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herbzet · Guest

John Jones wrote:
[quote]herbzet wrote:
herbzet wrote:

From an algebraic point of view, the variables of a Boolean algebra
could take as arguments and values propositions, truth-values, or
any one of many other sorts of things, e.g., switches (open or closed),
sets of points in a plane, etc.)

I meant, from an algebraic point of view, the /operators/ of a Boolean
algebra take as arguments and values ... etc.

Arguments, values, operators and variables look like another thing to
look at.
[/quote]
Yes, also functions.

Witt, btw, distinguishes between an operator and a function, in
5.23, 5.251, 5.3, which I believe many people do, but the discussion
gets pretty abstruse. I use "operator" in a pretty loose way.

--
hz

herbzet · Guest

Mitch wrote:
[quote]herbzet wrote:
Mitch wrote:

I>m surprised you>re picking on 'and'. 'or' is much more problematic.
Cripes, what about 'if then'?

It is precisely because 'and' is *less* problematic than 'or' or
'if then' that it is the place for a really fundamental critique
of (symbolic?) logic.

Yes, I see that.

About the only thing that is about as un-problematic as 'and'
in logic is 'not'...

There I>d say there>s quite a bit more problematic about
'not' than 'and'.

(p and p) <-> p is a lot more obvious and uncontestable than --p -> p
[/quote]
Well, I guess you have a point there.

What I was thinking was that in ordinary (non-mathematical) discourse
there is virtually no ambiguity in saying "It is not the case that X"
(or just "X is false") where X is a declarative sentence -- that is,
beyond any ambiguity that may attach to X itself.

--
hz

Alan Smaill · Guest

herbzet <herbzet@gmail.com> writes:

[quote]Mitch wrote:
herbzet wrote:
Mitch wrote:

I>m surprised you>re picking on 'and'. 'or' is much more problematic.
Cripes, what about 'if then'?

It is precisely because 'and' is *less* problematic than 'or' or
'if then' that it is the place for a really fundamental critique
of (symbolic?) logic.

Yes, I see that.

About the only thing that is about as un-problematic as 'and'
in logic is 'not'...

There I>d say there>s quite a bit more problematic about
'not' than 'and'.

(p and p) <-> p is a lot more obvious and uncontestable than --p -> p

Well, I guess you have a point there.

What I was thinking was that in ordinary (non-mathematical) discourse
there is virtually no ambiguity in saying "It is not the case that X"
(or just "X is false") where X is a declarative sentence -- that is,
beyond any ambiguity that may attach to X itself.
[/quote]
The simple existence of advocates of intuitionist logic,
on a philosophical basis, also says that there is
basic disagreement about the meaning of "not" in a way that
doesn>t happen about "and".

[quote]--
hz
[/quote]
--
Alan Smaill

herbzet · Guest

Alan Smaill wrote:
[quote]herbzet writes:
Mitch wrote:
herbzet wrote:
Mitch wrote:

I>m surprised you>re picking on 'and'. 'or' is much more problematic.
Cripes, what about 'if then'?

It is precisely because 'and' is *less* problematic than 'or' or
'if then' that it is the place for a really fundamental critique
of (symbolic?) logic.

Yes, I see that.

About the only thing that is about as un-problematic as 'and'
in logic is 'not'...

There I>d say there>s quite a bit more problematic about
'not' than 'and'.

(p and p) <-> p is a lot more obvious and uncontestable than --p -> p

Well, I guess you have a point there.

What I was thinking was that in ordinary (non-mathematical) discourse
there is virtually no ambiguity in saying "It is not the case that X"
(or just "X is false") where X is a declarative sentence -- that is,
beyond any ambiguity that may attach to X itself.

The simple existence of advocates of intuitionist logic,
on a philosophical basis, also says that there is
basic disagreement about the meaning of "not" in a way that
doesn>t happen about "and".
[/quote]
I don>t know if you>re saying anything different from what
Mitch has already suggested.

In any case, I think in ordinary discourse (which may
be of more concern to JJ) "not X" is not(!) used to say
"there is no proof X", which is how it seems to be used
in intuitionist logic.

