Modeling counterfactual conditionals in set theory

[Dan Christensen]

Consider this counterfactual conditional:

"If the release valve had been working properly, the explosion would have been contained."

Let T= the set of all points in time

Let F= the set of all times that the release value failed

Let E= the set of all times that an explosion occurred

Q: Does the following statement in the language of set theory adequately model the above counterfactual conditional:

F ⊂ T ∧ E ⊂ T ∧ F ⊂ E

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[Mostowski Collapse]

There is not much magic behind counterfactual conditionals.
Usally we have this inference rule:

G, A |- B
----------------
G |- A -> B

So to prove A -> B, we might add A to our background theory
G, and then try to prove from G, A the formula B.

In counter factual the idea is to remove also something
from G, to change some facts. So one could thing of

a certain negation -A as a counter factual, with the
following inference rule:

G \ {A} |- B
----------------
G |- -A -> B

Problem is that G \ {A} is a little ill defined concept.
So its not that trivial to deal with counter factuals.

In Prolog there is retract/1.

[Dan Christensen]

On Thursday, January 26, 2023 at 11:27:12 PM UTC-5, Dan Christensen wrote:
> Consider this counterfactual conditional:
>
> "If the release valve had been working properly, the explosion would have been contained."
>

From Wikipedia:

"The PROBLEM of counterfactuals

"According to the material conditional analysis, a natural language conditional, a statement of the form ‘if P then Q’, is true whenever its antecedent, P, is false. Since counterfactual conditionals are those whose antecedents are false, this analysis would wrongly predict that all counterfactuals are vacuously true."

https://en.wikipedia.org/wiki/Counterfactual_conditional#The_problem_of_counterfactuals

Is that really a problem?

Suppose: ValveFailure => Explosion (for logical propositions ValveFailure and Explosion)

This implication is vacuously true if ValveFailure is false, but so what? In that case, we cannot determine the truth value for Explosion. Without introducing any inconsistencies, Explosion could be either true or false (see lines 3 and 4 in the following truth table).

V E V=>E
T T T
T F F
F T T <--------
F F T <--------

So, NOT a problem in this case.

[Mostowski Collapse]

The valve example is very easy. But you need to use
biconditional and not condictional. So we have:

V, V <-> E |- E

And:

~V, V <-> E | ~E

Biconditional comes also into play in approaches
to so called "defaults", where one can model "abnormals",
This is all beyond FOL and DC Proof, since FOL and DC

Proof are montonic, but the corresponding logic
approaches are not anymore monotonic. So you have
to study non-monotonic logics:

"The term “non-monotonic logic” (in short, NML) covers a
family of formal frameworks devised to capture and represent
defeasible inference. Reasoners draw conclusions defeasibly
when they reserve the right to retract them in the
light of further information."
https://plato.stanford.edu/entries/logic-nonmonotonic

You see the word *retract* again in the above. It
all started maybe with advice taker, and early ChatGPT.
By John McCarthy:

PROGRAMS WITH COMMON SENSE
John McCarthy - 1959
http://jmc.stanford.edu/articles/mcc59/mcc59.pdf

Not sure how much defeasible reasoning the
paper already discusses. You have more papers
by John McCarthy and others about the topic.

[Mostowski Collapse]

Corr.: I wrote "all started maybe with advice taker,
and early ChatGPT". Well I guess ChatGPT would
be an advice taker and advice giver.

You can try ChatGPT, it remembers to some extend
what it gets told, and it can also revises to some extend
what it gets told.

BTW, what would PROF. Y. BAR-HILLEL say now.
Back in the times he thought:

"PROF. Y. BAR-HILLEL: When I uttered my doubts that a
machine working under the programme outlined by
Dr. McCarthy would be able to do what he expects it to do,
I was using ‘programme’ in the technical sense."

[Dan Christensen]

On Friday, January 27, 2023 at 1:34:37 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Freitag, 27. Januar 2023 um 19:21:21 UTC+1:
> > On Thursday, January 26, 2023 at 11:27:12 PM UTC-5, Dan Christensen wrote:
> > > Consider this counterfactual conditional:
> > >
> > > "If the release valve had been working properly, the explosion would have been contained."
> > >
> > From Wikipedia:
> >
> > "The PROBLEM of counterfactuals
> >
> > "According to the material conditional analysis, a natural language conditional, a statement of the form ‘if P then Q’, is true whenever its antecedent, P, is false. Since counterfactual conditionals are those whose antecedents are false, this analysis would wrongly predict that all counterfactuals are vacuously true."
> >
> > https://en.wikipedia.org/wiki/Counterfactual_conditional#The_problem_of_counterfactuals
> >
> > Is that really a problem?
> >
> > Suppose: ValveFailure => Explosion (for logical propositions ValveFailure and Explosion)
> >
> > This implication is vacuously true if ValveFailure is false, but so what? In that case, we cannot determine the truth value for Explosion. Without introducing any inconsistencies, Explosion could be either true or false (see lines 3 and 4 in the following truth table).
> >
> > V E V=>E
> > T T T
> > T F F
> > F T T <--------
> > F F T <--------
> >
> > So, NOT a problem in this case.

> The valve example is very easy. But you need to use
> biconditional and not condictional.

That would rule out other causes of an explosion, e.g. fire or sabotage. Not necessary if you use the material condition.

My point is that, with its principle of vacuous truth, the material condition will suffice here.

[Mostowski Collapse]

If you don't have a biconditional you have not
enough causality. Although you might have:

V, V -> E |- E

This does not hold:

~V, V -> E |- ~E

That would be a fallacious conclusion. The
explosion could also happen because there

was a second faulty valve, so if the first valve
isn't faulty this doesn't guarantee that there is
no explosion. So if somebody says if the

valve were not faulty there would be an
exposion, he can only make this counter
factual conclusion if there is also a strong

causality between V and E, i.e. V <-> E. Its
a very common fallacy, that one often thinks
A -> B, also implies ~A -> ~B. But this has

a simple counter model in general:

(A→B) → (¬A → ¬B) is invalid.
Countermodel:
A: false
B: true
https://www.umsu.de/trees/#(A~5B)~5(~3A~5~3B)

[Mostowski Collapse]

Corr.:

> no explosion. So if somebody says if the
> valve were not faulty there would be an

Should say:

> no explosion. So if somebody says if the

> valve were not faulty there would not be an

[Dan Christensen]

On Friday, January 27, 2023 at 1:57:45 PM UTC-5, Mostowski Collapse wrote:
> If you don't have a biconditional you have not
> enough causality. Although you might have:
> V, V -> E |- E

Yes.

