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herbzet · Guest

Jan Burse wrote:
[quote]John Jones schrieb:

So in 2) a conjunction is no more than a logically insignificant
translation. Also, 2) assumes that truths can be totalised, or assembled
in truth tables. Truth tables are of no value when the objecthood or
descriptions of their elements is in doubt.

Ernst Specker constructed a paradox out of that.
He invented the operator |, which stands for
lets say "comensurable". And then a and b is only
defined when a|b holds.

Out of that it can be shown that the leibniz equality
(a and b) and (c and d) = (a and c) and (b and d)
does not necessarely hold. There is also a physical
interpretation of |.

Its all quite fun.
[/quote]
It does sound like fun. Got a reference?

--
hz

herbzet · Guest

John Jones wrote:

[...]

[quote]I don>t think identity is involved. I>m asking if A is true, and B is
true, then what is "'A and B' is true"?
[/quote]
If

A is true, and B is true, <------------
|
then |
|
"'A and B' is true" |
|
is |
|
"A is true" and "B is true" >----------

You already know what 'A and B' is if you are to articulate
the premise of your question.

[quote]The problem is that 'A and B' can only be a valid term or sign if 'and'
is defined. But 'and' is not defined. So, how can an assertion be made
to the effect that "'A and B' is true/false", if 'A and B' is not
adequately defined (underdetermined)?
[/quote]
By defining it. But that>s hard without our already knowing what
'and' means. There *is* a bootstrap problem. We can>t define
everything from nothing. Something must be assumed as known.
All the words that make up definitions in the dictionary must
themselves have definitions, or the dictionary is useless, right?

(But see my concluding remark.)

[...]

[quote]How is 'and' understood? That is the question. 'And' is not defined or
described in logic.
[/quote]
It may be taken as primitive, and hence undefined. And if not
'and', then something else must be taken as primitive.

[quote]And its operations are for that reason,
unintelligible; what is the object A "and" B?
[/quote]
A proposition.

=================================
I>d just like to add that you might, or might not, want to distinguish
a definition of what 'and' "is" from a description of the circumstances
in which 'A and B' is true. Maybe one will serve for the other?

--
hz

Herbert Newman · Guest

Am Sat, 25 Oct 2008 23:01:29 -0400 schrieb herbzet:

[quote]
I have to put this as a challenge. What is the nature of the object that
IS 'A and B' in a conjunction or disjunction of the elements 'A' and 'B'?

If A and B are propositions, then the conjunction 'A and B' is a
proposition. A compound proposition.

I>d prefer to say: If A and B are sentences, then 'A and B' and 'A or B'[/quote]
are sentences too. [I>m using the Quinean corner-quotes here.]

For example, consider the two sentences (in the English language)

John Jones is a crank.
and
John Jones is a schizophrenic.

Clearly the "resulting object"

John Jones is a crank or John Jones is a schizophrenic.

is a sentence (in the English language) again. (Concerning its "nature" I>d
say a sentence is a sentence is a sentence.)

Actually, that>s the _reason_ why we call "and" and "or" _connectives_ in
logic.

Herb

herbzet · Guest

John Jones wrote:
[quote]
First a definition:
A 'disjunction' "A V B " read as "A or B " is false if both A and B are
false. In all other cases it is true.
[/quote]
Good. This is standard, i.e., classical. It is assumed, classically,
that true and false are the only possiblilities, and that they are
mutually exclusive. I>m happy to go with the classical, i.e. standard
presumption.

[quote]The problem that arose with the connective 'and' also arises with a
disjunction.
[/quote]
Insofar as there is a problem with 'and', yep.

[quote]Simply, for both disjunction and conjunction, whatever the truth values
of A and B ARE (plural case), the nature of the third element that IS
(singular case) both A and B is underdetermined whether or not that
element is true or false.
[/quote]
Since you have determined A and B as being possessors of truth-values,
I can see no alternative than to conclude that A and B are propositions.

Therefore the third element that IS (singular case) both A and B is ...
see below.

[quote]The third, unspecified element in conjunction and disjunction can be
seen in implicit form in the diagrams used to teach logical conjunction
and disjunction. Thus we have the Venn overlap
[/quote]
Venn diagrams are not really used for propositional logic. They are
used for the logic of classes (another interpretation of the Boolean
algebra, distinct from its interpretation as propositional logic) and
for some quantified predicate logic. And probably other things, but
not really for propositional logic.

[quote]and the gate of the logic gate.
[/quote]
I>m not sure how you think a logic gate is used to teach logical
conjunction and disjunction. Maybe something like: let 'D' be
an AND gate (it>s not a letter here, just a shape). Then

A ___________
|____________
B ________________________ D ______________ Z
_____________
C __________|

Then the output Z of the logic gate D will be high if and only if
all the inputs A, B, C are high, and low otherwise.