For that matter, I>m not sure that "and" in intuitionist
logic is wholly identical in meaning with the "and" of
classical logic.

--
hz

John Jones · Guest

Alan Smaill wrote:
[quote]herbzet <herbzet@gmail.com> writes:

Mitch wrote:
herbzet wrote:
Mitch wrote:

I>m surprised you>re picking on 'and'. 'or' is much more problematic.
Cripes, what about 'if then'?
It is precisely because 'and' is *less* problematic than 'or' or
'if then' that it is the place for a really fundamental critique
of (symbolic?) logic.
Yes, I see that.

About the only thing that is about as un-problematic as 'and'
in logic is 'not'...
There I>d say there>s quite a bit more problematic about
'not' than 'and'.

(p and p) <-> p is a lot more obvious and uncontestable than --p -> p
Well, I guess you have a point there.

What I was thinking was that in ordinary (non-mathematical) discourse
there is virtually no ambiguity in saying "It is not the case that X"
(or just "X is false") where X is a declarative sentence -- that is,
beyond any ambiguity that may attach to X itself.

The simple existence of advocates of intuitionist logic,
on a philosophical basis, also says that there is
basic disagreement about the meaning of "not" in a way that
doesn>t happen about "and".

--
hz

[/quote]
I>m not disagreeing with the non-intuitionist working definition of
'and'. I am saying that there is no coherent definition. The claim 'p
and q' is itself incomprehensible, whether or not it is somehow
"obtained" or taken from the "conjunction" of 'p' and 'q'.

John Jones · Guest

herbzet wrote:
[quote]
John Jones wrote:

It>s unclear from what you wrote just above whether you mean to
assert that by "not-P" whether you consider the propositional
operator 'not' as operating on P or 'P'.

[/quote]
'P' or P, offers only itself, while not-P, if P, or etc, offer others.

And, not, if, or, are convenient ways of expressing other objects or
proposals other than the proposal 'operated' on. In other words,
operators are objects or proposals. They are not logical powers or
agencies that act upon a proposal, object or proposition etc. I don>t
see any distinction between operators and what they operate on.

John Jones · Guest

herbzet wrote:
[quote]
John Jones wrote:
herbzet wrote:
herbzet wrote:

From an algebraic point of view, the variables of a Boolean algebra
could take as arguments and values propositions, truth-values, or
any one of many other sorts of things, e.g., switches (open or closed),
sets of points in a plane, etc.)
I meant, from an algebraic point of view, the /operators/ of a Boolean
algebra take as arguments and values ... etc.
Arguments, values, operators and variables look like another thing to
look at.

Yes, also functions.

Witt, btw, distinguishes between an operator and a function, in
5.23, 5.251, 5.3, which I believe many people do, but the discussion
gets pretty abstruse. I use "operator" in a pretty loose way.

--
hz
[/quote]
I don>t know about these arguments and values etc. The ideas never took
root with me. Every time I see the words I must look them up, time after
time. Something must be wrong. I suspect that they are convoluted ways
of presenting objects. The idea that there are objects that are acted
upon by logical force is a strange one.

herbzet · Guest

Alan Smaill wrote:
[quote]herbzet writes:
Mitch wrote:
herbzet wrote:
Mitch wrote:

I>m surprised you>re picking on 'and'. 'or' is much more problematic.
Cripes, what about 'if then'?

It is precisely because 'and' is *less* problematic than 'or' or
'if then' that it is the place for a really fundamental critique
of (symbolic?) logic.

Yes, I see that.

About the only thing that is about as un-problematic as 'and'
in logic is 'not'...

There I>d say there>s quite a bit more problematic about
'not' than 'and'.

(p and p) <-> p is a lot more obvious and uncontestable than --p -> p

Well, I guess you have a point there.

What I was thinking was that in ordinary (non-mathematical) discourse
there is virtually no ambiguity in saying "It is not the case that X"
(or just "X is false") where X is a declarative sentence -- that is,
beyond any ambiguity that may attach to X itself.