> This does not hold:
>
> ~V, V -> E |- ~E
>

[snip]

Why should it? You seem to be ruling out other possible causes of an explosion. There could be an explosion even without a release valve failure, e.g. a fire or sabotage.

[Mostowski Collapse]

It must hold according to the statement of
the counter factual reasoning example:

"If the release valve had been working properly,
the explosion would have been contained."

So you must daw the following conclusion:

~V, G |- ~E

Where G is your causality theory. You cannot
do that with G=V -> E only.

[Mostowski Collapse]

You braught up the example by yourself:

Dan Christensen schrieb am Freitag, 27. Januar 2023 um 05:27:12 UTC+1:
> Consider this counterfactual conditional:

> "If the release valve had been working properly, the explosion would have been contained."

https://groups.google.com/g/sci.logic/c/klsQDByVjTs/m/zspCLXQOAQAJ

Did you already forget?

[Dan Christensen]

On Friday, January 27, 2023 at 2:39:06 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Freitag, 27. Januar 2023 um 20:36:55 UTC+1:
> > On Friday, January 27, 2023 at 1:57:45 PM UTC-5, Mostowski Collapse wrote:
> > > If you don't have a biconditional you have not
> > > enough causality. Although you might have:
> > > V, V -> E |- E
> > Yes.
> > > This does not hold:
> > >
> > > ~V, V -> E |- ~E
> > >
> > [snip]
> >
> > Why should it? You seem to be ruling out other possible causes of an explosion. There could be an explosion even without a release valve failure, e.g. a fire or sabotage.

> It must hold according to the statement of
> the counter factual reasoning example:
> "If the release valve had been working properly,
> the explosion would have been contained."

[snip]

We have only ValveFailed => Explosion. You could, of course, add other assumptions, e.g. that Explosion => ValveFailed to see what would happen.

[Mostowski Collapse]

You are a complete moron writing nonsense like:

> You could, of course, add other assumptions

Its not a could, its a must to make your problem
true, to make this counterfactual statement
true. Whats wrong with you idiot?

Dan Christensen schrieb am Freitag, 27. Januar 2023 um 05:27:12 UTC+1:
> Consider this counterfactual conditional:

> "If the release valve had been working properly, the explosion would have been contained."

https://groups.google.com/g/sci.logic/c/klsQDByVjTs/m/zspCLXQOAQAJ

Your nonsense, that you don't understand the counterfactual
statement, what would make it true, only explains again
that you also don't understand "iff" and therefore 50% of

mathematics. We saw this already in your definition of even/odd.
Once a moron always a moron, that clearly displays in your
crank behaviour. Blind like a mole concerning the biconditional

<=> and its everyday use.

LMAO!

[Mostowski Collapse]

What definition makes this true?

If x is not a natural number then x is neither even nor odd.

You cannot argue one "could" add fix this and that
to your even odd. Because if you don't add fix something
you don't get the above true. So the answer would

be something along we "need" to add fix something,
to make it true, for example bla bla. You are using
the wrong words. You asked about this inference:

Dan Christensen schrieb am Freitag, 27. Januar 2023 um 05:27:12 UTC+1:
> Consider this counterfactual conditional:
> "If the release valve had been working properly, the explosion would have been contained."
> https://groups.google.com/g/sci.logic/c/klsQDByVjTs/m/zspCLXQOAQAJ

There is no "could" answer. The above "needs"
something different from ValveFailed => Explosion.
Also there is nowhere stated that we have

ValveFailed => Explosion only. When you analyse
a problem you don't stop at the first implication,
the first "if" that you find. Thats how you worked

in the past with DC Proof and all your nonsense
you posted here. Usually mathematicians don't stop
at the first "if". They try to model the full thing,

always strive for "iff". Otherwise you get an
incomplete definition, function spaces that don't work
function equality that doesn't work, all your nonsense.

[Dan Christensen]

On Thursday, January 26, 2023 at 11:27:12 PM UTC-5, Dan Christensen wrote:

> Consider this counterfactual conditional:
>
> "If the release valve had been working properly, the explosion would have been contained."
>

This statement suggests that the release valve had actually failed (ValveFailed is true) and an explosion actually occurred (Explosion is true). Correct?

Now (A & B) => (~A => ~B) is a tautology in ordinary propositional logic. (Verify with a truth table).

Therefore, we have: ValveFailed & Explosion => [~ValveFailed => ~Explosion] (behold, the consequent is our "counterfactual conditional!")

This result was obtained using ONLY ordinary propositional logic (an ordinary truth table), assuming ONLY that the release valve had actually failed (ValveFailed is true) and an explosion actually occurred (Explosion is true). Exactly why did we need a "counterfactual conditional" here? We didn't.

[Mostowski Collapse]

Yes of course, the causality of a some thing that
can explode is, that it is already exploded. Very
useful device. Especially for Los Alomos, they

didn't need to fly some Atomic Bob to Japan,
they just let explode while building it, let it explode
in the USA. Did you think about:

A & B => B

I only know about one lad who doens't know
this this rule, his name is ChatGPT. The thingy
recently stated:

Q: Does p and q imply p?

A: No, “p and q” does not imply “p.”

The statement “p and q” means that both p
and q are true. However, this does not
necessarily mean that p is true on its own.
For example, consider the following statements:

“It is raining and the sun is shining.”
“The earth is round and the moon is round.”
In both of these cases, “p and q” is true,
but neither “p” nor “q” is true on its own.