I don>t recall this as a method of teaching propositional logic,
but I suppose it could work, as logic gates are, by intent,
isomorphic (pretty much) in their functioning to propositional
logic (which is why they>re called "logic gates" I suppose).

A real AND gate need not have only two inputs (as shown above)
although in propositional logic the restriction of the 'and'
operator to binary scope necessitates some awkward work-arounds.

Ditto for OR gates.

[quote]I have to put this as a challenge. What is the nature of the object that
IS 'A and B' in a conjunction or disjunction of the elements 'A' and 'B'?
[/quote]
If A and B are propositions, then the conjunction 'A and B' is a
proposition. A compound proposition.

Ditto the disjunction 'A or B'.

--
hz

Herbert Newman · Guest

Am Sun, 26 Oct 2008 01:07:17 -0500 schrieb herbzet:

[quote]
Similarly for 'and'. We don>t define it; we give some axioms
[or rules] for using it, and away we go. [...]

Actually, I like this "approach" very much. The funny thing is that we can[/quote]
actually set up a logical system for, say, propositional logic (PC) this
way. In a decent system of _natural deduction_ for PC we have the RULES:

A B A and B A and B
------- (&I) ------- (&E) ------- (&E)
A and B A B

Those rules tell us how to manipulate formulas (or sentences) in connection
with the connective "and".

The first rule allows to "introduce" the conjunctive 'A and B' if two
sentences A and B have already occurred in the course of proof. While the
other rule(s) allows to "split up" the conjunctive into the two involved
conjuncts.

We might claim that this way "and" is "introduced" in an "operationalistic"
way. Pondering about these things, Wittgenstein comes to mind:

"For a large class of cases ¡X though not for all ¡X in which we employ the
word 'meaning' it can be defined thus: the meaning of a word is its use in
the language" (PI 43)

Herb

Jan Burse · Guest

herbzet schrieb:
[quote]
Jan Burse wrote:
John Jones schrieb:

So in 2) a conjunction is no more than a logically insignificant
translation. Also, 2) assumes that truths can be totalised, or assembled
in truth tables. Truth tables are of no value when the objecthood or
descriptions of their elements is in doubt.
Ernst Specker constructed a paradox out of that.
He invented the operator |, which stands for
lets say "comensurable". And then a and b is only
defined when a|b holds.

Out of that it can be shown that the leibniz equality
(a and b) and (c and d) = (a and c) and (b and d)
does not necessarely hold. There is also a physical
interpretation of |.

Its all quite fun.

It does sound like fun. Got a reference?

--
hz
[/quote]
Sure:

http://www.iumj.indiana.edu/IUMJ/dfulltext.php?year=1968&volume=17&artid=17004

Herbert Newman · Guest

Am Sun, 26 Oct 2008 01:07:17 -0500 schrieb herbzet:

[quote]
Similarly for 'and'. We don>t define it; we give some axioms
[or rules] for using it, and away we go. [...]

Actually, I like this "approach" very much. The funny thing is that we can[/quote]
actually set up a logical system for, say, propositional logic (PC) this
way. In a decent system of _natural deduction_ for PC we have the RULES:

A B A and B A and B
------- (&I) ------- (&E) ------- (&E)
A and B A B

Those rules tell us how to manipulate formulas (or sentences) in connection
with the connective "and".

The first rule allows to "introduce" the conjunction 'A and B' if two
sentences A and B have already occurred in the course of proof. While the
other rule(s) allows to "split up" the conjunction into the two involved
conjuncts.

We might claim that this way "and" is "introduced" in an "operationalistic"
way. Pondering about these things, Wittgenstein comes to mind:

"For a large class of cases ¡X though not for all ¡X in which we employ the
word 'meaning' it can be defined thus: the meaning of a word is its use in
the language" (PI 43)

Herb

Daryl McCullough · Guest

herbzet says...
[quote]Similarly for 'and'. We don>t define it; we give some axioms
for using it, and away we go. Some things must be accepted as
given just to kickstart the whole enterprise.
[/quote]
Yes, that>s right. However, it is interesting that in
some systems of higher-order logic, the only operators
that need to be introduced is universal quantification
and implication.

The other logical operators are then definable. For
example:

A and B == forall propositions C, (A implies (B implies C)) implies C

A or B == forall propositions C,
((A implies C) implies ((B implies C) implies C))

not A == forall propositions C, A implies C

exists x . B(x)
== forall propositions C, (forall x, B(x) implies C) implies C

--
Daryl McCullough
Ithaca, NY

John Jones · Guest

herbzet wrote:
[quote]
John Jones wrote:

[...]