The simple existence of advocates of intuitionist logic,
on a philosophical basis, also says that there is
basic disagreement about the meaning of "not" in a way that
doesn>t happen about "and".
[/quote]
I>m not sure that you>re saying something different from what
Mitch has already suggested.

In ordinary discourse "It is not the case that X" does not mean the
same as "There is no proof of X", which is what ~X seems to mean
in intuitionist logic.

I>m not sure that "X and Y" means exactly the same in intuitionist
logic as it does in classical logic, either.

(I thought I had posted this earlier, but it seems to have
disappeared into the ozone.)

--
hz

herbzet · Guest

Alan Smaill wrote:
[quote]herbzet writes:
Mitch wrote:
herbzet wrote:
Mitch wrote:

I>m surprised you>re picking on 'and'. 'or' is much more problematic.
Cripes, what about 'if then'?

It is precisely because 'and' is *less* problematic than 'or' or
'if then' that it is the place for a really fundamental critique
of (symbolic?) logic.

Yes, I see that.

About the only thing that is about as un-problematic as 'and'
in logic is 'not'...

There I>d say there>s quite a bit more problematic about
'not' than 'and'.

(p and p) <-> p is a lot more obvious and uncontestable than --p -> p

Well, I guess you have a point there.

What I was thinking was that in ordinary (non-mathematical) discourse
there is virtually no ambiguity in saying "It is not the case that X"
(or just "X is false") where X is a declarative sentence -- that is,
beyond any ambiguity that may attach to X itself.

The simple existence of advocates of intuitionist logic,
on a philosophical basis, also says that there is
basic disagreement about the meaning of "not" in a way that
doesn>t happen about "and".
[/quote]
I>m not sure that you>re saying anything different from what
Mitch has already suggested.

In ordinary discourse "It is not the case that X" does not
mean the same as "There is no proof of X", which is what
it seems to mean in intuitionist logic.

I>m not sure that "X and Y" means the same in intuitionist
logic as it does in classical logic either.

--
hz

Mitch · Guest

On Oct 15, 8:55 pm, herbzet <herb...@gmail.com> wrote:
[quote]John Jones wrote:
herbzet wrote:
Mitch wrote:
I>m surprised you>re picking on 'and'. 'or' is much more problematic..
Cripes, what about 'if then'?
....
Er...yes, I will take a look at those others soon.

Stick with "and" till you beat it death, is my advice. You
can>t get much more fundamental than that, and "not".
[/quote]
OK, but now I realize that all the substantive discussion so far
really has only said things about 'and' that are also the case for
other operators...um...logical connectors. That is, the discussion has
been about operators, what a proposition -is-, 'combining' values.
Anything special about that simplest of connectors 'and', different
from the others?

[quote]I think that 'and' in logic is a mysterious operation or power that
allows two truths to be reduced to one truth. How is that possible? It
breaks the link between truth and the object of truth.
[/quote]
(substituting 'or', or 'then' or 'xor' doesn>t make much difference in
the above)

Can anything be said about 'and' that is special? (-is- there a
difference between classical and intuitionistic 'and'? (I don>t think
so))

Mitch

herbzet · Guest

John Jones wrote:
[quote]herbzet wrote:
John Jones wrote:

It>s unclear from what you wrote just above whether you mean to
assert that by "not-P" whether you consider the propositional
operator 'not' as operating on P or 'P'.

'P' or P, offers only itself, while not-P, if P, or etc, offer others.

And, not, if, or, are convenient ways of expressing other objects or
proposals other than the proposal 'operated' on. In other words,
operators are objects or proposals. They are not logical powers or
agencies that act upon a proposal, object or proposition etc. I don>t
see any distinction between operators and what they operate on.
[/quote]
"what they operate on" = operand

It wouldn>t hurt, I think, to ascribe, for convenience, a purely
grammatical existence to operators, e.g., "the" is an operator
(in this case, an article) that takes a noun phrase as an argument
and returns a noun phrase.

In any case, it>s sometimes helpful to keep in mind the distinction
between the language and what is denoted by the language, if anything.

After all, the language is a pretty concrete object, while what,
presumably, is being denoted (especially in logic/mathematics)
is not, particularly.