[Dan Christensen]

On Friday, January 27, 2023 at 9:38:27 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Samstag, 28. Januar 2023 um 03:05:20 UTC+1:
> > On Thursday, January 26, 2023 at 11:27:12 PM UTC-5, Dan Christensen wrote:
> > > Consider this counterfactual conditional:
> > >
> > > "If the release valve had been working properly, the explosion would have been contained."
> > >
> > This statement suggests that the release valve had actually failed (ValveFailed is true) and an explosion actually occurred (Explosion is true). Correct?
> >
> > Now (A & B) => (~A => ~B) is a tautology in ordinary propositional logic. (Verify with a truth table).
> >
> > Therefore, we have: ValveFailed & Explosion => [~ValveFailed => ~Explosion] (behold, the consequent is our "counterfactual conditional!")
> >
> > This result was obtained using ONLY ordinary propositional logic (an ordinary truth table), assuming ONLY that the release valve had actually failed (ValveFailed is true) and an explosion actually occurred (Explosion is true). Exactly why did we need a "counterfactual conditional" here? We didn't.

> Yes of course, the causality of a some thing that
> can explode is, that it is already exploded. Very
> useful device. Especially for Los Alomos, they
>
> didn't need to fly some Atomic Bob to Japan,
> they just let explode while building it, let it explode
> in the USA. Did you think about:
>
> A & B => B
>
> I only know about one lad who doens't know
> this this rule, his name is ChatGPT. The thingy
> recently stated:
>
> Q: Does p and q imply p?
>
> A: No, “p and q” does not imply “p.”
>

It may not recognize either p or q as logical propositions.

> The statement “p and q” means that both p
> and q are true. However, this does not
> necessarily mean that p is true on its own.
> For example, consider the following statements:
>
> “It is raining and the sun is shining.”
> “The earth is round and the moon is round.”
> In both of these cases, “p and q” is true,
> but neither “p” nor “q” is true on its own.

This makes no sense. A & B => A. Always. Make a truth table.

[Mostowski Collapse]

ChatGPT makes no sense. Yes, similarly like your
ValveFailed & Explosion. Or let it write as:

ValveFailed, Explosion

So you have the fact ValveFailed. Recall that
counter factual means somewhere a "retract",
since it belongs to defeasible reasoning.

G \ {A} |- B
----------------
G |- -A -> B

Note the retract operator (-)/1 in the above.
But you don't show a retract, when you make
this proof, in fact you will have an inconsistency

ValveFailed & ~ValveFailed in the premisses:

> > > Therefore, we have: ValveFailed & Explosion =>
> > > [~ValveFailed => ~Explosion] (behold, the consequent is our "counterfactual conditional!")

ValveFailed is still in the premisse. Although you
prove a conditional. You don't prove a counterfactual
conditional, since a counterfactual conditional

is an inference process that modifies the premisses
in a non-monotonical way. The usual implication
is monotonic, this here adds a premisse:

G, A |- B
----------------
G |- A -> B

And if the logic itself is also monotonic, the G, A
can derive the same like G and a little bit more,
namely additionally what it can derive from G u {A},

i.e. proofs that also make use of A. But in
counterfactual situations, facts are countered,
but alternative state of a affairs, so what you

prove and claim is a counter factual conditional,
is by far the biggest nonsense claim that any sane
person could secrete. Its just dumb.

Your approach is so dumb, because you provoke
an inconsistency in the premisses, you can also
prove the following:

ValveFailed & Explosion => [~ValveFailed => Explosion]
(behold! The other counterfactual is also valid
according to the Dumbo Approach of Dan-O-Matik)

behold! Now the bomb is exploding and is not exploding,
which reminds me of this scene:

Dark Star - Talking to the bomb
https://www.youtube.com/watch?v=h73PsFKtIck

Since ~Explosion & Explosion we probably need to
conclude there is no Bomb, i.e. the domain of
discourse is empty. Right Dan-O-Matik?

LoL

[Dan Christensen]

I said nothing about any counterfactual conditional being valid or invalid. If all you know is that both propositions A and B are true, there are not many useful logical inferences you can draw from this:

A & B
B & A
A => B
B => A
~B => ~A <---- "counterfactual" conditional
~A => ~B <---- "counterfactual" conditional
~A => C
~B => C
etc.

where C is any logical proposition (vacuously true)

Counterfactual conditionals, as the name suggests, don't deal with facts so much as opinion and speculation about past events and what might of have been. This may be a useful device for storytellers, but not very useful at all for a logical analysis of the current state of the world.

If you want to talk about causality, you should use the language of causality. If A is said to cause B, what else causes B, etc.

[Mostowski Collapse]

You cannot list counter factual conditional with other
truth functional connectives. Its not a truth functional
connective. There are 16 truth functioncal connectives

in two variables A and B. Which you can list in that you
list the pattern x00,x01,x10,x11 and the you can give names:

0 0 0 0 0
0 0 0 1 A & B
0 0 1 0 ~A & B
0 0 1 1 B
0 1 0 0 A & ~B
0 1 0 1 A
0 1 1 0 ~(A <-> B)
0 1 1 1 A v B
1 0 0 0 ~(A v B)
1 0 0 1 A <-> B
1 0 1 0 ~A
1 0 1 1 A <- B
1 1 0 0 ~B
1 1 0 1 A -> B
1 1 1 0 A | B /* Sheffer Stroke */
1 1 1 1 1

None of these truth functional connectives are
counter factual conditionals. Because a counter
factual conditional makes references to

some facts, and modifies them first, before
evaluating some truth value.

[Mostowski Collapse]

I mean its defined here, its not only a truth functional
in its two arguments. It depends on a third value, the
modification of the circumstances:

Counterfactual conditionals (also subjunctive or X-marked) are
conditional sentences which discuss what would have been true
under **different circumstances**, e.g.
https://en.wikipedia.org/wiki/Counterfactual_conditional

Lets say A > B denotes a counter factual conditional,
then an inference rule for it would be:

G', A |- B
----------------
G |- A > B

Where G' is the modification that creates the different
circumstances. You cannot model that in FOL or in
DC Proof so easily. It took Artificial Intelligence quite

some decades to arrive at it, you find that in logic
programming. For example negation as failure can help
implement counterfactuals, because it is defeasible.

