In need of the all-out revision of symbolic logic
Eukie_M_SHIRAISHI
20th century),
“p implies q” has had been defined to mean “not-p or q”.
Here let P be “A lion is a mammal” and Q be “A lion and a whale are
both mammals”
It is clear that Q implies P but P does not imply Q.
Hence “P implies Q” is a falsity.
On the other hand, “not-P or Q” is a truth.
Therefore, it is a mistake to define “p implies q” to mean “not-p or
q”.
Since implication is one of the most fundamental concepts in logic,
all-out revision
is needed in the Fregean Theory of Logic.
Have a look at the following Web site:- <http://www.age.ne.jp/x/eurms/
PNG/Key/Honron-2.html#02-2>
sar...@gmail.com
> In the Fregean Theory of Logic (the standard theory of logic in the
> 20th century),
> “p implies q” has had been defined to mean “not-p or q”.
>
> Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> both mammals”
> It is clear that Q implies P but P does not imply Q.
> Hence “P implies Q” is a falsity.
> On the other hand, “not-P or Q” is a truth.
> Therefore, it is a mistake to define “p implies q” to mean “not-p or
> q”.
Sorry, but I don't see why is “not-P or Q” true. I mean, what if a
lion and a whale are both not mammals?
It is not the situation in real world, but when you compare logic
statements, only the possible truth vlues of p and q metter. In our
case, they both aree true, so the truth values of both “p implies q”
and “not-p or q” are T.
I'm not a logician, so I'd be dilighted to see an explanation on the
subject...
Eukie_M_SHIRAISHI
> On Apr 29, 4:59 am, Eukie_M_SHIRAISHI <ms.eu...@gmail.com> wrote:
>
> > In the Fregean Theory of Logic (the standard theory of logic in the
> > 20th century),
> > “p implies q” has had been defined to mean “not-p or q”.
>
> > Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> > both mammals”
> > It is clear that Q implies P but P does not imply Q.
> > Hence “P implies Q” is a falsity.
> > On the other hand, “not-P or Q” is a truth.
> > Therefore, it is a mistake to define “p implies q” to mean “not-p or
> > q”.
>
> Sorry, but I don't see why is “not-P or Q” true. I mean, what if a
> lion and a whale are both not mammals?
Thank you for your following.
"not-P or Q”"means "A lion is not a mammal or a lion and a whale are
both mammals". Therefore, it is a truth
ST
> Thank you for your following.
>
> "not-P or Q”"means "A lion is not a mammal or a lion and a whale are
>
> -הראה טקסט מצוטט-
Yes, but isn't "p->q" truth too?
I mean, if I remember correctly the truth value table of "implies", it
only gives F when p is F and p is T
Therefore, as here we have T->T, we have truth value T...
Or have I misunderstood something?
Eukie_M_SHIRAISHI
> > Thank you for your following.
>
> > "not-P or Q”"means "A lion is not a mammal or a lion and a whale are
> > both mammals". Therefore, it is a truth-הסתר טקסט מצוטט-
>
> > -הראה טקסט מצוטט-
>
> Yes, but isn't "p->q" truth too?
In the case where P is "A lion is a mammal." and Q is "A lion and a
whale
are both mammals", P does not implie Q. Therefore, in this case "P
implies Q"
is a falsity
> I mean, if I remember correctly the truth value table of "implies", it
> only gives F when p is F and p is T
> Therefore, as here we have T->T, we have truth value T...
It is shown in the case above that it is a mistake to define "p impies
q"
by the truth table.
Eukie_M_SHIRAISHI
> On 4月30日, 午後6:44, ST <sara...@gmail.com> wrote:
Sorry, I've made a typing mistake.
P does not "implie" Q ----> P does not "imply" Q
Eukie_M_SHIRAISHI
> Or have I misunderstood something?
It is truly a grave mistake to define “p implies q” by the truth-table.
Frederick Williams
>
> In the Fregean Theory of Logic (the standard theory of logic in the
> 20th century),
> “p implies q” has had been defined to mean “not-p or q”.
>
> Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> both mammals”
> It is clear that Q implies P but P does not imply Q.
> Hence “P implies Q” is a falsity.
