Q strange results when applying logic to some events

[cc]

Hi:

When applying logic rules of if-then relation to two events describing daily life situations, strange results occurred.

The truth table of the if-then relation (If P THEN Q) is given as:

P Q ->
T T T
T F F
F T T
F F T

Now, let P = the dog bites and Q = the child cries, and we have:

IF (the dog bites) THEN (the child cries) -> T, make sense
( does NOT cry) -> F, make sense
(the dog does NOT bite) (the child cries) -> T, ODD!
( does NOT cry) -> T, make sense
Why would we get an odd result?

[Mike Terry]

On 30/05/2020 11:59, cc wrote:
> Hi:
>
> When applying logic rules of if-then relation to two events describing daily life situations, strange results occurred.
>
> The truth table of the if-then relation (If P THEN Q) is given as:
>
> P Q ->
> T T T
> T F F
> F T T
> F F T

You are using -> which is representing the material conditional. The
truth of P -> Q depends on (and only on) the truth of P and Q. There is
no claim of P "causing" Q here, which is often quite different to the
english language of IF/THEN.

>
> Now, let P = the dog bites and Q = the child cries, and we have:
>

So to determine the truth of

(the dog bites) -> (the child cries)

we need only to consider the truth of (the dog bites) and (the child cries).

> IF (the dog bites) THEN (the child cries) -> T, make sense

But you are NOT considering the truth of the individual components here.
The (only) question is whether (the dog bites) is true, and whether
(the child cries) is true. So what you wrote simply DOES NOT make
sense! Also you seem to have changed your usage of -> and introduced
new notation IF/THEN. Maybe you mean IF P THEN Q to be another way of
saying P -> Q? But then what do you mean by -> T ??? You need to be
much more careful with your notation.

> ( does NOT cry) -> F, make sense
> (the dog does NOT bite) (the child cries) -> T, ODD!
> ( does NOT cry) -> T, make sense
> Why would we get an odd result?

Same as above, none of these are correct use of terminology, and so NONE
of them make sense. That is to say, they're all effectively gibberish
combinations of symbols.

I wonder if there is a further complication here? When you write

IF (the dog does NOT bite) (the child cries) -> T, ODD!

are you thinking that you're saying (the dog bites) is false? You are
not saying that in any shape or form! (the dog does not bite) is simply
the negation of (the dog bites), i.e. the former is true exactly when
the latter is false (but either could actually be true "in the real world").

Summary:
1. The material conditional only considers the truth/falseness of
P and Q. Sure, this can lead to all sorts of "odd results" if
you simply take an english sentence containing IF/THEN and
transcribe it into a propositional logic statement using ->.
When that happens, it is a problem with the transcription
process, not with ->.
(which after all, is simply what it's defined to be, no
more or less).
2. Your examples above are NOT examples of where such
transcription problems occur, because what you have
actually written is incoherent, given the meaning of ->.
3. Remember, P -> Q is NOT making any claim that Q is "caused"
somehow by P. It may help in many examples to read P -> Q
as (NOT P) OR Q, which is logically equivalent.

Regards,
Mike.

[cc]

Thank you for the detailed clarification.

Yes, I did mistakenly mix the material conditional (IF-THEN) with the '->'.

A further question is could we use the rules defined in math logics to analyze the material conditions? For example, to check the IF-THEN expressed in English? If yes, how do we do that?

[Dan Christensen]

Before addressing this example, consider another, hopefully less confusing example of material implication.

The statement "if it is raining, then it is cloudy" does NOT mean that rain causes cloudiness. It means ONLY that, AT THE MOMENT, it is not both raining and not cloudy.

Raining => Cloudy = ~[Raining and ~Clouding]

So, if "Raining" is false, then clearly "Raining => Cloudy" must be true regardless of whether it is cloudy or not.

Back to your example... Material implication simply does not apply in that case.

Material implication can only be applied in the real world to logical propositions that are unambiguously either true or false at some instant in time, e.g. to those apparent from a single still photo.

The dog's bite may well have caused the child to cry, but that would not be apparent from a single photo. You would need perhaps a rapid sequence of photos over time, say a high-quality video from a time before, during and after the dog biting the child, to reasonably establish a causal link.


Dan

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

[cc]

Thank you for clarifying the difference between the material conditional (or implication,) and the causal relation.

But then, how does one apply the rules of math logic for analyzing real-world situations, especially the causal relation? Determining if a causal relationship exits could be difficult, and thus I wonder if the math logic could aid by providing a systematic way for this determination.

Something like the use of algebra to convert a geometric problem into a systematic process of solving a set of algebraic equations.

[Dan Christensen]

On Saturday, May 30, 2020 at 3:33:59 PM UTC-4, cc wrote:
> Thank you for clarifying the difference between the material conditional (or implication,) and the causal relation.
>
> But then, how does one apply the rules of math logic for analyzing real-world situations, especially the causal relation?