[Mostowski Collapse]

Not everything that has "conditional" in its name is propositional
logic with simple truth functional semantics. Examples that
are not simple propositional logic:

- counterfactual conditional
https://en.wikipedia.org/wiki/Counterfactual_conditional

- strict conditional
https://en.wikipedia.org/wiki/Strict_conditional

- relevance conditional
https://en.wikipedia.org/wiki/Relevance_logic

- What else?

[Mostowski Collapse]

If the change G' has someting to do with A, i.e.
if G' = G * A, where * is some revision operator, the
inference rule boils down to

G * A |- B

----------------
G |- A > B

Also known as the Ramsey Test:
https://en.wikipedia.org/wiki/Belief_revision#The_Ramsey_test

[Dan Christensen]

On Saturday, January 28, 2023 at 2:39:50 PM UTC-5, Mostowski Collapse wrote:
[snip]

> Counterfactual conditionals (also subjunctive or X-marked) are
> conditional sentences which discuss what would have been true
> under **different circumstances**, e.g.

Like changing the parameters of a function. You don't need to concoct another form logic to do that.

[Mostowski Collapse]

Well you need if you want to mimic what humans do.
For example you have a "New Permise Button" in DC Proof.
This button builds your context chain,

until you close it by the "Conclusion Button" in DC Proof.

Why don't you have a "Toggle Premise Button"? This would
change an premise in the current context chain from is
formula A to the negation of the formula, i.e. ~A,

for the while of a proof? The "Conclusion Button" would
also undo the change.

Dan Christensen schrieb:

[Mostowski Collapse]

With your current version of DC Proof, you
cannot perform this change of premisses.
And if you would allow this change, the

result will be "Proofs" that a mathematician
will have a hard time to accept as a proof
from a math text book. Counter factuals is

usually not something a math text book uses.
In a math text books you can factor the proof,
so that you have case 1, case 2, with the

different circumstance. You done take case 1,
and modify it into case 2, on top of case 1.

Mostowski Collapse schrieb:

[Mostowski Collapse]

Historically there are some proof systems
that allow explicit substitution. But change of
parameter, what you call it, is not necessarely

a counter factual reasoning step. For example
in ths change of parameter, i.e. substitution
rule, which DC Proof also has:

A(x), x=y |- A(y)

A new copy of the formula A(x) is created. So
it doesn't count as counter factual reasoning.
Nothing is changed, its still monotonic.

Only if you would have some inference rule
of change of parameter, which is also non-monotonic,
you could claim you would also have

counter factual reasoning. Also it begs the
question whether x=y is really a change of parameter,
since equals are substituted for equals.

You would need some inference rule where
you go an arbitrary A(x) to another arbitrary
A(y). And if you do this, you will probably leave

the domain of mathematical proofs, which try
to avoid the usage of counterfactual reasoning.

P.S.: AUTOMATH had explicit substitution.
https://en.wikipedia.org/wiki/Automath

Mostowski Collapse schrieb:

[Mostowski Collapse]

In as far your title counterfactual conditionals
in set theory, the part "in set theory". Is utter
nonsense. Where did you get that title from?

Mostowski Collapse schrieb:

[Dan Christensen]

On Saturday, January 28, 2023 at 3:33:16 PM UTC-5, Mostowski Collapse wrote:
> Well you need if you want to mimic what humans do.
> For example you have a "New Permise Button" in DC Proof.
> This button builds your context chain,
>
> until you close it by the "Conclusion Button" in DC Proof.
>
> Why don't you have a "Toggle Premise Button"? This would
> change an premise in the current context chain from is
> formula A to the negation of the formula, i.e. ~A,
>
> for the while of a proof? The "Conclusion Button" would
> also undo the change.
>

Is this what you mean:

1. A
Premise

2. ~A
Premise

3. B
Premise

4. A & ~A
Join, 1, 2

5. ~B
Conclusion, 3

6. ~A => ~B
Conclusion, 2

7. A => [~A => ~B]
Conclusion, 1

[Mostowski Collapse]

Nope, nothing to do with counter factual. Already
syntactically you use =>. You cannot do counter
factual in DC Proof.

You only have:

1) For example you have a "New Permise Button" in DC Proof.

This button builds your context chain, until you close it by the
"Conclusion Button" in DC Proof.

What is missing:

2) Why don't you have a "Toggle Premise Button"? This would

change an premise in the current context chain from is
formula A to the negation of the formula, i.e. ~A, for the while
of a proof? The "Conclusion Button" would also undo the change.

The conclusion button would generate "A > B" and not "A -> B".

But this was more meant as a joke to implement this inference rule:

> G * A |- B

> ----------------
> G |- A > B
>

> Also known as the Ramsey Test:
> https://en.wikipedia.org/wiki/Belief_revision#The_Ramsey_test

You would clearly leave the terrain of math text book proofs.

[Dan Christensen]

On Saturday, January 28, 2023 at 3:44:17 PM UTC-5, Mostowski Collapse wrote:
> Historically there are some proof systems
> that allow explicit substitution. But change of
> parameter, what you call it, is not necessarely
>
> a counter factual reasoning step.

[snip]

Your "different circumstances" could be set by changing the parameters of the function in question. Using a parameter of time, for example, we might have something like:

[time = 0 => ... ] & [time = 1 => ... ] & [time = 2 => ... ]

Dan

[Mostowski Collapse]

Thats not counter factual. No facts are changed.

[Dan Christensen]

On Saturday, January 28, 2023 at 4:18:23 PM UTC-5, Mostowski Collapse wrote:
> Nope, nothing to do with counter factual. Already
> syntactically you use =>. You cannot do counter
> factual in DC Proof.
>

You can assume A is true and prove B. Then you can assume A is false and prove C. We consider two different circumstances: A and ~A. No need for some weird new form of "logic" to do this.