Not so. Since Q is true, P implies Q. If you think this is a silly
meaning for "implies" to have, you might wish to read "not-p or q" as "p
materially implies q" or (better) as "if p then q". For other accounts
of "implies" you may wish to start here:
http://en.wikipedia.org/wiki/Logical_implication or here:
http://plato.stanford.edu/entries/conditionals/
> On the other hand, “not-P or Q” is a truth.
> Therefore, it is a mistake to define “p implies q” to mean “not-p or
> q”.
>
> Since implication is one of the most fundamental concepts in logic,
> all-out revision
> is needed in the Fregean Theory of Logic.
>
> Have a look at the following Web site:- <http://www.age.ne.jp/x/eurms/
> PNG/Key/Honron-2.html#02-2>
--
Remove "antispam" and ".invalid" for e-mail address.
Colin
> In the Fregean Theory of Logic (the standard theory of logic in the
> 20th century),
> “p implies q” has had been defined to mean “not-p or q”.
>
> Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> both mammals”
Ok. That means (Ux)(Lion(x) --> Mammal(x)) and either (Ux)((Lion(x) v
Whale(x)) --> Mammal(x)), or (Ux)(Lion(x) --> Mammal(x)) & (Ux)
(Whale(x) --> Mammal(x)).
> It is clear that Q implies P but P does not imply Q.
> Hence “P implies Q” is a falsity.
Well, no: P does imply Q because both P and Q happen to be true.
Perhaps you meant Q is not a logical consequence of P?
> On the other hand, “not-P or Q” is a truth.
Correct, simply because Q is true.
> Therefore, it is a mistake to define “p implies q” to mean “not-p or
> q”.
No.
Eukie_M_SHIRAISHI
Williams"@antispamhotmail.co.uk.invalid> wrote:
> Eukie_M_SHIRAISHI wrote:
>
> > In the Fregean Theory of Logic (the standard theory of logic in the
> > 20th century),
> > “p implies q” has had been defined to mean “not-p or q”.
>
> > Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> > both mammals”
> > It is clear that Q implies P but P does not imply Q.
> > Hence “P implies Q” is a falsity.
>
> Not so. Since Q is true, P implies Q.
Do you think that "A lion is a mammal" implies "A lion and a whale are
both mammalds " ?
"A lion and a whale are both mammalds " does imply "A lion is a
mammal"
but "A lion is a mammal" does NOT imply "A lion and a whale are both
mammalds "
We found that it is truly a grave mistake to define "p implies q" by
the truth-table.
See:-<http://www.age.ne.jp/x/eurms/PNG/Key/Honron-3.html#02-3>
ST
> mammal"
> but "A lion is a mammal" does NOT imply "A lion and a whale are both
> mammalds "
There is a difference between the boolean "implies" and "being a
logical
consequence".
herbzet
I think it's a bad idea to distinguish between "A implies B" and
"B is a logical consequence of A". It's confusing.
Just my opinion.
--
hz
Eukie_M_SHIRAISHI
Quite agree with you.
The question is how to define "p implies q" and it is a serious
mistake to define it by the truth-table.
See:-<http://www.age.ne.jp/x/eurms/PNG/Key/Honron-3.html#02-3>
herbzet
Quite agree with you -- implication is not truth-functional:
premise | conclusion | implication
|-----------|--------------|---------------||-----------
| true | true | valid || possible
|-----------|--------------|---------------||-----------
| true | true | invalid || possible
|-----------|--------------|---------------||-----------
| true | false | valid || impossible
|-----------|--------------|---------------||-----------
| true | false | invalid || possible
|-----------|--------------|---------------||-----------
| false | true | valid || possible
|-----------|--------------|---------------||-----------
| false | true | invalid || possible
|-----------|--------------|---------------||-----------
| false | false | valid || possible
|-----------|--------------|---------------||-----------
| false | false | invalid || possible
|-----------|--------------|---------------||-----------
There is virtually no relationship between truth and validity.
That an argument have a true premise (or premises) and have
a false conclusion is a sufficient, but not necessary condition
for the invalidity of the argument.
The validity of an argument is not a function of the truth or
falsehood of its premises and conclusion. The relation of
truth to validity is quite tangential.
If we take "p implies q" to mean that the argument from p
to q is a valid argument, then implication is clearly not
truth-functional, and cannot be defined in terms of a
truth table.
