on, does not depend on any such explanation.

Right. But note that that isn't what I said: what I said involves
"a model".

----

Now in the _usual semantic_ of e, we can express a relation between some
x, and y as:

(*) y = { {{x}}, {{{x}}} }

So the x is a kind of mereological _component_ of y, but not an
_element_ of y in the usual semantic of 'e'. Iow, we don't have
xey in this case.

Note that in this case of "nested membership", the element of
y denoted as the "leftmost" element in (*) has total of 2 left-brackets
'{' on the left side of 'x'. So given an x and y, we could say
x is a "member of degree n" of y iff y contains at least one element
as:

y = { {{{...{x}...}}}, ... }

where the total number of the left brackets next to x is n.

By convention, if y = {x, ...} then there the membership-degree
of x is just 0.

So, if we interpret 'e' as such kind of "component/membership" such
that n is always greater than 0, then that would be the desired new
interpretation for 'e'.

Something like that.

[MoeBlee]

On Dec 24, 12:57 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:

> one can _cohesively interpret the epsilon_
> _symbol 'e' _ in such a way that the following formula is meaningless
> as a non-logical expression and would have no model:
>
> Axy[(x=y) <-> Az[zex <-> zey]]
>
> So much for the antique belief that every syntactically consistent
> formal system would have a model.

Axy(x=y <-> Az(zex <-> zey))

has tons of models.

Here's one:

Universe is {0}.

'e' stands for {<0 0>}.

'=' stands for the identity relation, viz. {<0 0>}.

MoeBlee

[Nam Nguyen]

On 24/12/2011 1:21 PM, MoeBlee wrote:
> On Dec 24, 12:57 pm, Nam Nguyen<namducngu...@shaw.ca> wrote:
>
>> one can _cohesively interpret the epsilon_
>> _symbol 'e' _ in such a way that the following formula is meaningless
>> as a non-logical expression and would have no model:
>>
>> Axy[(x=y)<-> Az[zex<-> zey]]
>>
>> So much for the antique belief that every syntactically consistent
>> formal system would have a model.
>
> Axy(x=y<-> Az(zex<-> zey))
>
> has tons of models.

But not per the new interpretation where the degree of membership is
always greater than 1. (Which is what I conveyed).

[Frederick Williams]

Nam Nguyen wrote:
>
> > [...] In an account of any particular theory one is told what the

> > logical symbols are, and they are usually chosen from a well-known
> > collection: 'and', 'for all', etc. But why _those_?
>
> Because the semantics of the logical symbols are invariant from
> one axiomatization to the other.

Why? Look

P v ~P

is logically true because whatever value is assigned to P, it comes out
to true. But why are we allowed to treat P as a variable but not v? If
we read v as and, the displayed formula is logically false.

[Frederick Williams]

MoeBlee wrote:

>
> The logical symbols are invariant in interpretation, while the non-
> logical symbols have different interpretations per different
> structures for the language.

Why? Here's my reply to Nam Nguyen:

"Why? Look

P v ~P

is logically true because whatever value is assigned to P, it comes out

to [be *] true. But why are we allowed to treat P as a variable but not

v? If
we read v as and, the displayed formula is logically false."

which may explain my problem.

* - sorry, missed out "be" in my post.

[Nam Nguyen]

On 24/12/2011 1:57 PM, Frederick Williams wrote:
> Nam Nguyen wrote:
>>
>>> [...] In an account of any particular theory one is told what the
>>> logical symbols are, and they are usually chosen from a well-known
>>> collection: 'and', 'for all', etc. But why _those_?
>>
>> Because the semantics of the logical symbols are invariant from
>> one axiomatization to the other.
>
> Why? Look
>
> P v ~P
>
> is logically true because whatever value is assigned to P, it comes out
> to true. But why are we allowed to treat P as a variable but not v? If
> we read v as and, the displayed formula is logically false.

Somewhere you have to use what we'd refer to as _formulas_
which are meant to be _ well-formed _ . Any in a wff, logical
symbols have to be _reserved_ symbols, so that _consistently_
they used and mentioned in _all formulas_ .

If you'd like to use 'P' as a logical symbol that would be fine,
technically speaking, as long as you use it _consistently throughout_ .

[I'm not quite sure what you're really trying to get at here. :-( ]

[Tony Orlow]

On Dec 24, 12:31 pm, Frederick Williams

<freddywilli...@btinternet.com> wrote:
> Aatu Koskensilta wrote:
> >   Yes, with the caveat that in general a formula doesn't express
> > anything unless its non-logical symbols are given some interpretation.
>
> How does one distinguish between logical symbols and non-logical
> symbols?  In an account of any particular theory one is told what the
> logical symbols are, and they are usually chosen from a well-known
> collection: 'and', 'for all', etc.  But why _those_?

Hi Frederick -

In general, there are n-place operators, where n is a member of N (0
included). Logical operators take n logical values and returna a
logical value, either 0 or 1, true or false. Given n parameters, there
are a number of possible operators equal to the base-2 tetration of 2.
The 0-place operators include true and false (2 possibilities). Of the
1-place operators, one is always true and one always false (no matter
the truth value of the one variable, x), one is x itself (either true
or false), and the fourth (2^2) is "not x" (1 if x is 0, and 0 if x is
1). So, "not" is the only non-trivial (not previously stated) operator
on the single parameter x. Of the 2-place logical operators on
parameters x and y, there are 2^2^2, or 16, possible operators. Among
these, there are, always true and always false, true if x and true if
not x, true if y and true if not y, and 10 other operators, the five
basic ones and their negations. The five non-trivial 2-place operators
are "and", "or", "implies", "is implied by", and "equals"(with their
negations).

Those are logical operators, which take logical truth value parameter
and return a logical truth value result. Non-logical operators have
either non-logical (not 0 or 1) parameters, or a non-logical result.
For instance, 'e' as a 2-place operator takes an object and a set as
parameters, returning a logical truth value, but nonetheless a
"nonlogical" operator.

That might have been a distraction, or it may be just the question Moe
was trying to ask. That's the foundation of propositional logic, upon
which we may either lay a layer of quantifiers over some universe, or
the 2-place membership operator and the universal object. That's the
whole point.

Peace,

Tonyveylb

[Tony Orlow]

You think extensionality is meaningless? Hmmm...

Peace,
Tony

[Tony Orlow]

On Dec 24, 2:33 pm, Uergil <Uer...@uer.net> wrote:
> In article
> <1788d530-ff86-4342-a49c-ef1576e6e...@n6g2000vbg.googlegroups.com>,

>  Tony Orlow <bonyto...@gmail.com> wrote:
>
> > Well, yes, as are 'A' and 'E'. The fact that "Ax f(x)" is equal to
> > "~Ex ~f(x)" and that "Ex f(x)" is equal to "~Ax ~f(x)" is generally
> > accepted, and essentially the same as Morgan's Law relating "and" and
> > "or".
>
> If you are referring to Augustus de Morgan, and his "laws" that say
> "Not (A and B)" is equivalent to "(not A) or (not B)" and so on, those
> are "de Morgan's laws"
> --
> "Ignorance is preferable to error, and he is less
>  remote from the- truth who believes nothing than
> he who believes what is wrong.
>                      Thomas Jefferson

Thanks, Virge.

Happy Holidays!

Tony

[Tony Orlow]

On Dec 24, 2:54 pm, MoeBlee <modem...@gmail.com> wrote:
> On Dec 24, 11:31 am, Frederick Williams
>
> <freddywilli...@btinternet.com> wrote:
> > How does one distinguish between logical symbols and non-logical
> > symbols?
>
> The logical symbols are the quantifiers and sentential connectives.

No. The quantifiers are ultimately non-logical. That's my whole point.
All the logical connectives are enumerated via Propositional Calculus
and the Truth Table.

> The non-logical symbols are the predicate and function symbols.
> However, often, the 1-place predicate symbol '=' is an exception and
> is regarded as a logical symbol.

That's a 2-place operator, and is **equal** to "<->". :D

>
> > In an account of any particular theory one is told what the
> > logical symbols are, and they are usually chosen from a well-known
> > collection: 'and', 'for all', etc.  But why _those_?
>
> (1) The logical symbols are common to all languages, but the non-
> logical symbols are different for different languages. (2) It is
> convenient in certain formlations to distinguish between the logical
> symbols and non-logical symbols.
>
> MoeBlee

Yes, sometimes. The logical operators are without fuzz. Extending them
begins to get fuzzy, when people cannot distinguish between what is
determined by the math of logic, and what constitutes a step outside
of those boundaries.

Peace,

Tony

[Frederick Williams]

Nam Nguyen wrote:
>
> On 24/12/2011 1:57 PM, Frederick Williams wrote:
> > Nam Nguyen wrote:
> >>
> >>> [...] In an account of any particular theory one is told what the
> >>> logical symbols are, and they are usually chosen from a well-known
> >>> collection: 'and', 'for all', etc. But why _those_?

[...]

> [I'm not quite sure what you're really trying to get at here. :-( ]

That'll be because I didn't (and probably can't) make my meaning clear.
Never mind.

[Nam Nguyen]

It's all context driven, Tony. The context is "membership with degree n"
where n is greater than 1.

Given:

e1 = { {{x}}, {{{x}}}, {{{{{x}}}}} }
e2 = { {{x}}, {{{x}}}, {{{{{{{x}}}}}}} }

In that interpretation of 'e', Az[xe1 <-> xe2] is true.

But would e1=e2 be true you? Why?

[Nam Nguyen]

Of course I meant "Ax[xe1 <-> xe2] is true."

[Tony Orlow]

On Dec 24, 2:48 pm, MoeBlee <modem...@gmail.com> wrote:
> > > My remarks don't depend on a premise that all knowledge comes from
> > > textbooks. I'll repeat what I said:
>
> > Moe: "What textbook in mathematical logic are you using to understand
> > these
> > basics?"
>
> I did not mean for that to imply that knowledge comes only from
> textbooks. So perhaps the question would have been better worded:
>
> Is there a textbook you're using to understand these basics? If so,
> what textbook is it?
>
> > > You are making certain claims about first order logic and a proposed
> > > change to it. As I said, if you tell me a specific textbook that you
> > > refer to for your understanding of first order logic, then I may be
> > > able to refer you to the appropriate passages in that textbook that
> > > explain the crucial particulars you're overlooking.
>
> > It's a little presumptuous to assume that I am overlooking details in
> > some text, when what I am doing is running past other minds an
> > alternative formulation of quantification that has occurred to me
> > while considering how basic propositional calculus is extended to
> > "ordered" logic.
>
> No, you're making claims about how you can "run past" an aspect of
> first order logic but you don't understand really what is actually
> involved in that claim, since your arguments don't take into account a
> number of the crucial details about the subject of first order logic.

Reread. "Run" is used as a transitive verb, with the aspect of FOL as
the direct object, and you as the indirect object. FOL involves
quantifiers over the domain of objects. The point is that they are
used in conjunction some specified universe of discourse, or not, in
which case they operate within the "entire" universe.

>
> > I have yet to see any objection that doesn't rely on
> > assuming that x is some particular object, rather than the variable
> > that it was obviously meant to represent.
>
> For about the thrid time: There is the symbol 'x'. It is a variable.
> Then, per a structure for the language and per an assignment for the
> variables, the symbol 'x' is mapped to some member of the universe for
> the structure.

You can repeat the same thing, but if it didn't make sense the first
time, that won't change.

>
> > > As far as I can tell, you are claiming that you are able to dispense
> > > with the universal quantifier as a primitive by instead adopting
> > > primitive symbols 'U' and 'e'. But you have not shown how that would
> > > work to express the actual semantics of universal quantification.
>
> > > It's as if you are claiming that you know how to dispense with one of
> > > the parts in the Fiat automobile engine and replace it with something
> > > else you've devised. But when you start to describe how you'd do that,
> > > it is clear that you don't know how a Fiat engine works.
>
> > That's a nice analogy, I suppose, but completely off point. If you can
> > give some kind of example where it doesn't work, that would be
> > helpful, but I haven't seen any such exception yet.
>
> I already explained what is lacking in your claim. You don't
> understand it, because you don't have a grasp of the basics of this
> subject.

I already explained the basics, and if you can't translate between
English and Endertonian, that's not my issue.

>
> To be clear: I do not claim that there cannot be a formulation of
> first order logic in which the universal quantifier is replaced with
> 'U' and 'e' while still preserving the ability to express, with an
> alternative method of semantics, that something holds for all members
> of the universe. Rather, my point is that you have not yourself shown
> how do accomplish such a thing. Most plainly, you have not specfied an
> actual and coherent alternative semantical method that you've shown to
> work. And I've given you some of the considerations that you would
> need to address in doing that. Unfortunately, my remarks are going
> right into the ditch, since you don't know the basics of syntax and
> semantics of first order languages.

Instead of concentrating on penmanship, you might rather try
addressing the subject matter.

>
> > Is
> > there a particular question you have about what I am suggesting?
>
> I've already mentioned the key problems with your claim that I find
> most salient. Morevover, again, you have not given a coherent and
> rigorous alternative semantics that accomplishes what you claim.
> You're handwaving while you don't understand the crucial and basic
> considerations you're waving right over.

Stop testing me and try having a conversation for a change.

>
> > > But, in brief, a structure for a language L is a function that assigns
> > > to the universal quantifier a non-empty set (this set we call 'the
> > > universe'), assigns to each n-place predicate symbol of L an n-place
> > > relation on the universe (where n=0, the assignment is to a truth
> > > value, and where n=1, the assignment is to a subset of the universe),
> > > and assigns to each n-place function symbol of L an n-place function
> > > on the universe (where n=0, i.e., the symbol is a constant, the
> > > assignment is to a member of universe).
>
> PLEASE do you understand what I wrote above or not?

Yes. I understand that when Dan said that generally the universal
quantifier is used within the context of some base set, you gave him a
hard time, but that now you are saying the same thing. Come down off
it.

>
> > > to show that one can dispense with the universal quantifier by
> > > instead using the symbols 'U' and 'e' requires specifying a specific
> > > semantics for those symbols and such that that semantics fulfills the
> > > same semantical properties for the universal quantifier.
>
> > > But you have not done that. Moreover, in my earlier posts I mentioned
> > > certain points that you would have to overcome in order to to supply a
> > > successful semantics as just described.
>
> > I rather think I have done that sufficiently. In ZF there is no
> > explanation of what 'e' *means*.
>
> You're completely missing the point and working in a fog as you're
> conflating differnent things. But it's hopeless if you continue to
> refuse to learn how the semantics for first order languages actually
> work.
>
> 'e' is a 2-place predicate symbol. It has no interpretation onto
> itself. It is given an interpretation with a structure for the
> language. The interpretation for 'e' per a given structure for the
> language is some 2 place relation on the universe for the structure.

The "structure" consists entirely of axioms that describe the behavior
of the predicate.

>
> But, more fundamentally, whether we ourselves mention any structure
> for the language, there is still the GENERAL definition of 'structure
> for a language' so that at least we know what a structure for the
> language is.

Yeah. That wasn't what I was talking about.

>
> Now, you are using 'U' and 'e' as special symbols to capture a FIXED
> notion of universal quantification. That is a different situation from
> 'e' ordinarily as a 2-place relation symbol that does not itself have
> a FIXED interpretation (in the sense that even though we generally
> intend 'e' to stand for membership, our semantics itself does not
> REQUIRE that interpretation).

Whatever. It's been obvious from the beginning that 'e' is meant with
the usual "interpretation".

>
> So your situation is more analogous to when we take '=' as having a
> FIXED interpretation, where we require that for ANY interpretation,
> '=' must map to the equality relation on the universe

'=' has a few "interpretations". Been there done that with you, not no
avail, already.

