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LOGICAL Illusions in Natural Language?
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Dan Christensen
Mar 9, 2021, 11:31:23 AM3/9/21
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In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
"Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
https://en.wikipedia.org/wiki/Material_conditional
Why "erroneously judged?" Using a form of natural deduction based on classical logic, we can formally prove that if a logical proposition A is false, then the implication A => B must be true regardless of the truth value of proposition B:
1 ~A (Premise, A is false)
2 A (Premise, A is true)
3 ~B (Premise, B is false)
4 ~A & A (Join, 1, 2)
5 ~~B (Conclusion, proof my contradiction, discharging line 3)
6 B (Eliminating ~~, line 5)
7 A => B (Conclusion, conditional proof, discharging line 2)
Why then can we not teach students that our brains may also have tricked us into thinking "If 8 is odd then 3 is prime" is false? By a simple, logical analysis, we can see that it is actually true.
Your comments?
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog a http://www.dcproof.wordpress.com
What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
"Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
https://en.wikipedia.org/wiki/Material_conditional
Why "erroneously judged?" Using a form of natural deduction based on classical logic, we can formally prove that if a logical proposition A is false, then the implication A => B must be true regardless of the truth value of proposition B:
1 ~A (Premise, A is false)
2 A (Premise, A is true)
3 ~B (Premise, B is false)
4 ~A & A (Join, 1, 2)
5 ~~B (Conclusion, proof my contradiction, discharging line 3)
6 B (Eliminating ~~, line 5)
7 A => B (Conclusion, conditional proof, discharging line 2)
Why then can we not teach students that our brains may also have tricked us into thinking "If 8 is odd then 3 is prime" is false? By a simple, logical analysis, we can see that it is actually true.
Your comments?
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog a http://www.dcproof.wordpress.com
Mostowski Collapse
Mar 9, 2021, 2:56:02 PM3/9/21
to
In an empty universe everything is possible.
Where is your assumption that the universe is non-empty?
In an empty universe ~A & A is not a contradiction.
Where is your assumption that the universe is non-empty?
In an empty universe ~A & A is not a contradiction.
Dan Christensen
Mar 9, 2021, 3:01:17 PM3/9/21
to
On Tuesday, March 9, 2021 at 2:56:02 PM UTC-5, Mostowski Collapse wrote:
> In an empty universe everything is possible.
>
> Where is your assumption that the universe is non-empty?
>
> In an empty universe ~A & A is not a contradiction.
I am talking about propositional logic. No quantifiers. Just propositions that are ambiguously either true or false. I hope this helps.
> In an empty universe everything is possible.
>
> Where is your assumption that the universe is non-empty?
>
> In an empty universe ~A & A is not a contradiction.
Peter
Mar 9, 2021, 5:38:27 PM3/9/21
to
Mostowski Collapse wrote:
> In an empty universe everything is possible.
>
> Where is your assumption that the universe is non-empty?
>
> In an empty universe ~A & A is not a contradiction.
>
> Dan Christensen schrieb am Dienstag, 9. März 2021 um 17:31:23 UTC+1:
>> In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
>>
>> What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
>>
>> "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
>> https://en.wikipedia.org/wiki/Material_conditional
>>
>> Why "erroneously judged?" Using a form of natural deduction based on classical logic, we can formally prove that if a logical proposition A is false, then the implication A => B must be true regardless of the truth value of proposition B:
... which is irrelevant. You quote Wikipedia as saying
> In an empty universe everything is possible.
>
> Where is your assumption that the universe is non-empty?
>
> In an empty universe ~A & A is not a contradiction.
>
> Dan Christensen schrieb am Dienstag, 9. März 2021 um 17:31:23 UTC+1:
>> In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
>>
>> What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
>>
>> "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
>> https://en.wikipedia.org/wiki/Material_conditional
>>
>> Why "erroneously judged?" Using a form of natural deduction based on classical logic, we can formally prove that if a logical proposition A is false, then the implication A => B must be true regardless of the truth value of proposition B:
the *natural language statement* "If 8 is odd then 3 is prime"
is typically judged false.
(Emphasis added.) You then turn to (in your words)
a form of natural deduction based on classical logic
planet knows that
(P & ~P) -> Q
in classical propositional calculus. But not everyone knows that
natural language's "if...then..." is logic's "...->...". Why don't they
know it? Because it's false. *Some* if-thens are arrows and some
*aren't*. Will you ever get that? No, I don't think you will.
>>
>> 1 ~A (Premise, A is false)
>>
>> 2 A (Premise, A is true)
>>
>> 3 ~B (Premise, B is false)
>>
>> 4 ~A & A (Join, 1, 2)
>>
>> 5 ~~B (Conclusion, proof my contradiction, discharging line 3)
>>
>> 6 B (Eliminating ~~, line 5)
>>
>> 7 A => B (Conclusion, conditional proof, discharging line 2)
>>
>> Why then can we not teach students that our brains may also have tricked us into thinking "If 8 is odd then 3 is prime" is false? By a simple, logical analysis, we can see that it is actually true.
>>
>> Your comments?
>>
>> Dan
>>
>> Download my DC Proof 2.0 freeware at http://www.dcproof.com
>> Visit my Math Blog a http://www.dcproof.wordpress.com
When, once, reference was made to a statesman almost universally
recognized as one of the villains of this century, in order to
induce him to a negative judgment, he replied: "My situation is
so different from his, that it is not for me to pass judgment".
Ernst Specker on Paul Bernays
Dan Christensen
Mar 9, 2021, 5:51:33 PM3/9/21
to
On Tuesday, March 9, 2021 at 5:38:27 PM UTC-5, Peter wrote:
> Mostowski Collapse wrote:
> > In an empty universe everything is possible.
> >
> > Where is your assumption that the universe is non-empty?
> >
> > In an empty universe ~A & A is not a contradiction.
> >
> > Dan Christensen schrieb am Dienstag, 9. März 2021 um 17:31:23 UTC+1:
> >> In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
> >>
> >> What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
> >>
> >> "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
> >> https://en.wikipedia.org/wiki/Material_conditional
> >>
> >> Why "erroneously judged?" Using a form of natural deduction based on classical logic, we can formally prove that if a logical proposition A is false, then the implication A => B must be true regardless of the truth value of proposition B:
> ... which is irrelevant. You quote Wikipedia as saying
>
> the *natural language statement* "If 8 is odd then 3 is prime"
> is typically judged false.
>
> (Emphasis added.) You then turn to (in your words)
> a form of natural deduction based on classical logic
> In other words you have left natural language behind. Everybody on the
> planet knows that
>
> (P & ~P) -> Q
>
> in classical propositional calculus. But not everyone knows that
> natural language's "if...then..." is logic's "...->...". Why don't they
> know it? Because it's false. *Some* if-thens are arrows and some
> *aren't*. Will you ever get that? No, I don't think you will.
Just as there are optical illusions that fool the brain's visual processing, there seems to be LOGICAL illusions that fools the brain's language processing. Are you ruling out that possibility? If so, why? These illusions are very minor defects IMHO, but defects nonetheless.
> Mostowski Collapse wrote:
> > In an empty universe everything is possible.
> >
> > Where is your assumption that the universe is non-empty?
> >
> > In an empty universe ~A & A is not a contradiction.
> >
> > Dan Christensen schrieb am Dienstag, 9. März 2021 um 17:31:23 UTC+1:
> >> In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
> >>
> >> What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
> >>
> >> "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
> >> https://en.wikipedia.org/wiki/Material_conditional
> >>
> >> Why "erroneously judged?" Using a form of natural deduction based on classical logic, we can formally prove that if a logical proposition A is false, then the implication A => B must be true regardless of the truth value of proposition B:
> ... which is irrelevant. You quote Wikipedia as saying
>
> the *natural language statement* "If 8 is odd then 3 is prime"
> is typically judged false.
>
> (Emphasis added.) You then turn to (in your words)
> a form of natural deduction based on classical logic
> In other words you have left natural language behind. Everybody on the
> planet knows that
>
> (P & ~P) -> Q
>
> in classical propositional calculus. But not everyone knows that
> natural language's "if...then..." is logic's "...->...". Why don't they
> know it? Because it's false. *Some* if-thens are arrows and some
> *aren't*. Will you ever get that? No, I don't think you will.
Mostowski Collapse
Mar 10, 2021, 2:37:49 AM3/10/21
to
So you assume existence of a set {true, false} ?
Mostowski Collapse
Mar 10, 2021, 2:49:23 AM3/10/21
to
Or maybe only existence of true and false?
If you prove:
~A & A |- B
You also use quantifiers, basically its QSAT
of this here:
∀A∀B(~A & A |- B)
Its not that you showed there are 8 bananas
on the table and there are 3 apples on the table,
i.e. true & true. You used so called "propositional
variables", and since you showed a tautology
you used so called "propositional quantifiers".
But where do you state that your universe is
non-empty? In an empty universe:
|- A v ~A
i.e. ~(~A & A) holds. LoL
If you prove:
~A & A |- B
You also use quantifiers, basically its QSAT
of this here:
∀A∀B(~A & A |- B)
Its not that you showed there are 8 bananas
on the table and there are 3 apples on the table,
i.e. true & true. You used so called "propositional
variables", and since you showed a tautology
you used so called "propositional quantifiers".
But where do you state that your universe is
non-empty? In an empty universe:
|- A v ~A
i.e. ~(~A & A) holds. LoL
Mostowski Collapse
Mar 10, 2021, 6:13:59 AM3/10/21
to
And even this here, in an empty universe, with
no true and false, because vacously true:
|- ~A & A.
no true and false, because vacously true:
|- ~A & A.
Mostowski Collapse
Mar 10, 2021, 6:57:54 AM3/10/21
to
But next to vacously true, there could be other "logical illusions
of natural language", LoL. I find, quite a long wikipedia article BTW:
Counterfactual conditional
https://en.wikipedia.org/wiki/Counterfactual_conditional
It contains original research from Donald Trump, aka Anita Recount:
"Fake aspect
Fake aspect often accompanies fake tense in languages that mark
aspect. In some languages (e.g. Modern Greek, Zulu, and the
Romance languages) this fake aspect is imperfective."
of natural language", LoL. I find, quite a long wikipedia article BTW:
Counterfactual conditional
https://en.wikipedia.org/wiki/Counterfactual_conditional
It contains original research from Donald Trump, aka Anita Recount:
"Fake aspect
Fake aspect often accompanies fake tense in languages that mark
aspect. In some languages (e.g. Modern Greek, Zulu, and the
Romance languages) this fake aspect is imperfective."
Dan Christensen
Mar 10, 2021, 9:30:58 AM3/10/21
to
On Wednesday, March 10, 2021 at 6:57:54 AM UTC-5, Mostowski Collapse wrote:
> But next to vacously true, there could be other "logical illusions
> of natural language", LoL. I find, quite a long wikipedia article BTW:
>
> Counterfactual conditional
> https://en.wikipedia.org/wiki/Counterfactual_conditional
>
Propositional logic deals with propositions that are (present tense) unambiguously either true or false. It is very useful in math, science and engineering. Counterfactual logic, not so much. It is probably best left to philosophers.
> But next to vacously true, there could be other "logical illusions
> of natural language", LoL. I find, quite a long wikipedia article BTW:
>
> Counterfactual conditional
> https://en.wikipedia.org/wiki/Counterfactual_conditional
>
Visit my Math Blog at http://www.dcproof.wordpress.com
Dan Christensen
Mar 10, 2021, 9:34:03 AM3/10/21
to
On Wednesday, March 10, 2021 at 2:37:49 AM UTC-5, Mostowski Collapse wrote:
> So you assume existence of a set {true, false} ?
Not necessary.
> So you assume existence of a set {true, false} ?
Visit my Math Blog at http://www.dcproof.wordpress.com
Message has been deleted
Dan Christensen
Mar 10, 2021, 12:24:12 PM3/10/21
to
https://www.quora.com/Is-language-processing-in-the-brain-subject-to-logical-illusions/answer/Dan-Christensen-8
Dan
FredJeffries
Mar 10, 2021, 12:53:06 PM3/10/21
to
Dan Christensen
Mar 10, 2021, 1:42:19 PM3/10/21
to
On Wednesday, March 10, 2021 at 12:53:06 PM UTC-5, FredJeffries wrote:
> https://en.wikipedia.org/wiki/Procrustes
I take it you would favour a new form of geometry that would compensate for the effects any OPTICAL illusions. If most people, when polled, say Line A is longer than Line B in some "optical illusion," it must somehow be true, right?
Dan
> https://en.wikipedia.org/wiki/Procrustes
I take it you would favour a new form of geometry that would compensate for the effects any OPTICAL illusions. If most people, when polled, say Line A is longer than Line B in some "optical illusion," it must somehow be true, right?
Dan
Peter
Mar 10, 2021, 2:09:31 PM3/10/21
to
I don't think you will.
Dan Christensen
Mar 10, 2021, 4:18:56 PM3/10/21
to
Visit my Math Blog at http://www.dcproof.wordpress.com
Peter
Mar 10, 2021, 5:22:15 PM3/10/21
to
Dan Christensen
Mar 10, 2021, 10:02:16 PM3/10/21
to
You must has seen that optical illusion with two lines, one seeming to be longer than the other. When you measure the two lines with a ruler, you can verify that, despite appearances, they are both the same length.
Just because most people seeing that illusion for the first time judge one line to be longer than the other does not mean it is true.. There also seem to be "illusions" in the case in language processing. We have both logical and optical illusions.