> See:-<http://www.age.ne.jp/x/eurms/PNG/Key/Honron-3.html#02-3>
While I agree with you that there is a similarity in the relation
of implication to set inclusion, I differ from you in that I think
implication is most simply understood as the recognition that what
the premises of a valid argument assert, whether true or false,
include (in full) what is asserted by the conclusion. To assert
the premise is already to have asserted the conclusion, however
disguised this assertion may be.
In english, the etymological root of "implies" is the Latin word
"implicare" -- to enfold, involve; from "in-" + "plicare" -- to fold;
akin to Latin "plectere", Greek "plekein" -- to braid. My dictionary
gives as the earliest ascertainable meaning for "imply" -- to "enfold,
entwine"; for "implicate" -- "to fold or twist together: entwine".
In an implication, the conclusion of a valid argument is enfolded
or entwined in the premises. Logic is the careful unfolding of
meaning which reveals the conclusion embedded in the premises.
I believe I am in agreement with Frege on this point.
--
hz
Eukie_M_SHIRAISHI
Thank you for your detailed article.
I do not quite understand on what point you agree with Frege.
It was Frege who defined "if p, then q" to mean "not-p or q".
Charlie-Boo
> In the Fregean Theory of Logic (the standard theory of logic in the
> 20th century),
> “p implies q” has had been defined to mean “not-p or q”.
>
> Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> both mammals”
> It is clear that Q implies P but P does not imply Q.
> Hence “P implies Q” is a falsity.
> On the other hand, “not-P or Q” is a truth.
You're mixing meanings - being inconsistent in how you interpret these
sentences.
It is not true than in all worlds if P is true then Q is true.
It is true that in our world if P is true then Q is true.
You could still formalize this distinction, however.
C-B
herbzet
Charlie-Boo wrote:
> Eukie_M_SHIRAISHI wrote:
>
> > In the Fregean Theory of Logic (the standard theory of logic in the
> > 20th century),
> > “p implies q” has had been defined to mean “not-p or q”.
> >
> > Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> > both mammals”
> > It is clear that Q implies P but P does not imply Q.
> > Hence “P implies Q” is a falsity.
> > On the other hand, “not-P or Q” is a truth.
>
> You're mixing meanings - being inconsistent in how you interpret these
> sentences.
Do you mean that the OP is using a word in different senses in
different sentences? Which word in which sentences?
> It is not true than in all worlds if P is true then Q is true.
Do you mean "It is false that P implies Q"?
> It is true that in our world if P is true then Q is true.
Do you mean "It is true that P implies Q"?
> You could still formalize this distinction, however.
What name would you use in the two cases to describe the relation
between P and Q?
What would you call the case when it is not true that in our
world if P is true then Q is true, but it is true in some
worlds that if P is true then Q is true?
--
hz
herbzet
Eukie_M_SHIRAISHI wrote:
> herbzet wrote:
> > Eukie_M_SHIRAISHI wrote:
Actually, this definition goes back to the Stoic philosophers of
ancient Greece. It is usually attributed to Philo of Megara, and
is sometimes referred to as a "Philonian conditional" or "Philonian
implication."
According to http://www.humboldt.edu/~essays/parikhrev.html
"Frege re-discovered the material conditional but did not know that it had already
been studied by Philo."
I don't have any books by Frege handy, but here is a paraphrase by
"Tron" <tron...@frizurf.no> in
news:uIGdnZt7ao-...@telenor.com :
<quote>
"Frege made much of the definition of what a concept is, including
discussions of whether it is important or not for a concept whether there
are things that fall under it or not [...]
"If one considers premise-like propositions to state something about the
extension of a concept A (the number of properties that make up the concept,
not the number of items that fall under the concept), and a sub-statement to
state about another concept B that it either includes or excludes one or
more of the properties of the first concept A, the inclusion or
non-inclusion of B under A becomes to some extent (and AFAICS and IMHO and
all the other acronymic reservations) analytic, to be decided solely on the
basis of the law of contradiction; whether any entity actually falls under
the concept - i.e. the interpretation, semantics or empirical content of the
concept - is an entirely different issue which does not touch on the first
question at all.
"That more or less follows from the whole classical concept of formality,
where truth is conveyed from one type of proposition to another type solely
by virtue of the form of these propositions and their conjunction, with
content never entering the picture."
<end quote>
My only emendation to this is to say that ALL of the properties that
make up concept B must be included in A for "A implies B" to be
analytic.