>
> > When I said U is the collection of
> > all conceivable objects, then xeU is vacuously true, as soon as x is
> > conceived.
>
> That is handwaving waffle.

I'll pass the syrup and butter.

>
> What you need to do is give a rigorous specification of your
> alternative semantics. But you can't do that, because you don't have
> the tools to do it, because you refuse to learn the basics of this
> subject.

What YOU need to do is communicate less like a machine, and more like
a mind with a capacity for interpretation of new material.

>
> > > Please refer to a textbook in basic mathematical logic, such as, I
> > > would suggest, Enderton's 'A Mathematical Introduction To Logic'.
>
> > Again, that is not an objection, but an instruction.
>
> Correct. But I've given you specific objections. Your responses to
> those objections are confusions. That is why I recommend that you find
> out how the Fiat engine actually runs before you start claiming that
> it doesn't need a fuel system.

If you had any idea about your Fiat, you wouldn't ask if I'm changing
the oil "in" or "out" of the engine.

>
> > If you have any
> > actual logical objection to the replacement of 'A' with 'e' and 'U',
> > please do state it succinctly.
>
> I HAVE.

You have many times claimed to have given clear objections without
having done so.

>
> > > The interpretation is given by a structure for the language. If the
> > > language has constants, then the structure (which is a certain kind of
> > > function), among other things, assigns to each constant a member of
> > > the universe for the structure.
>
> > If I had included a statement, "xeX", by itself, then 'x' may be taken
> > as some sort of constant, but I did not.
>
> You raised a point about variables and constants. I merely answered
> it.

No, YOU raise the spectre of variable assignments, which are
irrelevant in this conversation.

>
> > I used 'x' as a variable in
> > the condition for an implication, as per the basic logical 2-place
> > operator "->".
>
> I don't know what you refer to in ordinary first order logic with "x
> as a variable in condition for implication" where 'x' is an individual
> variable.

I don't know what Enderton taught you, but I can see it's more
confusion than information.

Any logical statement of implication includes a statement of
conditions under which the conclusion can be held to be true. In this
case, 'x' is a variable used in the statement of the condition, which
also includes the non-logical operator 'e' and the postulated
collection 'U'.

> You seem to have built up your own very personal hodge podge
> of notions about first order logic, based on various bits and pieces
> of you've read here and there. That is hurting you terribly. You need
> to get a good book and read a systematic account of the syntax and
> semantics for first order languages.

So, you put down what I've read, and then tell me to read some more.

Think some more, and get your nose out of a book for a while.

>
> > > The next step is to give, in addition to a structure, an assignment
> > > for the variables, which is a function that assigns to each variable a
> > > member of the universe for the structure.
>
> > So, if I say "AneN (n+1)eN", then I have to specify which natural
> > number n refers to?
>
> No, I said no such thing.
>
> What I said is that IF you give a structure for the language and an
> assignment for the variables, and if 'n' is a variable, then n is
> assigned to some member of the universe. I did not say that is
> required just to utter the formula you mentioned. Structures and
> assignments for the variables are used for the DEFINTIONS of
> 'satisfied' and 'true'; but I did not claim that they are needed
> merely to utter formulas.

Uh, yeah, whatever.

>
> PLEASE, the notions of structure for a language and assignment for the

> variables are basic notions in beginning...
>
> read more »

Google cuts it off here, which is fine. You clearly cannot grasp the
simple flower without throwing glitter and perfume all over it.

Take care,

TOny

[MoeBlee]

On Dec 24, 2:55 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 24/12/2011 1:21 PM, MoeBlee wrote:
>
> > On Dec 24, 12:57 pm, Nam Nguyen<namducngu...@shaw.ca>  wrote:
>
> >> one can _cohesively interpret the epsilon_
> >> _symbol 'e' _ in such a way that the following formula is meaningless
> >> as a non-logical expression and would have no model:
>
> >> Axy[(x=y)<->  Az[zex<->  zey]]
>
> >> So much for the antique belief that every syntactically consistent
> >> formal system would have a model.
>
> > Axy(x=y<->  Az(zex<->  zey))
>
> > has tons of models.
>
> But not per the new interpretation where the degree of membership is
> always greater than 1. (Which is what I conveyed).

I was responding to this post of yours:

[begin post]

On 24/12/2011 2:03 AM, Aatu Koskensilta wrote:

> Yes, with the caveat that in general a formula doesn't express
> anything unless its non-logical symbols are given some interpretation.

And with that, a Pandora box has come into existence in the FOL

model-theoretical world; for, one can _cohesively interpret the

epsilon_
_symbol 'e' _ in such a way that the following formula is meaningless
as a non-logical expression and would have no model:

Axy[(x=y) <-> Az[zex <-> zey]]

So much for the antique belief that every syntactically consistent
formal system would have a model.

[end post]

If you added later something about "degree of membership", then I
hadn't seen it by the time I got the above post. And I can't make
sense of your later remarks:

[begin portion of another post]

Now in the _usual semantic_ of e, we can express a relation between
some
x, and y as:

(*) y = { {{x}}, {{{x}}} }

So the x is a kind of mereological _component_ of y, but not an
_element_ of y in the usual semantic of 'e'. Iow, we don't have
xey in this case.

Note that in this case of "nested membership", the element of
y denoted as the "leftmost" element in (*) has total of 2 left-
brackets
'{' on the left side of 'x'. So given an x and y, we could say
x is a "member of degree n" of y iff y contains at least one element
as:

y = { {{{...{x}...}}}, ... }

where the total number of the left brackets next to x is n.

By convention, if y = {x, ...} then there the membership-degree
of x is just 0.

So, if we interpret 'e' as such kind of "component/membership" such
that n is always greater than 0, then that would be the desired new
interpretation for 'e'.

Something like that.

[end post\

Yeah, something like that. I can't make sense of it. But I'm sure
that's only because I'm stupid and a "dishonest debator" as you've
observed so many times.

In any case, 'e' is a 2-place relation symbol and may be interpreted
as any 2-place relation on the universe. And, of course, with some
structures Axy(x=y<-> Az(zex<-> zey)) is true and with certain other
structures it is false. That is, of course, 'e' may be interpreted so
that Axy(x=y<-> Az(zex<-> zey)) is false.

MoeBlee

[MoeBlee]

On Dec 24, 2:57 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:

>    P v ~P
>
> is logically true because whatever value is assigned to P, it comes out
> to true.  But why are we allowed to treat P as a variable but not v?

Actually 'P' there is not a variable but is 0-place predicate symbol.
However, yes, for different interpretations, 'P' may map to 0 (true)
or 1 (false) depending on the interpretation, while the recursive
definitions of 'satisfied in the structure per the assignment for the
variables' and 'true in the structure' have built in clauses that
mandate how 'v' (the disjunction symbol) determines the satisfaction/
non-satisfaction of formulas and truth/falsehood of sentences per
structures and assignments for the variables.

Of course, anyone is free to stipulate an alternative method of
semantics for the sentential connectives and such that, conceivably,
for whatever purpose, the connectives work differently for different
structures. But, meanwhile, the ordinary method of a fixed semantical
role of each connective is useful for our ordinary purposes of
evaluating formulas and sentences.

MoeBlee

[MoeBlee]

On Dec 24, 3:07 pm, Tony Orlow <bonyto...@gmail.com> wrote:

> or it may be just the question Moe
> was trying to ask.

No, your remarks are not any question I've asked.

MoeBlee

[MoeBlee]

On Dec 24, 3:16 pm, Tony Orlow <bonyto...@gmail.com> wrote:
> On Dec 24, 2:54 pm, MoeBlee <modem...@gmail.com> wrote:

> > The logical symbols are the quantifiers and sentential connectives.
>
> No. The quantifiers are ultimately non-logical.

Why are you bickering with me about ordinary stipulative definitions?
I'm just reporting a certain definition that is widely used in
mathematical logic. I really don't see why you think it worthy for you
to argue about what is just a plain matter of stipulative definition.

These are just techinical terms: 'logical symbols' and 'non-logical
symbols'. I correctly reported their usual definitions in ordinary
mathematical logic. You would be incorrect to read into that that the
word "logical" determines how the terms are used. If that confuses you
then consider each term as one word 'logicalsymbols' and
'nonlogicalsymbols' with no special meaning attached to the letters 'l-
o-g-i-c-a-l' as part of the larger words.

The logical symbols (or logicalsymbols if it would make you happier)
are the connectives and the quantifiers.

The non-logical symbols (or nonlogicalsymbols if is would make you
happier) are the predicate symbols and function symbols.

Again, that's just a report of ordinary stipulative definitions in
mathematical logic. There's no need to argue about such things.

MoeBlee

[MoeBlee]

Orlow, as I said, you would reply with yet more confusions. I can't
perpetually disabuse you or your own confusions that you manufacture
in the vacuum of your knowledge on the subject.

But a few points:

> > > > But, in brief, a structure for a language L is a function that assigns
> > > > to the universal quantifier a non-empty set (this set we call 'the
> > > > universe'), assigns to each n-place predicate symbol of L an n-place
> > > > relation on the universe (where n=0, the assignment is to a truth
> > > > value, and where n=1, the assignment is to a subset of the universe),
> > > > and assigns to each n-place function symbol of L an n-place function
> > > > on the universe (where n=0, i.e., the symbol is a constant, the
> > > > assignment is to a member of universe).
>
> > PLEASE do you understand what I wrote above or not?
>
> Yes. I understand that when Dan said that generally the universal
> quantifier is used within the context of some base set, you gave him a
> hard time, but that now you are saying the same thing. Come down off
> it.

YOU asked me for a definition. I obliged you by giving the definition.
It would help if you'd say whether you understand it or not. Not a
bunch of other business about Christensen. Rather, would you at least
just say whether you understand the definition I gave you.

> > Is there a textbook you're using to understand these basics? If so,
> > what textbook is it?

No reply from you. May I take it that the answer is no? So there is
not a textbook that you have been sourcing so that I can point you to
the relevant sections for you to read yourself?

> > For about the thrid time: There is the symbol 'x'. It is a variable.
> > Then, per a structure for the language and per an assignment for the
> > variables, the symbol 'x' is mapped to some member of the universe for
> > the structure.
>
> You can repeat the same thing, but if it didn't make sense the first
> time, that won't change.

It makes perfect sense. Granted, it would not be simple to understand
if you didn't know its basic context, which is found in many an
ordinary textbook on mathematical logic.

> > I've already mentioned the key problems with your claim that I find
> > most salient. Morevover, again, you have not given a coherent and
> > rigorous alternative semantics that accomplishes what you claim.
> > You're handwaving while you don't understand the crucial and basic
> > considerations you're waving right over.
>
> Stop testing me and try having a conversation for a change.

Perhaps you'd stop testing my PATIENCE.

You've not supported your claim. You have not given a coherent method
of semantics by which universal generalization is expressed without a
quantifier but instead using 'U' and 'e'.

I have not ruled out that one might do it, but the fact is that you
have not, so far, done it. And I suggest that you'd be very unlikely
to do it while you still don't know the very basics of the subject.

> The "structure" consists entirely of axioms that describe the behavior
> of the predicate.

No, that is not what a structure for a language is. I'm using ordinary
terminology of mathematical logic. I'm not using this terminology
according to what you think it should be according to your own
personal whim.

MoeBlee

[Jesse F. Hughes]

Aatu Koskensilta <aatu.kos...@uta.fi> writes:

> Tony Orlow <bony...@gmail.com> writes:
>
>> Okay, thank you, Aatu :) The difference being, I think, that formulas
>> are actually strings of characters with a syntax that expresses a
>> relation between variables, but not all relations can be so expressed.

>
> Yes, with the caveat that in general a formula doesn't express
> anything unless its non-logical symbols are given some interpretation.

You actually understood what he means when he says "formulas are
actually *strings of characters* that expresses a relation between
variables"? And you agree that this is correct?

What *does* it mean, then? I don't get it.

--
"I've been thinking about my problems with getting any kind of
admission that my math arguments showing the core error in mathematics
are correct, so I've gone to marketing books."
-- James S. Harris, on when mathematics isn't enough

[Dan Christensen]

On Dec 24, 11:13 am, Tony Orlow <bonyto...@gmail.com> wrote:
> On Dec 24, 1:39 am, Dan Christensen <Dan_Christen...@sympatico.ca>

> wrote:
>
> > On Dec 23, 8:37 am, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > [snip]
>
> > > So, back to the original point. If we say, "AxeX f(x)", we can easily
> > > rewrite that as, "xeX -> f(x)".
>
> > They don't mean the same thing. "AxeX f(x)" is a statement about ALL
> > elements of X. If we have yeX, then we can infer that f(y).
>
> > "xeX -> f(x)", on the other hand, is a statement about a particular
> > element of X. If we have yeX and ~y=x, we cannot infer that f(y).
>
> What? x is a variable. What "xeX -> f(x)" means is, "if x is an
> element of X, then f(x) is true", for ANY x.
> This sounds like
> MoeBlee's insistence on variable value assignment. I don't understand

> why you think that 'x' is some particular object in this context.
>

You need to be able to make statements about BOTH general and
particular cases. Remember, too, that implication is not about any
causal or temporal relationships. "xeX -> f(x)" simply means ~(xeX &
~f(x))

>
>
> > > If no universe of discourse is
> > > specified, as in "Ax f(x)", then one can surmise that f(x) is true of
> > > ALL objects.
>
> > As you said in your original posting here, "'Ax' is usually followed
> > by the condition of inclusion in some predefined set of objects, the
> > universe of discourse." So, the "f(x)"  above would be an expression
> > of the form g(x) -> h(x), where the "g(x)" is your "condition of
> > inclusion." It could be membership in a set (as in most mathematical
> > theory) or some other logical expression in x.
>
> What I am saying here is that, if there is *no* condition specified
> over which the quantification holds, then "Ax" means that the
> statement following that quantification is *universally", or always,
> true.

[snip]

You can still have the implication g(x) -> h(x) being true for any
object x. In every branch of mathematics (with the possible exception
of set theory itself), the "g(x)" can be used to restrict the
universal quantifier to the desired domain of discussion.

Dan
Download my DC Proof 2.0 software at http://www.dcproof.com
Also see "The Barber Paradox Video"

[MoeBlee]

On Dec 24, 9:57 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:

> In every branch of mathematics (with the possible exception
> of set theory itself), the "g(x)" can be used to restrict the
> universal quantifier to the desired domain of discussion.

In such basic matters, first order logic works in the same way for
whatever "branch of mathematics". I don't know where you get the
notion that set theory is "possibly" different in this way.

It is built into the semantics of first order logic that

Ax(Gx -> Hx) is true per a given structure
iff
every member of the subset named by 'G' per the structure is a member
of the subset named by 'H' per the structure.

So, in that sense, the universal quantifier is
"relativized" ("restricted") to G.

The above holds no matter what theory or branch of mathematics
(including set theory).

MoeBlee

[MoeBlee]

On Dec 24, 9:57 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
wrote:

> You can still have the implication g(x) -> h(x) being true for any
> object x.

In at least some common terminologies, we say that formulas (including
open formulas) are satisfied or not satisfied by a structure and
variable assignment, and that a sentence is true or false by a
structure.

I know that I'm being somewhat pedantic in mentioning that
distinction, since usually there's no harm in refering to an open
formula being "true for an object x", but being mindful of the
distinction can help to avoid confusions that may occur as we get
deeper into the subject.

MoeBlee

[Nam Nguyen]

No big harm here, MoBlee, to the essence of what I mentioned to Aatu.
I used the normal set concept to convey a special kind of mereological
part-hood, and it was a bit glossing: I should have put thing in quotes
like:

"y = { {{{...{x}...}}}, ... }"

to indicate we could go only so far to exemplify this kind of part-hood
with the familiar set-hood concept.