Dan

[Mostowski Collapse]

But proof by cases is nothing counter factual.
For counter factual, you need:

i) You cannot use the sign => of material implication
for counter factual. It is a different sign, like for example > .
So when you ask, whether this is counter factual:

[time = 0 => ... ] & [time = 1 => ... ] & [time = 2 => ... ]

This is anyway moot. A counter factual conditional in
formula syntax is the appearance of a conditional different
from material implication =>, thus using a differnet sign.

For example > .

ii) Then you have to support this new sign, this new
connective in your proof tool. So if the new sign is A > B,
and your proof tool follows some natural deduction style
proof approach, you possibly must have somewhere:

An I Rule: An introducion rule for A > B

An E Rule: An elimination rule for A > B.

iii) What should the elimination rule for A > B be? The introduction
rule is usually this here:

> G * A |- B
> ----------------
> G |- A > B
>
> Also known as the Ramsey Test:
> https://en.wikipedia.org/wiki/Belief_revision#The_Ramsey_test

But what would be an elimination rule. Material implication A => B
can be eliminated via modus ponens:

G |- A D |- A => B
----------------------------------
G, D |- B

How do you want to eliminate counter factual conditional A > B ?

[Dan Christensen]

On Saturday, January 28, 2023 at 5:11:01 PM UTC-5, Mostowski Collapse wrote:

> > G * A |- B
> > ----------------
> > G |- A > B
> >
> > Also known as the Ramsey Test:
> > https://en.wikipedia.org/wiki/Belief_revision#The_Ramsey_test
> But what would be an elimination rule. Material implication A => B
> can be eliminated via modus ponens:
>
> G |- A D |- A => B
> ----------------------------------
> G, D |- B
>
> How do you want to eliminate counter factual conditional A > B ?
> Dan Christensen schrieb am Samstag, 28. Januar 2023 um 22:27:56 UTC+1:
> > On Saturday, January 28, 2023 at 4:18:23 PM UTC-5, Mostowski Collapse wrote:
> > > Nope, nothing to do with counter factual. Already
> > > syntactically you use =>. You cannot do counter
> > > factual in DC Proof.
> > >
> > You can assume A is true and prove B. Then you can assume A is false and prove C. We consider two different circumstances: A and ~A. No need for some weird new form of "logic" to do this.
> >

> But proof by cases is nothing counter factual.

Huh??? No proof by cases here. Note that the consequents differ. You can think of A as one "circumstance," and ~A as another. I hope this helps.

Dan

[Mostowski Collapse]

Is the internet broken in Big Foot land? Whats the
reason that you have no clue about counterfactual
conditionals? Educate yourself:

Key Semantic Theses about Counterfactuals
Counterfactuals are not truth-functional.
Counterfactuals have context-sensitive truth-conditions.
Counterfactual antecedents are interpreted non-monotonically.
https://plato.stanford.edu/entries/counterfactuals/

Natural deduction calculi and sequent calculi for
counterfactual logics - Francesca Poggiolesi, 2018
https://hal.science/hal-01271556/file/strictconditional10.pdf

But the calculus propose in the above paper is not formulate
with a believe revision operator as a Ramsey test for the
introduction rule. And the elimination rule is also not somehow

in resemblance with modus ponens. Maybe there are other papers around?
But clear they suggest a syntax with separate material implication
A => B and conterfactual conditional A > B.

[Dan Christensen]

On Saturday, January 28, 2023 at 5:25:50 PM UTC-5, Mostowski Collapse wrote:
> Is the internet broken in Big Foot land? Whats the
> reason that you have no clue about counterfactual
> conditionals?

I have yet to find a need for any conditionals other than those in math textbooks (i.e. material conditionals). Do we really need to formalize the bizarre opinion that "If Oswald had not killed JFK, then someone else would have?" Or that "If the original Nazis had won WW2, there would be universal peace and prosperity?" What insights might such formalizations provide? None.

Dan

[Mostowski Collapse]

How about this counter factual:

"If Dan Christensen would plough books he would have a DC Proof of Zorns Lemma"

[Mostowski Collapse]

Usually in the example of a valve, or lets say your
bicycle, I guess this counter factual is not bizzare:

"If you had a flat tire you would need to find the
puncture, tape over it and inflate the tire"

[Jeffrey Rubard]

On Thursday, January 26, 2023 at 8:27:12 PM UTC-8, Dan Christensen wrote:
> Consider this counterfactual conditional:
>
> "If the release valve had been working properly, the explosion would have been contained."
>

> Let T= the set of all points in time
>
> Let F= the set of all times that the release value failed
>
> Let E= the set of all times that an explosion occurred
>
> Q: Does the following statement in the language of set theory adequately model the above counterfactual conditional:
>
> F ⊂ T ∧ E ⊂ T ∧ F ⊂ E

>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

"So this is the metatheory of that kind of bullshitter?"

[Dan Christensen]

On Saturday, January 28, 2023 at 6:22:35 PM UTC-5, Mostowski Collapse wrote:

> Mostowski Collapse schrieb am Sonntag, 29. Januar 2023 um 00:01:40 UTC+1:

[snip]

> > > I have yet to find a need for any conditionals other than those in math textbooks (i.e. material conditionals). Do we really need to formalize the bizarre opinion that "If Oswald had not killed JFK, then someone else would have?" Or that "If the original Nazis had won WW2, there would be universal peace and prosperity?" What insights might such formalizations provide? None.

> Usually in the example of a valve, or lets say your
> bicycle, I guess this counter factual is not bizzare:
>
> "If you had a flat tire you would need to find the
> puncture, tape over it and inflate the tire"

FlatTire -> RepairTire

Dan

[Mostowski Collapse]

A very easy and drastic way to model counterfactual conditionals,
is to use provability theory. So if you have a modal operator
Bew() as in Gödels incompletness theorem, then you could define

/* A > B == A |- B */
A > B :<=> Bew('A -> B')

You would get an inconsistency anymore as with normal
material implication, since Bew() is a modal operator and
it opens a new scope for circumstances:

/* Provable */
Valve => (~Valve => false)

/* Not Provable */
Valve => (~Valve > false)

You could use a provability theory where the modal operator
communicates a little bit with its environment, concerning the
current variable assignment, so that you could also have:

/* Provable */
Valve(x) => (~Valve(x) => false)
Valve(x) => (x = y => Valve(y))

/* Not Provable */
Valve(x) => (~Valve(x) > false)
/* Provable */
Valve(x) => (x = y > Valve(y))

Dan Christensen schrieb:
> EXAMPLE
>
> A counterfactual conditional: "If the release valve had been working properly, the explosion would have been contained."
>
> Let t = the set of all points in time
> Let f = the set of all times that the release value failed
> Let e = the set of all times that an explosion occurred
>
> In the notation of DC Proof, we have:
>
> ALL(x):[x in t => [x in f => x in s]]
>
> Where "in" = "is an element of"
>
> Your comments?