Peter
Mar 11, 2021, 4:49:54 AM3/11/21
to
used to convey a meaning other than that of the arrow of classical
logic. But I think I may have been wrong, and I am unable to decide between
i) you really do think that every if-then in natural language is really
a ->, and those who don't use it that way are making an error; and
ii) you don't think that, but you pretend to.
Which is it?
> When you measure the two lines with a ruler, you can verify that,
> despite appearances, they are both the same length.
>
> Just because most people seeing that illusion for the first time
> judge one line to be longer than the other does not mean it is true..
> There also seem to be "illusions" in the case in language processing.
> We have both logical and optical illusions.
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com Visit my
> Math Blog at http://www.dcproof.wordpress.com
>
Dan Christensen
Mar 11, 2021, 9:41:01 AM3/11/21
to
On Thursday, March 11, 2021 at 4:49:54 AM UTC-5, Peter wrote:
> >> You quote Wikipedia as saying
> >>
> >> the *natural language statement* "If 8 is odd then 3 is prime" is
> >> typically judged false.
> >>
> >
> > ERRONENOUSLY judged false IMHO!
> >
> > You must has seen that optical illusion with two lines, one seeming
> > to be longer than the other.
> I have. And I was just about to write that you must have seen if-then
> used to convey a meaning other than that of the arrow of classical
> logic. But I think I may have been wrong, and I am unable to decide between
> i) you really do think that every if-then in natural language is really
> a ->
Certainly when we are dealing with logical propositions that are (present tense) unambiguously either true or false. Otherwise, maybe not.
> >> You quote Wikipedia as saying
> >>
> >> the *natural language statement* "If 8 is odd then 3 is prime" is
> >> typically judged false.
> >>
> >
> > ERRONENOUSLY judged false IMHO!
> >
> > You must has seen that optical illusion with two lines, one seeming
> > to be longer than the other.
> I have. And I was just about to write that you must have seen if-then
> used to convey a meaning other than that of the arrow of classical
> logic. But I think I may have been wrong, and I am unable to decide between
> i) you really do think that every if-then in natural language is really
> a ->
Peter
Mar 11, 2021, 12:17:50 PM3/11/21
to
Dan Christensen wrote:
> On Thursday, March 11, 2021 at 4:49:54 AM UTC-5, Peter wrote:
>
>>>> You quote Wikipedia as saying
>>>>
>>>> the *natural language statement* "If 8 is odd then 3 is prime" is
>>>> typically judged false.
>>>>
>>>
>>> ERRONENOUSLY judged false IMHO!
>>>
>>> You must has seen that optical illusion with two lines, one seeming
>>> to be longer than the other.
>
>> I have. And I was just about to write that you must have seen if-then
>> used to convey a meaning other than that of the arrow of classical
>> logic. But I think I may have been wrong, and I am unable to decide between
>> i) you really do think that every if-then in natural language is really
>> a ->
>
> Certainly when we are dealing with logical propositions that are (present tense) unambiguously either true or false. Otherwise, maybe not.
>
> Dan
How about: "If I put a knob of butter in my hand it will melt."? Maybe
> On Thursday, March 11, 2021 at 4:49:54 AM UTC-5, Peter wrote:
>
>>>> You quote Wikipedia as saying
>>>>
>>>> the *natural language statement* "If 8 is odd then 3 is prime" is
>>>> typically judged false.
>>>>
>>>
>>> ERRONENOUSLY judged false IMHO!
>>>
>>> You must has seen that optical illusion with two lines, one seeming
>>> to be longer than the other.
>
>> I have. And I was just about to write that you must have seen if-then
>> used to convey a meaning other than that of the arrow of classical
>> logic. But I think I may have been wrong, and I am unable to decide between
>> i) you really do think that every if-then in natural language is really
>> a ->
>
> Certainly when we are dealing with logical propositions that are (present tense) unambiguously either true or false. Otherwise, maybe not.
>
> Dan
it's not logical, which leads me to this question: what is a logical
proposition?
Dan Christensen
Mar 11, 2021, 12:31:34 PM3/11/21
to
On Thursday, March 11, 2021 at 12:17:50 PM UTC-5, Peter wrote:
> Dan Christensen wrote:
> > On Thursday, March 11, 2021 at 4:49:54 AM UTC-5, Peter wrote:
> >
> >>>> You quote Wikipedia as saying
> >>>>
> >>>> the *natural language statement* "If 8 is odd then 3 is prime" is
> >>>> typically judged false.
> >>>>
> >>>
> >>> ERRONENOUSLY judged false IMHO!
> >>>
> >>> You must has seen that optical illusion with two lines, one seeming
> >>> to be longer than the other.
> >
> >> I have. And I was just about to write that you must have seen if-then
> >> used to convey a meaning other than that of the arrow of classical
> >> logic. But I think I may have been wrong, and I am unable to decide between
> >> i) you really do think that every if-then in natural language is really
> >> a ->
> >
> > Certainly when we are dealing with logical propositions that are (present tense) unambiguously either true or false. Otherwise, maybe not.
> >
> > Dan
> How about: "If I put a knob of butter in my hand it will melt."? Maybe
> it's not logical, which leads me to this question: what is a logical
> proposition?
For the purposes of this discussion, by a logical proposition, I mean a statement that at the moment is unambiguously either true or false.
> Dan Christensen wrote:
> > On Thursday, March 11, 2021 at 4:49:54 AM UTC-5, Peter wrote:
> >
> >>>> You quote Wikipedia as saying
> >>>>
> >>>> the *natural language statement* "If 8 is odd then 3 is prime" is
> >>>> typically judged false.
> >>>>
> >>>
> >>> ERRONENOUSLY judged false IMHO!
> >>>
> >>> You must has seen that optical illusion with two lines, one seeming
> >>> to be longer than the other.
> >
> >> I have. And I was just about to write that you must have seen if-then
> >> used to convey a meaning other than that of the arrow of classical
> >> logic. But I think I may have been wrong, and I am unable to decide between
> >> i) you really do think that every if-then in natural language is really
> >> a ->
> >
> > Certainly when we are dealing with logical propositions that are (present tense) unambiguously either true or false. Otherwise, maybe not.
> >
> > Dan
> How about: "If I put a knob of butter in my hand it will melt."? Maybe
> it's not logical, which leads me to this question: what is a logical
> proposition?
Peter
Mar 11, 2021, 12:56:49 PM3/11/21
to
Dan Christensen wrote:
> On Thursday, March 11, 2021 at 12:17:50 PM UTC-5, Peter wrote:
>> Dan Christensen wrote:
>>> On Thursday, March 11, 2021 at 4:49:54 AM UTC-5, Peter wrote:
>>>
>>>>>> You quote Wikipedia as saying
>>>>>>
>>>>>> the *natural language statement* "If 8 is odd then 3 is prime" is
>>>>>> typically judged false.
>>>>>>
>>>>>
>>>>> ERRONENOUSLY judged false IMHO!
>>>>>
>>>>> You must has seen that optical illusion with two lines, one seeming
>>>>> to be longer than the other.
>>>
>>>> I have. And I was just about to write that you must have seen if-then
>>>> used to convey a meaning other than that of the arrow of classical
>>>> logic. But I think I may have been wrong, and I am unable to decide between
>>>> i) you really do think that every if-then in natural language is really
>>>> a ->
>>>
>>> Certainly when we are dealing with logical propositions that are (present tense) unambiguously either true or false. Otherwise, maybe not.
>>>
>>> Dan
>> How about: "If I put a knob of butter in my hand it will melt."? Maybe
>> it's not logical, which leads me to this question: what is a logical
>> proposition?
>
> For the purposes of this discussion, by a logical proposition, I mean a statement that at the moment is unambiguously either true or false.
>
Is "If I put a knob of butter in my hand it will melt." one, and if it
> On Thursday, March 11, 2021 at 12:17:50 PM UTC-5, Peter wrote:
>> Dan Christensen wrote:
>>> On Thursday, March 11, 2021 at 4:49:54 AM UTC-5, Peter wrote:
>>>
>>>>>> You quote Wikipedia as saying
>>>>>>
>>>>>> the *natural language statement* "If 8 is odd then 3 is prime" is
>>>>>> typically judged false.
>>>>>>
>>>>>
>>>>> ERRONENOUSLY judged false IMHO!
>>>>>
>>>>> You must has seen that optical illusion with two lines, one seeming
>>>>> to be longer than the other.
>>>
>>>> I have. And I was just about to write that you must have seen if-then
>>>> used to convey a meaning other than that of the arrow of classical
>>>> logic. But I think I may have been wrong, and I am unable to decide between
>>>> i) you really do think that every if-then in natural language is really
>>>> a ->
>>>
>>> Certainly when we are dealing with logical propositions that are (present tense) unambiguously either true or false. Otherwise, maybe not.
>>>
>>> Dan
>> How about: "If I put a knob of butter in my hand it will melt."? Maybe
>> it's not logical, which leads me to this question: what is a logical
>> proposition?
>
> For the purposes of this discussion, by a logical proposition, I mean a statement that at the moment is unambiguously either true or false.
>
is, how do you formalize that "if"?
Mostowski Collapse
Mar 11, 2021, 1:12:25 PM3/11/21
to
Maybe such LOGICAL Illusions in Natural Language are
best left to philosophers and not to math bloggers.
LoL
Dan Christensen schrieb am Dienstag, 9. März 2021 um 17:31:23 UTC+1:
best left to philosophers and not to math bloggers.
LoL
Dan Christensen schrieb am Dienstag, 9. März 2021 um 17:31:23 UTC+1:
Dan Christensen
Mar 11, 2021, 1:48:54 PM3/11/21
to
On Thursday, March 11, 2021 at 12:56:49 PM UTC-5, Peter wrote:
> Dan Christensen wrote:
> > On Thursday, March 11, 2021 at 12:17:50 PM UTC-5, Peter wrote:
> >> Dan Christensen wrote:
> >>> On Thursday, March 11, 2021 at 4:49:54 AM UTC-5, Peter wrote:
> >>>
> >>>>>> You quote Wikipedia as saying
> >>>>>>
> >>>>>> the *natural language statement* "If 8 is odd then 3 is prime" is
> >>>>>> typically judged false.
> >>>>>>
> >>>>>
> >>>>> ERRONENOUSLY judged false IMHO!
> >>>>>
> >>>>> You must has seen that optical illusion with two lines, one seeming
> >>>>> to be longer than the other.
> >>>
> >>>> I have. And I was just about to write that you must have seen if-then
> >>>> used to convey a meaning other than that of the arrow of classical
> >>>> logic. But I think I may have been wrong, and I am unable to decide between
> >>>> i) you really do think that every if-then in natural language is really
> >>>> a ->
> >>>
> >>> Certainly when we are dealing with logical propositions that are (present tense) unambiguously either true or false. Otherwise, maybe not.
> >>>
> >>> Dan
> >> How about: "If I put a knob of butter in my hand it will melt."? Maybe
> >> it's not logical, which leads me to this question: what is a logical
> >> proposition?
> >
> > For the purposes of this discussion, by a logical proposition, I mean a statement that at the moment is unambiguously either true or false.
> >
> Is "If I put a knob of butter in my hand it will melt." one, and if it
> is, how do you formalize that "if"?
Predictions about the future are always ambiguous in the present. Can you restate it using the present tense?
> Dan Christensen wrote:
> > On Thursday, March 11, 2021 at 12:17:50 PM UTC-5, Peter wrote:
> >> Dan Christensen wrote:
> >>> On Thursday, March 11, 2021 at 4:49:54 AM UTC-5, Peter wrote:
> >>>
> >>>>>> You quote Wikipedia as saying
> >>>>>>
> >>>>>> the *natural language statement* "If 8 is odd then 3 is prime" is
> >>>>>> typically judged false.
> >>>>>>
> >>>>>
> >>>>> ERRONENOUSLY judged false IMHO!
> >>>>>
> >>>>> You must has seen that optical illusion with two lines, one seeming
> >>>>> to be longer than the other.
> >>>
> >>>> I have. And I was just about to write that you must have seen if-then
> >>>> used to convey a meaning other than that of the arrow of classical
> >>>> logic. But I think I may have been wrong, and I am unable to decide between
> >>>> i) you really do think that every if-then in natural language is really
> >>>> a ->
> >>>
> >>> Certainly when we are dealing with logical propositions that are (present tense) unambiguously either true or false. Otherwise, maybe not.
> >>>
> >>> Dan
> >> How about: "If I put a knob of butter in my hand it will melt."? Maybe
> >> it's not logical, which leads me to this question: what is a logical
> >> proposition?
> >
> > For the purposes of this discussion, by a logical proposition, I mean a statement that at the moment is unambiguously either true or false.
> >
> Is "If I put a knob of butter in my hand it will melt." one, and if it
> is, how do you formalize that "if"?
My favourite example is: "If it raining, then it is cloudy." This is false at the moment iff is raining and not cloudy. It is true at the moment iff (1) it is raining and cloudy, or (2) it is not raining.
Mostowski Collapse
Mar 11, 2021, 1:50:44 PM3/11/21
to
For example take equality = between two Cauchy sequences
s=(sj) and t=(tj) defined as follows:
s = t :<=> lim n->oo (sn-tn) = 0
Is this vacously true constructively?
x = y & x ≠ y -> P
s=(sj) and t=(tj) defined as follows:
s = t :<=> lim n->oo (sn-tn) = 0
Is this vacously true constructively?
x = y & x ≠ y -> P
Message has been deleted
Dan Christensen
Mar 11, 2021, 2:00:14 PM3/11/21
to
On Thursday, March 11, 2021 at 1:12:25 PM UTC-5, Mostowski Collapse wrote:
> Maybe such LOGICAL Illusions in Natural Language are
> best left to philosophers and not to math bloggers.