The key thing I conveyed to Aatu (and Tony) is:

>> So much for the antique belief that every syntactically consistent
>> formal system would have a model.

And if you interpret 'e' as this kind of part-hood where, say, 2
parts e1, e2 having exactly 3 same subparts would not necessarily be
identical, then the theory T = {x=y <-> Az[zex <-> zey]} would
not have a model. That is, not in the sense of model as we currently
define what a model is. But neither does it mean there can be no
different model definition for _this_ "set" semantics.

Any rate, what I mentioned there to Aatu and others is correct.

> But I'm sure
> that's only because I'm stupid and a "dishonest debator" as you've
> observed so many times.

For a change, can you be a little more constructive in technical
debate? There have been many posts in the ng's I don't recall
referring you as "stupid" and I really think this conversation
here shouldn't have anything do to with any past. There's such
a thing as "in good faith" you know. If I have a glossing mistake
(or any technical error) just point it out and move on.

Can you do this without being negative and acidic all the time?

> In any case, 'e' is a 2-place relation symbol and may be interpreted
> as any 2-place relation on the universe.

True. But I don't see what you're trying to convey here.

> And, of course, with some
> structures Axy(x=y<-> Az(zex<-> zey)) is true and with certain other
> structures it is false.

Can you give an example where it's false?

> That is, of course, 'e' may be interpreted so
> that Axy(x=y<-> Az(zex<-> zey)) is false.

That doesn't sound different from what I said to Aatu.
Would you agree with me then the following?

>> So much for the antique belief that every syntactically consistent
>> formal system would have a model.

Would that be a yes, or no, to you? (If "No, I don't agree" please
explain why).

[Nam Nguyen]

On 24/12/2011 7:45 PM, Jesse F. Hughes wrote:
> Aatu Koskensilta<aatu.kos...@uta.fi> writes:
>
>> Tony Orlow<bony...@gmail.com> writes:
>>
>>> Okay, thank you, Aatu :) The difference being, I think, that formulas
>>> are actually strings of characters with a syntax that expresses a
>>> relation between variables, but not all relations can be so expressed.
>>
>> Yes, with the caveat that in general a formula doesn't express
>> anything unless its non-logical symbols are given some interpretation.
>
> You actually understood what he means when he says "formulas are
> actually *strings of characters* that expresses a relation between
> variables"? And you agree that this is correct?
>
> What *does* it mean, then? I don't get it.

Let's recall that constants and terms are variables, thus standing for
individuals in a model. If model relations are about individuals then
formulas would be about variables: predicates (possibly with
non-logical symbols) of variables, connected by logical symbols,
naturally.

[Dan Christensen]

On Dec 25, 12:44 am, MoeBlee <modem...@gmail.com> wrote:
> On Dec 24, 9:57 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
> wrote:
>
> > In every branch of mathematics (with the possible exception
> > of set theory itself), the "g(x)" can be used to restrict the
> > universal quantifier to the desired domain of discussion.
>
> In such basic matters, first order logic works in the same way for
> whatever "branch of mathematics". I don't know where you get the
> notion that set theory is "possibly" different in this way.
>

It depends on whether or not your set theory has an "is a set"
predicate. If everything is a set (i.e. there is no "is as set"
predicate), the domain of discussion for set theory is the
unrestricted. If you do have an "is a set" predicate, the domain of
discussion is restricted to those things designated as sets and their
elements.

[Nam Nguyen]

I'm not in physics but I think the number of possible "fundamental"
sub-atomic particles that could exist in principle would be infinite,
with one consisting the others, each of which in turn consisting yet
others, etc...

So somewhere along the same line I think it seems plausible this
different semantic of 'e' could be used to formalize sub-atomic
particles.

Now there was this conversation in the other thread:

On 24/12/2011 11:03 AM, Nam Nguyen wrote:
> On 24/12/2011 5:58 AM, David C. Ullrich wrote:
>> On Sat, 24 Dec 2011 00:51:51 +0200, Aatu Koskensilta
>> <aatu.kos...@uta.fi> wrote:
>>
>>> "INFINITY POWER"<infi...@limited.com> writes:
>>>
>>>> STUPID PEOPLE LIKE AATU SAY
>>>>
>>>> "WE CAN FORMALISE ANYTHING SIMPLY BY ADDING SOME MORE RIGOUR!"
>>>
>>> I'm famous for my stupidity, but I don't think I've ever said that.
>>
>> I don't recall you ever saying that either. But I bet you wish you
>> had, eh? It's inspiring. We can formalize anything by adding more
>> rigour.
>>
>> Just yesterday I was eating this grilled-cheese sandwich.
>> I wanted to formalize it, didn't see how.
>
> Just wonder if one could start by formalizing neutrinos?

So it looks like after all we might be able to formalize neutrinos!

Anyone cares to take it over from here? :-)

[Graham Cooper]

On Dec 25, 3:44 pm, MoeBlee <modem...@gmail.com> wrote:
> On Dec 24, 9:57 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
> wrote:
>
> > In every branch of mathematics (with the possible exception
> > of set theory itself), the "g(x)" can be used to restrict the
> > universal quantifier to the desired domain of discussion.
>
> In such basic matters, first order logic works in the same way for
> whatever "branch of mathematics". I don't know where you get the
> notion that set theory is "possibly" different in this way.
>

We don't! You do, by calling proofs of 2nd Order Logic Formula
"simple elegant proofs in 1st Order Logic!"



CANTOR'S THEOREM
ALL(f):N->R .... For all function mappings from N to R
E(r):R ......... there exists a real r
!E(n):N ........ such that there is no index n
f(n)=r ......... that maps to r

[Tony Orlow]

On Dec 24, 7:55 pm, MoeBlee <modem...@gmail.com> wrote:
> Orlow, as I said, you would reply with yet more confusions. I can't
> perpetually disabuse you or your own confusions that you manufacture
> in the vacuum of your knowledge on the subject.

I do have a bachelor's in Computer Science, which consists of logical
structures as well as other informational representations and
structures. If that is a vacuum of knowledge, then I suppose I have to
be a published PhD for you to consider my thoughts as being remotely
worth consideration. A rather pompous position for you to take
IMHO....

>
> But a few points:
>
> > > > > But, in brief, a structure for a language L is a function that assigns
> > > > > to the universal quantifier a non-empty set (this set we call 'the
> > > > > universe'), assigns to each n-place predicate symbol of L an n-place
> > > > > relation on the universe (where n=0, the assignment is to a truth
> > > > > value, and where n=1, the assignment is to a subset of the universe),
> > > > > and assigns to each n-place function symbol of L an n-place function
> > > > > on the universe (where n=0, i.e., the symbol is a constant, the
> > > > > assignment is to a member of universe).
>
> > > PLEASE do you understand what I wrote above or not?
>
> > Yes. I understand that when Dan said that generally the universal
> > quantifier is used within the context of some base set, you gave him a
> > hard time, but that now you are saying the same thing. Come down off
> > it.
>
> YOU asked me for a definition. I obliged you by giving the definition.
> It would help if you'd say whether you understand it or not. Not a
> bunch of other business about Christensen. Rather, would you at least
> just say whether you understand the definition I gave you.

I didn't ask you for anything, but I understand that your statement
above includes exactly the same statement as you gave Dan problems
about, that the universal quantifier is generally used in the context
of some non-empty set as the universe of discourse and the domain of
the quantification. I don't particularly feel any obligation to
comment on every clause of your verbose over-complication of the
question.

>
> > > Is there a textbook you're using to understand these basics? If so,
> > > what textbook is it?
>
> No reply from you. May I take it that the answer is no? So there is
> not a textbook that you have been sourcing so that I can point you to
> the relevant sections for you to read yourself?

You said you should have stated it that way instead of your original
language, and I had already answered the question, so what are you on
about?

>
> > > For about the thrid time: There is the symbol 'x'. It is a variable.
> > > Then, per a structure for the language and per an assignment for the
> > > variables, the symbol 'x' is mapped to some member of the universe for
> > > the structure.
>
> > You can repeat the same thing, but if it didn't make sense the first
> > time, that won't change.
>
> It makes perfect sense. Granted, it would not be simple to understand
> if you didn't know its basic context, which is found in many an
> ordinary textbook on mathematical logic.

Any particular instance of x involves its assignment to an element of
the universe of discourse, but evaluating the general statement using
the variable x does not require such a particular assignment.

>
> > > I've already mentioned the key problems with your claim that I find
> > > most salient. Morevover, again, you have not given a coherent and
> > > rigorous alternative semantics that accomplishes what you claim.
> > > You're handwaving while you don't understand the crucial and basic
> > > considerations you're waving right over.
>
> > Stop testing me and try having a conversation for a change.
>
> Perhaps you'd stop testing my PATIENCE.
>
> You've not supported your claim. You have not given a coherent method
> of semantics by which universal generalization is expressed without a
> quantifier but instead using 'U' and 'e'.

What do you want beyond saying that "Ax f(x)" is the same as "xeU ->
f(x)", where 'e' is the usual membership operator, f(x) is a statement
using x as a variable, and U is the universe of discourse? What detail
of the interpretation do you not understand? If you had any patience,
you wouldn't be making straw-man arguments.

>
> I have not ruled out that one might do it, but the fact is that you
> have not, so far, done it. And I suggest that you'd be very unlikely
> to do it while you still don't know the very basics of the subject.
>
> > The "structure" consists entirely of axioms that describe the behavior
> > of the predicate.
>
> No, that is not what a structure for a language is. I'm using ordinary
> terminology of mathematical logic. I'm not using this terminology
> according to what you think it should be according to your own
> personal whim.
>
> MoeBlee

Nice out-of-context response. Whatever, Moe.

Take it easy,

TOny

[Tony Orlow]

Then you might want to stop arguing. My whole original point is that
the quantifiers are NOT logical operators, which are normally defined
as operators that take some natural number of logical truth values (0
or 1, false or true) and return a logical value as a result. If 'A' is
taken as an operator, then it is a 2-place operator, the variables
being an object, and a statement about that object, neither of which
are logical truth values, even though the value of the entire
expression is either true or false.

When you use the term "logical operator", is the word "logical"
completely superfluous and meaningless? If so, don't include it, as it
serves only to confuse your communication.

Peace,

Tony

[Barb Knox]

In article
<956f09a9-e075-4305...@e2g2000vbb.googlegroups.com>,

Tony Orlow <bony...@gmail.com> wrote:

> On Dec 24, 7:55 pm, MoeBlee <modem...@gmail.com> wrote:
> > Orlow, as I said, you would reply with yet more confusions. I can't
> > perpetually disabuse you or your own confusions that you manufacture
> > in the vacuum of your knowledge on the subject.
>
> I do have a bachelor's in Computer Science, which consists of logical
> structures as well as other informational representations and
> structures. If that is a vacuum of knowledge, then I suppose I have to
> be a published PhD for you to consider my thoughts as being remotely
> worth consideration. A rather pompous position for you to take
> IMHO....

He did say a vacuum of knowledge *on the subject*, which happens to be
mathematical logic. An encyclopaedic knowledge of (say) Etruscan
pottery would be useless here, and IMO a knowledge of computer "science"
(i.e., computer programming) is *worse* than useless.

Yes, you have worked with certain logical structures and certain finite
representations and structures. But trying to force all of mathematics
onto that Procrustean bed causes no end of mutilation.

Perhaps you have heard the old CS saying that if you know Fortran then
you can learn Lisp in 3 weeks, and if you don't know Fortran you can
learn Lisp in 2 weeks. (To modernise this, replace "Fortran" with "C".)

The point is that some kinds of knowledge induce a worldview that
actively interferes with some other kinds of knowledge. In your case
(and many others), knowledge of one-step-at-a-time stateful finite
processes and structures actively interferes with all-at-once stateless
infinite mathematics.

If some publisher wanted it, I would be happy to write a book on
Remedial Mathematics for Computer Programmers. But the demand would be
low, since so many competent computer programmers confuse their godlike
powers of calling things into existence by mere thought (and typing)
with actual omniscience. Oh well.

</rant>

[snip]

[added comp.edu]

--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum videtur.
| BBB aa a r bbb |
-----------------------------

[K_h]

"Tony Orlow" <bony...@gmail.com> wrote in message
news:b420245f-db20-4612...@h13g2000vbn.googlegroups.com...

Why not just browse the web and get the definitions right? We see that
`logical operator', `logical symbol', and `logical connective' are largely
synonymous:

http://en.wikipedia.org/wiki/Logical_connective

http://en.wikipedia.org/wiki/List_of_logic_symbols

It is also clear that quantifiers are categorized differently (not as logical
symbols):

http://en.wikipedia.org/wiki/Quantification

_

[K_h]

"K_h" <KHo...@SX729.com> wrote in message
news:mIWdnakbzIZZn2TT...@giganews.com...

Correction to a typo: I meant to write (but also as logical symbols) not (not

[Tony Orlow]

On Dec 26, 6:57 pm, "K_h" <KHol...@SX729.com> wrote:
> "K_h" <KHol...@SX729.com> wrote in message
>
> news:mIWdnakbzIZZn2TT...@giganews.com...
>
>
>
>
>
> > "Tony Orlow" <bonyto...@gmail.com> wrote in message

Yes, but not as logical operators, that is, operators that take
logical parameters, rather than an object and a statement about that
object.

TOny

[MoeBlee]

On Dec 26, 3:02 pm, Tony Orlow <bonyto...@gmail.com> wrote:
> On Dec 24, 7:55 pm, MoeBlee <modem...@gmail.com> wrote:
>
> > Orlow, as I said, you would reply with yet more confusions. I can't
> > perpetually disabuse you or your own confusions that you manufacture
> > in the vacuum of your knowledge on the subject.
>
> I do have a bachelor's in Computer Science, which consists of logical
> structures as well as other informational representations and
> structures. If that is a vacuum of knowledge, then I suppose I have to
> be a published PhD for you to consider my thoughts as being remotely
> worth consideration. A rather pompous position for you to take
> IMHO....

I said "ON THE SUBJECT" [emphasis added]. A PhD is not required. But
what is required at least is an understanding of the basics of first
order logic.

> > YOU asked me for a definition. I obliged you by giving the definition.
> > It would help if you'd say whether you understand it or not. Not a
> > bunch of other business about Christensen. Rather, would you at least
> > just say whether you understand the definition I gave you.
>
> I didn't ask you for anything,

You asked for a definition of 'structure for a language'. I gave you a
definition. I'm asking whether you understand that definition.
Understanding that definition would be a step in understanding the
basics of the subject under discussion. I don't see why you won't say
whether or not you understand the definition. And I don't know why you
would ask me for the definition if you don't have a purpose to
understand it.

> but I understand that your statement
> above includes exactly the same statement as you gave Dan problems
> about,

No, in the abysmal fog of your ignorance and confusion on the basics
of this subject, you conflate the definition I gave with something
about what I said about Christensen's notions.

> that the universal quantifier is generally used in the context
> of some non-empty set as the universe of discourse and the domain of
> the quantification.

Again, now for the tenth time, a universe of discourse is given with a
structure for the langauge. What I gave is a definition of 'structure
for a language'. And that does not contradict anything I said to
Christensen.

> > > > Is there a textbook you're using to understand these basics? If so,
> > > > what textbook is it?
>
> > No reply from you. May I take it that the answer is no? So there is
> > not a textbook that you have been sourcing so that I can point you to
> > the relevant sections for you to read yourself?
>
> You said you should have stated it that way instead of your original
> language, and I had already answered the question, so what are you on
> about?

> Any particular instance of x involves its assignment to an element of
> the universe of discourse, but evaluating the general statement using
> the variable x does not require such a particular assignment.

You say that Ax Fx can be written instead as xeU -> Fx

And, if I guess correctly (since I can only guess, since you have not
given an explicit semantics), xeU always evaluates as satisfied. But
then, per any interpretation and assignment for the variables, and for
ANY formula P, we have that xeU -> P is satsified iff P is satisfied.