--
hz
Eukie_M_SHIRAISHI
> On Apr 28, 10:59 pm, Eukie_M_SHIRAISHI <ms.eu...@gmail.com> wrote:
>
> > In the Fregean Theory of Logic (the standard theory of logic in the
> > 20th century),
> > “p implies q” has had been defined to mean “not-p or q”.
>
> > Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> > both mammals”
> > It is clear that Q implies P but P does not imply Q.
> > Hence “P implies Q” is a falsity.
> > On the other hand, “not-P or Q” is a truth.
>
> You're mixing meanings - being inconsistent in how you interpret these
> sentences.
>
> It is not true than in all worlds if P is true then Q is true.
>
> It is true that in our world if P is true then Q is true.
>
> You could still formalize this distinction, however.
I understand that you are quoting Model Theory.
However, Model Theory is a product of misunderstanding
the logical law.
The logical law is not a formal truth but it is a second-order
material truth.
See;<-http://www.age.ne.jp/x/eurms/PNG/Key/Honron-3.html#02-3>
Eukie_M_SHIRAISHI
>
> "Frege re-discovered the material conditional but did not know that it had already
> been studied by Philo."
>
> I don't have any books by Frege handy, but here is a paraphrase by
Eukie_M_SHIRAISHI
The true relations between the pilonian, implication, and conditional
are given at the end of the following Web site:-
<http://www.age.ne.jp/x/eurms/PNG/Key/Musubi.html#03>
Jan Burse
> The true relations between the pilonian, implication, and conditional
> are given at the end of the following Web site:-
> <http://www.age.ne.jp/x/eurms/PNG/Key/Musubi.html#03>
You write somewhere:
If P(x), then Q(x) is defined to mean Sx{P(x)} subset
Sx{Q(x)}, and we denote this by P(x) =>x Q(x), the
case in which Sx{P(x)} consists of only one member is
excluded.
So => is reserved for predicate??
Furthermore we write P(x) |=>x Q(x) to mean
(P(x) =>x Q(x)) & ~(Q(x) =>x P(x))
So |=> should express proper entailment??
We denote P(a) implies Q(a) by P(a) -} Q(a), and define
it to mean P(x) =>x Q(x) & x:a in Ux.
So the curly implication is reserved for propositions??
abolished concepts: propositional & predicate logic, etc..
Aha.
Theorem 4: (P(x) =>x Q(x)) <=>P,Q forall x(~P(x) v Q(x))
What if P(x) is a monon?? is P(x) =>x Q(x) false or true?
forall x(~P(x) v Q(x)) has no such restriction.
Theorem 5: P(a) -} Q(a) |=>P,Q,a ~P(a) v Q(a)
No this is not true. The problem is that the definition
of |=> is bad. It should be something like
(P(x) =>x Q(x)) & exists x ~(Q(x) => P(x)), instead
of (P(x) =>x Q(x)) & ~(Q(x) => P(x)). Because
it is quite possible to have ~P(a) v Q(a) =>
P(x) =>x Q(x). For example when P=Q.
Theorem 10: P(a) -} Q(a) |=>P,Q P(x)=>x Q(x).
Same remark as for theorem 5. Very problematic statement.
I think the essence of what you wanted to say, and
somehow failed to say, is the following, which
is true:
|/= P(a) -> forall x P(x) (I)
Note the above use of |/= (not |=), which means is
not tautology. But of saying that something is not
a tautology, you tried to formulate it by sayin
something is a antilogie. But unfortunately the
following is not true:
|= ~(P(a) -> forall x P(x)) (II)
You should write your paper again, when the difference
between (I) and (II) has become clear to you.
Bye
Eukie_M_SHIRAISHI
>
> Theorem 4: (P(x) =>x Q(x)) <=>P,Q forall x(~P(x) v Q(x))
>
> What if P(x) is a monon??
A monon is not a predicare.
Eukie_M_SHIRAISHI
>
> You write somewhere:
>
> P(x) =>x Q(x), the
> case in which Sx{P(x)} consists of only one member is
> excluded.
You mean the N.B.
We found it it was a mistake, so that we decided to erase the N.B.
