# Justifying the truth table for the IMPLIES-operator

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### Dan Christensen

Sep 21, 2022, 1:22:18 AM9/21/22
to
Here is the truth table for the IMPLIES-operator:

A B A=>B
T T T
T F F
F T T
F F T

Judging by frequent questions at MSE (at least once every day) on this topic, there seems to be much confusion about this one, simple table.

This table is often used to "define" the IMPLIES-operator in introductory textbooks. No real explanation is usually given, just a few examples. There is a good reason for this, namely that to truly understand why it works, you must understand some basic methods of proof, i.e. the following rules of inference in propositional logic:

1. Premise
2. Conclusion (intro =>, intro ~)
3. Join (intro &)
4. Split (elim &)
5. Detachment (elim =>)
6. Remove ~~

(For an excellent introduction to these rules of inference, you need only work your way through only the first 3 (of 13) examples in the DC Proof tutorial.)

The above truth table is really just a table of the following 4 theorems in propositional logic:

1. A & B => [A => B]

2. A & ~B => ~[A => B]

3. ~A & B => [A => B]

4. ~A & ~B => [A => B]

*****************************************************************

Thm 1: A & B => [A => B]

1. A & B
Premise

2. A
Premise

3. B
Split, 1

4. A => B
Conclusion, 2

5. A & B => [A => B]
Conclusion, 1

*****************************************************************

Thm 2: A & ~B => ~[A => B]

1. A & ~B
Premise

2. A => B
Premise

3. A
Split, 1

4. ~B
Split, 1

5. B
Detach, 2, 3

6. ~B & B
Join, 4, 5

7. ~[A => B]
Conclusion, 2

8. A & ~B => ~[A => B]
Conclusion, 1

*****************************************************************

Thm 3: ~A & B => [A => B]

1. ~A & B
Premise

2. A
Premise

3. B
Split, 1

4. A => B
Conclusion, 2

5. ~A & B => [A => B]
Conclusion, 1

*****************************************************************

Thm 4: ~A & ~B => [A => B]

1. ~A & ~B
Premise

2. A
Premise

3. ~B
Premise

4. ~A
Split, 1

5. A & ~A
Join, 2, 4

6. ~~B
Conclusion, 3

7. B
Rem DNeg, 6

8. A => B
Conclusion, 2

9. ~A & ~B => [A => B]
Conclusion, 1

Dan

Visit my Math Blog at http://www.dcproof.wordpress.com

### Mostowski Collapse

Sep 21, 2022, 3:04:48 AM9/21/22
to
If you know how material implications works, why do you write
nonsense like this here:

> > y=\=x => ~Bijection(id_y,x,x)
> Not true in general. What is your point, Jan Burse?

Do you have a pair y,x where the above statement
y=\=x => ~Bijection(id_y,x,x) is not true?

Hint: Must be a statement of the second row:

A B A=>B
T F F

### Dan Christensen

Sep 21, 2022, 9:56:33 AM9/21/22
to
> If you know how material implications works, why do you write
> nonsense like this here:
>
> > > y=\=x => ~Bijection(id_y,x,x)
> > Not true in general. What is your point, Jan Burse?
>
> Do you have a pair y,x where the above statement
> y=\=x => ~Bijection(id_y,x,x) is not true?
>

Suppose x = {0} and y = {0, 1}. Then x=/=y and for all z in x, id_y(z) =z, i.e. id_y is a bijection on set x.

### Ross A. Finlayson

Sep 21, 2022, 11:47:39 AM9/21/22
to
What's 0^0 again?
Message has been deleted

### Fritz Feldhase

Sep 21, 2022, 12:08:55 PM9/21/22
to
On Wednesday, September 21, 2022 at 5:47:39 PM UTC+2, Ross A. Finlayson wrote:

> What's 0^0 again?

1 .