But it is a plain fact of first order semantics that it is not the
case that in general Ax P is equivalent with P, or more specifically
that Ax Fx is equivalent with Fx.

And you have not given a specification of an alternative semantics for
first order languages that provides for expressing universal
generalization with 'e' and 'U' instead of the universal quantifer.

No matter your college degree in this or that, your claim is not
substantiated. You could have a PhD in mathematics and be the chairman
of the Institute for Advanced Studies, and your claim would still be
unsubstantiated unless you specify a semantics that provides for
expressing universal generalization with 'e' and 'U' instead of the
universal quantifer.

And just saying use 'xeU -> Fx' in place of Ax Fx is just IGNORANT, as
I explained in previous posts and as I just explained again.

One more time: IF xeU is always satisfied (does it, according to your
semantics?) then

per any structure and assignment for the variables,

xeU -> Fx

is satistied iff Fx is satisfied.

And if you have some semantical method other than structures and
assignments for the variables, then you need to specify your
semantical method, hopefully with inductive definitions on the
complexity of formulas.

MoeBlee

[Graham Cooper]

On Dec 27, 3:52 pm, MoeBlee <modem...@gmail.com> wrote:
> On Dec 26, 3:02 pm, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > On Dec 24, 7:55 pm, MoeBlee <modem...@gmail.com> wrote:
>
> > > Orlow, as I said, you would reply with yet more confusions. I can't
> > > perpetually disabuse you or your own confusions that you manufacture
> > > in the vacuum of your knowledge on the subject.
>
> > I do have a bachelor's in Computer Science, which consists of logical
> > structures as well as other informational representations and
> > structures. If that is a vacuum of knowledge, then I suppose I have to
> > be a published PhD for you to consider my thoughts as being remotely
> > worth consideration. A rather pompous position for you to take
> > IMHO....
>
> I said "ON THE SUBJECT" [emphasis added]. A PhD is not required. But
> what is required at least is an understanding of the basics of first
> order logic.

WARNING! MoeBlee thinks this is 1st Order Logic!

ALL(f):N->R .... For all function mappings from N to R
E(r):R ......... there exists a real r
!E(n):N ........ such that there is no index n
f(n)=r ......... that maps to r







>

MoeBlee actually got this right, let me paraphrase:

In a set theory with U, xeU is true for all x.

So xeU -> f(x)

is TRUE -> f(x)

is f(x)

****

This doesn't mean your FORALL Quantifier elimination theory is wrong,
it just supports that:

ALL(x): f(x)

is

f(x)

i.e. you can just remove all instances of ALL(?)
and it's the same logical formula, ALL(?) is redundant.


Herc

[MoeBlee]

On Dec 26, 2:49 pm, Tony Orlow <bonyto...@gmail.com> wrote:

> Then you might want to stop arguing. My whole original point is that
> the quantifiers are NOT logical operators, which are normally defined
> as operators that take some natural number of logical truth values (0
> or 1, false or true) and return a logical value as a result.

No that is NOT what you said as this tangent began:

On Dec 24, 3:16 pm, Tony Orlow <bonyto...@gmail.com> wrote:
> On Dec 24, 2:54 pm, MoeBlee <modem...@gmail.com> wrote:
>

> > On Dec 24, 11:31 am, Frederick Williams
>
> > <freddywilli...@btinternet.com> wrote:
> > > How does one distinguish between logical symbols and non-logical
> > > symbols?

>
> > The logical symbols are the quantifiers and sentential connectives.
>
> No. The quantifiers are ultimately non-logical

I said that the "logical SYMBOLS" [emphasis added] are the quantifiers
and sentential connectives.

And I said nothing about "operators" or "operators on truth values"

And you said, "No" to CONTRADICT my correct report of the ordinary
terminology "logical symbols".

And I've told you that this is merely a STIPULATIVE definition. It
happens to be the way certain authors use the terminology 'logical
symbols". Other authors may use the terminology differently, and
indeed I mentioned that Enderton has his own terminology that is
different. But it is a plain fact that it is an ordinary terminology
in which the logical symbols are the quantifiers and connectives
(actually, sometimes the variables too, which I should have mentioned)
and the non-logical symbols are the function and predicate symbols.

You were incorrect to say "No" when I merely correctly reported
certain ordinary terminology.

> When you use the term "logical operator",

I said NOTHING about the term "logical operator".

I said "logical symbol".

> When you use the term "logical operator", is the word "logical"
> completely superfluous and meaningless? If so, don't include it, as it
> serves only to confuse your communication.

A butterfly is neither a fly nor is it made of butter. So I wish
people would call it something else.

I could guess why the word 'logical' is used in 'logical symbols' is
used. But whatever the ETYMOLOGICAL reason for using the word
'logical' in the particular term 'logical symbols', the plain fact is
that it is an ordinary terminology in which the universal quantifier
is counted as a logical symbol.

And whether that is confusing or not, I don't opine. If it were up to
me, lots of ordinary terminology would be different. But that's not
what was at issue. Rather, I merely correctly reported ordinary
terminology, and you incorrectly disputed me.

MoeBlee

[MoeBlee]

On Dec 26, 6:10 pm, Tony Orlow <bonyto...@gmail.com> wrote:

> Yes, but not as logical operators, that is, operators that take
> logical parameters,

I said NOTHING about "operators".

YOU even included that quote in your own post to which I'm replying
now:

> > >> > > On Dec 24, 2:54 pm, MoeBlee <modem...@gmail.com> wrote:
> > >> > > > The logical symbols are the quantifiers and sentential connectives.
>
> > >> > > No. The quantifiers are ultimately non-logical.

Again, I gave a correct report of certain ordinary terminology in
mathematical logic. I did not claim that it is the best terminology or
that it is not confusing or even that it is terminology always used or
even always used in the manner I reported. Rather, Aatu had stated the
basics of the terminology, and his use of the terminology is quite
ordinary in mathematical logic, and I added some comments about the
notion of logical symbols vs. non-logical symbols in accordance with
the ordinary usage Aatu had first mentioned. Then you posted "No" to
dispute me. And you're STILL arguing after I made clear this is a
mater merely of STIPULATIVE definition, and further you changed the
subject from "logical SYMBOLS" to "logical OPERATORS".

Now please just shut up about it and move on if you don't have the
intellectual honesty to admit you were wrong.

MoeBlee

[MoeBlee]

On Dec 26, 11:57 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:

> WARNING!  MoeBlee thinks this is 1st Order Logic!
>
> ALL(f):N->R .... For all function mappings from N to R
> E(r):R ......... there exists a real r
> !E(n):N ........ such that there is no index n
> f(n)=r ......... that maps to r

WARNING! Graham Cooper is lying.

I never said that the formulations above are first order logic.

What I said is that the diagonal argument can be formalized in first
order set theory.

Graham Cooper may write formulas that he proposes as formalizations of
the diagonal argument in whatever language. That does not contradict
my correct claim that the diagonal argument can be formalized in first
order set theory.

> > IF xeU is always satisfied (does it, according to your semantics?) then per any structure and
> > assignment for the variables,

> > xeU -> Fx

> > is satistied iff Fx is satisfied.

> MoeBlee actually got this right, let me paraphrase:

> In a set theory with U, xeU is true for all x.

NO, that is NOT a paraphrase of what I said.

Neither in, for example, Z set theories nor in NBG class theory is
there a U such that every x is in U.

What I DID say is that if Orlow specifies that xeU is always
satisfied, then

no matter the structure or variable assignment, xeU -> Fx is satisifed
iff Fx is satisfied.

Graham Cooper, I've asked you before not to put words in my mouth.

MoeBlee

[Graham Cooper]

You're a stupid idiot, somehow you managed to parrot the decore of a
logicians typings but the nonsense behind your thought processes
glares ever more obvious the more you try to correct yourself.

Herc

[Graham Cooper]

On Dec 27, 4:37 pm, MoeBlee <modem...@gmail.com> wrote:

> On Dec 26, 11:57 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > WARNING!  MoeBlee thinks this is 1st Order Logic!
>
> > ALL(f):N->R .... For all function mappings from N to R
> > E(r):R ......... there exists a real r
> > !E(n):N ........ such that there is no index n
> > f(n)=r ......... that maps to r
>
> WARNING! Graham Cooper is lying.
>
> I never said that the formulations above are first order logic.
>
> What I said is that the diagonal argument can be formalized in first
> order set theory.
>
> Graham Cooper may write formulas that he proposes as formalizations of
> the diagonal argument in whatever language. That does not contradict
> my correct claim that the diagonal argument can be formalized in first
> order set theory.

You're so stupid it's not funny!

*I* am not formalising Cantor's Theorem in anything.

*CANTOR'S THEOREM* __IS__ a formula of 2OL.

Here it is MORON MOEBLEE!

[MoeBlee]

Whatever you're doing, or think you're doing, when you write:

> ALL(f):N->R .... For all function mappings from N to R
> E(r):R ......... there exists a real r
> !E(n):N ........ such that there is no index n
> f(n)=r ......... that maps to r

it's not anything that I said is first order logic.

MoeBlee

[Graham Cooper]

Is Cantor's Theorem a theorem in FOL?

Is the following true MoeBlee?

"MoeBlee can not prove this statement is true."


Herc

[Graham Cooper]

On Dec 27, 4:37 pm, MoeBlee <modem...@gmail.com> wrote:

> On Dec 26, 11:57 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > WARNING!  MoeBlee thinks this is 1st Order Logic!
>
> > ALL(f):N->R .... For all function mappings from N to R
> > E(r):R ......... there exists a real r
> > !E(n):N ........ such that there is no index n
> > f(n)=r ......... that maps to r
>
> WARNING! Graham Cooper is lying.
>
> I never said that the formulations above are first order logic.
>
> What I said is that the diagonal argument can be formalized in first
> order set theory.
>
> Graham Cooper may write formulas that he proposes as formalizations of
> the diagonal argument in whatever language. That does not contradict
> my correct claim that the diagonal argument can be formalized in first
> order set theory.
>
> > > IF xeU is always satisfied (does it, according to your  semantics?) then per any structure and
> > > assignment for the variables,
> > > xeU -> Fx
> > >  is satistied iff Fx is satisfied.
> > MoeBlee actually got this right, let me paraphrase:
> > In a set theory with U, xeU is true for all x.
>
> NO, that is NOT a paraphrase of what I said.
>
> Neither in, for example, Z set theories nor in NBG class theory is
> there a U such that every x is in U.
>

YOU'RE A STUPID IDIOT!

IT'S NOT A DEFINITION OF U THEN IS IT!

> What I DID say is that if Orlow specifies that xeU is always
> satisfied, then
>
> no matter the structure or variable assignment, xeU -> Fx is satisifed
> iff Fx is satisfied.
>
> Graham Cooper, I've asked you before not to put words in my mouth.
>
> MoeBlee

YOU SIR! ARE THE BIGGEST FUCKING IDIOT I HAVE EVER KNOWN.

A PARAPHRASE YOU FUCKHEAD

EVEN IF I AGREE WITH THE ONLY POINT YOU'VE EVER GOT RIGHT YOU STILL
SPEW RUBBISH ABOUT IT

[MoeBlee]

On Dec 27, 2:04 am, Graham Cooper <grahamcoop...@gmail.com> wrote:

> YOU SIR!  ARE THE BIGGEST FUCKING IDIOT I HAVE EVER KNOWN.

At least it's nice that you call me 'sir'.

MoeBlee

[Graham Cooper]

sure you talk the talk,

but you balk the balk.

Herc

[MoeBlee]

On Dec 27, 1:29 am, Graham Cooper <grahamcoop...@gmail.com> wrote:

> Is Cantor's Theorem a theorem in FOL?

(1) "Cantor's theorem" refers to the result that there is no function
from a set onto its power set. It can be formalized and proven in
first order set theory.

(2) Cantor also gave an argument that there is no function from the
set of natural numbers onto the set of denumerable binary sequences.
It can be formalized and proven in first order set theory.

(3) From (2) we infer (also in the manner of Cantor) that there is no
function from the set of natural numbres onto the set of real numbers,
so, since there is an injection from the set of natural numbers into
the set of real numbers, we infer that the set of real numbers
strictly dominates the set of natural numbers. All of that too can be
formalized and proven in first order set theory.

In any case,

> > > ALL(f):N->R .... For all function mappings from N to R
> > > E(r):R ......... there exists a real r
> > > !E(n):N ........ such that there is no index n
> > > f(n)=r ......... that maps to r

whatever it is, it is not anything that I said is first order logic.

MoeBlee

[Tony Orlow]

Hi Moe -

I'm going to skip over the above, since it doesn't seem at all
productive, but the following looks to be part of a reasonable
discussion, so I'll try to comment as much to your satisfaction as I
can manage. :)

>
> You say that Ax Fx can be written instead as xeU -> Fx
>
> And, if I guess correctly (since I can only guess, since you have not
> given an explicit semantics), xeU always evaluates as satisfied. But
> then, per any interpretation and assignment for the variables, and for
> ANY formula P, we have that xeU -> P is satsified iff P is satisfied.

That is a reasonable statement, except for the "But then" part. If one
says "Ax Fx", then is that statement also not only to be held true
where Fx is true "for all x"? "Ax x=x" is false, if you can provide
even a single x which is not equal to itself.

>
> But it is a plain fact of first order semantics that it is not the
> case that in general Ax P is equivalent with P, or more specifically
> that Ax Fx is equivalent with Fx.

What is the difference between "Ax Fx" and "Fx", in your mind, where x
is quantified over an unspecified universe? If the universe of
discourse is specified, then certainly the universal quantifier
applies only over that domain, but in the absence of any base set, the
base set can be considered to be the entire universe.

For instance, what is the difference between "all pigs like truffles"
and "if an animal is a pig, then it likes truffles", or for that
matter, simply, "pigs like truffles"? If I say, "things are what they
are." what does it add to prefix it by "for all things"?

>
> And you have not given a specification of an alternative semantics for
> first order languages that provides for expressing universal
> generalization with 'e' and 'U' instead of the universal quantifer.
>
> No matter your college degree in this or that, your claim is not
> substantiated. You could have a PhD in mathematics and be the chairman
> of the Institute for Advanced Studies, and your claim would still be
> unsubstantiated unless you specify a semantics that provides for
> expressing universal generalization with 'e' and 'U' instead of the
> universal quantifer.

Your objection is unsubstantiated until you can express what the
difference in meaning is between the two expressions. You have not
done so, and so you have not really made any point.

>
> And just saying use 'xeU -> Fx' in place of Ax Fx is just IGNORANT, as
> I explained in previous posts and as I just explained again.

That''s just another ad hominem without justification. You have not
explained what you think the difference in meaning is between the two
statements. Is there some symbol I am using that you don't understand?
They all look pretty standard to me, except for the universal
collection U, which is replaced with a base set whenever one is
specified.

>
> One more time: IF xeU is always satisfied (does it, according to your
> semantics?) then
>
> per any structure and assignment for the variables,
>
> xeU -> Fx
>
> is satistied iff Fx is satisfied.

Right. Same for "Ax Fx", no? I really can't discern any difference in
this respect. It's kind of like you are objecting because my pig likes
truffles, because it is a pig, and there is no pig that doesn't like
truffles, so my pig HAS to like truffles. I just don't see your
distinction or objection.

>
> And if you have some semantical method other than structures and
> assignments for the variables, then you need to specify your
> semantical method, hopefully with inductive definitions on the
> complexity of formulas.
>
> MoeBlee

Peace,

Tony

[Tony Orlow]

That's pretty much all I am saying, and I haven't seen a particular
objection or counter-example to it yet.