>
I just think it is misleading to suggest that we need more than one kind of implication in classical logic. If we are talking about logical propositions that are, at the moment, unambiguously either true or false, then you need only what is called material implication.
> Maybe such LOGICAL Illusions in Natural Language are
> best left to philosophers and not to math bloggers.
>
Visit my Math Blog at http://www.dcproof.wordpress.com
Dan Christensen
Mar 11, 2021, 2:12:39 PM3/11/21
to
On Thursday, March 11, 2021 at 1:50:44 PM UTC-5, Mostowski Collapse wrote:
> For example take equality = between two Cauchy sequences
> s=(sj) and t=(tj) defined as follows:
>
> s = t :<=> lim n->oo (sn-tn) = 0
>
> Is this vacously true constructively?
>
> x = y & x ≠ y -> P
In DC Proof:
> For example take equality = between two Cauchy sequences
> s=(sj) and t=(tj) defined as follows:
>
> s = t :<=> lim n->oo (sn-tn) = 0
>
> Is this vacously true constructively?
>
> x = y & x ≠ y -> P
1 x=y & ~x=y
Premise
2 ~P
Premise
3 x=y & ~x=y
Copy, 1
4 ~~P
Conclusion, 2
5 P
Rem DNeg, 4
6 ALL(x):ALL(y):[x=y & ~x=y => P]
Conclusion, 1
If you don't like the quantifiers, you can introduce x and y as constants in an axiom at the the beginning.
Visit my Math Blog at http://www.dcproof.wordpress.com
Mostowski Collapse
Mar 11, 2021, 2:20:20 PM3/11/21
to
And whats "constructively" about it?
Mostowski Collapse
Mar 11, 2021, 2:39:42 PM3/11/21
to
If we take P = ⊥, x = 0, then you just proved:
0 = y v 0 ≠ y
Now assume:
y = 0.y1y2....
where yj = 0 if all even integer <j are sums of two primes otherwise 1
If your proof 0 = y v 0 ≠ y were constructive then it would either prove the
Goldbach Conjecture or give an even integer not the sum of two prime.
See also:
Constructive Mathenatics - Bridges
https://core.ac.uk/download/pdf/82492373.pdf
0 = y v 0 ≠ y
Now assume:
y = 0.y1y2....
where yj = 0 if all even integer <j are sums of two primes otherwise 1
If your proof 0 = y v 0 ≠ y were constructive then it would either prove the
Goldbach Conjecture or give an even integer not the sum of two prime.
See also:
Constructive Mathenatics - Bridges
https://core.ac.uk/download/pdf/82492373.pdf
Peter
Mar 11, 2021, 2:53:25 PM3/11/21
to
> My favourite example is: "If it raining, then it is cloudy." This is false at the moment iff is raining and not cloudy. It is true at the moment iff (1) it is raining and cloudy, or (2) it is not raining.
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>
Dan Christensen
Mar 11, 2021, 4:23:44 PM3/11/21
to
H = a knob of butter is in my hand
M = a knob of butter is melting in my hand
Then we have: H=>M <=> ~[H & ~M].
Now what?
Mostowski Collapse
Mar 17, 2021, 4:06:58 PM3/17/21
to
Ha Ha, LOGAX compared to Metamath
Generation and Use of Hints and Feedback in a Hilbert-Style Axiomatic Proof Tutor
Josje Lodder1 · Bastiaan Heeren1 · Johan Jeuring1,2 · Wendy Neijenhuis1
Accepted: 27 September 2020
International Journal of Artificial Intelligence in Education
https://link.springer.com/content/pdf/10.1007/s40593-020-00222-2.pdf
How does DC Proof compare?
Generation and Use of Hints and Feedback in a Hilbert-Style Axiomatic Proof Tutor
Josje Lodder1 · Bastiaan Heeren1 · Johan Jeuring1,2 · Wendy Neijenhuis1
Accepted: 27 September 2020
International Journal of Artificial Intelligence in Education
https://link.springer.com/content/pdf/10.1007/s40593-020-00222-2.pdf
How does DC Proof compare?
Dan Christensen
Mar 17, 2021, 9:53:45 PM3/17/21
to
On Wednesday, March 17, 2021 at 4:06:58 PM UTC-4, Mostowski Collapse wrote:
> Ha Ha, LOGAX compared to Metamath
>
> Generation and Use of Hints and Feedback in a Hilbert-Style Axiomatic Proof Tutor
What a waste. Who uses Hilbert-style axioms in proofs? No one who writes proofs for a living.
> Ha Ha, LOGAX compared to Metamath
>
> Generation and Use of Hints and Feedback in a Hilbert-Style Axiomatic Proof Tutor
Dan
Mostowski Collapse
Mar 18, 2021, 6:46:29 AM3/18/21
to
You also got Hilbert style in DC Proof, you only call your
modus ponens "Detach". Why do you call it "Detach"?
Normally it is called (=>) Elimination:
1 Foo
Axiom
2 Foo => Bar
Axiom
3 Bar
Detach, 2, 1
Here is a screenshot:
Dan-O-Matik doesn't know that Natural Deduction covers Hilbert Style:
https://gist.github.com/jburse/dbd654073621f21333e3e4dbd330a30d#gistcomment-3669482
Dan Christensen lost his marbles, 18. März 2021 um 02:53:45 UTC+1:
> What a waste
Yes, what a waste DC Proof is, acccording to your own logic.
LoL
modus ponens "Detach". Why do you call it "Detach"?
Normally it is called (=>) Elimination:
1 Foo
Axiom
2 Foo => Bar
Axiom
3 Bar
Detach, 2, 1
Here is a screenshot:
Dan-O-Matik doesn't know that Natural Deduction covers Hilbert Style:
https://gist.github.com/jburse/dbd654073621f21333e3e4dbd330a30d#gistcomment-3669482
Dan Christensen lost his marbles, 18. März 2021 um 02:53:45 UTC+1:
> What a waste
Yes, what a waste DC Proof is, acccording to your own logic.
LoL
Dan Christensen
Mar 18, 2021, 9:56:52 AM3/18/21
to
On Thursday, March 18, 2021 at 6:46:29 AM UTC-4, Mostowski Collapse wrote:
> You also got Hilbert style in DC Proof, you only call your
> modus ponens "Detach". Why do you call it "Detach"?
Short for Detachment used in many books in English. Really quite basic stuff, Jan.
> You also got Hilbert style in DC Proof, you only call your
> modus ponens "Detach". Why do you call it "Detach"?
See https://www.basic-mathematics.com/law-of-detachment.html
> Normally it is called (=>) Elimination:
>
> 1 Foo
> Axiom
>
> 2 Foo => Bar
> Axiom
>
> 3 Bar
> Detach, 2, 1
>
> Here is a screenshot:
>
> Dan-O-Matik doesn't know that Natural Deduction covers Hilbert Style
[A => [B => C]] => [[A => B] => [A => C]]
Why is that an axiom? It rarely comes up in practice, as is easily derived using my built-in Premise, Detachment and Conclusion Rules for the few times it might come up. No wonder his system didn't catch on in mathematics.
Mostowski Collapse
Mar 18, 2021, 10:09:46 AM3/18/21
to
You confuse Hilberts **system** with Hilbert **style system**.
Hilbert style system is a family of calculi.
https://en.wikipedia.org/wiki/List_of_Hilbert_systems
You cannot derive it in DC Proof? This here?
[A => [B => C]] => [[A => B] => [A => C]] (S)
Hilbert style doesnt mean you use axiom schema (S). Frege
also used axiom schema (S). This is nothing specific to Hilbert.
Hilbert **style system** only means you have a set of axiom A, can be
the instances of (S) or anything else. These axioms all mean you can
derive |- A. And then you have modus ponens:
|- A |- A -> B
-------------------------------
|- B
You confuse Hilberts system with Hilbert style system.
Hilbert style system is a family of calculi.
https://en.wikipedia.org/wiki/List_of_Hilbert_systems
You cannot derive it in DC Proof? This here?
[A => [B => C]] => [[A => B] => [A => C]] (S)
Hilbert style doesnt mean you use axiom schema (S). Frege
also used axiom schema (S). This is nothing specific to Hilbert.
Hilbert **style system** only means you have a set of axiom A, can be
the instances of (S) or anything else. These axioms all mean you can
derive |- A. And then you have modus ponens:
|- A |- A -> B
-------------------------------
|- B
You confuse Hilberts system with Hilbert style system.
Dan Christensen
Mar 18, 2021, 10:18:45 AM3/18/21
to
On Thursday, March 18, 2021 at 10:09:46 AM UTC-4, Mostowski Collapse wrote:
> You confuse Hilberts **system** with Hilbert **style system**.
> Hilbert style system is a family of calculi.
>
> https://en.wikipedia.org/wiki/List_of_Hilbert_systems
>
See: http://www.diag.uniroma1.it/liberato/planning/hilbert/hilbert.html
> You confuse Hilberts **system** with Hilbert **style system**.
> Hilbert style system is a family of calculi.
>
> https://en.wikipedia.org/wiki/List_of_Hilbert_systems
>
> You cannot derive it in DC Proof? This here?
>
> [A => [B => C]] => [[A => B] => [A => C]] (S)
>
Premise
2 A => B
Premise
3 A
Premise
4 B
Detach, 2, 3
5 B => C
Detach, 1, 3
6 C
Detach, 5, 4
7 A => C
Conclusion, 3
8 A => B => [A => C]
Conclusion, 2
9 A => [B => C] => [A => B => [A => C]]
Conclusion, 1
No need to have this as an axiom.
Mostowski Collapse
Mar 18, 2021, 10:23:01 AM3/18/21
to
The paper explicitily mentions Hilbert **style**, when
it says in the intro. "student who stepwise constructs a
Hilbert-style axiomatic proof in propositional logic"
But you did read the paper past a flys leg. They later also
discuss "Bolotov’s algorithm for natural deduction proofs"
I admit their paper title isn't wise.
Generation and Use of Hints and Feedback in a Hilbert-Style Axiomatic Proof Tutor
it says in the intro. "student who stepwise constructs a
Hilbert-style axiomatic proof in propositional logic"
But you did read the paper past a flys leg. They later also
discuss "Bolotov’s algorithm for natural deduction proofs"
I admit their paper title isn't wise.
Generation and Use of Hints and Feedback in a Hilbert-Style Axiomatic Proof Tutor
Josje Lodder1 · Bastiaan Heeren1 · Johan Jeuring1,2 · Wendy Neijenhuis1
Accepted: 27 September 2020
International Journal of Artificial Intelligence in Education
https://link.springer.com/content/pdf/10.1007/s40593-020-00222-2.pdf
Accepted: 27 September 2020
International Journal of Artificial Intelligence in Education
https://link.springer.com/content/pdf/10.1007/s40593-020-00222-2.pdf
Mostowski Collapse
Mar 18, 2021, 10:29:29 AM3/18/21
to
You make the same error again when you post this link:
http://www.diag.uniroma1.it/liberato/planning/hilbert/hilbert.html
The link says "**An** Hilbert system". So its a family of systems
and Dan-O-Matik thinks its THE Hilbert style system.
What are you, a moron?
You can also use this Hilbert style system:
"B": (B → C) → ((A → B) → (A → C)),
"C": (A → (B → C)) → (B → (A → C)),
"K": A → (B → A),
"W": (A → (A → B)) → (A → B).
No S rule.
http://www.diag.uniroma1.it/liberato/planning/hilbert/hilbert.html
The link says "**An** Hilbert system". So its a family of systems
and Dan-O-Matik thinks its THE Hilbert style system.
What are you, a moron?
You can also use this Hilbert style system:
"B": (B → C) → ((A → B) → (A → C)),
"C": (A → (B → C)) → (B → (A → C)),
"K": A → (B → A),
"W": (A → (A → B)) → (A → B).
No S rule.
Peter
Mar 18, 2021, 11:04:31 AM3/18/21
to
Dan Christensen wrote:
> On Thursday, March 18, 2021 at 6:46:29 AM UTC-4, Mostowski Collapse
> wrote:
>> You also got Hilbert style in DC Proof, you only call your modus
>> ponens "Detach". Why do you call it "Detach"?
>
> Short for Detachment used in many books in English. Really quite
> basic stuff, Jan.
>
> See https://www.basic-mathematics.com/law-of-detachment.html
>
>
>> Normally it is called (=>) Elimination:
>>
>> 1 Foo Axiom
>>
>> 2 Foo => Bar Axiom
>>
>> 3 Bar Detach, 2, 1
>>
>> Here is a screenshot:
>>
>> Dan-O-Matik doesn't know that Natural Deduction covers Hilbert
>> Style
>
> Of course, the Hilbert axioms are derivable in DC Proof, but there is
> no built-in rule for his axiom:
>
> [A => [B => C]] => [[A => B] => [A => C]]
>
> Why is that an axiom? It rarely comes up in practice, as is easily
> derived using my built-in Premise, Detachment and Conclusion Rules
That's why it's not an axiom in your system, but in one that doesn't
> On Thursday, March 18, 2021 at 6:46:29 AM UTC-4, Mostowski Collapse
> wrote:
>> You also got Hilbert style in DC Proof, you only call your modus
>> ponens "Detach". Why do you call it "Detach"?
>
> Short for Detachment used in many books in English. Really quite
> basic stuff, Jan.