By two different definitions:

a.) ^ defined recursively on IN:

Let m e IN:

m^0 = 1
m^(n+1) = m^n * m (for all n e IN)

b.) "set-theoretically" on IN (where IN is defined due to von Neumann):

m^n = card({f : n --> m}) (for all n, m e IN)

### Dan Christensen

Sep 21, 2022, 1:28:57 PM9/21/22
to
On Wednesday, September 21, 2022 at 1:22:18 AM UTC-4, Dan Christensen wrote:
> Here is the truth table for the IMPLIES-operator:
>
> A B A=>B
> T T T
> T F F
> F T T
> F F T
>
> Judging by frequent questions at MSE (at least once every day) on this topic, there seems to be much confusion about this one, simple table.
>
> This table is often used to "define" the IMPLIES-operator in introductory textbooks. No real explanation is usually given, just a few examples. There is a good reason for this, namely that to truly understand why it works, you must understand some basic methods of proof, i.e. the following rules of inference in propositional logic:
>
> 1. Premise
> 2. Conclusion (intro =>, intro ~)
> 3. Join (intro &)
> 4. Split (elim &)
> 5. Detachment (elim =>)
> 6. Remove ~~
>
> (For an excellent introduction to these rules of inference, you need only work your way through only the first 3 (of 13) examples in the DC Proof tutorial.)
>
> The above truth table is really just a table of the following 4 theorems in propositional logic:
>
> 1. A & B => [A => B]
>
> 2. A & ~B => ~[A => B]
>
> 3. ~A & B => [A => B]
>
> 4. ~A & ~B => [A => B]
>

Started working on an update to the DC Proof tutorial to include these theorems as exercises with hints and full solutions. Should have thought of this years ago! The perfect proofs for the beginner--no more than 9 lines each and you don't even need that much logic. And IMHO quite profound when taken as a whole.

### Dan Christensen

Sep 21, 2022, 2:01:08 PM9/21/22
to
On Wednesday, September 21, 2022 at 11:47:39 AM UTC-4, Ross A. Finlayson wrote:

[snip]

> What's 0^0 again?

Huh? Oh, I see... You would rather change the subject. Can't blame you, really.

See: https://dcproof.wordpress.com/2013/10/09/oh-the-ambiguity-2/

Start a new thread if you want to rehash it after all these years. Could be fun.

### Dan Christensen

Sep 21, 2022, 7:55:31 PM9/21/22
to
On Wednesday, September 21, 2022 at 1:28:57 PM UTC-4, Dan Christensen wrote:
> On Wednesday, September 21, 2022 at 1:22:18 AM UTC-4, Dan Christensen wrote:
> > Here is the truth table for the IMPLIES-operator:
> >
> > A B A=>B
> > T T T
> > T F F
> > F T T
> > F F T
> >
> > Judging by frequent questions at MSE (at least once every day) on this topic, there seems to be much confusion about this one, simple table.
> >
> > This table is often used to "define" the IMPLIES-operator in introductory textbooks. No real explanation is usually given, just a few examples. There is a good reason for this, namely that to truly understand why it works, you must understand some basic methods of proof, i.e. the following rules of inference in propositional logic:
> >
> > 1. Premise
> > 2. Conclusion (intro =>, intro ~)
> > 3. Join (intro &)
> > 4. Split (elim &)
> > 5. Detachment (elim =>)
> > 6. Remove ~~
> >
> > (For an excellent introduction to these rules of inference, you need only work your way through only the first 3 (of 13) examples in the DC Proof tutorial.)
> >
> > The above truth table is really just a table of the following 4 theorems in propositional logic:
> >
> > 1. A & B => [A => B]
> >
> > 2. A & ~B => ~[A => B]
> >
> > 3. ~A & B => [A => B]
> >
> > 4. ~A & ~B => [A => B]
> >
> Started working on an update to the DC Proof tutorial to include these theorems as exercises with hints and full solutions. Should have thought of this years ago! The perfect proofs for the beginner--no more than 9 lines each and you don't even need that much logic. And IMHO quite profound when taken as a whole.

Now available at my homepage.

Dan

### Fritz Feldhase

Sep 21, 2022, 8:00:20 PM9/21/22
to
In the context of set theory, we may define addition, multiplication and the power operation on IN quite naturally using set theoretic means:

Let n, m e IN. Then

n + m = card(n x {0} u m x {1}) ,
n * m = card(n x m) ,
n ^ m = card({f | f : m --> n}) .

n = 0 and/or m = 0 is no "special case" here.

Actually, we would "artificially" have to treat 0^0 as a special case to "avoid" the result 1. But there is absolutely no reason to do so (in this context).