Take care,

Tony

[Tony Orlow]

On Dec 27, 1:15 am, MoeBlee <modem...@gmail.com> wrote:
> On Dec 26, 2:49 pm, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > Then you might want to stop arguing. My whole original point is that
> > the quantifiers are NOT logical operators, which are normally defined
> > as operators that take some natural number of logical truth values (0
> > or 1, false or true) and return a logical value as a result.
>
> No that is NOT what you said as this tangent began:
>
> On Dec 24, 3:16 pm, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > On Dec 24, 2:54 pm, MoeBlee <modem...@gmail.com> wrote:
>
> > > On Dec 24, 11:31 am, Frederick Williams
>
> > > <freddywilli...@btinternet.com> wrote:
> > > > How does one distinguish between logical symbols and non-logical
> > > > symbols?
>
> > > The logical symbols are the quantifiers and sentential connectives.
>
> > No. The quantifiers are ultimately non-logical
>
> I said that the "logical SYMBOLS" [emphasis added] are the quantifiers
> and sentential connectives.
>
> And I said nothing about "operators" or "operators on truth values"
>
> And you said, "No" to CONTRADICT my correct report of the ordinary
> terminology "logical symbols".

I think you misread that, and I don't care to dredge up every
conversation. I'm not going through marriage counseling with you. I
made it clear from the beginning that I was talking about the initial
extension of basic propositional calculus, where symbols denote only
objects and logical connectives, and suggesting that adding
quantifiers over objects to achieve FOL is equivalent to adding the
non-logical operator 'e' and the collection 'U', which of course may
be replaced with any specified universe of discourse. If that wasn't
clear to you then, let it be so now, and move on.

>
> And I've told you that this is merely a STIPULATIVE definition. It
> happens to be the way certain authors use the terminology 'logical
> symbols". Other authors may use the terminology differently, and
> indeed I mentioned that Enderton has his own terminology that is
> different. But it is a plain fact that it is an ordinary terminology
> in which the logical symbols are the quantifiers and connectives
> (actually, sometimes the variables too, which I should have mentioned)
> and the non-logical symbols are the function and predicate symbols.

So, you're arguing about the terminology of various authors, and
*don't* see this as a distraction and derailment of what was
originally a pretty simple topic? I am not interested in your
obsession with knowing the quirks of expression with every author.
Most of us here uses his own flavor of logical language, and to be
honest, some people's formulations are easier for me to grasp than
others', but I am not going to claim that one is better than another
without some evidence of benefit in efficiency, simplicity, or clarity
of expression. I simply try to grasp what the underlying logical point
is that one is trying to make. Maybe growing up in Manhattan with a
lot of broken English was helpful in that respect...

>
> You were incorrect to say "No" when I merely correctly reported
> certain ordinary terminology.

I'm not doing counseling with you. You can argue about that with your
wife. I have no interest in anything but the topic at hand.
Quantifiers are not logical operators. They are not represented in
computer instructions or logic gates, and yet, they are expressible
nonetheless, due to the interpretation of membership in some
particular universe or domain of discourse. The very introduction of
quantifiers is equivalent to the introduction of the non-logical set
membership operator.

>
> > When you use the term "logical operator",
>
> I said NOTHING about the term "logical operator".
>
> I said "logical symbol".

Well, you didn't start the thread, and should not try to derail and
monopolize it with your own irrelevant ruminations concerning
"symbols".

>
> > When you use the term "logical operator", is the word "logical"
> > completely superfluous and meaningless? If so, don't include it, as it
> > serves only to confuse your communication.
>
> A butterfly is neither a fly nor is it made of butter. So I wish
> people would call it something else.

It used to be called a "flutterby", which is precisely what it does
(since you are bringing up history and all). :)

The point remains that you seem to be sing the word "logical" for
essentially no reason that you can explain except with gibberish like
the above.

>
> I could guess why the word 'logical' is used in 'logical symbols' is
> used. But whatever the ETYMOLOGICAL reason for using the word
> 'logical' in the particular term 'logical symbols', the plain fact is
> that it is an ordinary terminology in which the universal quantifier
> is counted as a logical symbol.

I don't care what is ordinary. I care what is correct. And, the fact
remains, that the universal quantifier implies inclusion in some
domain of discourse, and so, is equivalent to the introduction of a
non-logical operator. Do you disagree?

>
> And whether that is confusing or not, I don't opine. If it were up to
> me, lots of ordinary terminology would be different. But that's not
> what was at issue. Rather, I merely correctly reported ordinary
> terminology, and you incorrectly disputed me.
>
> MoeBlee

Nobody asked you what the "ordinary terminology" is for whatever
you're on about. That's irrelevant, but have a nice day.

Tony

[Tony Orlow]

Despite his vitriolic delivery, I think Herc makes an interesting
point, actually. As soon as you start talking about a mapping from one
set to another, you are bringing in relations between objects, and you
really are talking on what is considered a second-order level, no? The
generation of that anti-diagonal involves a mapping from a set of
strings to a single string via a formula, if you will. I am not saying
you are necessarily wrong, Moe. But, it may be that such a FOL
formulation of the diagonal argument rests on a usage of quantifiers
that masks a second-order nature to the statement. After all, Herc's
SOL statement above seems to capture the argument. Of course, while
that may an interesting topic, and somewhat related to my point, I'm
not sure it's germane.

Peace,

TOny

[Tonico]

I can't but notice the interesting yet queer coincidence that you
write "...pretty much ALL I am saying..." immediately after the one
that flew over the cuckoo's nest wrote that ALL can be removed "in ALL
instances"...again all!

Would you say that was paradoxical...?

Tonio

[Frederick Williams]

Aatu Koskensilta wrote:

>
> Frederick Williams <freddyw...@btinternet.com> writes:
>
> > How does one distinguish between logical symbols and non-logical

> > symbols? In an account of any particular theory one is told what the
> > logical symbols are, and they are usually chosen from a well-known
> > collection: 'and', 'for all', etc. But why _those_?
>
> The basic idea is that they are logical in the sense that they "mean
> the same" in any structure, on any interpretation. Giving a principled
> mathematical (or philosophical) account of this notion is a classical
> problem -- on which old Alfred T. himself had a few things to say. You
> will find illuminating discussion (and helpful references for further
> study) in Sol Feferman's recent paper /Which Quantifiers are Logical?/:
>
> http://math.stanford.edu/~feferman/papers/WhichQsLogical(text).pdf
>
> and in the _Standford Encyclopedia of Philosophy_ article on logical
> constants:
>
> http://plato.stanford.edu/entries/logical-constants/
>
> Quantifier here means generalized quantifier in the sense of Lindström,
> so the usual propositional connectives in particular count as
> quantifiers.

Thank you for the references. The matter seems to be rather
non-trivial.

--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

[Tony Orlow]

On Dec 27, 9:47 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:
> Aatu Koskensilta wrote:

I second that appreciation. Those are good references. Thanks, Aatu.
And, yes, Frederick, it is non-trivial, actually rather crucial, and
not conclusively resolved.

Peace,

TOny

[Frederick Williams]

Tony Orlow wrote:

> > > Frederick Williams <freddywilli...@btinternet.com> writes:
> >
> > > > How does one distinguish between logical symbols and non-logical
> > > > symbols?

[...]

>
> I second that appreciation. Those are good references. Thanks, Aatu.
> And, yes, Frederick, it is non-trivial, actually rather crucial, and
> not conclusively resolved.

Crucial indeed, for if one cannot recognize a logical symbol as such,
how can one decide that (say) P v ~P is a validity? And yet I don't
think I've ever seen a discussion of the matter in a text; rather, one
is presented with a list in a "take or leave it" manner.

(Let it be noted, my example of P v ~P is of no great significance;
substitute P <-> P or anything else that takes your fancy.)

[Tony Orlow]

I don't think "paradoxical" is the right term, but it rather seems
somewhat circuitously self-referential. Herc has a pretty correct and
succinct understanding of my point, that quantifiers imply set
membership, and can be eliminated by replacement with the inevitable
set membership operator. You *will* find that I occasionally, as a
matter of literary style, like to use the same word that is bouncing
around in a slightly different sense, often with the hope of
elucidating by example to point at hand. However, in this case, it was
not deliberate, although it does indicate that there is a set (or
sequence) of what I have said, and that a large portion ("pretty
much") of it was succinctly expressed by Herc (truth value relative to
original point ~= 0.9431628).

Peace,

Tony

[Tony Orlow]

On Dec 27, 10:31 am, Frederick Williams

Would you like such a summary? I think I spewed it earlier in this
thread, but would be happy to reiterate my basic take.

First, let me say I have always taken "logical" constant to be a
member of the set {0,1} or {F,T}, and that those constant values of
true or false are the 0-place operators within basic propositional
logic, that is, they take no arguments upon which their evaluation
depends. I define a logical operator as an n-ary mapping from an
ordered set of n logical values to a single logical value. Thus T and
F are 0-ary operators.

Of the unary operators, those that take a single truth value parameter
and return a single truth value, there is the one that always returns
T no matter the input, the one that always returns false, the one that
always returns the same value as the parameter, and the one that
returns the opposite of the parameter (T if F and F if T). Since we
already have T and F as 0-ary operators, those are redundant, and
since we already have our single parameter (say,x) returning the same
value adds no functionality. So, the only non-trivial unary logical
operator is "not", that which inverts whatever truth value is given as
input.

Enumerating the binary logical operators is almost as simple. As we
had 2=2 possible 0-ary operators, and 2^2=4 possible unary operators,
we have 2^2^2=16 (the third tetration of 2) possible binary operators.
Again, we have one operator always yield true no matter the parameters
and one always false, both redundant. We have, for input parameters x
and y, the operators corresponding to true if x and true if y, and to
not x and not y. Those six we can consider trivial, as they can be
expressed with a 0-ary and unary operators. The remaining ten consist
of five binary operators represented by five 4-bit strings (one bit
for each of the possible 2-digit binary combinations of true and
false), and their bitwise negations. Thus, we have "and" (0001) and
its negation (1110), "or" (0111) and its negation, "equals" or
"mutually implies" (1001) and its negation, "implies" (1101) and its
negation, and finally, "is implied by (1011) and its negation. There
are no other possible combinations of return values, given the four
possible combinations of input value, except for these sixteen, so the
set of logical operators up to the binary is fully enumerated. Beyond
that, it's not hard to show that every ternary or higher operator is
trivial, in that it can already be expressed with binary and lower
operators, and so, the set of all non-trivial logical operators has
been enumerated.

Hope that helps.

Peace,

Tony

[MoeBlee]

On Dec 25, 5:16 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> On 24/12/2011 5:24 PM, MoeBlee wrote:

> > with some
> > structures Axy(x=y <-> Az(zex <-> zey)) is true and with certain other
> > structures it is false.
>
> Can you give an example where it's false?

Given '=' as standing for the identity relation, I'll give two
strutures, such that the is sentence true in one structure and false
in the other structure.

Axy(x=y <-> Az(zex <-> zey)) is true in this structure:

universe is {0}
'e' stands for the empty relation

Axy(x=y <-> Az(zex <-> zey)) is false in this structure:

universe is {0 1}
'e' stands for the empty relation

MoeBlee

[Nam Nguyen]

Not quite sure I understand your example here. Are you saying 0=1?

(Also, for clarity can you spell out 0 and 1 exactly? I take 0 = {},
what is 1?)



--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

[LudovicoVan]

"Frederick Williams" <freddyw...@btinternet.com> wrote in message
news:4EF9E4C2...@btinternet.com...

> Crucial indeed, for if one cannot recognize a logical symbol as such,
> how can one decide that (say) P v ~P is a validity? And yet I don't
> think I've ever seen a discussion of the matter in a text; rather, one
> is presented with a list in a "take or leave it" manner.

A very good one is Strawson's Introduction to Logical Theory, which brings
the reader from the very foundations of the notion of logical validity to
the development of the various types of logic (propositional, predicate-,
class-based logic, etc.) with at least one chapter specifically devoted to
induction and the fallacious quest for a "principle behind induction".
Strawson emphasizes the relationship of logic to natural language, which is
maybe not the only possible approach to logic, in any case and not only in
my personal opinion this a book really worth reading.

-LV

[MoeBlee]

On Dec 27, 7:39 am, Tony Orlow <bonyto...@gmail.com> wrote:
> On Dec 27, 12:52 am, MoeBlee <modem...@gmail.com> wrote:

> I'm going to skip over the above, since it doesn't seem at all
> productive,

I do think it would be productive for you to tell me whether or not
you understand the definition I gave you in response to your asking
for a definition.

Unless you understand what a structure for a language is, it is
virtually impossible that you'll understand the technical specifics of
why your proposal does not work to the extent that you've defended it
so far.

> but the following looks to be part of a reasonable
> discussion, so I'll try to comment as much to your satisfaction as I
> can manage. :)

It's not a matter of my satisfaction; rather it really comes down to
whether you'll ever do yourself the favor of learning the basics of
this subject.

> > You say that Ax Fx can be written instead as xeU -> Fx
>
> > And, if I guess correctly (since I can only guess, since you have not
> > given an explicit semantics), xeU always evaluates as satisfied. But
> > then, per any interpretation and assignment for the variables, and for
> > ANY formula P, we have that xeU -> P is satsified iff P is satisfied.
>
> That is a reasonable statement, except for the "But then" part.

Stop right there. Unless you understand the definition of 'structure
for a language' and then the recursive definition of 'formula
satisfied per a structure and assignment for the variables', it would
be silly for me to try to explain to you why

xeU always evaluates as satisfied
implies
per any interpretation and assignment for the variables, and for any

formula P, we have that xeU -> P is satsified iff P is satisfied

except to say that it follows from our basic result that for any
formulas Q and P

If Q is always satisfied, then Q -> P is satisfied exactly when P is
satisfied.

If Q and P are sentences, then instead of 'satisfied' we may say
'true'.

If Q is always true (i.e., true in every structure for the language)
then Q -> P is true exactly when P is true (i.e., Q -> P is true in
exactly those structures in which P is true).

But, alas, you won't tell me whether you understand the definition of
'structure for a language' and, I would bet Obama's war chest that you
don't know or understand the definition of ' 'formula satisfied per a
structure and assignment for the variables'.

> If one
> says "Ax Fx", then is that statement also not only to be held true
> where Fx is true "for all x"?

I really don't understand why you won't get a good textbook and learn
the material systematically?

Ax Fx is true iff for every assignment for the variables, Fx is
satisfied.

Since 'x' is the only free variable in the matrix Fx, we can reduce
to:

Ax Fx is true iff for every assignment fwhose domain is the set of
variables {x}, we have that Fx is satisfied.

In more plain terms, Ax Fx is true iff Fx is satisfied no matter what
member of the universe for the structure we take 'x' to
("temporarily") stand for.

> "Ax x=x" is false, if you can provide
> even a single x which is not equal to itself.

Roughly speaking, yes.

However, if the interpretation of '=' is fixed as the identity
relation, then Ax x=x is not false in any structure.

> > But it is a plain fact of first order semantics that it is not the
> > case that in general Ax P is equivalent with P, or more specifically
> > that Ax Fx is equivalent with Fx.
>
> What is the difference between "Ax Fx" and "Fx", in your mind, where x
> is quantified over an unspecified universe?

It's not just in my mind. It's a plain fact of ordinary mathematical
logic that it is not the case that in general Ax Fx is equivalent with
Fx.

Proof. Here's a structure and variable assignment in which Fx is
satsified but Ax Fx is not satisfied:

the universe is {0 1}
F stands for {0}
x is assigned to 0

With that structure and variable assignment, Fx is satisfied but Ax Fx
is not satisfied.