AFAIK the laws of causality cannot be formalized. If X causes Y, and X is true at the moment, Y may still never be be true. It is well known, for example, that smoking causes cancer, but many smokers never get cancer and will die of other causes.

[cc]

What if we somehow relax the requirement of causality?
Originally, we require that causality exists if all the following are satisfied:

If the cause does not exist, then the result must not happen.
If the cause exists, then the result must exist.

Now we weaken the requirements as:

If the cause does not exist, then the result must not happen.
If the cause exists, then the result will happen most of the times

[Sergio]

the table is not if-then, it is straight logic table


Q

0 1
. . . . . . .
0 . 1 1
P .
1 . 0 1
.



or

P Q ->
0 0 1
0 1 1
1 0 0
1 1 1

which is => (not P) or Q

[graham...@gmail.com]

consider the last 2 as a pair

> F T T
> F F T

if the dog does not bite
then either the child cries or not

ie. whether the child cries is independant

[Dan Christensen]

Whether X causes Y cannot be answered with any truth table. It is not "truth functional."

[Mike Terry]

The "material conditional" is what you represented with the -> symbol,
i.e. it is the situation analysed with the truth table for the result:

P Q P -> Q
----------------------

T T T
T F F
F T T
F F T

the IF/THEN of normal english language is tricky, because there are many
nuances as to how it should be interpreted. If you are interested in
the language issues, you need to find a book that discusses all that.
That is not something that interests me very much.

There is an intoductory book: An Introduction to Formal Logic (Peter
Smith), which is very gentle mathematically, with a lot of focus on the
process of interpreting real language statements into formal language
statements, and covers the different problems associated with the
material conditional. I'd say it's a book aimed more at philosophy
students rather than maths students, and that might be what you want.
Not that I'm recommending the book! (It's the only one like this I have
access to, so I mention it by default.)

Also, if you are interested in causality or necessity (P is NECESSARY
for Q, etc.) then there are other "modal" logics you could research, but
perhaps it's best to get the basics of "classical" logic nailed first. :)

Finally, the classical logic with material conditional is likely to be
sufficient if your interest is doing/analysing general mathematical
proofs. The reason is that these proofs use conditionals much more
precisely than general arguments in english - they use conditionals to
step from true statements to further true statements. I.e. if P and
(P->Q) are true, then we conclude Q is true (Modus Ponens). For this
use, material conditionals suffice, and it does not really matter
whether P somehow mysteriously "causes" Q, as long as the conclusion Q
is reliable. :)

Mike.

[Julio Di Egidio]

On Sunday, 31 May 2020 05:03:51 UTC+2, Mike Terry wrote:

> the IF/THEN of normal english language is tricky, because there are many
> nuances as to how it should be interpreted. If you are interested in
> the language issues, you need to find a book that discusses all that.

There is also LOGIC proper, which is neither mathematics, nor linguistics,
and not even philosophy.

*** Mathematical logic is not logic (proper),
and material implication is not logical consequence! ***

That is the first distinction to mention when asked about "the paradoxes of
material implication".

> That is not something that interests me very much.

You and Dan Christensen should learn not to pontificate on things you know
less than nothing about...

Julio

[Dan Christensen]

On Sunday, May 31, 2020 at 10:39:41 AM UTC-4, Julio Di Egidio wrote:
> On Sunday, 31 May 2020 05:03:51 UTC+2, Mike Terry wrote:
>
> > the IF/THEN of normal english language is tricky, because there are many
> > nuances as to how it should be interpreted. If you are interested in
> > the language issues, you need to find a book that discusses all that.
>
> There is also LOGIC proper, which is neither mathematics, nor linguistics,
> and not even philosophy.
>

So, your "logic proper" it is NONE of the above. Can it be applied to anything at all, other than, say, angels dancing on the head of a pin? If so, can you give an example?


Dan

[graham...@gmail.com]

> >
> > If the cause does not exist, then the result must not happen.
> > If the cause exists, then the result will happen most of the times

CAUSE is formalizable with ->

its tricky because there are 2 types of CAUSE



PAST TENSE FACT <<>> FUTURE TENSE RULE

reverses the implication


1

clouds ONLY-CAN-SOMETIMES-CAUSE rain

clouds <- rain

if it_rained
then it_was_cloudy

PAST TENSE FACT


2

dog_bite ALWAYS-CAUSES cry

dog_bite -> cry

if A-HAPPENS then B-HAPPENS

FUTURE TENSE RULE



So the DIRECTION of the -> changes whether cause is

MANY:ONE
or
ONE:MANY

[Me]

On Saturday, May 30, 2020 at 8:07:11 PM UTC+2, cc wrote:

> Thank you for the detailed clarification.
>

> Yes, I did mistakenly mix [some things].