### Ross A. Finlayson

Sep 21, 2022, 9:54:01 PM9/21/22
to
No you brickhead clod, that was changing _your_ subject,
just pointing out that your stipulations and

I know you "have to have it your way", but when it comes
to usual set-theoretic disambiguation, you're just a usual
shill and fool.

Showing that now your "implies is inference" also fails
in truth tables and no "false antecedents",
you false antecedent.

Just change the order of stipulations of your proofs they
change what they say. Don't you think inference should
be a little more reliable than assuming what you set out to prove?

No offense, it's a little "show your work" tool for some reasoning,
but it's weak and compounded by "false antecedents in a world
of truth tables", it's further "garbage-in garbage-out".

Then, really I do not care what you think "0^0" is,
just pointing out that you think you do,
but there's more than one answer, so,
one of yours must be wrong.

### Dan Christensen

Sep 21, 2022, 10:37:33 PM9/21/22
to
On Wednesday, September 21, 2022 at 9:54:01 PM UTC-4, Ross A. Finlayson wrote:
> On Wednesday, September 21, 2022 at 11:01:08 AM UTC-7, Dan Christensen wrote:
> > On Wednesday, September 21, 2022 at 11:47:39 AM UTC-4, Ross A. Finlayson wrote:
> >
> > [snip]
> >
> > > What's 0^0 again?
> >
> > Huh? Oh, I see... You would rather change the subject. Can't blame you, really.
> >
> > See: https://dcproof.wordpress.com/2013/10/09/oh-the-ambiguity-2/
> >
> > Start a new thread if you want to rehash it after all these years. Could be fun.

[snip]

So, you would rather not. Very wise.

> Don't you think inference should
> be a little more reliable than assuming what you set out to prove?
>

[snip]

Where do you imagine that I assume ~A & B => [A => B] ?

Here again is my proof:

1. ~A & B
Premise

2. A
Premise

3. B
Split, 1

4. A => B
Conclusion, 2

5. ~A & B => [A => B]
Conclusion, 1

### Ross A. Finlayson

Sep 22, 2022, 11:30:55 AM9/22/22
to
You have no concept of time, the modal, or change.

For example, A as "A is premise 1", i.e. whether A or not A,
then "A is premise 2", it's an example of a generic premise
that according to its introduction, and what so follows,
doesn't admit false antecedents, at all.

What otherwise you have there as a template for "wrong".

### Ross A. Finlayson

Sep 22, 2022, 11:59:57 AM9/22/22
to
It's like implication is a bullet, and you are leaving bullets around in
all these guns that shoot backward.

Then you forget which ones you put where, and none of them are safe.

You got no reliable reverse "material" implication.

What you've written there is "not-A not-implies-B", not, "not-A not not-implies-not-B",
that a false antecedent is always "not implies".

Which is not necessarily "implies".

Leave "material" "implication" out of things, direct implication only. It's about
one of the stupidest kinds of notational laziness.

Don't you have any kind of object in your theory that changes?
Can't a _change_ even possibly be an object in your little theory?

Then it's like you say "whu all these are constants" then it's like

### Mostowski Collapse

Sep 22, 2022, 12:42:51 PM9/22/22
to
Nope, id_y isn't a bijection x -> x, it has missing

dom(id_y) = x

id_y is not member of this function space:

{ f : x -> x | f bijective }

Whats wrong with you?

### Dan Christensen

Sep 22, 2022, 12:47:29 PM9/22/22
to
On Thursday, September 22, 2022 at 11:30:55 AM UTC-4, Ross A. Finlayson wrote:
> On Wednesday, September 21, 2022 at 7:37:33 PM UTC-7, Dan Christensen wrote:
> > On Wednesday, September 21, 2022 at 9:54:01 PM UTC-4, Ross A. Finlayson wrote:
> > > On Wednesday, September 21, 2022 at 11:01:08 AM UTC-7, Dan Christensen wrote:
> > > > On Wednesday, September 21, 2022 at 11:47:39 AM UTC-4, Ross A. Finlayson wrote:
> > > >
> > > > [snip]
> > > >
> > > > > What's 0^0 again?
> > > >
> > > > Huh? Oh, I see... You would rather change the subject. Can't blame you, really.
> > > >
> > > > See: https://dcproof.wordpress.com/2013/10/09/oh-the-ambiguity-2/
> > > >
> > > > Start a new thread if you want to rehash it after all these years. Could be fun.
> > [snip]
> >
> > So, you would rather not. Very wise.
> > > Don't you think inference should
> > > be a little more reliable than assuming what you set out to prove?
> > >
> > [snip]
> >
> > Where do you imagine that I assume ~A & B => [A => B] ?
> >

No comment?