[MoeBlee]

On Dec 27, 8:04 am, Tony Orlow <bonyto...@gmail.com> wrote:
> On Dec 27, 1:15 am, MoeBlee <modem...@gmail.com> wrote:

> > On Dec 26, 2:49 pm, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > > Then you might want to stop arguing. My whole original point is that
> > > the quantifiers are NOT logical operators, which are normally defined
> > > as operators that take some natural number of logical truth values (0
> > > or 1, false or true) and return a logical value as a result.
>
> > No that is NOT what you said as this tangent began:
>
> > On Dec 24, 3:16 pm, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > > On Dec 24, 2:54 pm, MoeBlee <modem...@gmail.com> wrote:
>
> > > > On Dec 24, 11:31 am, Frederick Williams
>
> > > > <freddywilli...@btinternet.com> wrote:
> > > > > How does one distinguish between logical symbols and non-logical
> > > > > symbols?
>
> > > > The logical symbols are the quantifiers and sentential connectives.
>
> > > No. The quantifiers are ultimately non-logical
>
> > I said that the "logical SYMBOLS" [emphasis added] are the quantifiers
> > and sentential connectives.
>
> > And I said nothing about "operators" or "operators on truth values"
>
> > And you said, "No" to CONTRADICT my correct report of the ordinary
> > terminology "logical symbols".
>
> I think you misread that,

No, if you meant something other than "No" then you miswrote it.

> I
> made it clear from the beginning that I was talking about the initial
> extension of basic propositional calculus, where symbols denote only
> objects and logical connectives,

Whatever YOU were talking about. When you said "No" regarding what *I*
(and two other posters were talking about), you were incorrect.

> > And I've told you that this is merely a STIPULATIVE definition. It
> > happens to be the way certain authors use the terminology 'logical
> > symbols". Other authors may use the terminology differently, and
> > indeed I mentioned that Enderton has his own terminology that is
> > different. But it is a plain fact that it is an ordinary terminology
> > in which the logical symbols are the quantifiers and connectives
> > (actually, sometimes the variables too, which I should have mentioned)
> > and the non-logical symbols are the function and predicate symbols.
>
> So, you're arguing about the terminology of various authors, and
> *don't* see this as a distraction and derailment of what was
> originally a pretty simple topic?

You really are an all-time piece of work.

Aatu responded to something you wrote. In doing so, he used the
terminology 'logical symbols'. Then another poster asked the
distinction between the logical symbols and the non-logical symbols.
And Aatu answered that. And I added some more comments to expand on
Aatu's answer. And then YOU said to me, INcorrectly, "No", INcorrectly
that the universal quantifier is not a logical symbol. And then I took
my own time to explain to you, since you won't read a damn textbook on
the subject, more about the stipulative definition in which indeed the
universal quantifier is among the logical symbols.

Whether any given tangent is a distraction or not, I don't opine, and
I certainly don't hold myself or other posters to adhere to one
certain topic in a thread. For that matter, I've never asked YOU to
adhere to only one topic in a thread, as indeed YOU have come into so
MANY threads to start jabbering about your own tangential
preoccupations with your own psuedo-math.

> I am not going to claim that one is better than another
> without some evidence of benefit in efficiency, simplicity, or clarity
> of expression.

And I did not claim any superiority of the usage. I merely reported
that it is a common terminology and that my remarks about it are
correct as to that common terminology.

> > You were incorrect to say "No" when I merely correctly reported
> > certain ordinary terminology.

> Quantifiers are not logical operators.

For the fifth time, I did not say that quantifiers are ANY kind of
"operators".

You keep saying you don't want to argue about this, well then you
should not have STARTED an argument about it when you INcorrectly said
"No" to my correct report of common use of certain terminology. And
then CONTINUED to arguen by switching to terminology I DID mention -
logical SYMBOLS - to terminology I did NOT mention - logical
OPERATORS.

> > I said NOTHING about the term "logical operator".
>
> > I said "logical symbol".
>
> Well, you didn't start the thread, and should not try to derail and
> monopolize it with your own irrelevant ruminations concerning
> "symbols".

You're REALLY are an incorrigible hole

(1) Anyone can see that threads COMMONLY go into various tangents,
and in about ten years of posting and reading, including reading
archives from decades past, I can't recall anyone, even the worst
cranks, ever being so petty as to demand "It's my thread, now you
should talk only about what I want to talk about." (Maybe someone has
done that but I don't recall ever seeing it.)

(2) YOU yourself often participate in various tangents, and a plenty
of times you've STARTED those tangents.

(3) In this thread itself, the thread when into different tangents
started by OTHER posters.

(4) And the particular tangent now at hand was NOT started by me. And
it only became less than informative when YOU jumped in, INcorrectly
to say I was wrong in my use of the common terminology.

(5) Basically the same (1) and (2), mutatis mutandis, to "monopolize".

> The point remains that you seem to be sing the word "logical" for
> essentially no reason that you can explain except with gibberish like
> the above.

Right, you want to move on, get away from the tangent. How? By posting
more about it! You really are a case.

Again, the terminology 'logical symbols' is common in mathematical
logic, with a stipulative definition. I merely added some additional
and correct remarks about the distinction between 'logical symbols'
and 'non-logical symbols'. Whether it is best to use such terminology
or not, I don't opine. Rather, I merely added remarks about the
terminology as it happens to be commonly used.

> I don't care what is ordinary. I care what is correct.

Do you really STILL not understand what a STIPULATIVE definition is?

> Nobody asked you what the "ordinary terminology" is for whatever
> you're on about.

Another poster asked Aatu about Aatu's use of the terminology 'logical
symbols'. Aatu was using that terminology in an ordinary way in
mathematical logic. I commented also, and my remarks were in keeping
with that ordinary terminology. Then YOU made an ass out of yourself
(big surprise) by saying I was wrong. Then, you don't even have the
intellectual integrity, the intellectual courtesy to say, "Okay, I see
now" but instead you go on with even more specious arguments about it,
and even keep claiming that you want to move on from the subject while
you keep posting about it!

MoeBlee

[MoeBlee]

P.S. Again, for a guy who wants the thread to stick to the subject,
you sure have a funny way of showing it.

[MoeBlee]

On Dec 27, 8:20 am, Tony Orlow <bonyto...@gmail.com> wrote:

> it may be that such a FOL
> formulation of the diagonal argument rests on a usage of quantifiers
> that masks a second-order nature to the statement.

Whatever about "masking" and the "nature" of this or that, my plain,
clear, and correct statement was that the diagonal argument can be
formalized in first order set theory.

> After all, Herc's
> SOL statement above seems to capture the argument.

His notation is, at best, ersatz.

I never claimed that the diagonal argument cannot be done in a second
order theory or in all kinds of theories. Rather, I merely stated that
the diagonal argument can be done in first order set theory. And that
is true: the diagonal argument can be done in first order set theory,
and that can be said without prejudice as to whether the diagonal
argument can be done in many other different ways.

MoeBlee

[MoeBlee]

On Dec 27, 9:14 am, Tony Orlow <bonyto...@gmail.com> wrote:

> I second that appreciation. Those are good references. Thanks, Aatu.
> And, yes, Frederick, it is non-trivial, actually rather crucial, and
> not conclusively resolved.

You said you wanted to move on from the subject. But you're STILL
posting about it.

And what is not "conclusively resolved"? Whatever philosophical
matters mentioned by Aatu as to MOTIVATIONS for a certain stipulative
definition, they do not contradict that we have a STIPULATIVE
definition of 'logical sybmols' and by that STIPULATIVE definition,
the universal quantifier is a logical symbol.

MoeBlee

[MoeBlee]

On Dec 27, 11:39 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
> > Axy(x=y<->  Az(zex<->  zey)) is false in this structure:
>
> > universe is {0 1}
> > 'e' stands for the empty relation
>
> Not quite sure I understand your example here.

What is there not to understand?

I gave a structure for the language in which Axy(x=y <-> Az(zex <->
zey)) is false.

It's false because the universal generalization is false. The
universal generalization is false because the biconditional does not
hold for all x and y.

When x is 0 and y is 1, we have x=y is not satisfied but Az(zex <->
zey) is satisfied, so the biconditional is not satisfied.

> Are you saying 0=1?

No. Quite the contrary.

> (Also, for clarity can you spell out 0 and 1 exactly? I take 0 = {},
> what is 1?)

As a von Neumann, 1 is {0}. But that is not crucial to the example.
Just let 0 and 1 be any two objects that are not identical to each
other.

If 0 and 1 confuse you as members of a universe (though it shouldn't)
just choose any two objects different from one another.

Anyway, it should not be confusing. It is common to give models in
which the members of the universe are natural numbers and as it is
well enough understood (either formally or informally) that 0 and 1
are not identical to each other.

MoeBlee

[MoeBlee]

On Dec 27, 9:31 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:

> Tony Orlow wrote:
> > > > Frederick Williams <freddywilli...@btinternet.com> writes:
>
> > > > > How does one distinguish between logical symbols and non-logical
> > > > > symbols?
> [...]
>
> > I second that appreciation. Those are good references. Thanks, Aatu.
> > And, yes, Frederick, it is non-trivial, actually rather crucial, and
> > not conclusively resolved.
>
> Crucial indeed, for if one cannot recognize a logical symbol as such,
> how can one decide that (say) P v ~P is a validity?

The questionf P v ~P being a validity does not depend very much on the
definition of 'logical symbol'. We could dispense with the terminology
'logical symbol' and still stipulate a semantics such that P v ~P is a
validity.

> And yet I don't
> think I've ever seen a discussion of the matter in a text; rather, one
> is presented with a list in a "take or leave it" manner.

You may be right that authors don't usually go into the reason for
using the word 'logical' in the terminology 'logical symbols'. And,
yes, it is a stipulative definition (so I guess "take it or leave it"
in that sense).

MoeBlee

[Frederick Williams]

Tony Orlow wrote:
>
> On Dec 27, 10:31 am, Frederick Williams
> <freddywilli...@btinternet.com> wrote:
> > Tony Orlow wrote:
> > > > > Frederick Williams <freddywilli...@btinternet.com> writes:
> >
> > > > > > How does one distinguish between logical symbols and non-logical
> > > > > > symbols?
> > [...]
> >
> > > I second that appreciation. Those are good references. Thanks, Aatu.
> > > And, yes, Frederick, it is non-trivial, actually rather crucial, and
> > > not conclusively resolved.
> >
> > Crucial indeed, for if one cannot recognize a logical symbol as such,
> > how can one decide that (say) P v ~P is a validity? And yet I don't
> > think I've ever seen a discussion of the matter in a text; rather, one
> > is presented with a list in a "take or leave it" manner.
> >
> > (Let it be noted, my example of P v ~P is of no great significance;
> > substitute P <-> P or anything else that takes your fancy.)
>

> Would you like such a summary?

[Summary snipped]

Your summary is about the propositional connectives. Far more
interesting is (e.g.)

for all x, x = x.

That is (if I understand correctly) a logically valid formula because

x = x

is true whatever x is. I.e., we can _vary_ (the denotation of) 'x', but
we may not vary (the denotations of) 'for all' and '='. Sez who, and
why?

[Tony Orlow]

On Dec 27, 12:48 pm, MoeBlee <modem...@gmail.com> wrote:
> On Dec 27, 7:39 am, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > On Dec 27, 12:52 am, MoeBlee <modem...@gmail.com> wrote:
> > I'm going to skip over the above, since it doesn't seem at all
> > productive,
>
> I do think it would be productive for you to tell me whether or not
> you understand the definition I gave you in response to your asking
> for a definition.
>
> Unless you understand what a structure for a language is, it is
> virtually impossible that you'll understand the technical specifics of
> why your proposal does not work to the extent that you've defended it
> so far.

I rather imagine you are referring to the recursive definition of a
language when you talk about its structure, but I don't think there is
much question about the structure of the language I am using.

>
> > but the following looks to be part of a reasonable
> > discussion, so I'll try to comment as much to your satisfaction as I
> > can manage. :)
>
> It's not a matter of my satisfaction; rather it really comes down to
> whether you'll ever do yourself the favor of learning the basics of
> this subject.

As far as our discussion,. it comes down to whether we understand what
each other is saying/meaning.

>
> > > You say that Ax Fx can be written instead as xeU -> Fx
>
> > > And, if I guess correctly (since I can only guess, since you have not
> > > given an explicit semantics), xeU always evaluates as satisfied. But
> > > then, per any interpretation and assignment for the variables, and for
> > > ANY formula P, we have that xeU -> P is satsified iff P is satisfied.
>
> > That is a reasonable statement, except for the "But then" part.
>
> Stop right there. Unless you understand the definition of 'structure
> for a language' and then the recursive definition of 'formula
> satisfied per a structure and assignment for the variables', it would
> be silly for me to try to explain to you why

I don't need you to explain what you mean. Start your own thread and I
will try as hard as possible to comprehend where you are going. Since
I started this discussion without reference to the concept of a
language structure, but with generally accepted syntax and semantics,
there should be no question what I mean.

>
> xeU always evaluates as satisfied
> implies
> per any interpretation and assignment for the variables, and for any
> formula P, we have that xeU -> P is satsified iff P is satisfied
>
> except to say that it follows from our basic result that for any
> formulas Q and P
>
> If Q is always satisfied, then Q -> P is satisfied exactly when P is
> satisfied.

Right, so where Q is always satisfied, Q->P evaluates simply to P.

>
> If Q and P are sentences, then instead of 'satisfied' we may say
> 'true'.
>
> If Q is always true (i.e., true in every structure for the language)
> then Q -> P is true exactly when P is true (i.e., Q -> P is true in
> exactly those structures in which P is true).
>
> But, alas, you won't tell me whether you understand the definition of
> 'structure for a language' and, I would bet Obama's war chest that you
> don't know or understand the definition of ' 'formula satisfied per a
> structure and assignment for the variables'.

Luckily for you his war chest is less than empty (implying negative
set membership).

>
> > If one
> > says "Ax Fx", then is that statement also not only to be held true
> > where Fx is true "for all x"?
>
> I really don't understand why you won't get a good textbook and learn
> the material systematically?
>
> Ax Fx is true iff for every assignment for the variables, Fx is
> satisfied.

Satisfied, or as you say, true. Yes, Fx is true whenever Ax Fx is
true.

>
> Since 'x' is the only free variable in the matrix Fx, we can reduce
> to:
>
> Ax Fx is true iff for every assignment fwhose domain is the set of
> variables {x}, we have that Fx is satisfied.

If we call that domain U, then that is the same as xeU -> Fx. Agreed?

>
> In more plain terms, Ax Fx is true iff Fx is satisfied no matter what
> member of the universe for the structure we take 'x' to
> ("temporarily") stand for.
>
> > "Ax x=x" is false, if you can provide
> > even a single x which is not equal to itself.
>
> Roughly speaking, yes.
>
> However, if the interpretation of '=' is fixed as the identity
> relation, then Ax x=x is not false in any structure.

Of course.

>
> > > But it is a plain fact of first order semantics that it is not the
> > > case that in general Ax P is equivalent with P, or more specifically
> > > that Ax Fx is equivalent with Fx.
>
> > What is the difference between "Ax Fx" and "Fx", in your mind, where x
> > is quantified over an unspecified universe?
>
> It's not just in my mind. It's a plain fact of ordinary mathematical
> logic that it is not the case that in general Ax Fx is equivalent with
> Fx.
>
> Proof. Here's a structure and variable assignment in which Fx is
> satsified but Ax Fx is not satisfied:
>
> the universe is {0 1}
> F stands for {0}
> x is assigned to 0

Stop! F is a one-place operator, a predicate of some sort taking
parameter x, and returning a result in {0,1} if it is to be used as a
parameter to ANY logical operator, such as "->". Otherwise, "blahblah -
>Fx", or "Ax Fx", has no logical meaning. "{0}" is not a logical
value, nor a statement with a logical value. It's not a predicate or
proposition of any sort, but a singleton set that is neither true nor
false.