[Mostowski Collapse]

In Prolog you get that very easily, since it is
homoiconic. You don't need a Gödel numbering as in
Bew() from Gödels incompletness theorem.

https://en.wikipedia.org/wiki/Homoiconicity

You can define a meta predicate prove/2, lets
say for a propositional logic prover, and could
then define, where G is the current prove context:

prove(G, A > B) :- prove([A], B).

So this is basically this drastic Ramsey test
inference rule. Where we would set G*A = A
as the believe revision:

A |- B
--------------
G |- A > B

Mostowski Collapse schrieb:

[Mostowski Collapse]

Corr.: The example with equality should read:

/* Provable */
x = y => (Valve(x) => Valve(y))

/* Provable */
x = y => (Valve(x) > Valve(y))

[Dan Christensen]

On Sunday, January 29, 2023 at 3:32:01 AM UTC-5, Mostowski Collapse wrote:
> A very easy and drastic way to model counterfactual conditionals,
> is to use provability theory. So if you have a modal operator
> Bew() as in Gödels incompletness theorem, then you could define
>
> /* A > B == A |- B */
> A > B :<=> Bew('A -> B')
>

Again, what insights might such formalizations provide? Mostly, counterfactuals seem to be nothing more than opinion or speculation about the past, not about actually happened, but what MIGHT have been. In themselves, even if you manage to somehow formalize them, they cannot be the basis for any formal logical analysis of the state of the world at any point in time. At most, they might stimulate discussions about the events of the past.

[Mostowski Collapse]

Often what might have been now, might have been
also tomorrow. Take this example:

"If Dan Christensen would have ploughed books he would have a DC Proof of Zorns Lemma now"

"If Dan Christensen will ploughed books he will have a DC Proof of Zorns Lemma by the end of year"

Conterfactuals are not restricted to the present tense.
Since factuals i.e. circumstances are not points
in time, they are rather points that extend into the future.

Why do you only give past tense examples?

Anyway, we are now right into the frame problem:

"To most AI researchers, the frame problem is the challenge
of representing the effects of action in logic without having
to represent explicitly a large number of intuitively obvious non-effects. "
https://plato.stanford.edu/entries/frame-problem/

[Mostowski Collapse]

"The challenge, then, is to find a way to capture the non-effects
of actions more succinctly in formal logic. What we need, it
seems, is some way of declaring the general rule-of-thumb
that an action can be assumed not to change a given property
of a situation unless there is evidence to the contrary. This
default assumption is known as the common sense law of inertia.
The (technical) frame problem can be viewed as the task
of formalising this law."
https://plato.stanford.edu/entries/frame-problem/

I guess your vacuous truth is not very useful. It just takes
the inertia, like for example ValveFailed , doesn't change
it, and adds ~ValveFailed on top of it. What can go wrong?

LoL

[Dan Christensen]

On Sunday, January 29, 2023 at 11:20:31 AM UTC-5, Mostowski Collapse wrote:

[snip]

> > > Again, what insights might such formalizations provide? Mostly, counterfactuals seem to be nothing more than opinion or speculation about the past, not about actually happened, but what MIGHT have been. In themselves, even if you manage to somehow formalize them, they cannot be the basis for any formal logical analysis of the state of the world at any point in time. At most, they might stimulate discussions about the events of the past.

No comment, Mr. Collapse?

[...]

> "The challenge, then, is to find a way to capture the non-effects
> of actions more succinctly in formal logic. What we need, it
> seems, is some way of declaring the general rule-of-thumb
> that an action can be assumed not to change a given property
> of a situation unless there is evidence to the contrary. This
> default assumption is known as the common sense law of inertia.
> The (technical) frame problem can be viewed as the task
> of formalising this law."

How is this relevant?

> https://plato.stanford.edu/entries/frame-problem/
>
> I guess your vacuous truth is not very useful.

It is useful in mathematical proofs and it does not seem to lead to any inconsistencies in what is true or false in the "real world" in the present timeframe.

[Mostowski Collapse]

Well your vacous truth doesn't allow any conclusions
about the future. For example if you would build a
logical expert system that assists some repair engineer.

If you model this here:

~ValveFailed on top ValveFailed

Your system gets inconsistent, you can now prove:

ValveFailed => (~ValveFailed => false)

Or written differently, you can now prove both:

ValveFailed => (~ValveFailed => ~Explosion)
ValveFailed => (~ValveFailed => Explosion)

What makes you think vacous truth is not related
to inconsistency? The false premisse is an inconsistency.
A false premisse, a violated axiom or whatever, is

an inconsistency. What do you think is an inconsistency?

[Mostowski Collapse]

Even your DC Proof tool agrees with my observation:

7 ValveFailed => [~ValveFailed => ~Explosion]
Conclusion, 1

15 ValveFailed => [~ValveFailed => Explosion]
Conclusion, 8

------------------------- begin proof -------------------------------------------

1 ValveFailed
Premise

2 ~ValveFailed
Premise

3 Explosion
Premise

4 ValveFailed & ~ValveFailed
Join, 1, 2

5 ~Explosion
Conclusion, 3

6 ~ValveFailed => ~Explosion
Conclusion, 2

7 ValveFailed => [~ValveFailed => ~Explosion]
Conclusion, 1

8 ValveFailed
Premise

9 ~ValveFailed
Premise

10 ~Explosion
Premise

11 ValveFailed & ~ValveFailed
Join, 8, 9

12 ~~Explosion
Conclusion, 10

13 Explosion
Rem DNeg, 12

14 ~ValveFailed => Explosion
Conclusion, 9

15 ValveFailed => [~ValveFailed => Explosion]
Conclusion, 8

------------------------- end proof -------------------------------------------

[Jim Burns]

On 1/29/2023 3:42 AM, Mostowski Collapse wrote:

> In Prolog you get that very easily, since it is
> homoiconic. You don't need a Gödel numbering as in
> Bew() from Gödels incompletness theorem.
>
> https://en.wikipedia.org/wiki/Homoiconicity

"Homoiconicity" is a lovely word.
I don't remember seeing it before.
|
| This property is often summarized by saying that
| the language treats "code as data".