--
hz

herbzet · Guest

John Jones wrote:
[quote]herbzet wrote:
John Jones wrote:
herbzet wrote:
herbzet wrote:

From an algebraic point of view, the variables of a Boolean algebra
could take as arguments and values propositions, truth-values, or
any one of many other sorts of things, e.g., switches (open or closed),
sets of points in a plane, etc.)
I meant, from an algebraic point of view, the /operators/ of a Boolean
algebra take as arguments and values ... etc.
Arguments, values, operators and variables look like another thing to
look at.

Yes, also functions.

Witt, btw, distinguishes between an operator and a function, in
5.23, 5.251, 5.3, which I believe many people do, but the discussion
gets pretty abstruse. I use "operator" in a pretty loose way.

--
hz

I don>t know about these arguments and values etc. The ideas never took
root with me. Every time I see the words I must look them up, time after
time. Something must be wrong. I suspect that they are convoluted ways
of presenting objects. The idea that there are objects that are acted
upon by logical force is a strange one.
[/quote]
Just think of them as grammatical devices. Like "John>s wife" is a
way of indicating a particular person. Or "the (positive) square root
of two" indicates a particular real number.

Of course, adjectives are objects too, but different sorts of objects than nouns.

But still, when it comes to differences and similarities between things,
recall (or google) "herbzet>s axiom".

--
hz

herbzet · Guest

Mitch wrote:
[quote]herbzet wrote:
John Jones wrote:
herbzet wrote:
Mitch wrote:

I>m surprised you>re picking on 'and'. 'or' is much more problematic.
Cripes, what about 'if then'?
...
Er...yes, I will take a look at those others soon.

Stick with "and" till you beat it death, is my advice. You
can>t get much more fundamental than that, and "not".

OK, but now I realize that all the substantive discussion so far
really has only said things about 'and' that are also the case for
other operators...um...logical connectors. That is, the discussion has
been about operators, what a proposition -is-, 'combining' values.
Anything special about that simplest of connectors 'and', different
from the others?
[/quote]
From a mathematical point of view, no, it>s just an unremarkable
Boolean operator. From a natural language point-of-view, it>s
about as unambiguous a sentential connective as you>re gonna get.

If we go sub-sentential, even the copula "is" has at least three distinct
meanings:

1) Identity -- Santa Claus is Kris Kringle.
2) Membership -- Smith is a physician.
3) Subset -- A mammal is a vertebrate.

[quote]I think that 'and' in logic is a mysterious operation or power that
allows two truths to be reduced to one truth. How is that possible? It
breaks the link between truth and the object of truth.

(substituting 'or', or 'then' or 'xor' doesn>t make much difference in
the above)

Can anything be said about 'and' that is special?
[/quote]
The main point is that, from a natural language point of view, which
is, I think, part of JJ>s concern, it>s meaning is less ambiguous than
practically any other sentential connective.

I think "not" is similarly unambiguous as a sentential connective.
It takes some mathematical sophistication to see "not" with anything
other than its natural language meaning.

[quote](-is- there a
difference between classical and intuitionistic 'and'? (I don>t think
so))
[/quote]
Intuitionist proposititional operators are (mostly) not interdefinable
as they are in classical logic. Also, there is no finite-valued
semantics for the intuitionist propositional calculus, whereas a two-valued
semantics {true, false} is available for classical propositional logic.

It could be argued that the intuitionist "and" is thus subtly different
from the classical "and".

A good discussion is in "Formal Logic" [1962, OUP] by Arthur N. Prior

--
hz

Mitch · Guest

On Oct 20, 12:33 pm, John Jones <jonescard...@aol.com> wrote:
[quote]herbzet wrote:
John Jones wrote:
And, not, if, or, are convenient ways of expressing other objects or
proposals other than the proposal 'operated' on. In other words,
operators are objects or proposals. They are not logical powers or
agencies that act upon a proposal, object or proposition etc. I don>t
see any distinction between operators and what they operate on.