If F is {0} and x is 0, then what is Fx, "{0}0"? That is the least
decipherable string I have seen this entire thread, and perhaps, in my
entire sci.math/logic experience.

>
> With that structure and variable assignment, Fx is satisfied but Ax Fx
> is not satisfied.

Q: With that inherently incoherent structure, what does "Fx" mean?
A: Nil. :(

>
> As I said:
>
> > > if you have some semantical method other than structures and
> > > assignments for the variables, then you need to specify your
> > > semantical method, hopefully with inductive definitions on the
> > > complexity of formulas.
>
> MoeBlee

You should stop quoting yourself. Remember what Einstein said about
the definition of insanity.

Peace,

Tony

[MoeBlee]

On Dec 27, 1:02 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:

> but
> we may not vary (the denotations of) 'for all' and '='.  Sez who, and
> why?

Says whatever author is setting forth his system and definition.
You're free to do otherwise for your own writings, but when an author
gives certain stipulative definitions (such as the stipulative
definition of 'satisfied per a structure and variable assignment' in
which the universal quantifier and identity symbol have certain
clauses about them that "fix" their interpretation) then there's not a
lot of sense in fighting with the author's freedom to give his
definitions in his own treatment of the subject. Then, enough authors
happen to treat the quantifier and identity symbol in that same way
and we observe that this is ordinary mathematical usage.

By the way though, note that some authors do allow that '=' may be
interepreted differently by different structures. An author would
choose between two different approaches:

(1) '=' is always interpreted as the identity relation.

(2) '=' is just another 2-place relation symbol and may be interpreted
as any 2-place relation on the universe. However, even then, we may go
on, syntactically, to describe an identity theory, with axioms for '='
in the usual manner (e.g., axiomatized by reflexivity of identity and,
roughly, "Axy(x=y -> (P(x) <-> P(y)))".

Also, as I mentioned, Enderton has a somewhat different approach to
the universal quantifier than most authors (though, in the wash, it
comes out the same). For him, a structure is purely a certain kind of
function, and the universal quantifier is in the domain of that
function (by that, I don't mean the domain of discourse or the
universe for the structure), so, per a structure, the universal
quantifier is mapped to a particular non-empty set that is the
universe for the structure. But, then in the 'satisfaction' inductive
definition, the universal quantifier is treated by Enderton just as
with more ordinary treatments.

MoeBlee

[Frederick Williams]

Frederick Williams wrote:

>
> Your summary is about the propositional connectives.

So let's consider my problem from a propositional point of view.

In virtue of what is

P & Q
-----
P

a logically valid inference? Answer (I think): because whatever
statements one takes P and Q to be, the conclusion is true whenever the
premiss is true, and the premiss is false whenever the conclusion is
false. One can try it out:

Queen Elizabeth is mortal & The pope is German
------------------------------------------------
Queen Elizabeth is mortal


Queen Elizabeth is a redhead & The pope is German
---------------------------------------------------
Queen Elizabeth is a redhead

In the first case truth is transmitted downwards (one might say) and in
the second falsity is transmitted upwards. But, note, while we may
substitute various constant statements for the statement variables P and
Q, we may not substitute various constant connectives in place of &, we
may only read that as 'and'. Try it with 'or':

Queen Elizabeth is a redhead or The pope is German
----------------------------------------------------
Queen Elizabeth is a redhead

Now the premiss is true but the conclusion is false. _Why_ are P and Q
allowed to vary, but & not?

If _that_ doesn't make my problem clear, then I shall give up.

[Nam Nguyen]

On 27/12/2011 11:55 AM, MoeBlee wrote:
> On Dec 27, 11:39 am, Nam Nguyen<namducngu...@shaw.ca> wrote:
>>> Axy(x=y<-> Az(zex<-> zey)) is false in this structure:
>>
>>> universe is {0 1}
>>> 'e' stands for the empty relation
>>
>> Not quite sure I understand your example here.
>
> What is there not to understand?

If we talk about set and you mentioned say, 1 (which is)
in the L(ZF), without a definition then there's at least
one thing people wouldn't understand.

Especially we're in the context that the semantic of 'e' might not
be the "normal" one.

>
> I gave a structure for the language in which Axy(x=y<-> Az(zex<->
> zey)) is false.
>
> It's false because the universal generalization is false. The
> universal generalization is false because the biconditional does not
> hold for all x and y.
>
> When x is 0 and y is 1, we have x=y is not satisfied but Az(zex<->
> zey) is satisfied, so the biconditional is not satisfied.
>
>> Are you saying 0=1?
>
> No. Quite the contrary.
>
>> (Also, for clarity can you spell out 0 and 1 exactly? I take 0 = {},
>> what is 1?)
>
> As a von Neumann, 1 is {0}. But that is not crucial to the example.
> Just let 0 and 1 be any two objects that are not identical to each
> other.

But, how do you know {} and {{}} are 2 different sets, given that your
new interpretation of "'e' stands for the empty relation" would have to
be uniformly interpreted about _all_ or _every_ sets? Can you elaborate?

[Tony Orlow]

On Dec 27, 1:28 pm, MoeBlee <modem...@gmail.com> wrote:
> On Dec 27, 8:04 am, Tony Orlow <bonyto...@gmail.com> wrote:
>
>
>
> > On Dec 27, 1:15 am, MoeBlee <modem...@gmail.com> wrote:
> > > On Dec 26, 2:49 pm, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > > > Then you might want to stop arguing. My whole original point is that
> > > > the quantifiers are NOT logical operators, which are normally defined
> > > > as operators that take some natural number of logical truth values (0
> > > > or 1, false or true) and return a logical value as a result.
>
> > > No that is NOT what you said as this tangent began:
>
> > > On Dec 24, 3:16 pm, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > > > On Dec 24, 2:54 pm, MoeBlee <modem...@gmail.com> wrote:
>
> > > > > On Dec 24, 11:31 am, Frederick Williams
>
> > > > > <freddywilli...@btinternet.com> wrote:
> > > > > > How does one distinguish between logical symbols and non-logical
> > > > > > symbols?
>
> > > > > The logical symbols are the quantifiers and sentential connectives.
>
> > > > No. The quantifiers are ultimately non-logical
>
> > > I said that the "logical SYMBOLS" [emphasis added] are the quantifiers
> > > and sentential connectives.
>
> > > And I said nothing about "operators" or "operators on truth values"
>
> > > And you said, "No" to CONTRADICT my correct report of the ordinary
> > > terminology "logical symbols".
>
> > I think you misread that,
>
> No, if you meant something other than "No" then you miswrote it.

Perhaps I did, and if so, I apologize, but I am not going to dredge my
way through these many posts to resolve who is "right" about what is
apparently a failed communication. It takes two to tango. Any good
therapist will tell you that. So, save it for your Thursday
appointments.

>
> > I
> > made it clear from the beginning that I was talking about the initial
> > extension of basic propositional calculus, where symbols denote only
> > objects and logical connectives,
>
> Whatever YOU were talking about. When you said "No" regarding what *I*
> (and two other posters were talking about), you were incorrect.

If you say so. "No" not even real sentence because no verb, or
subject, even.

>
> > > And I've told you that this is merely a STIPULATIVE definition. It
> > > happens to be the way certain authors use the terminology 'logical
> > > symbols". Other authors may use the terminology differently, and
> > > indeed I mentioned that Enderton has his own terminology that is
> > > different. But it is a plain fact that it is an ordinary terminology
> > > in which the logical symbols are the quantifiers and connectives
> > > (actually, sometimes the variables too, which I should have mentioned)
> > > and the non-logical symbols are the function and predicate symbols.
>
> > So, you're arguing about the terminology of various authors, and
> > *don't* see this as a distraction and derailment of what was
> > originally a pretty simple topic?
>
> You really are an all-time piece of work.

I didn't ask your opinion of me, but of the quality of your statement.

>
> Aatu responded to something you wrote. In doing so, he used the
> terminology 'logical symbols'. Then another poster asked the
> distinction between the logical symbols and the non-logical symbols.
> And Aatu answered that. And I added some more comments to expand on
> Aatu's answer. And then YOU said to me, INcorrectly, "No", INcorrectly
> that the universal quantifier is not a logical symbol. And then I took
> my own time to explain to you, since you won't read a damn textbook on
> the subject, more about the stipulative definition in which indeed the
> universal quantifier is among the logical symbols.

If 'e' (as ordinarily used) is a non-logical symbols (as it is
considered ordinarily), then any symbol which equates to it or some
combination of it with logical symbols, is not purely logical, but at
least partially non-logical. I was not INcorrect in such a statement,
but perhaps SUPERcorrect.

>
> Whether any given tangent is a distraction or not, I don't opine, and
> I certainly don't hold myself or other posters to adhere to one
> certain topic in a thread. For that matter, I've never asked YOU to
> adhere to only one topic in a thread, as indeed YOU have come into so
> MANY threads to start jabbering about your own tangential
> preoccupations with your own psuedo-math.

That is generally when other bring up older statements of mine. Note
that Barb (to whom I tried to respond unsuccessfully for reasons I
cannot explain) brought up infinitary considerations, when I have not
mentioned any such thing in this thread. I generally try to stick with
the thread's original topic, and have a couple of times started a new
separate thread, because I felt the offshoot conversation was
interfering with the OP's original point. It's called mutual respect.

>
> > I am not going to claim that one is better than another
> > without some evidence of benefit in efficiency, simplicity, or clarity
> > of expression.
>
> And I did not claim any superiority of the usage. I merely reported
> that it is a common terminology and that my remarks about it are
> correct as to that common terminology.

But, Moe, what did any of that have to do with the underlying meaning
of the logical quantifier symbols? Certainly I recognize that 'A' and
'E' are symbols used in statements using logic, but they are not
ultimately purely logical symbols. So, the distinction, I would think,
should be obvious.

>
> > > You were incorrect to say "No" when I merely correctly reported
> > > certain ordinary terminology.
> > Quantifiers are not logical operators.
>
> For the fifth time, I did not say that quantifiers are ANY kind of
> "operators".

I am glad to hear that, though the previous statements on your part
may not have been so clear. If they are not logical operators, then
what exactly are they? That is the subject of this discussion.

>
> You keep saying you don't want to argue about this, well then you
> should not have STARTED an argument about it when you INcorrectly said
> "No" to my correct report of common use of certain terminology. And
> then CONTINUED to arguen by switching to terminology I DID mention -
> logical SYMBOLS - to terminology I did NOT mention - logical
> OPERATORS.

I'm sorry, Dear. I didn't mean to make you feel dismissed, or to
negate any of your "feelings". I care about your input, and I value
our relationship, and never meant to make you feel unjustly rejected.
Please accept my apologies for what was probably just a lazy e-tongue
on my part (and these chocolates). I'll make it up to you, if only
you'll give me the chance. I want to make this work, for the children,
if no one else. Please give me one more chance. <3

>
> > > I said NOTHING about the term "logical operator".
>
> > > I said "logical symbol".
>
> > Well, you didn't start the thread, and should not try to derail and
> > monopolize it with your own irrelevant ruminations concerning
> > "symbols".
>
> You're REALLY are an incorrigible hole

Do holes need correcting? That is pretty hurtful, but I understand
you're under stress, and this is a very delicate topic, so I forgive
you. Still, a hole? You called me a hole? That's half a step before
the c-word (sniff). It's okay, I understand, and still want to make
this work between us. Please?

>
> (1)  Anyone can see that threads COMMONLY go into various tangents,
> and in about ten years of posting and reading, including reading
> archives from decades past, I can't recall anyone, even the worst
> cranks, ever being so petty as to demand "It's my thread, now you
> should talk only about what I want to talk about." (Maybe someone has
> done that but I don't recall ever seeing it.)

My original thread experience went on for about 10,000, in Lester
Zick's (pbuh) Epistemology 201 thread, where we discussed everything
from free will to...this. I love discussing all sorts of topics, but
when I am trying to explore a simple point and find it complicated to
the point where I can't get any answer that's not off-topic, I feel a
little frustrated. (note that I phrased that in terms of my own
feelings, and no actions of yours or anyone else's). <3

>
> (2) YOU yourself often participate in various tangents, and a plenty
> of times you've STARTED those tangents.

Somtimes I have if they seem relevant, but usually I am drawn into
them by others (at least in my intuitively statistical estimation).

>
> (3) In this thread itself, the thread when into different tangents
> started by OTHER posters.

Absolutely true. It takes AT LEAST two to tangle, or tango, and
usually involves more than 2.

>
> (4) And the particular tangent now at hand was NOT started by me. And
> it only became less than informative when YOU jumped in, INcorrectly
> to say I was wrong in my use of the common terminology.

I'm sorry. I wasn't referring to terminology, but to underlying
meaning, in which context it becomes clear that the quantifiers are
not purely logical, as are the operators restricted to {0,1} for input
and output. Honestly, Sweetie, I am not sure at this point what "no"
exactly referred to. I think I was thinking about whether you looked
fat in that dress...

You don't, by the way (though I think you could hit the gym once in a
while).

>
> (5) Basically the same (1) and (2), mutatis mutandis, to "monopolize".

Ah, "you all change what is to be changed", as far as my 32-year old
Latin training still holds. That's not necessarily deliberate, but in
the best of all worlds, is simply a matter of evolution.

>
> > The point remains that you seem to be sing the word "logical" for
> > essentially no reason that you can explain except with gibberish like
> > the above.
>
> Right, you want to move on, get away from the tangent. How? By posting
> more about it! You really are a case.
>
> Again, the terminology 'logical symbols' is common in mathematical
> logic, with a stipulative definition. I merely added some additional
> and correct remarks about the distinction between 'logical symbols'
> and 'non-logical symbols'. Whether it is best to use such terminology
> or not, I don't opine. Rather, I merely added remarks about the
> terminology as it happens to be commonly used.

Is 'e' non-logical? If 'A' entails 'e', is it not also non-logical?
Or, is it "partially logical", being a combination of logical and non-
logical operators?

>
> > I don't care what is ordinary. I care what is correct.
>
> Do you really STILL not understand what a STIPULATIVE definition is?

Do you understand that the establishment of a limited universe of
discourse, within which context the "universal" quantifier is applied,
is a stipulation?

>
> > Nobody asked you what the "ordinary terminology" is for whatever
> > you're on about.
>
> Another poster asked Aatu about Aatu's use of the terminology 'logical
> symbols'. Aatu was using that terminology in an ordinary way in
> mathematical logic. I commented also, and my remarks were in keeping
> with that ordinary terminology. Then YOU made an ass out of yourself
> (big surprise) by saying I was wrong. Then, you don't even have the
> intellectual integrity, the intellectual courtesy to say, "Okay, I see
> now" but instead you go on with even more specious arguments about it,
> and even keep claiming that you want to move on from the subject while
> you keep posting about it!
>
> MoeBlee

Alright Moe. I've refrained (I think) from calling you an ass so far,
but it takes one to know one. If I am trying to make the point that
"logical" quantifiers are not entirely logical, since they rely on
"non-logical" symbols like 'e', then I am perfectly justified in
saying "No" to the concept that they are "logical" symbols. The are,
at best, semi-logical, and at worst (and in fact) an obstruction to
logical clarity and general consensus on what is true.

So, go smoke something, look at the stars, ask an opossum about truth,
and sleep on it. Take your time. This is not meant to be a
competition.

Peace,

Tony

[Tony Orlow]

I didn't say I disagreed. You are most likely "correct". However, I'd
be interested to see your (succinct) formulation of the statement in
FOL, and see if my suspicion holds any water, that the "disagreement"
Herc has is related to the topic at hand regarding logical
quantifiers. I think it might be illuminating. That's all.

