## Correcting the definition of VALID inference

What is the basis for reason? And mathematics?

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PeteOlcott
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Joined: Mon Jul 25, 2016 6:55 pm

### Correcting the definition of VALID inference

Instead of defining valid inference this way:
It is impossible for the conclusion to be false and all of the premises are true.
¬(P ∧ ¬C) or this P → C

We define it this way:
The premises necessitate the conclusion: ◻(P ⊢ C)
p q ◻(P ⊢ C)
T T T
T F F
F T F
F F F

After this correction mathematical logic and deductive inference
correspond to the commonly understood meaning of the way provability
actually works.

This eliminates the Principle of Explosion and seems to eliminate the
need for relevance logic.

wtf
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Joined: Tue Sep 08, 2015 11:36 pm

### Re: Correcting the definition of VALID inference

Isn't that just logical AND?

So that if it's raining AND it's Tuesday you'd say that rain implies Tuesday?

PeteOlcott
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Joined: Mon Jul 25, 2016 6:55 pm

### Re: Correcting the definition of VALID inference

wtf wrote:
Thu Sep 05, 2019 5:26 pm
Isn't that just logical AND?

So that if it's raining AND it's Tuesday you'd say that rain implies Tuesday?
That seems to be the best that I can do for propositional logic.
It does screen out the principle of explosion and "paradoxes" of material implication.

https://en.wikipedia.org/wiki/Relevance_logic
Relevance logic aims to capture aspects of implication that are ignored by the "material implication" operator in classical truth-functional logic, namely the notion of relevance between antecedent and conditional of a true implication. This idea is not new: C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication such as the principle that a falsehood implies any proposition. Hence "if I'm a donkey, then two and two is four" is true when translated as a material implication, yet it seems intuitively false since a true implication must tie the antecedent and consequent together by some notion of relevance. And whether or not I'm a donkey seems in no way relevant to whether two and two is four.

When we get into predicate logic we begin to have a basis for ensuring a semantic
relevance connection between the premises and the conclusion.

Semantic connection
(A) All men are mortal.
(B) is a man.
(C) Therefore, Socrates is mortal.
Socrates ∈ men ⊆ mortals
∴ Socrates ∈ mortals

When we translate this back to the propositional logic form
it derives the same result as the predicate logic form:

A∧B---C---◻({A, B} ⊢ C)
--T----T----T
--T----F----F
--F----T----F
--F----F----F

Eodnhoj7
Posts: 5328
Joined: Mon Mar 13, 2017 3:18 am

### Re: Correcting the definition of VALID inference

PeteOlcott wrote:
Thu Sep 05, 2019 4:22 pm
Instead of defining valid inference this way:
It is impossible for the conclusion to be false and all of the premises are true.
¬(P ∧ ¬C) or this P → C

We define it this way:
The premises necessitate the conclusion: ◻(P ⊢ C)
p q ◻(P ⊢ C)
T T T
T F F
F T F
F F F

After this correction mathematical logic and deductive inference
correspond to the commonly understood meaning of the way provability
actually works.

This eliminates the Principle of Explosion and seems to eliminate the
need for relevance logic.
So if you negate the principle of explosion, which is negative by nature....are you actually just further justifying it with a continual string of infinitely variating true and false statements?

Actually it is possible for a concluding to be false and the premises to be true...it is called a contradiction.

Example: The sun is shining and the weather is nice. This makes me happy. Therefore the day is good.

False, the sun is shining and the whether is nice might make me happy... but this does not mean:

The statement is about today or anyother day.

That some tragedy has not happened which ruined the day, and the sun and weather just cheer me up...not make the day good.

One can have a good day without having all the things required for happiness present.

And the list goes on....

Your statement require variables that are empty in themselves, thus subject to not just equivocation but a myriad of other things.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Correcting the definition of VALID inference

Eodnhoj7 wrote:
Thu Sep 05, 2019 8:53 pm

So if you negate the principle of explosion, which is negative by nature....are you actually just further justifying it with a continual string of infinitely variating true and false statements?
We stop the principle of explosion before it gets started:

Eodnhoj7
Posts: 5328
Joined: Mon Mar 13, 2017 3:18 am

### Re: Correcting the definition of VALID inference

PeteOlcott wrote:
Thu Sep 05, 2019 11:22 pm
Eodnhoj7 wrote:
Thu Sep 05, 2019 8:53 pm

So if you negate the principle of explosion, which is negative by nature....are you actually just further justifying it with a continual string of infinitely variating true and false statements?
We stop the principle of explosion before it gets started:
Not really, because the principle of explosion observe reductio ad absurdum as applicable to all propositions and axioms of an argument or system.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Correcting the definition of VALID inference

Eodnhoj7 wrote:
Fri Sep 06, 2019 12:06 am
PeteOlcott wrote:
Thu Sep 05, 2019 11:22 pm
Eodnhoj7 wrote:
Thu Sep 05, 2019 8:53 pm

So if you negate the principle of explosion, which is negative by nature....are you actually just further justifying it with a continual string of infinitely variating true and false statements?
We stop the principle of explosion before it gets started:
Not really, because the principle of explosion observe reductio ad absurdum as applicable to all propositions and axioms of an argument or system.
Reductio ad absurdum proves that a proposition is not true because (for example) it derives a contradiction.
We just plug that same contradiction into the Principle of Explosion and this time we derive the original proposition.

When we do this the same proof that proves a proposition false proves that it is true on the subsequent POE step.