> > Here again is my proof:
> > 1. ~A & B
> > Premise
> >
> > 2. A
> > Premise
> >
> > 3. B
> > Split, 1
> >
> > 4. A => B
> > Conclusion, 2
> >
> > 5. ~A & B => [A => B]
> > Conclusion, 1

> You have no concept of time, the modal, or change.
>

I am looking the state of "the world" at some instant in time (usually the present). A snapshot. It may sound limiting, but it turns out to be extremely useful in the real world and in mathematics.

The statement, "If it is raining then it is cloudy" means only that, at present, it is not both raining and not cloudy. It does NOT mean that rain causes cloudiness. Or that, historically, whenever it was raining it was also cloudy.

> For example, A as "A is premise 1", i.e. whether A or not A,
> then "A is premise 2", it's an example of a generic premise
> that according to its introduction, and what so follows,
> doesn't admit false antecedents, at all.
>

A premise need not be consistent with previous ones if that is what you are getting at. A premise could even be a contradiction.

EXAMPLE

1 A & ~A
Premise

2 ~[A & ~A]

### Mostowski Collapse

Sep 22, 2022, 12:49:00 PM9/22/22
to
Usually in set theory this here should hold for product function spaces:

Definition: 4.2.6 The Exponentiation Axiom (abbreviated Exp) postulates
that for sets a; b the class of all functions from a to b forms a set:
∀a∀b∃c ∀f(f e c <-> (f : a -> b))
https://www1.maths.leeds.ac.uk/~rathjen/book.pdf

You can even strengthen it to ∃!c. Now it seems in
DC Spoiled, the notion of product function space is
shape shifting, and so are function spaces that are

subset of the product function space, like this here:

{ f : x -> x | f bijective }

If id_y is element of the above function space, this
would imply that x -> x and y -> y are not disjoint,
for x =\= y. But usually they are disjoint, even in the

set theoretic approach, and also in the Bourbaki approach.

LMAO!

### Dan Christensen

Sep 22, 2022, 1:13:28 PM9/22/22
to
On the contrary. If A => B, then we can infer ~B => ~A (the rule of contrapositive).

> What you've written there is "not-A not-implies-B", not, "not-A not not-implies-not-B",
> that a false antecedent is always "not implies".
>

Guessing at your meaning here: ~[A => B] <=> A & ~B

> Which is not necessarily "implies".
>
> Leave "material" "implication" out of things, direct implication only. It's about
> one of the stupidest kinds of notational laziness.
>

If you accept the "first principles" (rules of inference) that I list above, material implication is inevitable. Here they are again:

1. Premise
2. Conclusion (intro =>, intro ~)
3. Join (intro &)
4. Split (elim &)
5. Detachment (elim =>)
6. Remove ~~

Which do you propose we limit or eliminate altogether?

> Don't you have any kind of object in your theory that changes?

[snip]

You could interpret some independent variable t as time with different values of t corresponding to different points in space if that's what you mean? But that's getting into the realm of science.

### Dan Christensen

Sep 22, 2022, 1:29:14 PM9/22/22
to
On Thursday, September 22, 2022 at 12:49:00 PM UTC-4, Mostowski Collapse wrote:

[snip]

> Usually in set theory this here should hold for product function spaces:
>
> Definition: 4.2.6 The Exponentiation Axiom (abbreviated Exp) postulates
> that for sets a; b the class of all functions from a to b forms a set:
> ∀a∀b∃c ∀f(f e c <-> (f : a -> b))
> https://www1.maths.leeds.ac.uk/~rathjen/book.pdf
>

Not unlike the DC Proof Function Space axiom:

ALL(dom):ALL(cod):[Set(dom) & Set(cod)
=> EXIST(fsp):[Set(fsp) & ALL(f):[f in fsp <=> Function(f,dom,cod) & ALL(a1):[a1 in dom => f(a1) in cod]]]]

> You can even strengthen it to ∃!c. Now it seems in
> DC Proof, the notion of product function space is
> shape shifting, and so are function spaces that are
> subset of the product function space, like this here:
> { f : x -> x | f bijective }
> If id_y is element of the above function space, this
> would imply that x -> x and y -> y are not disjoint,
> for x =\= y.