"what they operate on" = operand

Or doer and done to. It>s strange that we describe the former as an
agency or operator but not the latter.
[/quote]
I>m sure one could rearrange things in a dual fashion so that operator
and operand reverse their roles. And that might be meaningful and
useful. But it is quite useful as it stands that 'and', 'or', 'not',
etc, are considered operators and their parameters are operands
(circularly).

But maybe you>re just want to say that somehow these operators/
functions on objects are objects themselves. That>s surely a
productive direction (just calling them boolean functions or operators
has already made objects out of them to be manipulated themselves).

[quote]After all, it>s always both that
change. The reason is that doer and done to are a priviliged binary.
[/quote]
Are you saying the distinction itself is privileged? With respect to
what outside that pair? What do you mean by 'privilege'?

[quote]This means that a privilege, or favoured status, is granted from outside
the binary to one of the pair. In this case the operator or doer is
classed as a superior sort of object - an agency, while the done to, or
operand, is not so favoured.
[/quote]
Oh. The oppressed operand. It does seem a bit unfair to the operands
that the operators get all the power. There are mathematical
situations where the operands are almost entirely ignored (the
execrable Category Theory). But at least the operands are happy.

[quote]All we have to do then is give a reason for
granting this privilege, which shouldn>t be hard to find.
[/quote]
Yes. The reason is that it is a convenient idea borrowed from other
parts of mathematics. (Note though that logical thinking can be done
quite apart from these valuations (operators applied to operands))

[quote]It wouldn>t hurt, I think, to ascribe, for convenience, a purely
grammatical existence to operators, e.g., "the" is an operator
(in this case, an article) that takes a noun phrase as an argument
and returns a noun phrase.

I would have to see some examples. I know latin doesn>t have a word for
'the'.
[/quote]
- example? He>s giving an example of operator/operand, thing to do
work/thing work to be done with. NP -- the -> NP, a function on noun
phrases that takes a noun phrase and returns a noun phrase with
modified meaning. In the context of this thread, it is convenient to
think of the boolean connectors as (multivariate) functions from
boolean values to boolean values (or if you prefer operators on
booleans).

- Whether a particular language (like Latin) can be described as
having such operators or not is not the issue. herbzet is just saying
'there>s one way of looking at things' for English.

- 'ille' is etymologically the source for the definite article in all
Romance languages. It>s a bit more emphatic than 'the', more like
'that-there', but it supplies a very similar function (pointing out a
specific item). But anyway, the existence of a language without a
definite article (of which there are many) is a bit of a red-herring.
Can one think productively about certain grammatical constructs as
meaning operators? Yes.

Mitch

John Jones · Guest

herbzet wrote:
[quote]
John Jones wrote:
herbzet wrote:
John Jones wrote:
It>s unclear from what you wrote just above whether you mean to
assert that by "not-P" whether you consider the propositional
operator 'not' as operating on P or 'P'.

'P' or P, offers only itself, while not-P, if P, or etc, offer others.

And, not, if, or, are convenient ways of expressing other objects or
proposals other than the proposal 'operated' on. In other words,
operators are objects or proposals. They are not logical powers or
agencies that act upon a proposal, object or proposition etc. I don>t
see any distinction between operators and what they operate on.

"what they operate on" = operand
[/quote]
Or doer and done to. It>s strange that we describe the former as an
agency or operator but not the latter. After all, it>s always both that
change. The reason is that doer and done to are a priviliged binary.
This means that a privilege, or favoured status, is granted from outside
the binary to one of the pair. In this case the operator or doer is
classed as a superior sort of object - an agency, while the done to, or
operand, is not so favoured. All we have to do then is give a reason for
granting this privilege, which shouldn>t be hard to find.

[quote]It wouldn>t hurt, I think, to ascribe, for convenience, a purely
grammatical existence to operators, e.g., "the" is an operator
(in this case, an article) that takes a noun phrase as an argument
and returns a noun phrase.
[/quote]
I would have to see some examples. I know latin doesn>t have a word for
'the'.

[quote]In any case, it>s sometimes helpful to keep in mind the distinction
between the language and what is denoted by the language, if anything.

After all, the language is a pretty concrete object, while what,
presumably, is being denoted (especially in logic/mathematics)
is not, particularly.

--
hz[/quote]


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