>
> See https://www.basic-mathematics.com/law-of-detachment.html
>
>
>> Normally it is called (=>) Elimination:
>>
>> 1 Foo Axiom
>>
>> 2 Foo => Bar Axiom
>>
>> 3 Bar Detach, 2, 1
>>
>> Here is a screenshot:
>>
>> Dan-O-Matik doesn't know that Natural Deduction covers Hilbert
>> Style
>
> Of course, the Hilbert axioms are derivable in DC Proof, but there is
> no built-in rule for his axiom:
>
> [A => [B => C]] => [[A => B] => [A => C]]
>
> Why is that an axiom? It rarely comes up in practice, as is easily
> derived using my built-in Premise, Detachment and Conclusion Rules
have your rules but uses if-then as a primitive connective it is a
perfectly respectable axiom.
> for the few times it might come up. No wonder his system didn't catch
> on in mathematics.
dismissive attitude. Here are some well-known(but see footnote) places
where it is used or mentioned -
Mendelson, /Introduction to mathematical logic/, page 27 of the fifth
edition.
Frege, /Begriffsschrift/, various editions and translations.
Lukasiewicz, referred to in the first appendix of Prior, /Formal logic/,
OUP.
I would suppose that it was Frege's use of the axiom that inspired
others to use it. To say it didn't catch on is silly.
Footnote - Well-known, that is, to those with a moderate interest in
logic. One should not expect those who are unimaginative, dogmatic and
narrow-minded to know of them.
Dan Christensen
Mar 18, 2021, 12:33:02 PM3/18/21
to
On Thursday, March 18, 2021 at 11:04:31 AM UTC-4, Peter wrote:
> >
> > Of course, the Hilbert axioms are derivable in DC Proof, but there is
> > no built-in rule for his axiom:
> >
> > [A => [B => C]] => [[A => B] => [A => C]]
> >
> > Why is that an axiom? It rarely comes up in practice, as is easily
> > derived using my built-in Premise, Detachment and Conclusion Rules
> That's why it's not an axiom in your system, but in one that doesn't
> have your rules but uses if-then as a primitive connective it is a
> perfectly respectable axiom.
You can concoct any axioms you like. Some are just more useful in practice than others.
> >
> > Of course, the Hilbert axioms are derivable in DC Proof, but there is
> > no built-in rule for his axiom:
> >
> > [A => [B => C]] => [[A => B] => [A => C]]
> >
> > Why is that an axiom? It rarely comes up in practice, as is easily
> > derived using my built-in Premise, Detachment and Conclusion Rules
> That's why it's not an axiom in your system, but in one that doesn't
> have your rules but uses if-then as a primitive connective it is a
> perfectly respectable axiom.
Mostowski Collapse
Mar 18, 2021, 1:48:48 PM3/18/21
to
You don't use axioms in your system. So what do you know
about axioms for classical propositional logic?
You just post dozen times some dubious claims about
material implication, reducible to DC Proof.
Never occured to you that there also other reductions?
about axioms for classical propositional logic?
You just post dozen times some dubious claims about
material implication, reducible to DC Proof.
Never occured to you that there also other reductions?
Dan Christensen
Mar 18, 2021, 2:12:39 PM3/18/21
to
On Thursday, March 18, 2021 at 1:48:48 PM UTC-4, Mostowski Collapse wrote:
> Dan Christensen schrieb am Donnerstag, 18. März 2021 um 17:33:02 UTC+1:
> > On Thursday, March 18, 2021 at 11:04:31 AM UTC-4, Peter wrote:
> >
> > > >
> > > > Of course, the Hilbert axioms are derivable in DC Proof, but there is
> > > > no built-in rule for his axiom:
> > > >
> > > > [A => [B => C]] => [[A => B] => [A => C]]
> > > >
> > > > Why is that an axiom? It rarely comes up in practice, as is easily
> > > > derived using my built-in Premise, Detachment and Conclusion Rules
> > > That's why it's not an axiom in your system, but in one that doesn't
> > > have your rules but uses if-then as a primitive connective it is a
> > > perfectly respectable axiom.
> > You can concoct any axioms you like. Some are just more useful in practice than others.
> Dan Christensen schrieb am Donnerstag, 18. März 2021 um 17:33:02 UTC+1:
> > On Thursday, March 18, 2021 at 11:04:31 AM UTC-4, Peter wrote:
> >
> > > >
> > > > Of course, the Hilbert axioms are derivable in DC Proof, but there is
> > > > no built-in rule for his axiom:
> > > >
> > > > [A => [B => C]] => [[A => B] => [A => C]]
> > > >
> > > > Why is that an axiom? It rarely comes up in practice, as is easily
> > > > derived using my built-in Premise, Detachment and Conclusion Rules
> > > That's why it's not an axiom in your system, but in one that doesn't
> > > have your rules but uses if-then as a primitive connective it is a
> > > perfectly respectable axiom.
> > You can concoct any axioms you like. Some are just more useful in practice than others.
> You don't use axioms in your system. So what do you know
> about axioms for classical propositional logic?
>
See the Logic Menu in DC Proof. It includes the axioms of propositional logic, predicated logic and some shortcuts that I have found useful.
> about axioms for classical propositional logic?
>
> You just post dozen times some dubious claims about
> material implication, reducible to DC Proof.
>
> Never occurred to you that there also other reductions?
Sure, but how useful are they in practice?
Mostowski Collapse
Mar 18, 2021, 2:17:54 PM3/18/21
to
"the axioms of propositional logic" nice try. LoL
You are quite a moron, aren't you.
See other thread about truth tables.
You are quite a moron, aren't you.
See other thread about truth tables.
Dan Christensen
Mar 18, 2021, 2:33:42 PM3/18/21
to
On Thursday, March 18, 2021 at 2:17:54 PM UTC-4, Mostowski Collapse wrote:
> "the axioms of propositional logic" nice try. LoL
> You are quite a moron, aren't you.
>
The kind of insightful analysis that we have come to expect from you, Jan Burse.
> "the axioms of propositional logic" nice try. LoL
> You are quite a moron, aren't you.
>
Been taking lessons from AP and JG?
Dan
Peter
Mar 18, 2021, 2:42:34 PM3/18/21
to
Mostowski Collapse wrote:
> You don't use axioms in your system. So what do you know
> about axioms for classical propositional logic?
Given his dismissive remarks about [A => [B => C]] => [[A => B] => [A =>
> You don't use axioms in your system. So what do you know
> about axioms for classical propositional logic?
C]] (which occurs in Frege's Begriffsschrift no less!) Dan knows nothing
about axioms for classical propositional logic. I have noticed that if
something is unknown to him, or of no interest to him, he often treats
it with disdain.
Peter
Mar 18, 2021, 2:46:00 PM3/18/21
to
Dan Christensen wrote:
> On Thursday, March 18, 2021 at 1:48:48 PM UTC-4, Mostowski Collapse
> wrote:
>
>> Dan Christensen schrieb am Donnerstag, 18. März 2021 um 17:33:02
>> UTC+1:
>>> On Thursday, March 18, 2021 at 11:04:31 AM UTC-4, Peter wrote:
>>>
>>>>>
>>>>> Of course, the Hilbert axioms are derivable in DC Proof, but
>>>>> there is no built-in rule for his axiom:
>>>>>
>>>>> [A => [B => C]] => [[A => B] => [A => C]]
>>>>>
>>>>> Why is that an axiom? It rarely comes up in practice, as is
>>>>> easily derived using my built-in Premise, Detachment and
>>>>> Conclusion Rules
>>>> That's why it's not an axiom in your system, but in one that
>>>> doesn't have your rules but uses if-then as a primitive
>>>> connective it is a perfectly respectable axiom.
>
>>> You can concoct any axioms you like. Some are just more useful in
>>> practice than others.
>
>> You don't use axioms in your system. So what do you know about
>> axioms for classical propositional logic?
>>
>
> See the Logic Menu in DC Proof. It includes the axioms of
> propositional logic, predicated logic and some shortcuts that I have
> found useful.
I don't understand _the_ axioms of propositional logic. There are
> On Thursday, March 18, 2021 at 1:48:48 PM UTC-4, Mostowski Collapse
> wrote:
>
>> Dan Christensen schrieb am Donnerstag, 18. März 2021 um 17:33:02
>> UTC+1:
>>> On Thursday, March 18, 2021 at 11:04:31 AM UTC-4, Peter wrote:
>>>
>>>>>
>>>>> Of course, the Hilbert axioms are derivable in DC Proof, but
>>>>> there is no built-in rule for his axiom:
>>>>>
>>>>> [A => [B => C]] => [[A => B] => [A => C]]
>>>>>
>>>>> Why is that an axiom? It rarely comes up in practice, as is
>>>>> easily derived using my built-in Premise, Detachment and
>>>>> Conclusion Rules
>>>> That's why it's not an axiom in your system, but in one that
>>>> doesn't have your rules but uses if-then as a primitive
>>>> connective it is a perfectly respectable axiom.
>
>>> You can concoct any axioms you like. Some are just more useful in
>>> practice than others.
>
>> You don't use axioms in your system. So what do you know about
>> axioms for classical propositional logic?
>>
>
> See the Logic Menu in DC Proof. It includes the axioms of
> propositional logic, predicated logic and some shortcuts that I have
> found useful.
numerous formulations of propositional logic with different axioms.
Whose are yours, or are they your own?
>
>> You just post dozen times some dubious claims about material
>> implication, reducible to DC Proof.
>>
>
> Nothing "dubious" about it, Jan Burse. I have proven that A=>B <=>
> ~[A & ~B] using only the rules for conditional proof (intro =>),
> proof by contradiction (intro ~), join (intro &), split (elim &),
> detachment (elim =>) and removing ~~. See
> http://www.dcproof.com/DeriveImplies.html (only 19 lines)
>
>> Never occurred to you that there also other reductions?
>
> Sure, but how useful are they in practice?
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com Visit my
> Math Blog at http://www.dcproof.wordpress.com
>
>
>
Dan Christensen
Mar 18, 2021, 3:10:04 PM3/18/21
to
On Thursday, March 18, 2021 at 2:46:00 PM UTC-4, Peter wrote:
> > See the Logic Menu in DC Proof. It includes the axioms of
> > propositional logic, predicated logic and some shortcuts that I have
> > found useful.
> I don't understand _the_ axioms of propositional logic. There are
> numerous formulations of propositional logic with different axioms.
> Whose are yours, or are they your own?
They are based largely on my own experience in reading and writing proofs over the years. Nothing very original, I'm sure, but I can't point to any particular technical sources. Hofstadter's "Godel, Escher, Bach" inspired me. I went back to that delightful book years later and was surprised by how much it had influenced the design of DC Proof. I tried to interest him in a collaboration but he had moved on from GEB by that time.
> > See the Logic Menu in DC Proof. It includes the axioms of
> > propositional logic, predicated logic and some shortcuts that I have
> > found useful.
> I don't understand _the_ axioms of propositional logic. There are
> numerous formulations of propositional logic with different axioms.
> Whose are yours, or are they your own?
Whatever works for teaching the basic methods of mathematical proof.
Mostowski Collapse
Mar 18, 2021, 3:30:28 PM3/18/21
to
Your own experience is your own illusion.
Try proving in DC Proof:
(p <=> (q <=> (r <=> s))) <=> (((s <=> r) <=> q) <=> p)
Easy to prove?
Try proving in DC Proof:
(p <=> (q <=> (r <=> s))) <=> (((s <=> r) <=> q) <=> p)
Easy to prove?
Peter
Mar 18, 2021, 4:03:37 PM3/18/21
to
Dan Christensen wrote:
> On Thursday, March 18, 2021 at 2:46:00 PM UTC-4, Peter wrote:
>
>>> See the Logic Menu in DC Proof. It includes the axioms of
>>> propositional logic, predicated logic and some shortcuts that I have
>>> found useful.
>> I don't understand _the_ axioms of propositional logic. There are
>> numerous formulations of propositional logic with different axioms.
>> Whose are yours, or are they your own?
>
> They are based largely on my own experience in reading and writing proofs over the years.
Have you proved your axioms for propositional logic sound and complete?
> On Thursday, March 18, 2021 at 2:46:00 PM UTC-4, Peter wrote:
>
>>> See the Logic Menu in DC Proof. It includes the axioms of
>>> propositional logic, predicated logic and some shortcuts that I have
>>> found useful.
>> I don't understand _the_ axioms of propositional logic. There are
>> numerous formulations of propositional logic with different axioms.
>> Whose are yours, or are they your own?
>
> They are based largely on my own experience in reading and writing proofs over the years.
> Nothing very original, I'm sure, but I can't point to any particular technical sources. Hofstadter's "Godel, Escher, Bach"
lay my hands on it at the moment.
> inspired me. I went back to that delightful book years later and was surprised by how much it had influenced the design of DC Proof. I tried to interest him in a collaboration but he had moved on from GEB by that time.
>
> Whatever works for teaching the basic methods of mathematical proof.
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>
Peter
Mar 18, 2021, 4:11:55 PM3/18/21
to
Hilbert and Bernays, /Grundlagen der Mathematik/
which might have been guessed at, but the historical notes in
Church, /Introduction to mathematical logic/, 1956 edition
confirm it. Church himself uses it in both his P_1 and P_2. I conclude
that the axioms can hardly be dismissed as rarely coming up in practice.
Dan Christensen
Mar 18, 2021, 7:04:12 PM3/18/21
to
On Thursday, March 18, 2021 at 3:30:28 PM UTC-4, Mostowski Collapse wrote:
> Your own experience is your own illusion.