If, however, x is a proper subset of y, we can say that id_y (the identity function on set y) is also a bijection on set x .

### Mostowski Collapse

Sep 22, 2022, 4:06:24 PM9/22/22
to
Dan Christensen halucinated on Thursday, 22. September 2022:
> If, however, x is a proper subset of y, we can say that id_y (the identity function on set y) is also a bijection on set x .

No you cannot say that. People don't say that.

What does exist in mathematics, is the operation restriction:

f | A = { (x,y) e f | x e A }

And you can for example say:

id_X = id_Y | X

for X ⊆ Y. And subsequetly you can also say id_Y | X is a X -> X bijection.

This is all explained in the very first chapters here:

Basic Set Theory (Dover Books on Mathematics)
Azriel Levy - Revised Edition (13. August 2002)
https://www.amazon.com/dp/0486420795

In mathematics, the restriction of a function f is a new function, denoted f | A
https://en.wikipedia.org/wiki/Restriction_%28mathematics%29

### Dan Christensen

Sep 22, 2022, 5:09:31 PM9/22/22
to
On Thursday, September 22, 2022 at 4:06:24 PM UTC-4, Mostowski Collapse wrote:

> > If, however, x is a proper subset of y, we can say that id_y (the identity function on set y) is also a bijection on set x .

> No you cannot say that. People don't say that.
>
> What does exist in mathematics, is the operation restriction:
>
> f | A = { (x,y) e f | x e A }
>
> And you can for example say:
>
> id_X = id_Y | X
>
> for X ⊆ Y. And subsequetly you can also say id_Y | X is a X -> X bijection.
>

In other words, like I said, "id_y is a bijection on set x."

Thanks for clearing that up, Jan Burse. (Hee, hee!)

### Mostowski Collapse

Sep 22, 2022, 5:32:17 PM9/22/22
to
Nope, id_y | x is a bijection on x.
id_y isn't a bijection on x.

Whats wrong with you?

Dan Christensen schrieb:

### Dan Christensen

Sep 22, 2022, 6:56:36 PM9/22/22
to
On Thursday, September 22, 2022 at 5:32:17 PM UTC-4, Mostowski Collapse wrote:
> Nope, id_y | x is a bijection on x.
> id_y isn't a bijection on x.
>

It is. Deal with it, Jan Burse.

### Mostowski Collapse

Sep 22, 2022, 8:12:35 PM9/22/22
to

Not in math text books. Did your mama
notice that you are brain damaged?

The word "is a" translates to membership,
so the two sentences translate to:

id_y | x e { f : x -> x | f bijective }
id_y e { f : x -> x | f bijective }

For x ⊆ y & x =\= y only the first membership
is true, the second membership is false.

Dan Christensen schrieb:

### Dan Christensen

Sep 22, 2022, 10:30:51 PM9/22/22
to
On Thursday, September 22, 2022 at 8:12:35 PM UTC-4, Mostowski Collapse wrote:

> For x ⊆ y & x =\= y only the first membership
> is true, the second membership is false.
>
> Dan Christensen schrieb:
> > On Thursday, September 22, 2022 at 5:32:17 PM UTC-4, Mostowski Collapse wrote:
> >> Nope, id_y | x is a bijection on x.
> >> id_y isn't a bijection on x.
> >>
> >
> > It is. Deal with it, Jan Burse.
> >

> Not in math text books.

[snip childish abuse]

>
> The word "is a" translates to membership,
> so the two sentences translate to:
>
> id_y | x e { f : x -> x | f bijective }
> id_y e { f : x -> x | f bijective }
>

I don't know why you are bringing function spaces into this other than to perhaps muddy the waters.

Here is a the usual definition/abbreviation for the bijectivity predicate:

ALL(f):ALL(a):ALL(b):[Bijective(f,a,b) <=>
[ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]]

It's use is optional. It is used only to improve readability. Instead of writing the abbreviation Bijective(f,a,b), you could write it out in full as:

ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]

From the definitions of x, y and id_y above, we have Bijective(id_y,x,x). Deal with it, Jan Burse. Just admit you were wrong.