I don>t think identity is involved. I>m asking if A is true, and B is
true, then what is "'A and B' is true"?

If

A is true, and B is true, <------------
|
then |
|
"'A and B' is true" |
|
is |
|
"A is true" and "B is true" >----------

You already know what 'A and B' is if you are to articulate
the premise of your question.
[/quote]
Ah, but I don>t. Look: I know what "A" is. I know what "B" is. I know
what "A" and "B" ARE. But what IS "A and B"?

I really have no idea. Is it a new object? an emergent property? Or is
it an odd way of saying '"A" and "B"'. It could be any of these or
something else.

[quote]
The problem is that 'A and B' can only be a valid term or sign if 'and'
is defined. But 'and' is not defined. So, how can an assertion be made
to the effect that "'A and B' is true/false", if 'A and B' is not
adequately defined (underdetermined)?

By defining it. But that>s hard without our already knowing what
'and' means.
[/quote]
Yes, we all know what "and" means. We know what "and" means in "A" and
"B". But what does "and" mean in "A and B"? Obviously the term is being
used in a different way. It looks like a synthesis of some sort.

[quote]There *is* a bootstrap problem. We can>t define
everything from nothing. Something must be assumed as known.
All the words that make up definitions in the dictionary must
themselves have definitions, or the dictionary is useless, right?
[/quote]
Yes, we don>t have to go that far. Hope the above explains it, and that
it can be seen that "and" is being used in an unknown way.

[quote](But see my concluding remark.)

[...]

How is 'and' understood? That is the question. 'And' is not defined or
described in logic.

It may be taken as primitive, and hence undefined. And if not
'and', then something else must be taken as primitive.

And its operations are for that reason,
unintelligible; what is the object A "and" B?

A proposition.

=================================
I>d just like to add that you might, or might not, want to distinguish
a definition of what 'and' "is" from a description of the circumstances
in which 'A and B' is true. Maybe one will serve for the other?
[/quote]
But logic is claiming that "A and B" is distinct from '"A" and "B"'. So
the word 'and' is being used in an unknown way - we are not told how.

John Jones · Guest

Herbert Newman wrote:
[quote]Am Sat, 25 Oct 2008 23:01:29 -0400 schrieb herbzet:

I have to put this as a challenge. What is the nature of the object that
IS 'A and B' in a conjunction or disjunction of the elements 'A' and 'B'?

If A and B are propositions, then the conjunction 'A and B' is a
proposition. A compound proposition.

I>d prefer to say: If A and B are sentences, then 'A and B' and 'A or B'
are sentences too. [I>m using the Quinean corner-quotes here.]

For example, consider the two sentences (in the English language)

John Jones is a crank.
and
John Jones is a schizophrenic.

Clearly the "resulting object"

John Jones is a crank or John Jones is a schizophrenic.

is a sentence (in the English language) again. (Concerning its "nature" I>d
say a sentence is a sentence is a sentence.)

Actually, that>s the _reason_ why we call "and" and "or" _connectives_ in
logic.

Herb
[/quote]
Try this then.

IF

"It is true that the number of deaths caused by drink is 1000, and it is
true that the number of deaths caused by driving is 2000",

THEN, how can I make any claim regarding the truth of

"It is true that the number of deaths caused by drinking and driving is
....."

Clearly, '"A" and "B"' is not the same as '"A and B"'. There>s no
connection between them.

I can>t make any claims regarding the truth of "A and B" from the truths
of "A" and "B". Yes?

John Jones · Guest

Mitch Harris wrote:
[quote]On Oct 24, 3:43 pm, John Jones <jonescard...@aol.com> wrote:
First a definition:
A 'disjunction' "A V B " read as "A or B " is false if both A and B are
false. In all other cases it is true.

The problem that arose with the connective 'and' also arises with a
disjunction.

Simply, for both disjunction and conjunction, whatever the truth values
of A and B ARE (plural case), the nature of the third element that IS
(singular case) both A and B is underdetermined whether or not that
element is true or false.

How are the total boolean functions for 'and' and 'or'
underdetermined? All the output values are specified for all possible
inputs. That>s pretty determined to me.

The third, unspecified element in conjunction and disjunction can be
seen in implicit form in the diagrams used to teach logical conjunction
and disjunction. Thus we have the Venn overlap and the gate of the logic
gate.

I have to put this as a challenge. What is the nature of the object that
IS 'A and B' in a conjunction or disjunction of the elements 'A' and 'B'?