> Try proving in DC Proof:
>
> (p <=> (q <=> (r <=> s))) <=> (((s <=> r) <=> q) <=> p)
>
> Easy to prove?
Very tedious, but doable. If I had thought that the commutativity of <=> (as well as that of & and |) was commonly used in proofs, I would have built in a shortcut. Apart from textbook exercises, I didn't see much call for it. I have built in some shortcuts.
> Your own experience is your own illusion.
> Try proving in DC Proof:
>
> (p <=> (q <=> (r <=> s))) <=> (((s <=> r) <=> q) <=> p)
>
> Easy to prove?
Mostowski Collapse
Mar 19, 2021, 3:48:30 AM3/19/21
to
I dont think DC Proof can do P_1. Since P_1 is built from implication →
and falsum f. But DC Proof doesn't support falsum.
P_1 is Hilbert style system with these axioms:
A → (B → A)
(A → (B → C)) → ((A → B) → (A → C))
((A → f) → f) → A
and falsum f. But DC Proof doesn't support falsum.
P_1 is Hilbert style system with these axioms:
A → (B → A)
(A → (B → C)) → ((A → B) → (A → C))
((A → f) → f) → A
Message has been deleted
Dan Christensen
Mar 19, 2021, 6:51:42 AM3/19/21
to
On Friday, March 19, 2021 at 3:48:30 AM UTC-4, Mostowski Collapse wrote:
> I dont think DC Proof can do P_1. Since P_1 is built from implication →
> and falsum f. But DC Proof doesn't support falsum.
>
You won't see it in any math textbook. Why do you think that is?
> I dont think DC Proof can do P_1. Since P_1 is built from implication →
> and falsum f. But DC Proof doesn't support falsum.
>
> P_1 is Hilbert style system with these axioms:
>
> A → (B → A)
1 A
Premise
2 B => A
Arb Ante, 1
3 A => [B => A]
Conclusion, 1
Also, making use of the Arbitrary Consequent Rule (vacuous truth):
1 A
Premise
2 ~A => B
Arb Cons, 1
3 A => [~A => B]
Conclusion, 1
> (A → (B → C)) → ((A → B) → (A → C))
1 A => [B => C]
Premise
2 A => B
Premise
3 A
Premise
4 B
Detach, 2, 3
5 B => C
Detach, 1, 3
6 C
Detach, 5, 4
7 A => C
Conclusion, 3
8 A => B => [A => C]
Premise
2 A => B
Premise
3 A
Premise
4 B
Detach, 2, 3
5 B => C
Detach, 1, 3
6 C
Detach, 5, 4
7 A => C
Conclusion, 3
Conclusion, 2
9 A => [B => C] => [A => B => [A => C]]
Conclusion, 1
> ((A → f) → f) → A
1 B
Premise
2 A => ~B => ~B
Premise
3 ~~B => ~[A => ~B]
Contra, 2
4 B => ~[A => ~B]
Rem DNeg, 3
5 ~[A => ~B]
Detach, 4, 1
6 ~~[A & ~~B]
Imply-And, 5
7 A & ~~B
Rem DNeg, 6
8 A & B
Rem DNeg, 7
9 A
Split, 8
10 A => ~B => ~B => A
Conclusion, 2
11 B => [A => ~B => ~B => A]
Conclusion, 1
Mostowski Collapse
Mar 19, 2021, 7:16:35 AM3/19/21
to
Not the same. As I said, there is no falsum ⊥ in DC Proof.
DC Proof does not properly support falsum ⊥ . You cannot
do such classical propositional calculi for the language
consisting of only { → , ⊥ } as depicted here:
Implication and falsum
https://en.wikipedia.org/wiki/List_of_Hilbert_systems#Implication_and_falsum
Because your language doesn't have falsum ⊥.
Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
> > ((A → f) → f) → A
DC Proof does not properly support falsum ⊥ . You cannot
do such classical propositional calculi for the language
consisting of only { → , ⊥ } as depicted here:
Implication and falsum
https://en.wikipedia.org/wiki/List_of_Hilbert_systems#Implication_and_falsum
Because your language doesn't have falsum ⊥.
Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
> > ((A → f) → f) → A
Dan Christensen
Mar 19, 2021, 7:42:23 AM3/19/21
to
On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
> > > ((A → f) → f) → A
> > 11 B => [A => ~B => ~B => A]
> > Conclusion, 1
> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
> > > ((A → f) → f) → A
> > 11 B => [A => ~B => ~B => A]
> > Conclusion, 1
> Not the same. As I said, there is no falsum ⊥ in DC Proof.
> DC Proof does not properly support falsum ⊥ .
Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
> DC Proof does not properly support falsum ⊥ .
[snip]
Mostowski Collapse
Mar 19, 2021, 8:20:17 AM3/19/21
to
You can use 0=1 for falsum. I have seen at least one
math book using 0=1 for falsum.
math book using 0=1 for falsum.
Dan Christensen
Mar 19, 2021, 9:04:23 AM3/19/21
to
On Friday, March 19, 2021 at 8:20:17 AM UTC-4, Mostowski Collapse wrote:
> Dan Christensen schrieb am Freitag, 19. März 2021 um 12:42:23 UTC+1:
> > On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
> >
> > > Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
> > > > > ((A → f) → f) → A
> > > > 11 B => [A => ~B => ~B => A]
> > > > Conclusion, 1
> > > Not the same. As I said, there is no falsum ⊥ in DC Proof.
> > > DC Proof does not properly support falsum ⊥ .
> > Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
> >
> You can use 0=1 for falsum. I have seen at least one
> math book using 0=1 for falsum.
Why bother? It would only cause needless confusion. Thanks anyway.
> Dan Christensen schrieb am Freitag, 19. März 2021 um 12:42:23 UTC+1:
> > On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
> >
> > > Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
> > > > > ((A → f) → f) → A
> > > > 11 B => [A => ~B => ~B => A]
> > > > Conclusion, 1
> > > Not the same. As I said, there is no falsum ⊥ in DC Proof.
> > > DC Proof does not properly support falsum ⊥ .
> > Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
> >
> You can use 0=1 for falsum. I have seen at least one
> math book using 0=1 for falsum.
Mostowski Collapse
Mar 19, 2021, 9:07:22 AM3/19/21
to
You are already confused... Like Archie Poo who thinks
there are only up to 10^600 natural numbers. LoL
there are only up to 10^600 natural numbers. LoL
Peter
Mar 19, 2021, 9:13:39 AM3/19/21
to
Dan Christensen wrote:
> On Friday, March 19, 2021 at 3:48:30 AM UTC-4, Mostowski Collapse wrote:
>> I dont think DC Proof can do P_1. Since P_1 is built from implication →
>> and falsum f. But DC Proof doesn't support falsum.
>>
>
> You won't see it in any math textbook. Why do you think that is?
Let us divide maths textbooks into two classes, those that are logic
> On Friday, March 19, 2021 at 3:48:30 AM UTC-4, Mostowski Collapse wrote:
>> I dont think DC Proof can do P_1. Since P_1 is built from implication →
>> and falsum f. But DC Proof doesn't support falsum.
>>
>
> You won't see it in any math textbook. Why do you think that is?
texts and those that aren't. To object to some particular feature of
logic (it might be f, it might be (A -> (B -> C)) -> ((A -> B) -> (A ->
C)) ) because it doesn't appear in any maths books of the second class
is *silly* because no feature of formal propositional logic appears in
them. If you claim that those two features don't appear in any
textbooks you are just exhibiting your *ignorance* because they appear
in many texts of the first class. It may well be that they don't appear
in the one logic text that you've read, but so what? What is it with
you, silliness or ignorance?
Peter
Mar 19, 2021, 9:19:06 AM3/19/21
to
Dan Christensen wrote:
> On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
>
>> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
>>>> ((A → f) → f) → A
>>> 11 B => [A => ~B => ~B => A]
>>> Conclusion, 1
>
>> Not the same. As I said, there is no falsum ⊥ in DC Proof.
>> DC Proof does not properly support falsum ⊥ .
>
> Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
Please cite those maths textbooks in which B => [A => ~B => ~B => A]
> On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
>
>> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
>>>> ((A → f) → f) → A
>>> 11 B => [A => ~B => ~B => A]
>>> Conclusion, 1
>
>> Not the same. As I said, there is no falsum ⊥ in DC Proof.
>> DC Proof does not properly support falsum ⊥ .
>
> Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
appears. Such formulae appear in those maths textbooks which are logic
textbooks, but in them ⊥ and [A => [B => C]] => [[A => B] => [A => C]]
appear too.
If you want to simulate ((A → f) → f) → A the obvious thing to do is to
define f to be ~(p -> p) for some atomic formula p and then prove
((A -> ~(p -> p)) -> ~(p -> p)) -> A.
>
> [snip]
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>
Dan Christensen
Mar 19, 2021, 9:49:34 AM3/19/21
to
On Friday, March 19, 2021 at 9:19:06 AM UTC-4, Peter wrote:
> Dan Christensen wrote:
> > On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
> >
> >> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
> >>>> ((A → f) → f) → A
> >>> 11 B => [A => ~B => ~B => A]
> >>> Conclusion, 1
> >
> >> Not the same. As I said, there is no falsum ⊥ in DC Proof.
> >> DC Proof does not properly support falsum ⊥ .
> >
> > Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
> Please cite those maths textbooks in which B => [A => ~B => ~B => A]
> appears.
Now you are being silly. I never claimed that "B => [A => ~B => ~B => A]" appears in any math textbook. You asked me to prove the equivalent of ((A → f) → f) → A in DC Proof. You got it.
> Dan Christensen wrote:
> > On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
> >
> >> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
> >>>> ((A → f) → f) → A
> >>> 11 B => [A => ~B => ~B => A]
> >>> Conclusion, 1
> >
> >> Not the same. As I said, there is no falsum ⊥ in DC Proof.
> >> DC Proof does not properly support falsum ⊥ .
> >
> > Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
> Please cite those maths textbooks in which B => [A => ~B => ~B => A]
> appears.
Peter
Mar 19, 2021, 11:09:39 AM3/19/21
to
Dan Christensen wrote:
> On Friday, March 19, 2021 at 9:19:06 AM UTC-4, Peter wrote:
>> Dan Christensen wrote:
>>> On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
>>>
>>>> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
>>>>>> ((A → f) → f) → A
>>>>> 11 B => [A => ~B => ~B => A]
>>>>> Conclusion, 1
>>>
>>>> Not the same. As I said, there is no falsum ⊥ in DC Proof.
>>>> DC Proof does not properly support falsum ⊥ .
>>>
>>> Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
>> Please cite those maths textbooks in which B => [A => ~B => ~B => A]
>> appears.
>
> Now you are being silly. I never claimed that "B => [A => ~B => ~B => A]" appears in any math textbook.
You have claimed that it *doesn't* appear in any maths textbook.
> On Friday, March 19, 2021 at 9:19:06 AM UTC-4, Peter wrote:
>> Dan Christensen wrote:
>>> On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
>>>
>>>> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
>>>>>> ((A → f) → f) → A
>>>>> 11 B => [A => ~B => ~B => A]
>>>>> Conclusion, 1
>>>
>>>> Not the same. As I said, there is no falsum ⊥ in DC Proof.
>>>> DC Proof does not properly support falsum ⊥ .
>>>
>>> Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
>> Please cite those maths textbooks in which B => [A => ~B => ~B => A]
>> appears.
>
> Now you are being silly. I never claimed that "B => [A => ~B => ~B => A]" appears in any math textbook.
Similarly f. But if that counts against those then it counts against
all of propositional calculus.
They *do* appear in those maths textbooks that are logic textbooks.
(You haven't seen them? That's because you've only looked in one logic
text.)
You reject vast swathes of logic because, you claim, they don't appear
in maths textbooks. That is true if you exclude logic texts, but then
by the same argument *none* of formal logic appears in maths textbooks
and it should all be rejected. If you include logic textbooks then
*all* you can claim is that you haven't seen those vast swathes in the
one textbook you've read. That doesn't mean that those things haven't
appeared in other logic texts. f appears in Church which is one of the
best-know logic texts (and it appears in many others as well). If you
haven't heard of Church and his text book then you are rather ignorant
about logic. (A -> (B -> C)) -> ((A -> B) -> (A -> C)) appears (and is
made essential use of) in Frege's Begriffsschrift. Begriffsschrift is
widely claimed to be the most important logic text since Aristotle.
That you don't know that just means that you are rather ignorant about
logic.
When will you get out of the habit of thinking that if you haven't heard
of something, it is to be sneered at?
If something doesn't interest you then leave it alone. I have no
interest in any of ballroom dancing, fly fishing and pop music. What I
don't do is post about the little that I know about ballroom dancing,
fly fishing and pop music in ballroom dancing, fly fishing and pop music
newsgroups and dismiss the rest for no better reason that I've not heard
of it. Another thing I don't do is make some fallacious claim in any of
those newsgroups, have my error explained to me, and then go on
repeating the error for the next four hundred million years.
> You asked me to prove the equivalent of ((A → f) → f) → A in DC Proof. You got it.
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>
Peter
Mar 19, 2021, 11:16:26 AM3/19/21
to
Dan Christensen wrote:
> On Friday, March 19, 2021 at 8:20:17 AM UTC-4, Mostowski Collapse wrote:
>
>> Dan Christensen schrieb am Freitag, 19. März 2021 um 12:42:23 UTC+1:
>>> On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
>>>
>>>> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
>>>>>> ((A → f) → f) → A
>>>>> 11 B => [A => ~B => ~B => A]
>>>>> Conclusion, 1
>>>> Not the same. As I said, there is no falsum ⊥ in DC Proof.