[Tony Orlow]

On Dec 27, 1:36 pm, MoeBlee <modem...@gmail.com> wrote:
> P.S. Again, for a guy who wants the thread to stick to the subject,
> you sure have a funny way of showing it.

Humor is the best medicine, and battleships don't stop on a dime.

<3

Tony

[Nam Nguyen]

On 27/12/2011 12:41 PM, Nam Nguyen wrote:
> On 27/12/2011 11:55 AM, MoeBlee wrote:
>> On Dec 27, 11:39 am, Nam Nguyen<namducngu...@shaw.ca> wrote:
>>>> Axy(x=y<-> Az(zex<-> zey)) is false in this structure:
>>>
>>>> universe is {0 1}
>>>> 'e' stands for the empty relation
>>>
>>> Not quite sure I understand your example here.
>>
>> What is there not to understand?
>
> If we talk about set and you mentioned say, 1 (which is)
> in the L(ZF), without a definition then there's at least
> one thing people wouldn't understand.

It should be read (which is supposed to be)

[Tony Orlow]

On Dec 27, 1:55 pm, MoeBlee <modem...@gmail.com> wrote:
> On Dec 27, 11:39 am, Nam Nguyen <namducngu...@shaw.ca> wrote:
>
> > > Axy(x=y<->  Az(zex<->  zey)) is false in this structure:
>
> > > universe is {0 1}
> > > 'e' stands for the empty relation
>
> > Not quite sure I understand your example here.
>
> What is there not to understand?
>
> I gave a structure for the language in which Axy(x=y <-> Az(zex <->
> zey)) is false.
>
> It's false because the universal generalization is false. The
> universal generalization is false because the biconditional does not
> hold for all x and y.
>
> When x is 0 and y is 1, we have x=y is not satisfied but Az(zex <->
> zey) is satisfied, so the biconditional is not satisfied.

That is because you replaced the "ordinary interpretation" of 'e' with
the "empty relation". You could just as well replaced 'e' with "is an
orange member", or almost anything besides the "ordinary" membership
operation, and you know it.

You think, like Uergilistico, that this is a game?

>
> > Are you saying 0=1?
>
> No. Quite the contrary.

No, you are saying 'e' <>'e'.

>
> > (Also, for clarity can you spell out 0 and 1 exactly? I take 0 = {},
> > what is 1?)
>
> As a von Neumann, 1 is {0}. But that is not crucial to the example.
> Just let 0 and 1 be any two objects that are not identical to each
> other.
>
> If 0 and 1 confuse you as members of a universe (though it shouldn't)
> just choose any two objects different from one another.
>
> Anyway, it should not be confusing. It is common to give models in
> which the members of the universe are natural numbers and as it is
> well enough understood (either formally or informally) that 0 and 1
> are not identical to each other.
>
> MoeBlee

It SHOULDN'T be confusing. It should be clear. So, stop confusing
every issue, as if symbols don't have widely-accepted and well-
understood meanings in this context.You are deliberately confounding
things, and feigning innocence in that regard.

Peace. Peace. Peace, O Moe!!

Tony

[Tony Orlow]

Then define the reason for the word. It's simple. It deals only with
logical values, in and out. That's simple. So, go into that. It only
took me 15 seconds here.

BTW, I didn't really answer Frederick's question, in retrospect,
because I didn't explain in detail how to apply the bit strings I
mentioned to truth tables that produce them, and how to compound truth
tables.

Frederick, please ask, if you would like further explanation.

Peace,

Tony

[MoeBlee]

On Dec 27, 12:59 pm, Tony Orlow <bonyto...@gmail.com> wrote:
> On Dec 27, 12:48 pm, MoeBlee <modem...@gmail.com> wrote:

> I rather imagine you are referring to the recursive definition of a
> language when you talk about its structure,

No, wrong. Competely off. Completely ignorant of the basics of this
subject.

I DEFINED 'structure for a language' for you, and it is NOTHING like a
"recursive definition of a lanaguage".

You wouldn't even tell me whether you understood that definition, but
now I see that you did not.

> As far as our discussion,. it comes down to whether we understand what
> each other is saying/meaning.

Yeah, well the subject involves certain technical aspects. And you
don't understand them, because you REFUSE to understand them, though,
you could understand them if got a textbook on the sujbect and read it
systematically.

> > Stop right there. Unless you understand the definition of 'structure
> > for a language' and then the recursive definition of 'formula
> > satisfied per a structure and assignment for the variables', it would
> > be silly for me to try to explain to you why
>
> I don't need you to explain what you mean.

Correct. Because you could, if you were a reasonable person, get a
good textbook on the subject instead.

> > xeU always evaluates as satisfied
> > implies
> > per any interpretation and assignment for the variables, and for any
> > formula P, we have that xeU -> P is satsified iff P is satisfied
>
> > except to say that it follows from our basic result that for any
> > formulas Q and P
>
> > If Q is always satisfied, then Q -> P is satisfied exactly when P is
> > satisfied.
>
> Right, so where Q is always satisfied, Q->P evaluates simply to P.

Yes. And P is NOT in general equivalent to Ax P.

Moreover, since you stipulate that xeU is always satisfied, you
wouldn't even NEED it, since you stipulate that Ax P is expressed as
xeU -> P, but, as you agree, xeU -> P evaluates simply to P. So Ax P
would be expressed just as well, in your terms, by P.

But P is NOT in general equivalent to Ax P.

> > If Q and P are sentences, then instead of 'satisfied' we may say
> > 'true'.
>
> > If Q is always true (i.e., true in every structure for the language)
> > then Q -> P is true exactly when P is true (i.e., Q -> P is true in
> > exactly those structures in which P is true).
>
> > But, alas, you won't tell me whether you understand the definition of
> > 'structure for a language' and, I would bet Obama's war chest that you
> > don't know or understand the definition of ' 'formula satisfied per a
> > structure and assignment for the variables'.
>
> Luckily for you his war chest is less than empty (implying negative
> set membership).

What are you talking about? Obama's war chest is greater than any
Republican now running? I think Obama's war chest will be the largest
ever in history? (I should check that.)

Anyway, you earlier in your post showed that I would win that bet. You
clearly did not understand the definition I posted.

> Satisfied, or as you say, true. Yes, Fx is true whenever Ax Fx is
> true.

No, I use 'satisfied' instead of 'true' for formulas in general and
'true' for sentences. But, as long as later, if needed, we keep the
distinction, I don't mind using 'true' for formulas in general too.

And yes Fx is entailed by Ax Fx. But Ax Fx is NOT entailed by Fx. So
Ax Fx is not equivalent with Fx.

> > Since 'x' is the only free variable in the matrix Fx, we can reduce
> > to:
>
> > Ax Fx is true iff for every assignment fwhose domain is the set of
> > variables {x}, we have that Fx is satisfied.
>
> If we call that domain U, then that is the same as xeU -> Fx. Agreed?

The domain itself is not, in general, referred to in the object
language itself. The domain per an interpretation is referred to in
the meta-language that talks about the object language.

Your 'U' as a special kind of constant is your own method. In that
context, it's not for me to opine about it specifically unless you
give me its semantics.

> > It's not just in my mind. It's a plain fact of ordinary mathematical
> > logic that it is not the case that in general Ax Fx is equivalent with
> > Fx.
>
> > Proof. Here's a structure and variable assignment in which Fx is
> > satsified but Ax Fx is not satisfied:
>
> > the universe is {0 1}
> > F stands for {0}
> > x is assigned to 0
>
> Stop! F is a one-place operator, a predicate of some sort taking
> parameter x, and returning a result in {0,1} if it is to be used as a
> parameter to ANY logical operator, such as "->". Otherwise, "blahblah ->Fx", or "Ax Fx",  has no logical
> meaning.

F is a 1-place predicate symbol. An interpretation for a 1-place
predicate symbol is a subset (1-place relation) on the universe for
the interpretation.

You are in the basic ballpark though when you mention that Fx maps to
either 0 or 1 (to index 'satisfied' or 'not satisfied') if we add that
that depends on what x maps to in the variable-interpretation. That
comes from one of the clauses in the recursive definition of
'satisfied per structure per variable-assignment'.

That is ordinary mathematical logic.

If you have some other formal semantics, then you need to post a
treatment of it.

> "{0}" is not a logical
> value, nor a statement with a logical value. It's not a predicate or
> proposition of any sort, but a singleton set that is neither true nor
> false.

I'm giving the interpretation ALONE of the 1-place predicate F. I'm
not giving the truth value of Fx. The truth value (satisfaction,
really) of Fx is determined by whether what x maps to is a member of
what F maps to. If x maps to a member of the set that F maps to, then
Fx is satisfied.

In other words, F maps to a subset of the universe. x maps to a member
of the universe. Fx is satisfied by those mappings if and only the
object x maps to is a member of the set F maps to.

For example,

Let the universe be the set of natural numbers.

Map the 1-place predicate symbol F to the set of even numbers.

Map the variable x to 2.

Then Fx is satisfied by those mappings.

On the other hand:

Let the universe be the set of natural numbers.

Map the 1-place predicate symbol F to the set of even numbers.

Map the variable x to 3.

Then Fx is not satisfied by those mappings.

Really, Orlow, why do you refuse to get a good book so you can see how
this works? It's not convenient for me to explain it all in ad hoc
posts.

> If F is {0} and x is 0,

No, F is not {0} and x is not 0. Rather I MAPPED F to {0} and x to 0.

> then what is Fx, "{0}0"?

No, as you say, that's nonsense.

'Fx' itself is a formula. The satisfaction-VALUE of 'Fx' per a
structure and variable-assignment is either 'satisfied' (indexed, say,
by 1) or 'not satisfied (indexed, say, by 0).

> That is the least
> decipherable string I have seen this entire thread, and perhaps, in my
> entire sci.math/logic experience.

And you should not have even written it in response to anything I
said.

PLEASE, why don't you just get a good book on symbolic logic and then
one on mathematical logic? I

MoeBlee

[Tony Orlow]

On Dec 27, 1:41 pm, MoeBlee <modem...@gmail.com> wrote:
> On Dec 27, 9:14 am, Tony Orlow <bonyto...@gmail.com> wrote:
>
> > I second that appreciation. Those are good references. Thanks, Aatu.
> > And, yes, Frederick, it is non-trivial, actually rather crucial, and
> > not conclusively resolved.
>
> You said you wanted to move on from the subject. But you're STILL
> posting about it.

Aatu's references, to no surprise on my part, were remarkably
appropriate to the original topic. His and Frederick's remarks, while
concise, say more that most others.

>
> And what is not "conclusively resolved"? Whatever philosophical
> matters mentioned by Aatu as to MOTIVATIONS for a certain stipulative
> definition, they do not contradict that we have a STIPULATIVE
> definition of 'logical sybmols' and by that STIPULATIVE definition,
> the universal quantifier is a logical symbol.
>
> MoeBlee

I can only recommend that you look that the sites Aatu suggested, and
see if that makes anything clearer, or at least illuminates the
disclarities in the discussion.

That's just a STIPULATIVE suggestion.

Love,

Tony

[MoeBlee]

On Dec 27, 1:32 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:

> In virtue of what is
>
>    P & Q
>    -----
>      P
>
> a logically valid inference?

In informal terms, I think by virtue as you mentioned later in your
post. In formal terms, by virtue of the definition of 'valid
inference' along with the semantics for the '&' symbol.

> _Why_ are P and Q
> allowed to vary, but & not?

Because it is convenient for working in logic.

Say I'm an author and I am giving a system for logic. In doing that,
it tremendously helps to stipulate an interpretation of '&' and also
to allow different interpretions of sentence letters. It works to
express certain notions about logic.

MoeBlee

[Tony Orlow]

On Dec 27, 2:02 pm, Frederick Williams <freddywilli...@btinternet.com>

wrote:
> Tony Orlow wrote:
>
> > On Dec 27, 10:31 am, Frederick Williams
> > <freddywilli...@btinternet.com> wrote:
> > > Tony Orlow wrote:
> > > > > > Frederick Williams <freddywilli...@btinternet.com> writes:
>
> > > > > > > How does one distinguish between logical symbols and non-logical
> > > > > > > symbols?
> > > [...]
>
> > > > I second that appreciation. Those are good references. Thanks, Aatu.
> > > > And, yes, Frederick, it is non-trivial, actually rather crucial, and
> > > > not conclusively resolved.
>
> > > Crucial indeed, for if one cannot recognize a logical symbol as such,
> > > how can one decide that (say) P v ~P is a validity?  And yet I don't
> > > think I've ever seen a discussion of the matter in a text; rather, one
> > > is presented with a list in a "take or leave it" manner.
>
> > > (Let it be noted, my example of P v ~P is of no great significance;
> > > substitute P <-> P or anything else that takes your fancy.)
>
> > Would you like such a summary?
>
> [Summary snipped]
>
> Your summary is about the propositional connectives.  Far more
> interesting is (e.g.)
>
>    for all x, x = x.

Okay. I know I left out the details about compounding the bit-strings
representing the logical connectives, so that's a valid question...

>
> That is (if I understand correctly) a logically valid formula because
>
>    x = x
>
> is true whatever x is.  I.e., we can _vary_ (the denotation of) 'x', but
> we may not vary (the denotations of) 'for all' and '='.  Sez who, and
> why?

In logical operators, which take only true or false as parameters, and
return true or false as a result, '=' or "<->" is true iff both input
parameters are always equal, either both true or both false. So, in
the context of pure logic '=' is the same as "<->".

>
> --
> When a true genius appears in the world, you may know him by
> this sign, that the dunces are all in confederacy against him.
> Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

Peace,

Tony

[Graham Cooper]

On Dec 28, 6:06 am, Tony Orlow <bonyto...@gmail.com> wrote:
>
> In logical operators, which take only true or false as parameters, and
> return true or false as a result, '=' or "<->" is true iff both input
> parameters are always equal, either both true or both false. So, in
> the context of pure logic '=' is the same as "<->".
>
>

This only holds in a very simple base level BOOLEAN LOGIC THEORY.

In set theory there are OBJECTS, {} is neither true nor false.

Herc

[MoeBlee]

On Dec 27, 2:13 pm, MoeBlee <modem...@gmail.com> wrote:

> I DEFINED 'structure for a language' for you, and it is NOTHING like a
> "recursive definition of a lanaguage".

Though, of course, it does PARALLEL the recursive definition of 'well
formed formula' (as do many inductive definitions pertaining to
logic), though, still they are two very different definitions.

Correction:

> Obama's war chest is greater than any
> Republican now running?

That should not have a question mark but rather a period.

MoeBlee

[Frederick Williams]

Tony Orlow wrote:
>
> [...] So, in

> the context of pure logic '=' is the same as "<->".

Indeed, Frege thought that <-> was a case of =, but that is not an idea
much taken up nowadays.

Was it just good luck that Frege's choice of logical constants was the
right one? The apparent variations in later authors are just that:
apparent (because of interdefinability). (Supposing that it _was_
right; and if it was, how do we know that?)

Moeblee is right of course about = sometimes being a logical constant
and sometimes just another binary predicate. (I seem to recall that
Abraham Robinson was fond of the second point of view.)

Why not treat is-an-element-of as logical? I may know the answer to
that: is it because we don't know the "logic" that governs
is-an-element-of, but we do know that of the usual logical constants--or
rather, we have done since Gentzen showed us the introduction and
elimination rules?

Warning: just because Frege's name is used in my first and second
paragraphs, that doesn't mean the two paragraphs are otherwise
connected.

[Graham Cooper]

On Dec 28, 7:07 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:

> Why not treat is-an-element-of as logical?  I may know the answer to
> that: is it because we don't know the "logic" that governs
> is-an-element-of

What DRIVEL is this?

Is "cat" in { "cat", "dog", "horse" }

If you don't know what logic governs the answer being yes then you
don't know anything at all.

Herc