"Homoiconicity" is the word I was looking for when
I was answering Hermann Weyl (as quoted by WM)
in defense of the use of infinite sets.

On 1/18/2023 5:16 AM, WM wrote:
> Note:
> "classical logic was abstracted from the
> mathematics of finite sets and their subsets.
> (The word finite is here to be taken in the
> precise sense that the members of such set
> are explicitly exhibited one by one.)
> Forgetful of this limited origin, one afterwards
> mistook that logic for something above and
> prior to all mathematics, and finally applied it,
> without justification, to the mathematics of
> infinite sets. This is the Fall and original sin
> of set-theory, for which it is justly punished
> by the antinomies. Not that such contradictions
> showed up is surprising, but that they showed up
> at such a late stage of the game!
> [H. Weyl: "Levels of infinity: Selected writings
> on mathematics and philosophy", Peter Pesic (ed.),
> Dover Publications (2012) p. 140f]
[sci.math,
"Who recognizes these true pioneers of dark numbers?"]

A set may be infinite: consider it the data.

But the code,
descriptive claims about one of the elements of
the set we mean, no matter which element,
augmented with visibly not-first-possibly-false
claims about one of the elements,
can be finite.

It is enough for the claims to be finitely-many
and linearly ordered. From that we can conclude,
the same as we could conclude if claims were sheep,
that, if
each claim is not first with a property
(such as the property of being possibly false)
then
each claim does not have that property
(in this example, not being possibly false).

_Because claims are like sheep_
if
each claim about one of the ones we mean
in a sequence is visibly not-first-possibly-false
(for example, q preceded by p and p->q)
then
each claim about one of the ones we mean
in that sequence is not-possibly-false.

Because language in general,
including the language we use with infinite sets,
is homoiconic,
we can reason finitely about infinite sets.
Which is a Neat Trick.

[Dan Christensen]

On Sunday, January 29, 2023 at 12:30:36 PM UTC-5, Mostowski Collapse wrote:
> Even your DC Proof tool agrees with my observation:
>
> 7 ValveFailed => [~ValveFailed => ~Explosion]
> Conclusion, 1
>
> 15 ValveFailed => [~ValveFailed => Explosion]
> Conclusion, 8
>

[snip]

An example of vacuous truth. The general principle is that if proposition A is false, then, for any proposition B, it is true that A implies B.

Anyway, since my original posting above, I have come to the conclusion that, even if you could formalize counterfactual conditionals, such a construct would be not be very useful in any logic analysis of the state of the world at ANY point in time. They might be useful in stimulating informal discussion about the events of the past, discussions based on actual facts.

[Mostowski Collapse]

A reason why the composed theory is useless:

T = {ValveFailed , ~ValveFailed}

because it is inconsistent. Please lookup inconsistent.

P.S.:
Since (X => (Y => Z)) <=> (X & Y => Z), we can
rewrite the two formulas as:

ValveFailed & ~ValveFailed => ~Explosion
ValveFailed & ~ValveFailed => Explosion

And by deduction theorem (|- X => Y) <=> (X |- Y), we
can rewrite the two formulas as:

ValveFailed & ~ValveFailed |- ~Explosion
ValveFailed & ~ValveFailed |- Explosion

We can also use the indentity (X & Y |- Z) <=> (X,Y |- Z),
we can rewrite the two formulas as:

ValveFailed, ~ValveFailed |- ~Explosion
ValveFailed, ~ValveFailed |- Explosion

So Dan Christensense "vacuous truth" approach to counterfactual
conditionals amounts to nothing else than the proposal to
create an inconsistent theory T = {ValveFailed , ~ValveFailed},

and start working with this theory T. Seriously?

Dan Christensen schrieb:

[Dan Christensen]

On Sunday, January 29, 2023 at 1:15:43 PM UTC-5, Mostowski Collapse wrote:

>
> Dan Christensen schrieb:
> > On Sunday, January 29, 2023 at 12:30:36 PM UTC-5, Mostowski Collapse wrote:
> >> Even your DC Proof tool agrees with my observation:
> >>
> >> 7 ValveFailed => [~ValveFailed => ~Explosion]
> >> Conclusion, 1
> >>
> >> 15 ValveFailed => [~ValveFailed => Explosion]
> >> Conclusion, 8
> >>
> > [snip]
> >
> > An example of vacuous truth. The general principle is that if proposition A is false, then, for any proposition B, it is true that A implies B.
> >
> > Anyway, since my original posting above, I have come to the conclusion that, even if you could formalize counterfactual conditionals, such a construct would be not be very useful in any logic analysis of the state of the world at ANY point in time. They might be useful in stimulating informal discussion about the events of the past, discussions based on actual facts.
> >

> A reason why the composed theory is useless:
>
> T = {ValveFailed , ~ValveFailed}
>

[snip]

Pay attention, Mr. Collapse. I have moved past that. Again, I have come to the conclusion that, even if you could formalize counterfactual conditionals, such a construct would not be very useful in any logic analysis of the state of the world at ANY point in time. They might be useful in stimulating informal discussion about the events of the past, discussions based on actual facts.

[Mostowski Collapse]

I don't think you have moved past that, i.e. done a
counterfactual move, and we need now apply counterfactual
reasoning to all you are saying. You are just frickled.

Five minutes ago you wrote:
"He also never quite understood the notion of vacuous truth."