>>>> DC Proof does not properly support falsum ⊥ .
>>> Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
>>>
>
>> You can use 0=1 for falsum. I have seen at least one
>> math book using 0=1 for falsum.
>
> Why bother? It would only cause needless confusion. Thanks anyway.
In my minds ear I can hear a lecturer saying "I shall define a constant
> On Friday, March 19, 2021 at 8:20:17 AM UTC-4, Mostowski Collapse wrote:
>
>> Dan Christensen schrieb am Freitag, 19. März 2021 um 12:42:23 UTC+1:
>>> On Friday, March 19, 2021 at 7:16:35 AM UTC-4, Mostowski Collapse wrote:
>>>
>>>> Dan Christensen schrieb am Freitag, 19. März 2021 um 11:51:42 UTC+1:
>>>>>> ((A → f) → f) → A
>>>>> 11 B => [A => ~B => ~B => A]
>>>>> Conclusion, 1
>>>> Not the same. As I said, there is no falsum ⊥ in DC Proof.
>>>> DC Proof does not properly support falsum ⊥ .
>>> Neither does any math textbook I have ever seen. DC Proof was designed to teach the basic methods of MATHEMATICAL proof.
>>>
>
>> You can use 0=1 for falsum. I have seen at least one
>> math book using 0=1 for falsum.
>
> Why bother? It would only cause needless confusion. Thanks anyway.
falsehood, denoted 'f', to be 0=1." A buzz goes round the lecture
theatre - "What's he talking about? 0=1 isn't false. Oh lackaday I am
needlessly confused."
>
> 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
Jan 31, 2023, 2:57:37 AM1/31/23
to
So this was 2021, and now is 2023. Still
Dan Christensen struggles with counter factuals
and proposes vacuous truth.
ChatGPT isn't better off, by way of Fritz:
Fritz Feldhase schrieb:
> No, "P and ~P implies Q" is a contradiction and it is
always false. A contradiction can never imply anything. The
fact that P and ~P cannot both be true at the same time
makes this statement always false.
Spot the error.
Dan Christensen schrieb am Mittwoch, 10. März 2021 um 15:30:58 UTC+1:
> On Wednesday, March 10, 2021 at 6:57:54 AM UTC-5, Mostowski Collapse wrote:
> > But next to vacously true, there could be other "logical illusions
> > of natural language", LoL. I find, quite a long wikipedia article BTW:
> >
> > Counterfactual conditional
> > https://en.wikipedia.org/wiki/Counterfactual_conditional
> >
> Propositional logic deals with propositions that are (present tense) unambiguously either true or false. It is very useful in math, science and engineering. Counterfactual logic, not so much. It is probably best left to philosophers.
Dan Christensen struggles with counter factuals
and proposes vacuous truth.
ChatGPT isn't better off, by way of Fritz:
Fritz Feldhase schrieb:
> No, "P and ~P implies Q" is a contradiction and it is
always false. A contradiction can never imply anything. The
fact that P and ~P cannot both be true at the same time
makes this statement always false.
Spot the error.
Dan Christensen schrieb am Mittwoch, 10. März 2021 um 15:30:58 UTC+1:
> On Wednesday, March 10, 2021 at 6:57:54 AM UTC-5, Mostowski Collapse wrote:
> > But next to vacously true, there could be other "logical illusions
> > of natural language", LoL. I find, quite a long wikipedia article BTW:
> >
> > Counterfactual conditional
> > https://en.wikipedia.org/wiki/Counterfactual_conditional
> >
> Propositional logic deals with propositions that are (present tense) unambiguously either true or false. It is very useful in math, science and engineering. Counterfactual logic, not so much. It is probably best left to philosophers.
Dan Christensen
Jan 31, 2023, 10:33:04 AM1/31/23
to
On Tuesday, January 31, 2023 at 2:57:37 AM UTC-5, Mostowski Collapse wrote:
> So this was 2021, and now is 2023. Still
> Dan Christensen struggles with counter factuals
> and proposes vacuous truth.
>
I see you are still trying to figure out what might have been had unicorns existed. What are your latest findings, Mr. Collapse?
> So this was 2021, and now is 2023. Still
> Dan Christensen struggles with counter factuals
> and proposes vacuous truth.
>
And it seems you are still in denial about vacuous truth, and by extension, the standard truth table for logical implication.
A B A=>B
T T T
T F F
F T T <---------- Antecedent A is false, implication A=>B is true
F F T <---------- Antecedent A is false, implication A=>B is true, regardless of the truth value of B
If you never have to write rigorous mathematical proofs, I guess you can pretty much ignore such considerations.
Julio Di Egidio
Jan 31, 2023, 10:43:32 AM1/31/23
to
On Tuesday, 9 March 2021 at 17:31:23 UTC+1, Dan Christensen wrote:
> In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
>
> What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
>
> "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
> https://en.wikipedia.org/wiki/Material_conditional
>
> Why "erroneously judged?"
Because you are ass then a fraud.
> Using a form of natural deduction based on classical logic,
Where you just insist in missing the point: that material implication
is just *not* what one means when uttering those words in natural
language. Not the other way round.
Write on your wall: Mathematical logic is not Logic.
> Visit my Math Blog a http://www.dcproof.wordpress.com
Dan Christensen, the Pentcho Valev of logic.
Julio
> In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
>
> What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
>
> "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
> https://en.wikipedia.org/wiki/Material_conditional
>
> Why "erroneously judged?"
Because you are ass then a fraud.
> Using a form of natural deduction based on classical logic,
Where you just insist in missing the point: that material implication
is just *not* what one means when uttering those words in natural
language. Not the other way round.
Write on your wall: Mathematical logic is not Logic.
> Visit my Math Blog a http://www.dcproof.wordpress.com
Dan Christensen, the Pentcho Valev of logic.
Julio
Mostowski Collapse
Jan 31, 2023, 11:21:24 AM1/31/23
to
You don't make any sense, why do you list material implication truth tables?
Obviously vacuous truth is not a principle that holds in counterfactual
conditional. Let A => B denote material implication, and let A > B
denote couterfactual conditional:
/* Is a Tautology */
A => (~A => B)
/* is not generally valid */
A > (~A > B)
Whats your prololololoblem? Why do you always list truth tables for
material implication. They wont show couterfactual conditional.
At least not a counterfactual conditional that would
have this definition:
A > B :<=> Bew('A => B')
Where ' ' some Gödel numbering and Bew (german Beweis, english Proof)
is as in Gödels incompletness theorem.
Obviously vacuous truth is not a principle that holds in counterfactual
conditional. Let A => B denote material implication, and let A > B
denote couterfactual conditional:
/* Is a Tautology */
A => (~A => B)
/* is not generally valid */
A > (~A > B)
Whats your prololololoblem? Why do you always list truth tables for
material implication. They wont show couterfactual conditional.
At least not a counterfactual conditional that would
have this definition:
A > B :<=> Bew('A => B')
Where ' ' some Gödel numbering and Bew (german Beweis, english Proof)
is as in Gödels incompletness theorem.
Dan Christensen
Jan 31, 2023, 11:31:39 AM1/31/23
to
On Tuesday, January 31, 2023 at 10:43:32 AM UTC-5, ju...@diegidio.name wrote:
> On Tuesday, 9 March 2021 at 17:31:23 UTC+1, Dan Christensen wrote:
> > In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
> >
> > What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
> >
> > "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
> > https://en.wikipedia.org/wiki/Material_conditional
> >
> > Why "erroneously judged?"
>
[snip childish abuse]
> On Tuesday, 9 March 2021 at 17:31:23 UTC+1, Dan Christensen wrote:
> > In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
> >
> > What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
> >
> > "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
> > https://en.wikipedia.org/wiki/Material_conditional
> >
> > Why "erroneously judged?"
>
>
> > Using a form of natural deduction based on classical logic,
>
> Where you just insist in missing the point: that material implication
> is just *not* what one means when uttering those words in natural
> language. Not the other way round.
>
1. A & [A => B] => B (Rule of Detachment)
2. [A & ~B] => ~[A => B] (Rule of the Counter-example)
3. ~A => [A => B] (Rule of Vacuous Truth)
In natural language, we implicitly make constant use of (1), occasionally we use (2), and rarely if ever (3).
Vacuous truth (3) is pretty much confined to very technical arguments, e.g. mathematical proofs.
I see no need to concoct some other form of logic that will exclude the principle of vacuous proof simply because some may find it somehow offensive.
Dan
Visit my Math Blog at http://www.dcproof.wordpress.com
Dan Christensen
Jan 31, 2023, 11:48:17 AM1/31/23
to
On Tuesday, January 31, 2023 at 11:21:24 AM UTC-5, Mostowski Collapse wrote:
> Dan Christensen schrieb am Dienstag, 31. Januar 2023 um 16:33:04 UTC+1:
> > On Tuesday, January 31, 2023 at 2:57:37 AM UTC-5, Mostowski Collapse wrote:
> > > So this was 2021, and now is 2023. Still
> > > Dan Christensen struggles with counter factuals
> > > and proposes vacuous truth.
> > >
> > I see you are still trying to figure out what might have been had unicorns existed. What are your latest findings, Mr. Collapse?
> >
> > And it seems you are still in denial about vacuous truth, and by extension, the standard truth table for logical implication.
> >
> > A B A=>B
> > T T T
> > T F F
> > F T T <---------- Antecedent A is false, implication A=>B is true
> > F F T <---------- Antecedent A is false, implication A=>B is true, regardless of the truth value of B
> >
> > If you never have to write rigorous mathematical proofs, I guess you can pretty much ignore such considerations.
> Dan Christensen schrieb am Dienstag, 31. Januar 2023 um 16:33:04 UTC+1:
> > On Tuesday, January 31, 2023 at 2:57:37 AM UTC-5, Mostowski Collapse wrote:
> > > So this was 2021, and now is 2023. Still
> > > Dan Christensen struggles with counter factuals
> > > and proposes vacuous truth.
> > >
> > I see you are still trying to figure out what might have been had unicorns existed. What are your latest findings, Mr. Collapse?
> >
> > And it seems you are still in denial about vacuous truth, and by extension, the standard truth table for logical implication.
> >
> > A B A=>B
> > T T T
> > T F F
> > F T T <---------- Antecedent A is false, implication A=>B is true
> > F F T <---------- Antecedent A is false, implication A=>B is true, regardless of the truth value of B
> >
> > If you never have to write rigorous mathematical proofs, I guess you can pretty much ignore such considerations.
> You don't make any sense, why do you list material implication truth tables?
>
> Obviously vacuous truth is not a principle that holds in counterfactual
> conditional.
[snip]
>
> Obviously vacuous truth is not a principle that holds in counterfactual
> conditional.
If you want to determine what might have been had unicorns ever existed, counterfactual conditionals MIGHT be useful. Otherwise, not so much. BTW how is your unicorn project coming along, Mr. Collapse? Maybe AP can help you. (Hee, hee!)
Mostowski Collapse
Jan 31, 2023, 11:55:58 AM1/31/23
to
BTW: To have:
A > (~A > B)
You need a meta-circular Bew() definition, that can
also handle Bew(). What is meta-circular?
See also:
https://en.wikipedia.org/wiki/Meta-circular_evaluator
A > (~A > B)
You need a meta-circular Bew() definition, that can
also handle Bew(). What is meta-circular?
See also:
https://en.wikipedia.org/wiki/Meta-circular_evaluator
Julio Di Egidio
Feb 1, 2023, 4:56:42 AM2/1/23
to
On Tuesday, 31 January 2023 at 17:31:39 UTC+1, Dan Christensen wrote:
<snip>
> Dan
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
Dan Christensen, the nazi-Pentcho Valev of logic.
*Plonk*
Julio
<snip>
> I see no need to concoct some other form of logic that will exclude
> the principle of vacuous proof simply because some may find it
> somehow offensive.
You are simply full of shit.
> the principle of vacuous proof simply because some may find it
> somehow offensive.
> Dan
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
*Plonk*
Julio
Message has been deleted
Dan Christensen
Feb 1, 2023, 9:05:01 AM2/1/23
to
On Tuesday, January 31, 2023 at 11:55:58 AM UTC-5, Mostowski Collapse wrote:
> BTW: To have:
>
> A > (~A > B)
>
> You need a meta-circular Bew() definition, that can
> also handle Bew(). What is meta-circular?
>
Why bother formalizing counterfactuals? What new insights do we gain about the state of the world, past, present or future? AFAICT it is good only for presenting opinions or speculation about the past -- essentially, a non-logical appeal to authority.
> BTW: To have:
>
> A > (~A > B)
>
> You need a meta-circular Bew() definition, that can
> also handle Bew(). What is meta-circular?
>
Dan Christensen
Feb 1, 2023, 10:44:40 AM2/1/23
to
On Wednesday, February 1, 2023 at 4:56:42 AM UTC-5, ju...@diegidio.name wrote:
> On Tuesday, 31 January 2023 at 17:31:39 UTC+1, Dan Christensen wrote:
> <snip>
> > I see no need to concoct some other form of logic that will exclude
> > the principle of vacuous proof simply because some may find it
> > somehow offensive.
> You are simply full [snip childish abuse]
> On Tuesday, 31 January 2023 at 17:31:39 UTC+1, Dan Christensen wrote:
> <snip>
> > I see no need to concoct some other form of logic that will exclude
> > the principle of vacuous proof simply because some may find it
> > somehow offensive.