Right. Btw. consider the sentence

"He hit the man on the head with a hammer and the man died." (*)

If we let A = ""He hit the man on the head with a hammer" and B = "the man died", we might be tempted to represent (*) by the symbolic statement

A & B

where "&" represents the logical connective "conjcunction" (AND). Now in propositional logic we would have

"A & B" is true iff "B & A" is true,

while on the other hand, we might not accept the claim

"The man died and he hit the man on the head with a hammer." (**)

as being true even if we assumed that (*) is true.

Moral: Natural language is richer "in meaning" than simple predicate logic can represent.

[Me]

On Sunday, May 31, 2020 at 1:37:05 AM UTC+2, cc wrote:

> If the cause does not exist, then the result must not happen.

The problem with this is, that the certain "result states" may occur, even if a *certain* "cause" did not occure.

Example. Consider the claim:

It's warm, because the sun is shining.

Cause: the shining sun.

Result: it is warm.

But it might also be warm because an atomic bomb exploded.

In THIS case we would have:

It's warm, because an atomic bomb exploded.

So we have at least two possible causes for one and the same "result" (that it is warm).

So you would have to modify your approach (imho) the following way:

"If none of the possible causes exists, then the result must not happen."

So if the sun is not shining, no atomic bomb exploded, etc. etc
then is should not be warm.

If on the other hand (exactly) one of these "causes" occured, the result should occure also.

So if the sun is shining, an atomic bomb exploded, etc. etc
then is should be warm (some time after the root cause occured).

It seems to me that we also have to take into account certain "time delays" if we are considering (possible) causes and their result(s).

[Sergio]

yup,

logic is a structure for solving or simplifying complex problems of
simple inputs. (computer logic chips etc)

[Sergio]

On 5/31/2020 9:15 PM, Me wrote:
> On Sunday, May 31, 2020 at 1:37:05 AM UTC+2, cc wrote:
>
>> If the cause does not exist, then the result must not happen.
>
> The problem with this is, that the certain "result states" may occur, even if a *certain* "cause" did not occure.
>
> Example. Consider the claim:
>
> It's warm, because the sun is shining.
>
> Cause: the shining sun.
>
> Result: it is warm.
>
> But it might also be warm because an atomic bomb exploded.
>
> In THIS case we would have:
>
> It's warm, because an atomic bomb exploded.
>
> So we have at least two possible causes for one and the same "result" (that it is warm).

you have expanded the problem here, although there can be many causes,
one must stay bounded in the given information, one cause. or into
conjecture land.

if expanded then the results could also be expanded...

>
> So you would have to modify your approach (imho) the following way:
>
> "If none of the possible causes exists, then the result must not happen."

there are results where the causes are unknown too.

>
> So if the sun is not shining, no atomic bomb exploded, etc. etc
> then is should not be warm.

just negating (or inverting logic) sentences should be ok,

>
> If on the other hand (exactly) one of these "causes" occured, the result should occure also.
>
> So if the sun is shining, an atomic bomb exploded, etc. etc
> then is should be warm (some time after the root cause occured).
>
> It seems to me that we also have to take into account certain "time delays" if we are considering (possible) causes and their result(s).
>

that adds more complexity, which opens the door to miss understanding,
or miss interperting

[cc]

Well, now it seems that causality would be better expressed by the relation of math logic of <->.

Since in English, considering only single cause situations, we would say that causality exists if the two of the following statement are satisfied.

If the cause does not exist, then the result must not happen.

If the cause exists, then the result will happen most of the times

That is, we have

the cause <-> the result

By checking the truth table, <-> is T only when the case and the result are either both T or F, which seems to be what we described in the preceding paragraph.

[cc]

Indeed there would be multiple causes resulting single consequence.

cause1 = The dog bites
cause2 = The boy hits
effect = the baby cries

How are we going to describe this kind of causality both in terms of math logic and English?

[Me]

On Monday, June 1, 2020 at 8:14:12 AM UTC+2, cc wrote:

> cause1 = The dog bites
> cause2 = The boy hits
> effect = the baby cries
>

> How are we going to describe this kind of causality [...] in [...] English?

Well, we might have

The baby cries because the dog has bitten it.
or
The baby cries because the boy has hit it.

One of the two statements may hold depending on what has actually happend.

[Me]

On Monday, June 1, 2020 at 8:07:54 AM UTC+2, cc wrote:

> That is, we have
>
> the cause <-> the result
>
> By checking the truth table, <-> is T only when the case and the result are

> either both T or [both] F, which ...

of course doesn't "represent" a causal relation between "the cause" and "the result" "that well" (to say the least).

Imho the "time delay" between "cause" and "result" is crucial and should not be neglected here.

We might formulate it as a _necessary_ condition for the truth of the statement

A because B.