Why would I need to understand the notion of vacuous
truth in the contect of counterfactuals, if you have retracted
your proposal. Or didn't you make this counterfactual move?

LoL

[Mostowski Collapse]

Also it begs the question, if you make this counterfactual
move, which amounts to claiming there are no counterfatual
moves, then we might conclude that Dan Christensen is inconsistent.

[Dan Christensen]

On Sunday, January 29, 2023 at 2:39:35 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Sonntag, 29. Januar 2023 um 19:55:40 UTC+1:

> > Pay attention, Mr. Collapse. I have moved past that. Again, I have come to the conclusion that, even if you could formalize counterfactual conditionals, such a construct would not be very useful in any logic analysis of the state of the world at ANY point in time. They might be useful in stimulating informal discussion about the events of the past, discussions based [not on opinion and speculation but] on the facts.

> If you have retracted
> your proposal.

Yes. That should be obvious from the above.

[Mostowski Collapse]

Also it begs the question, if you make this counterfactual
move, which amounts to claiming there are no counterfatual

move, then we might conclude that Dan Christense is inconsistent.

Or maybe its the beginning of a great adventure, into a
brand new paradox, the Christensen Counterfactual Paradox.
Very moving moment in history of sci.logic, just like

Kate and Leonard on the Titanic.

[Dan Christensen]

On Sunday, January 29, 2023 at 2:51:38 PM UTC-5, Mostowski Collapse wrote:
> Also it begs the question, if you make this counterfactual

> move, which amounts to claiming there are no counterfactual
> moves

[snip]

Again, my position now is that counterfactual conditionals, by any definition, are based on opinion and speculation about past events, had things been different. Therefore, they are not very useful in any logical analysis of the state of the world at ANY point in time, past or present.

[Mostowski Collapse]

Dan Christensen and Counterfactuals, definitively a match
made in Heaven. Question is only, will it have a happy ending?

BTW: How was it with Kate and Leanardo
on the Titanic, did they survive the Ice Berg?

[Mostowski Collapse]

Here is a recommendation: I would also retract the
mathematical induction schema from your Peano Axioms.
I mean its not right in the light of your arguments and

your doubts about predictions into the future, that
when you only have made manually some computation
like for example:

2 + 3 = 5

That a mathematical induction suddently predicts:

n + m = m + n

What are these n, m which to not exist in the present
tense, in particular any present tense calculation
will never reach all n,m, there are always dark numbers,

so mathematical induction is to be rejected.

[Dan Christensen]

On Sunday, January 29, 2023 at 3:46:46 PM UTC-5, Mostowski Collapse wrote:
> Here is a recommendation: I would also retract the
> mathematical induction schema from your Peano Axioms.
> I mean its not right in the light of your arguments and
>
> your doubts about predictions into the future, that
> when you only have made manually some computation
> like for example:
>
> 2 + 3 = 5
>
> That a mathematical induction suddently predicts:
>
> n + m = m + n
>
> What are these n, m which to not exist in the present
> tense, in particular any present tense calculation
> will never reach all n,m, there are always dark numbers,
>
> so mathematical induction is to be rejected.

You have been spending too much time with your little buddy, AP!

HA, HA, HA!

[Mostowski Collapse]

"if Dan Chistensens brain would work properly, counterfactuals wouldn't be much a problem."

Once you have understood counterfactuals, or imagination,
conditionals inside this imagination, are rather a piece of cake.
Except that a counterfactual conditional gives also a hint how

this imagination looks like. About counterfactuals itself:

Semantic How do we communicate and reason about possibilities
which are remote from the way things actually are?
Epistemic How can our experience in the actual world justify thought
and talk about remote possibilities?
Metaphysical Do these remote possibilities exist independently from
the actual world, or are they grounded in things that actually exist?
https://plato.stanford.edu/entries/counterfactuals/

I guess you lack some imagination? Try him:

Joan Casas-Roma - Universitat Oberta de Catalunya
Unfolding the Dynamics of Imagination Acts
https://www.researchgate.net/publication/325335453

Dan Christensen schrieb am Freitag, 27. Januar 2023 um 05:27:12 UTC+1:
> Consider this counterfactual conditional:

>
> "If the release valve had been working properly, the explosion would have been contained."
>

> Let T= the set of all points in time
>
> Let F= the set of all times that the release value failed
>
> Let E= the set of all times that an explosion occurred
>
> Q: Does the following statement in the language of set theory adequately model the above counterfactual conditional:
>
> F ⊂ T ∧ E ⊂ T ∧ F ⊂ E
>

[Dan Christensen]

On Sunday, January 29, 2023 at 7:03:56 PM UTC-5, Mostowski Collapse wrote:
> "if Dan Chistensens brain would work properly, counterfactuals wouldn't be much a problem."
>
> Once you have understood counterfactuals, or imagination,
> conditionals inside this imagination, are rather a piece of cake.

[snip]

Suppose X is a historical fact. What if X was NOT a historical fact? Is it my imagination, or is that a silly question?

A better question would be, "What were the ACTUAL consequences of X?"

Dan

[Mostowski Collapse]

Ha Ha,

The perfect marriage Dan Christensen and Counter Factuals.
Embarking on a battle between Actualism and Possibilism.

"As if" somebody would care? LoL

Bye

Dan Christensen schrieb:

[Mostowski Collapse]

You never played this game when you were young?

Battleship (game)
https://en.wikipedia.org/wiki/Battleship_%28game%29

How would you model this game in DC Proof?

P.S.: A players move is a counter factual, before the
move to location (x,y) you had:

~Explode(x,y)

After the move you will have:

Explode(x,y)

If you would apply your vacuous truth approach,
you would get an inconsistency.

So you said you moved past vacuous truth, whats
your new approach? Give totally up and yeet logic?

LMAO!

[Mostowski Collapse]

Or better, show us Tic Tac Toe in DC Proof.

[Jeffrey Rubard]

On Monday, January 30, 2023 at 12:16:27 AM UTC-8, Mostowski Collapse wrote:
> Or better, show us Tic Tac Toe in DC Proof.

"You mean like the Booker T. and the MG's song?"