Just the kind of insightful analysis we have come to expect from you, Julio. (Ha, ha!)
Mostowski Collapse
Feb 1, 2023, 12:23:28 PM2/1/23
to
Still clueless what counterfactuals conditionals are?
Lets say you tell your girlfriend:
If we take the tube to piccadilly we can go to a chines restaurant.
Then your girlfriend responds:
No, at piccadilly we could go to a cinema.
Poor Dan-O-Matik will not understand his girlfriend.
LoL
Lets say you tell your girlfriend:
If we take the tube to piccadilly we can go to a chines restaurant.
Then your girlfriend responds:
No, at piccadilly we could go to a cinema.
Poor Dan-O-Matik will not understand his girlfriend.
LoL
Jeffrey Rubard
Feb 1, 2023, 3:15:58 PM2/1/23
to
On Tuesday, March 9, 2021 at 8:31:23 AM UTC-8, Dan Christensen wrote:
> In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
>
> What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
>
> "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
> https://en.wikipedia.org/wiki/Material_conditional
>
> Why "erroneously judged?" Using a form of natural deduction based on classical logic, we can formally prove that if a logical proposition A is false, then the implication A => B must be true regardless of the truth value of proposition B:
> In school, we learn about optical illusions in science class. We are told that our brains can be tricked. Even when we know how an optical illusion works, the illusion persists. (See examples and their explanations at https://science.howstuffworks.com/optical-illusions.htm)
>
> What about LOGICAL illusions, those that may occur in natural language? From Wikipedia:
>
> "Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd then 3 is prime" is typically [erroneously] judged false." (My comment in [ ]'s.)
> https://en.wikipedia.org/wiki/Material_conditional
>
>
> 1 ~A (Premise, A is false)
>
> 2 A (Premise, A is true)
>
> 3 ~B (Premise, B is false)
>
> 4 ~A & A (Join, 1, 2)
>
> 5 ~~B (Conclusion, proof my contradiction, discharging line 3)
>
> 6 B (Eliminating ~~, line 5)
>
> 7 A => B (Conclusion, conditional proof, discharging line 2)
>
> Why then can we not teach students that our brains may also have tricked us into thinking "If 8 is odd then 3 is prime" is false? By a simple, logical analysis, we can see that it is actually true.
>
> Your comments?
> Visit my Math Blog a http://www.dcproof.wordpress.com
"Logical illusion" is very arguably no part of formal logic, and musing the idea indicates a basic lack of familiarity with logic.
Dan Christensen
Feb 1, 2023, 4:33:03 PM2/1/23
to
On Wednesday, February 1, 2023 at 12:23:28 PM UTC-5, Mostowski Collapse wrote:
> Still clueless what counterfactuals conditionals are?
>
> Lets say you tell your girlfriend:
> If we take the tube to piccadilly we can go to a chines restaurant.
>
> Then your girlfriend responds:
> No, at piccadilly we could go to a cinema.
>
[snip]
> Still clueless what counterfactuals conditionals are?
>
> Lets say you tell your girlfriend:
> If we take the tube to piccadilly we can go to a chines restaurant.
>
> Then your girlfriend responds:
> No, at piccadilly we could go to a cinema.
>
Oddly put. I take it that we could also go to a cinema there. Now, where is the counterfactual in all this?
Dan Christensen
Feb 1, 2023, 4:48:04 PM2/1/23
to
From ordinary propositional logic, we have the following features of logical implication (among others):
2. [A & ~B] => ~[A => B] (Rule of the Counter-example)
3. ~A => [A => B] (Rule of Vacuous Truth)
Visit my Math Blog at http://www.dcproof.wordpress.com
Fritz Feldhase
Feb 1, 2023, 5:32:26 PM2/1/23
to
On Wednesday, February 1, 2023 at 10:48:04 PM UTC+1, Dan Christensen wrote:
> 3. ~A => [A => B] (Rule of Vacuous Truth)
And its "dual" (my wording):
> 3. ~A => [A => B] (Rule of Vacuous Truth)
4. A => (B => A) (Trivial Truth)
The latter is actually an axiom in some systems for propositional logic. (Actually, the first axiom in Frege's system introduced in his /Begriffsschrift/).
Mostowski Collapse
Feb 1, 2023, 7:06:42 PM2/1/23
to
Hint the "No" might indicate the counter factual.
If you want to have both options, she would say:
At piccadilly we could also go to a cinema.
Arent you familiar with the english language?
BTW: How would you formalize:
"piccadilly circus is not actually a circus"
Just joking. LoL
If you want to have both options, she would say:
At piccadilly we could also go to a cinema.
Arent you familiar with the english language?
BTW: How would you formalize:
"piccadilly circus is not actually a circus"
Just joking. LoL
Dan Christensen
Feb 1, 2023, 7:13:32 PM2/1/23
to
1 A
Premise
2 B
3 A
Copy, 1
4 B => A
Conclusion, 2
5 A => [B => A]
Conclusion, 1
If you don't like the Copy Rule (line 3):
1 A
Premise
2 B
3 A & A
Join, 1, 1
4 A
Split, 3
5 B => A
Conclusion, 2
6 A => [B => A]
Conclusion, 1
One extra line required (line 3).
Dan Christensen
Feb 1, 2023, 7:43:21 PM2/1/23
to
On Wednesday, February 1, 2023 at 7:06:42 PM UTC-5, Mostowski Collapse wrote:
> Dan Christensen schrieb am Mittwoch, 1. Februar 2023 um 22:33:03 UTC+1:
> > On Wednesday, February 1, 2023 at 12:23:28 PM UTC-5, Mostowski Collapse wrote:
> > > Still clueless what counterfactuals conditionals are?
> > >
> > > Lets say you tell your girlfriend:
> > > If we take the tube to piccadilly we can go to a chines restaurant.
> > >
> > > Then your girlfriend responds:
> > > No, at piccadilly we could go to a cinema.
> > >
> > [snip]
> >
> > Oddly put. I take it that we could also go to a cinema there. Now, where is the counterfactual in all this?
> Dan Christensen schrieb am Mittwoch, 1. Februar 2023 um 22:33:03 UTC+1:
> > On Wednesday, February 1, 2023 at 12:23:28 PM UTC-5, Mostowski Collapse wrote:
> > > Still clueless what counterfactuals conditionals are?
> > >
> > > Lets say you tell your girlfriend:
> > > If we take the tube to piccadilly we can go to a chines restaurant.
> > >
> > > Then your girlfriend responds:
> > > No, at piccadilly we could go to a cinema.
> > >
> > [snip]
> >
> > Oddly put. I take it that we could also go to a cinema there. Now, where is the counterfactual in all this?
> Hint the "No" might indicate the counter factual.
Counterfactual conditionals are used to speculate or opine about what MIGHT have been. It is based on the negation of a known historical fact, e.g. What if, contrary to historical fact, X hadn't happened? Nothing like that here. Sorry.
Mostowski Collapse
Feb 2, 2023, 4:34:25 AM2/2/23
to
Not only, this here is terminological knowledge:
"piccadilly circus is not actually a circus"
Not some speculation.
And this here is also not a speculation, but a communicated
intention. Forming of a joint plan. Its not about speculating
whether Hilbert hotels exists. LoL:
> > > > Lets say you tell your girlfriend:
> > > > If we take the tube to piccadilly we can go to a chines restaurant.
> > > > Then your girlfriend responds:
> > > > No, at piccadilly we could go to a cinema.
You confuse speculation with intention. What do
you want to speculate about, whether the chines
restaurant or the cinema exists? Whether the
tube works and some can walk the ramaining
distance? You are hilarious Dan-O-Matik. Also I
guess you confuse the goals of logic. Like you
postulate that logic serves "insight" on the objects
the logic talk about. You can do this also informally.
Many people see logic as a way to get "insight" on
the meta level, how language works.
Thats why Carnap wrote this:
Meaning and Necessity: A Study in Semantics and Modal Logic
https://en.wikipedia.org/wiki/Meaning_and_Necessity
I guess it is not about elicidating some set theory
theorems with the help of logic, rather applying some
concepts to better understand semantics of modal logic,
which has other use cases than only mathematics.
"piccadilly circus is not actually a circus"
And this here is also not a speculation, but a communicated
intention. Forming of a joint plan. Its not about speculating
whether Hilbert hotels exists. LoL:
> > > > Lets say you tell your girlfriend:
> > > > If we take the tube to piccadilly we can go to a chines restaurant.
> > > > Then your girlfriend responds:
> > > > No, at piccadilly we could go to a cinema.
you want to speculate about, whether the chines
restaurant or the cinema exists? Whether the
tube works and some can walk the ramaining
distance? You are hilarious Dan-O-Matik. Also I
guess you confuse the goals of logic. Like you
postulate that logic serves "insight" on the objects
the logic talk about. You can do this also informally.
Many people see logic as a way to get "insight" on
the meta level, how language works.
Thats why Carnap wrote this:
Meaning and Necessity: A Study in Semantics and Modal Logic
https://en.wikipedia.org/wiki/Meaning_and_Necessity
I guess it is not about elicidating some set theory
theorems with the help of logic, rather applying some
concepts to better understand semantics of modal logic,
which has other use cases than only mathematics.
Dan Christensen
Feb 2, 2023, 12:07:48 PM2/2/23
to
On Thursday, February 2, 2023 at 4:34:25 AM UTC-5, Mostowski Collapse wrote:
[snip]
> You confuse speculation with intention. What do
> you want to speculate about, whether the chines
> restaurant or the cinema exists?
[snip]
You need a better example, Mr. Collapse. In our example, you give no "counter fact," e.g. nothing like "if event X (in the past) hadn't happened, then, all things being equal, Y would would now be true."
[snip]
> You confuse speculation with intention. What do
> you want to speculate about, whether the chines
> restaurant or the cinema exists?
You need a better example, Mr. Collapse. In our example, you give no "counter fact," e.g. nothing like "if event X (in the past) hadn't happened, then, all things being equal, Y would would now be true."
Mostowski Collapse
Feb 2, 2023, 1:42:00 PM2/2/23
to
Thats why its not called "counter fact", but "counterfactual":
factual: adjective
- actually occurring.
"cases mentioned are factual"
Mostowski Collapse
Feb 2, 2023, 1:48:58 PM2/2/23
to
French:
Synonyme : réel, attesté, observable
Contraire : imaginaire, inobservable
Do you deny that this is imaginaire at the moment it is uttered:
> > > > Then your girlfriend responds:
> > > > No, at piccadilly we could go to a cinema.
The only problem is that material implication can also
deal with the imaginaire. For example take Dan O
Matiks definition of Odd() [See evenodd thread]:
ALL(a):[Odd(a) <=> ~Even(a)]
We can then deduce:
ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Quite imaginary theorem. Or do you claim its vacously true?
If it were vacuously true, you would be able to prove
this here, but you cannot prove this here:
~EXIST(a):Unicorn(a)
LoL
Synonyme : réel, attesté, observable
Contraire : imaginaire, inobservable
Do you deny that this is imaginaire at the moment it is uttered:
> > > > Then your girlfriend responds:
> > > > No, at piccadilly we could go to a cinema.
deal with the imaginaire. For example take Dan O
Matiks definition of Odd() [See evenodd thread]:
ALL(a):[Odd(a) <=> ~Even(a)]
We can then deduce:
ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Quite imaginary theorem. Or do you claim its vacously true?
If it were vacuously true, you would be able to prove
this here, but you cannot prove this here:
~EXIST(a):Unicorn(a)
LoL
Jeffrey Rubard
Feb 2, 2023, 2:44:32 PM2/2/23
to
On Thursday, February 2, 2023 at 10:48:58 AM UTC-8, Mostowski Collapse wrote:
> French:
>
> Synonyme : réel, attesté, observable
> Contraire : imaginaire, inobservable
>
> Do you deny that this is imaginaire at the moment it is uttered:
> > > > > Then your girlfriend responds:
> > > > > No, at piccadilly we could go to a cinema.
> The only problem is that material implication can also
> deal with the imaginaire. For example take Dan O
> Matiks definition of Odd() [See evenodd thread]:
>
> ALL(a):[Odd(a) <=> ~Even(a)]
>
> We can then deduce:
>
> ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
>
> Quite imaginary theorem. Or do you claim its vacously true?
> If it were vacuously true, you would be able to prove
> this here, but you cannot prove this here:
>
> ~EXIST(a):Unicorn(a)
>
> LoL
Fuckin' fool.
> French:
>
> Synonyme : réel, attesté, observable
> Contraire : imaginaire, inobservable
>
> Do you deny that this is imaginaire at the moment it is uttered:
> > > > > Then your girlfriend responds:
> > > > > No, at piccadilly we could go to a cinema.
> The only problem is that material implication can also
> deal with the imaginaire. For example take Dan O
> Matiks definition of Odd() [See evenodd thread]:
>
> ALL(a):[Odd(a) <=> ~Even(a)]
>
> We can then deduce:
>
> ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
>
> Quite imaginary theorem. Or do you claim its vacously true?
> If it were vacuously true, you would be able to prove
> this here, but you cannot prove this here:
>
> ~EXIST(a):Unicorn(a)
>
> LoL
Message has been deleted
Dan Christensen
Feb 2, 2023, 3:00:25 PM2/2/23
to
On Thursday, February 2, 2023 at 1:42:00 PM UTC-5, Mostowski Collapse wrote:
> > On Thursday, February 2, 2023 at 4:34:25 AM UTC-5, Mostowski Collapse wrote:
> >
> > [snip]
> > > You confuse speculation with intention. What do
> > > you want to speculate about, whether the chines
> > > restaurant or the cinema exists?