What is the nature of just plain old 'A'? What does that question
mean? Well, whatever the answer I highly suspect it is the same answer
as for 'A and B'?
[/quote]
The objection I made to that idea is that if 'A and B' means 'A' and 'B'
then as a translation or rephrasing it is not logically significant.
Besides which, Logic deduces or claims to have an insight on the matter.
This suggests that it isn>t a rephrasing. If that>s the case, then what
is 'A and B'? Somebody tell me.

[quote]So you>re really not complaining about anything in particular about
'and' yet (as opposed to 'or'). You>re troubled by something about
connectives in general.
[/quote]
No, it>s the logical 'and' that is bothering me. What is "A and B" if it
can be deduced from, or is not the same as, "A" and "B"?

Ross A. Finlayson · Guest

[quote]
I think this is interesting this description, even relevant:

http://www.jfsowa.com/ontology/toplevel.htm

[/quote]
Now that>s a theory.

Now that>s A theory.

Thanks,

Ross F.

Ross A. Finlayson · Guest

John Jones wrote:
[quote]Mitch Harris wrote:
On Oct 24, 3:43 pm, John Jones <jonescard...@aol.com> wrote:
First a definition:
A 'disjunction' "A V B " read as "A or B " is false if both A and B are
false. In all other cases it is true.

The problem that arose with the connective 'and' also arises with a
disjunction.

Simply, for both disjunction and conjunction, whatever the truth values
of A and B ARE (plural case), the nature of the third element that IS
(singular case) both A and B is underdetermined whether or not that
element is true or false.

How are the total boolean functions for 'and' and 'or'
underdetermined? All the output values are specified for all possible
inputs. That>s pretty determined to me.

The third, unspecified element in conjunction and disjunction can be
seen in implicit form in the diagrams used to teach logical conjunction
and disjunction. Thus we have the Venn overlap and the gate of the logic
gate.

I have to put this as a challenge. What is the nature of the object that
IS 'A and B' in a conjunction or disjunction of the elements 'A' and
'B'?

What is the nature of just plain old 'A'? What does that question
mean? Well, whatever the answer I highly suspect it is the same answer
as for 'A and B'?

The objection I made to that idea is that if 'A and B' means 'A' and 'B'
then as a translation or rephrasing it is not logically significant.
Besides which, Logic deduces or claims to have an insight on the matter.
This suggests that it isn>t a rephrasing. If that>s the case, then what
is 'A and B'? Somebody tell me.

So you>re really not complaining about anything in particular about
'and' yet (as opposed to 'or'). You>re troubled by something about
connectives in general.

No, it>s the logical 'and' that is bothering me. What is "A and B" if it
can be deduced from, or is not the same as, "A" and "B"?

[/quote]
Jones, you>re apparently a genius, I wouldn>t sweat it too much if
people identify you with cranks. Not being a troll, don>t feed them
anyways. (Jones isn>t a crank nor troll.)

You apparently just want a management summary of technical and analytic
philosophy, so you can go about describing pretty much everything in
terms of ontological primitives.

So, as you go about categorizing mostly canon, (as you>ll note Jones
purely spouts dogma, generally) it makes sense to basically generalize.

Sooner or later you>ll be back to computer science, then having a
reasonable understanding of the context leads to nice 2x2 squares.

Way to go, Jones.

Thanks,

Ross F.

John Jones · Guest

Ross A. Finlayson wrote:
[quote]John Jones wrote:

You apparently just want a management summary of technical and analytic
philosophy, so you can go about describing pretty much everything in
terms of ontological primitives.

So, as you go about categorizing mostly canon, (as you>ll note Jones
purely spouts dogma, generally) it makes sense to basically generalize.

Sooner or later you>ll be back to computer science, then having a
reasonable understanding of the context leads to nice 2x2 squares.

Way to go, Jones.

Thanks,

Ross F.
[/quote]
Thankyou. But I would still like someone here to tell me what the
distinction is between '"A" and "B"' and '"A and B"'.

Herbert Newman · Guest

Am Sun, 26 Oct 2008 16:37:00 +0000 schrieb John Jones:

[quote]IF

"It is true that the number of deaths caused by drink is 1000, and it is
true that the number of deaths caused by driving is 2000",

THEN, how can I make any claim regarding the truth of

"It is true that the number of deaths caused by drinking and driving is
...."

Huh? Relevance? :-o[/quote]

[quote]
I can>t make any claims regarding the truth of "A and B" from the truths
of "A" and "B". Yes?

Sure you can. If -for example- the two sentences[/quote]

John Jones is an idiot
and
John Jones does not know what he>s talking about

are both true, then the "compound" sentence

John Jones is an idiot and John Jones does not know what he>s
talking about

is true too.

Herb


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