"A because B" only holds if B occurs before A.

We clearly would reject the claim "C because D" if C occured before D, no?

[Julio Di Egidio]

We add to our knowledge base that "(usually) if a dog bites a boy, the boy
cries". It is a correlation: getting to causality proper involves metaphysical
assumptions. Then, if we saw a boy cry, we could ask the question "why is the
boy crying?", and, if given the answer "because a dog bit it", we would find
the answer at least plausible...

That said, consider that Logic is not concerned with causality, and not even
with logical consequence per se: just with the discernment of what can be said
without self-contradiction. (That essentially is the definition of Logic.)

OTOH, as was hinting at upthread, "math logic" is simply the *calculator*:
it isn't Logic any more than math physics is physics, or math engineering is
engineering, or architecture, or economy, or what have you.

My advice would be get those basics straight or stay forever confused...

HTH,

Julio

[Walker Hill]

Dan Christensen wrote:

>> There is also LOGIC proper, which is neither mathematics, nor
>> linguistics, and not even philosophy.
>
> So, your "logic proper" it is NONE of the above. Can it be applied to
> anything at all, other than, say, angels dancing on the head of a pin?
> If so, can you give an example?

my friend, nice to meet you. Happy holiday. What on dickens is this for a
corporation, because it can't be a country. Yesterday was about to bomb
into *_liberal capitalist "democracy"_* and forcefully vaccinate the
entire world. Now, you don't even have food to put on their family,
female tranny bill gates hiding in his bunker with the male tranny
melinda, allegedly with food enough for 2 years.

George Floyd protests - live: US engulfed in protests for sixth night as
curfews ignored and thousands arrested while Trump remains out of sight
https://www.independent.co.uk/news/world/americas/george-floyd-protests-
live-riots-trump-police-looting-minneapolis-death-update-a9541556.html

Trump news – live: President was rushed to secret underground bunker to
avoid protests, as he moves to designate Antifa terrorist group
https://www.independent.co.uk/news/world/americas/us-politics/trump-news-
live-white-house-protests-coronavirus-update-press-briefing-today-latest-
a9541741.html

I am so depressed. All those fake moon landings, and now..

[Dan Christensen]

To give your quest some focus, it would help if you could formulate a real-world or scientific problem that might be solved by a formalization of causal logic.

[graham...@gmail.com]

On Tuesday, June 2, 2020 at 2:07:43 AM UTC+10, Dan Christensen wrote:
> On Monday, June 1, 2020 at 2:14:12 AM UTC-4, cc wrote:
> > Indeed there would be multiple causes resulting single consequence.
> >
> > cause1 = The dog bites
> > cause2 = The boy hits
> > effect = the baby cries
> >
> > How are we going to describe this kind of causality both in terms of math logic and English?
>
> To give your quest some focus, it would help if you could formulate a real-world or scientific problem that might be solved by a formalization of causal logic.
>

I found this automotive diagnostic expert system!



%%% Knowledge Base for simple automotive diagnostic expert system.
%%%
%%% This is one of the example programs from the textbook:
%%%
%%% Artificial Intelligence:
%%% Structures and strategies for complex problem solving
%%%
%%% by George F. Luger and William A. Stubblefield



% rule base:

% Top level goal, starts search.
rule((fix_car(Advice) :-
bad_component(Y), fix(Y,Advice)),100).

% rules to infer bad component:

rule((bad_component(starter) :-
bad_system(starter_system),lights(come_on)),50).
rule((bad_component(battery) :-
bad_system(starter_system),not(lights(come_on))),90).
rule((bad_component(timing) :-
bad_system(ignition_system), not(tuned_recently)),80).
rule((bad_component(plugs) :-
bad_system(ignition_system),plugs(dirty)),90).
rule((bad_component(ignition_wires) :-
bad_system(ignition_system),
not(plugs(dirty)), tuned_recently),80).

% Rules to infer basic system that failed.

rule((bad_system(starter_system) :-
not(car_starts), not(turns_over)),90).
rule((bad_system(ignition_system) :-
not(car_starts), turns_over,gas_in_carb),80).
rule((bad_system(ignition_system) :-
car_starts, runs(rough),gas_in_carb),80).
rule((bad_system(ignition_system) :-
car_starts, runs(dies),gas_in_carb),60).

% Rules to make reccommendation for repairs.

rule(fix(starter,'replace starter'),100).
rule(fix(battery,'replace or recharge battery'),100).
rule(fix(timing, 'get the timing adjusted'),100).
rule(fix(plugs, 'replace spark plugs'),100).
rule(fix(ignition_wires, 'check ignition wires'),100).

% askable descriptions

askable(car_starts).
askable(turns_over).
askable(lights(X)).
askable(runs(X)).
askable(gas_in_carb).
askable(tuned_recently).
askable(plugs(X)).