> > [snip]
> >
> > You need a better example, Mr. Collapse. In our example, you give no "counter fact," e.g. nothing like "if event X (in the past) hadn't happened, then, all things being equal, Y would would now be true."
> > On Thursday, February 2, 2023 at 4:34:25 AM UTC-5, Mostowski Collapse wrote:
> >
> > [snip]
> > > You confuse speculation with intention. What do
> > > you want to speculate about, whether the chines
> > > restaurant or the cinema exists?
> > [snip]
> >
> > You need a better example, Mr. Collapse. In our example, you give no "counter fact," e.g. nothing like "if event X (in the past) hadn't happened, then, all things being equal, Y would would now be true."
> Thats why its not called "counter fact", but "counterfactual":
>
> factual: adjective
> - actually occurring.
> "cases mentioned are factual"
We are STILL waiting for an example of the form: If (contrary to reality) event X (in the past) hadn't happened, then, all things being equal, Y would now be true. Your example here is NOT of that form. Try again.
>
> factual: adjective
> - actually occurring.
> "cases mentioned are factual"
Mostowski Collapse
Feb 2, 2023, 4:52:30 PM2/2/23
to
There is no difference in real/imaginary between material
implication and couterfactual conditional, they can both
do hypothetical reasoning. You don't believe me, right?
Here some hypothetical reasoning about Unicorns with
material implication only. I am using your "arithmetic
axiom" that your praised so much in the past:
1 ALL(a):[Odd(a) <=> ~Even(a)]
Axiom
https://dcproof.com/EvenNextOdd.htm
Now we can prove, doesn't look real to me:
23 ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Rem DNeg, 22
-------------------------------------- begin proof ------------------------------------
1 ALL(a):[Odd(a) <=> ~Even(a)]
Axiom
2 ~ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Premise
3 ~~EXIST(a):~[Unicorn(a) => Odd(a) | Even(a)]
Quant, 2
4 EXIST(a):~[Unicorn(a) => Odd(a) | Even(a)]
Rem DNeg, 3
5 ~[Unicorn(u) => Odd(u) | Even(u)]
E Spec, 4
6 ~[~Unicorn(u) | [Odd(u) | Even(u)]]
Imply-Or, 5
7 ~~[~~Unicorn(u) & ~[Odd(u) | Even(u)]]
DeMorgan, 6
8 ~~[Unicorn(u) & ~[Odd(u) | Even(u)]]
Rem DNeg, 7
9 Unicorn(u) & ~[Odd(u) | Even(u)]
Rem DNeg, 8
10 Unicorn(u)
Split, 9
11 ~[Odd(u) | Even(u)]
Split, 9
12 ~~[~Odd(u) & ~Even(u)]
DeMorgan, 11
13 ~Odd(u) & ~Even(u)
Rem DNeg, 12
14 ~Odd(u)
Split, 13
15 ~Even(u)
Split, 13
16 Odd(u) <=> ~Even(u)
U Spec, 1
17 [Odd(u) => ~Even(u)] & [~Even(u) => Odd(u)]
Iff-And, 16
18 Odd(u) => ~Even(u)
Split, 17
19 ~Even(u) => Odd(u)
Split, 17
20 Odd(u)
Detach, 19, 15
21 ~Odd(u) & Odd(u)
Join, 14, 20
22 ~~ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Conclusion, 2
23 ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Rem DNeg, 22
-------------------------------------- end proof ------------------------------------
implication and couterfactual conditional, they can both
do hypothetical reasoning. You don't believe me, right?
Here some hypothetical reasoning about Unicorns with
material implication only. I am using your "arithmetic
axiom" that your praised so much in the past:
1 ALL(a):[Odd(a) <=> ~Even(a)]
Axiom
https://dcproof.com/EvenNextOdd.htm
Now we can prove, doesn't look real to me:
23 ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Rem DNeg, 22
-------------------------------------- begin proof ------------------------------------
1 ALL(a):[Odd(a) <=> ~Even(a)]
Axiom
2 ~ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Premise
3 ~~EXIST(a):~[Unicorn(a) => Odd(a) | Even(a)]
Quant, 2
4 EXIST(a):~[Unicorn(a) => Odd(a) | Even(a)]
Rem DNeg, 3
5 ~[Unicorn(u) => Odd(u) | Even(u)]
E Spec, 4
6 ~[~Unicorn(u) | [Odd(u) | Even(u)]]
Imply-Or, 5
7 ~~[~~Unicorn(u) & ~[Odd(u) | Even(u)]]
DeMorgan, 6
8 ~~[Unicorn(u) & ~[Odd(u) | Even(u)]]
Rem DNeg, 7
9 Unicorn(u) & ~[Odd(u) | Even(u)]
Rem DNeg, 8
10 Unicorn(u)
Split, 9
11 ~[Odd(u) | Even(u)]
Split, 9
12 ~~[~Odd(u) & ~Even(u)]
DeMorgan, 11
13 ~Odd(u) & ~Even(u)
Rem DNeg, 12
14 ~Odd(u)
Split, 13
15 ~Even(u)
Split, 13
16 Odd(u) <=> ~Even(u)
U Spec, 1
17 [Odd(u) => ~Even(u)] & [~Even(u) => Odd(u)]
Iff-And, 16
18 Odd(u) => ~Even(u)
Split, 17
19 ~Even(u) => Odd(u)
Split, 17
20 Odd(u)
Detach, 19, 15
21 ~Odd(u) & Odd(u)
Join, 14, 20
22 ~~ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Conclusion, 2
23 ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Rem DNeg, 22
-------------------------------------- end proof ------------------------------------
Dan Christensen
Feb 2, 2023, 8:11:08 PM2/2/23
to
On Thursday, February 2, 2023 at 4:52:30 PM UTC-5, Mostowski Collapse wrote:
> There is no difference in real/imaginary between material
> implication and couterfactual conditional, they can both
> do hypothetical reasoning.
[snip]
> There is no difference in real/imaginary between material
> implication and couterfactual conditional, they can both
> do hypothetical reasoning.
Again, counterfactual conditionals are really just speculation about alternative histories: If (contrary to reality) event X hadn't happened, then, all things being equal, Y would now be true.
Unlike material conditionals, they are quite useless for any kind of logical reasoning.
Jim Burns
Feb 3, 2023, 12:15:22 AM2/3/23
to
On 2/2/2023 2:54 PM, Dan Christensen wrote:
https://plato.stanford.edu/entries/causation-counterfactual/
Counterfactual Theories of Causation
...
| The chief obstacle in empiricists’ minds to
| explaining causation in terms of counterfactuals
| was the obscurity of counterfactuals themselves,
| owing chiefly to their reference to unactualised
| possibilities. The true potential of the
| counterfactual approach to causation did not
| become clear until counterfactuals became better
| understood through the development of possible world
| semantics in the early 1970s.
...
| The central notion of a possible world semantics
| for counterfactuals is a relation of comparative
| similarity between worlds. One world is said to be
| closer to actuality than another if the first
| resembles the actual world more than the second does.
| In terms of this similarity relation, the truth
| condition for the counterfactual “If A were
| (or had been) the case, C would be (or have been)
| the case” is stated as follows:
|
| (1)
|| “If A were the case, C would be the case” is true
|| in the actual world if and only if either
|| (i) there are no possible A-worlds; or
|| (ii) some A-world where C holds is closer to
|| the actual world than is any A-world where C
|| does not hold.
|
| We shall ignore the first case in which the
| counterfactual is vacuously true. The fundamental
| idea of this analysis is that the counterfactual
| “If A were the case, C would be the case” is true
| just in case it takes less of a departure from
| actuality to make the antecedent true along with
| the consequent than to make the antecedent true
| without the consequent.
FYI...
| There are deep metaphysical issues at stake here,
| then: one might view the SEF [Structural Equations
| Framework] approach as offering a more sophisticated
| variant of Lewis’s approach that shares the
| reductionist aspirations of that approach. Or one
| might – especially if one is sceptical about the
| prospects for those reductionist aspirations – take
| the SEF approach in anti-reductionist spirit,
| viewing it not as a way of defining causation in
| non-causal terms but rather as a way of extracting
| useful and sophisticated causal information from an
| inherently causal model of a given complex situation.
https://plato.stanford.edu/entries/causation-counterfactual/
Counterfactual Theories of Causation
...
| The chief obstacle in empiricists’ minds to
| explaining causation in terms of counterfactuals
| was the obscurity of counterfactuals themselves,
| owing chiefly to their reference to unactualised
| possibilities. The true potential of the
| counterfactual approach to causation did not
| become clear until counterfactuals became better
| understood through the development of possible world
| semantics in the early 1970s.
...
| The central notion of a possible world semantics
| for counterfactuals is a relation of comparative
| similarity between worlds. One world is said to be
| closer to actuality than another if the first
| resembles the actual world more than the second does.
| In terms of this similarity relation, the truth
| condition for the counterfactual “If A were
| (or had been) the case, C would be (or have been)
| the case” is stated as follows:
|
| (1)
|| “If A were the case, C would be the case” is true
|| in the actual world if and only if either
|| (i) there are no possible A-worlds; or
|| (ii) some A-world where C holds is closer to
|| the actual world than is any A-world where C
|| does not hold.
|
| We shall ignore the first case in which the
| counterfactual is vacuously true. The fundamental
| idea of this analysis is that the counterfactual
| “If A were the case, C would be the case” is true
| just in case it takes less of a departure from
| actuality to make the antecedent true along with
| the consequent than to make the antecedent true
| without the consequent.
FYI...
| There are deep metaphysical issues at stake here,
| then: one might view the SEF [Structural Equations
| Framework] approach as offering a more sophisticated
| variant of Lewis’s approach that shares the
| reductionist aspirations of that approach. Or one
| might – especially if one is sceptical about the
| prospects for those reductionist aspirations – take
| the SEF approach in anti-reductionist spirit,
| viewing it not as a way of defining causation in
| non-causal terms but rather as a way of extracting
| useful and sophisticated causal information from an
| inherently causal model of a given complex situation.
Dan Christensen
Feb 3, 2023, 1:02:34 AM2/3/23
to
Mostowski Collapse
Feb 3, 2023, 6:35:08 AM2/3/23
to
It looks your DC Proof material implication is also
a counterfactual conditional. Why can I prove this
alternate theory:
23 ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Rem DNeg, 22
That is counterfactual, there are no Unicorns around.
Or do you see some Unicorn? What is an Odd Unicorn,
a Unicorn with an odd number of legs? What
about the horn of the Unicorn, do we count the legs
and the horn, or only the legs. Also DC Proof
is not able to prove this:
~EXIST(a):Unicorn(a)
So we can only conclude, according to the guidelines
of Dan Christensen, that the material implication
of DC Proof is also a couterfactual conditional?
LMAO!
a counterfactual conditional. Why can I prove this
alternate theory:
23 ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Rem DNeg, 22
Or do you see some Unicorn? What is an Odd Unicorn,
a Unicorn with an odd number of legs? What
about the horn of the Unicorn, do we count the legs
and the horn, or only the legs. Also DC Proof
is not able to prove this:
~EXIST(a):Unicorn(a)
So we can only conclude, according to the guidelines
of Dan Christensen, that the material implication
of DC Proof is also a couterfactual conditional?
LMAO!
Dan Christensen
Feb 3, 2023, 10:46:33 AM2/3/23
to
On Friday, February 3, 2023 at 6:35:08 AM UTC-5, Mostowski Collapse wrote:
> Dan Christensen schrieb am Freitag, 3. Februar 2023 um 02:11:08 UTC+1:
> > On Thursday, February 2, 2023 at 4:52:30 PM UTC-5, Mostowski Collapse wrote:
> > > There is no difference in real/imaginary between material
> > > implication and couterfactual conditional, they can both
> > > do hypothetical reasoning.
> > [snip]
> >
> > Again, counterfactual conditionals are really just speculation about alternative histories: If (contrary to reality) event X hadn't happened, then, all things being equal, Y would now be true.
> >
> > Unlike material conditionals, they are quite useless for any kind of logical reasoning.
> Dan Christensen schrieb am Freitag, 3. Februar 2023 um 02:11:08 UTC+1:
> > On Thursday, February 2, 2023 at 4:52:30 PM UTC-5, Mostowski Collapse wrote:
> > > There is no difference in real/imaginary between material
> > > implication and couterfactual conditional, they can both
> > > do hypothetical reasoning.
> > [snip]
> >
> > Again, counterfactual conditionals are really just speculation about alternative histories: If (contrary to reality) event X hadn't happened, then, all things being equal, Y would now be true.
> >
> > Unlike material conditionals, they are quite useless for any kind of logical reasoning.
> It looks your DC Proof material implication is also
> a counterfactual conditional. Why can I prove this
> alternate theory:
> 23 ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
> Rem DNeg, 22
[snip]
> a counterfactual conditional. Why can I prove this
> alternate theory:
> 23 ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
> Rem DNeg, 22
Wrong again, Mr. Collapse. You started by assuming...
1 ALL(a):[Odd(a) <=> ~Even(a)]
Axiom
2 ~ALL(a):[Unicorn(a) => Odd(a) | Even(a)]
Premise
You want start with
1. ~EXIST(a):Unicorn(a)
Axiom
Then you want to ask, what if EXIST(a):Unicorn(a) (your supposed "counterfactual").
2. EXIST(a):Unicorn(a)
Premise
Of course, in ordinary logic, for ANY logical proposition X, it would then be vacuously true that EXIST(a):Unicorn(a) => X
I hope this helps.
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