### Fritz Feldhase

Sep 22, 2022, 11:27:42 PM9/22/22
to
On Friday, September 23, 2022 at 4:30:51 AM UTC+2, Dan Christensen wrote:

> Here is a

nonsensical

> definition/abbreviation for the bijectivity predicate:
>
> ALL(f):ALL(a):ALL(b):[Bijective(f,a,b) <=>
> [ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]]

Yeah, it's nonsense.

### Dan Christensen

Sep 22, 2022, 11:50:47 PM9/22/22
to
On Thursday, September 22, 2022 at 11:27:42 PM UTC-4, Fritz Feldhase wrote:
> On Friday, September 23, 2022 at 4:30:51 AM UTC+2, Dan Christensen wrote:
>
> > Here is definition/abbreviation for the bijectivity predicate:
> >
> > ALL(f):ALL(a):ALL(b):[Bijective(f,a,b) <=>
> > [ALL(c):ALL(d):[c in a & d in a => [f(c)=f(d) => c=d]] & ALL(c):[c in b => EXIST(d):[d in a & f(d)=c]]]

> Yeah, it's nonsense.

Oh, really? How do YOU define Bijective(f,a,b), Fritz?

### Ross A. Finlayson

Oct 6, 2022, 9:49:48 PM10/6/22
to
For-any a in A, there exists unique b in B such that F(a) = b.

And, for any b in B, there exists unique a in A such that F^-1(b) = a.

There exists F.

Here the unique part of course means for-any a, exists b, not exists
c =/= b, s.t. F(a) = c, and vice versa.

These days often set theorists are lazy and let Cantor/Schroeder/Bernstein thm.
work out there "exists" such an F, while of course it must result the above.

### Ross A. Finlayson

Oct 6, 2022, 9:52:44 PM10/6/22
to
Yeah you brought a great game from home, but nobody's playing it,
and it's pretty clear the rules are absolutely arbitrary.

### Ross A. Finlayson

Oct 6, 2022, 10:56:08 PM10/6/22
to
Ah, that's function theory, where all functions have generalized inverses in terms of their value.

You did say "function", right?

You got DesCartes, and for DesCartes, it was the time of geometry defining function.
So, the Cartesian function, it's the space of all the single value mappings from domain
to codomain. (Domain, image, domain, range, domain, codomain, image.)

This is where in geometry the only "space" is supports here is a line in geometry. According
to, "Euclid", the line is what exists and there are points on it. So, that's "not" a Cartesian function.

(I.e. functions like it without otherwise the multivalent are "discontinuous" functions.)

Which among them are series, ....

### Ross A. Finlayson

Oct 6, 2022, 11:25:02 PM10/6/22
to
Maybe it'd be easier if you built a world where "my material implications with these
extra conditions will never reverse on me when taken the wrong way".

Also called "science".

Yeah, why don't you put "science" first in all your theorems.

The "science": part is only for the "non-logical",

Of course there's syllogism, where words are abstract non-logically.

Mostly "fulfills question words with respect to is or has".

### Ross A. Finlayson

Oct 6, 2022, 11:31:29 PM10/6/22
to
Hey no worries, Dan, just you know, in your face.

### Dan Christensen

Oct 6, 2022, 11:59:38 PM10/6/22
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Huh???

>
> Yeah, why don't you put "science" first in all your theorems.
>
[snip]

Can you give even a single example of an actual error in science, engineering or any other human endeavour arising strictly from the application of faulty rules of logic that you imagine? What were the consequences? How many lives were lost or ruined, etc.

### Ross A. Finlayson

Oct 7, 2022, 12:05:40 AM10/7/22
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If you ever forget that I'd feel sorry for you.

It's about the shortest way to say it, ....

(... insert Wiki ....)

### Dan Christensen

Oct 7, 2022, 12:18:15 AM10/7/22
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On Friday, October 7, 2022 at 12:05:40 AM UTC-4, Ross A. Finlayson wrote:

> >
> > Can you give even a single example of an actual error in science, engineering or any other human endeavour arising strictly from the application of faulty rules of logic that you imagine? What were the consequences? How many lives were lost or ruined, etc.

> If you ever forget that I'd feel sorry for you.
>
> It's about the shortest way to say it, ....
>
>
>
> (... insert Wiki ....)

So, not a single example. A tempest in a teacup, then. Whew!

Dan