Charlie-Boo
> Charlie-Boo wrote:
> > Eukie_M_SHIRAISHI wrote:
>
> > > In the Fregean Theory of Logic (the standard theory of logic in the
> > > 20th century),
> > > “p implies q” has had been defined to mean “not-p or q”.
>
> > > Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> > > both mammals”
> > > It is clear that Q implies P but P does not imply Q.
> > > Hence “P implies Q” is a falsity.
> > > On the other hand, “not-P or Q” is a truth.
>
> > You're mixing meanings - being inconsistent in how you interpret these
> > sentences.
>
> Do you mean that the OP is using a word in different senses in
> different sentences? Which word in which sentences?
Add quantifiers to show we are talking about our world vs. all worlds.
C-B
> > It is not true than in all worlds if P is true then Q is true.
>
> Do you mean "It is false that P implies Q"?
>
> > It is true that in our world if P is true then Q is true.
>
> Do you mean "It is true that P implies Q"?
>
> > You could still formalize this distinction, however.
>
> What name would you use in the two cases to describe the relation
> between P and Q?
>
> What would you call the case when it is not true that in our
> world if P is true then Q is true, but it is true in some
> worlds that if P is true then Q is true?
>
> --
> hz
>
>
>
> > C-B
>
> > > Therefore, it is a mistake to define “p implies q” to mean “not-p or
> > > q”.
>
> > > Since implication is one of the most fundamental concepts in logic,
> > > all-out revision
> > > is needed in the Fregean Theory of Logic.
>
> > > Have a look at the following Web site:- <http://www.age.ne.jp/x/eurms/
> > > PNG/Key/Honron-2.html#02-2>- Hide quoted text -
>
> - Show quoted text -
herbzet
Charlie-Boo wrote:
> herbzet wrote:
> > Charlie-Boo wrote:
> > > Eukie_M_SHIRAISHI wrote:
> >
> > > > In the Fregean Theory of Logic (the standard theory of logic in the
> > > > 20th century),
> > > > “p implies q” has had been defined to mean “not-p or q”.
> >
> > > > Here let P be “A lion is a mammal” and Q be “A lion and a whale are
> > > > both mammals”
> > > > It is clear that Q implies P but P does not imply Q.
> > > > Hence “P implies Q” is a falsity.
> > > > On the other hand, “not-P or Q” is a truth.
> >
> > > You're mixing meanings - being inconsistent in how you interpret these
> > > sentences.
> >
> > Do you mean that the OP is using a word in different senses in
> > different sentences? Which word in which sentences?
>
> Add quantifiers to show we are talking about our world vs. all worlds.
How is the OP being inconsistent in how he interprets these sentences?
> > > It is not true than in all worlds if P is true then Q is true.
> >
> > Do you mean "It is false that P implies Q"?
Do you mean "It is false that P implies Q"?
> > > It is true that in our world if P is true then Q is true.
> >
> > Do you mean "It is true that P implies Q"?
Do you mean "It is true that P implies Q"?
> > > You could still formalize this distinction, however.
> >
> > What name would you use in the two cases to describe the relation
> > between P and Q?
What name would you use in the two cases to describe the relation
between P and Q?
> > What would you call the case when it is not true that in our
> > world if P is true then Q is true, but it is true in some
> > worlds that if P is true then Q is true?
What would you call the case when it is not true that in our
world if P is true then Q is true, but it is true in some
worlds that if P is true then Q is true?
> > > > Therefore, it is a mistake to define “p implies q” to mean “not-p or
> > > > q”.
> >
> > > > Since implication is one of the most fundamental concepts in logic,
> > > > all-out revision
> > > > is needed in the Fregean Theory of Logic.
> >
> > > > Have a look at the following Web site:- <http://www.age.ne.jp/x/eurms/
> > > > PNG/Key/Honron-2.html#02-2>- Hide quoted text -
--
hz
Eukie_M_SHIRAISHI
> On May 3, 7:42 pm, herbzet <herb...@gmail.com> wrote:
>
> > Do you mean that the OP is using a word in different senses in
> > different sentences? Which word in which sentences?
>
> Add quantifiers to show we are talking about our world vs. all worlds.
In the Fregean Theory of Logic a logical law is considered as a
tautology or
a universally valid formula, so that Model Theory is needed.
In the Reformed Theory of Logic, a logical law is proved a second-
order material
truth. Therefore there is no place where Model Theory works.