Eodnhoj7
Posts: 5328
Joined: Mon Mar 13, 2017 3:18 am

### Re: Correcting the definition of VALID inference

PeteOlcott wrote:
Fri Sep 06, 2019 3:25 am
Eodnhoj7 wrote:
Fri Sep 06, 2019 12:06 am
PeteOlcott wrote:
Thu Sep 05, 2019 11:22 pm

We stop the principle of explosion before it gets started:
Not really, because the principle of explosion observe reductio ad absurdum as applicable to all propositions and axioms of an argument or system.
Reductio ad absurdum proves that a proposition is not true because (for example) it derives a contradiction.

Reductio ad absurdum is applicable to all propositions as all propostions are both composed of and compose further propositions leading to a continuum.

We just plug that same contradiction into the Principle of Explosion and this time we derive the original proposition.

And all the others that come with it, what you do is make loops upon loops.

So you have the proposition:

The cat exists and the cat does not exist, this infers I may walk my dog and water my garden,...

The cat does not exist, existence allows for all things, this infers the cat exists, as well as dogs, as well as trees, as well as catdogs,...

Or you can have the proposition x, with x representing either of the above propositions, and plug it in with the antithesis:

X and cats do not exist, therefore cat exist and dogs do not exist as well as trees, as well as cat dogs, etc.

You are assuming for every thetical proposition there is one antithetical one, but considering each propostion contains metapropositions there is not always a single antithetical statement...this occurs in thetical statements as well.

Anyhow give an example in English...considering the language of programming is a language of transition, if it does not transfer it is invalid.

When we do this the same proof that proves a proposition false proves that it is true on the subsequent POE step.
Not really.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Correcting the definition of VALID inference

Eodnhoj7 wrote:
Fri Sep 06, 2019 4:08 am
PeteOlcott wrote:
Fri Sep 06, 2019 3:25 am
Eodnhoj7 wrote:
Fri Sep 06, 2019 12:06 am

Not really, because the principle of explosion observe reductio ad absurdum as applicable to all propositions and axioms of an argument or system.
Reductio ad absurdum proves that a proposition is not true because (for example) it derives a contradiction.

Reductio ad absurdum is applicable to all propositions as all propostions are both composed of and compose further propositions leading to a continuum.

We just plug that same contradiction into the Principle of Explosion and this time we derive the original proposition.

And all the others that come with it, what you do is make loops upon loops.

So you have the proposition:

The cat exists and the cat does not exist, this infers I may walk my dog and water my garden,...

The cat does not exist, existence allows for all things, this infers the cat exists, as well as dogs, as well as trees, as well as catdogs,...

Or you can have the proposition x, with x representing either of the above propositions, and plug it in with the antithesis:

X and cats do not exist, therefore cat exist and dogs do not exist as well as trees, as well as cat dogs, etc.

You are assuming for every thetical proposition there is one antithetical one, but considering each propostion contains metapropositions there is not always a single antithetical statement...this occurs in thetical statements as well.

Anyhow give an example in English...considering the language of programming is a language of transition, if it does not transfer it is invalid.

When we do this the same proof that proves a proposition false proves that it is true on the subsequent POE step.
Not really.
You did that incorrectly. It was very hard to read the red because I have a blue blocker.
(1) Create a Reductio ad absurdum proof proving that P is false on the basis of a contradiction.
(2) Plug this contradiction into the principle of explosion and derive P.
Now you have just proven that Reductio ad absurdum can be overruled so it does not work.

POE and RAA contradict each other forming an inconsistent system.

Eodnhoj7
Posts: 5328
Joined: Mon Mar 13, 2017 3:18 am

### Re: Correcting the definition of VALID inference

PeteOlcott wrote:
Fri Sep 06, 2019 5:16 am
Eodnhoj7 wrote:
Fri Sep 06, 2019 4:08 am
PeteOlcott wrote:
Fri Sep 06, 2019 3:25 am

Reductio ad absurdum proves that a proposition is not true because (for example) it derives a contradiction.

Reductio ad absurdum is applicable to all propositions as all propostions are both composed of and compose further propositions leading to a continuum.

We just plug that same contradiction into the Principle of Explosion and this time we derive the original proposition.

And all the others that come with it, what you do is make loops upon loops.

So you have the proposition:

The cat exists and the cat does not exist, this infers I may walk my dog and water my garden,...

The cat does not exist, existence allows for all things, this infers the cat exists, as well as dogs, as well as trees, as well as catdogs,...

Or you can have the proposition x, with x representing either of the above propositions, and plug it in with the antithesis:

X and cats do not exist, therefore cat exist and dogs do not exist as well as trees, as well as cat dogs, etc.

You are assuming for every thetical proposition there is one antithetical one, but considering each propostion contains metapropositions there is not always a single antithetical statement...this occurs in thetical statements as well.

Anyhow give an example in English...considering the language of programming is a language of transition, if it does not transfer it is invalid.

When we do this the same proof that proves a proposition false proves that it is true on the subsequent POE step.
Not really.
You did that incorrectly. It was very hard to read the red because I have a blue blocker.
(1) Create a Reductio ad absurdum proof proving that P is false on the basis of a contradiction.
(2) Plug this contradiction into the principle of explosion and derive P.
Now you have just proven that Reductio ad absurdum can be overruled so it does not work.

POE and RAA contradict each other forming an inconsistent system.
If it is that hard to read, then maybe you read it wrong.

Actually It can't be read incorrectly as knowledge is grounded in assumption. How the theory or propostion is assumed it how it progresses, this is the chaotic element. All propositions must be adaptable to context and this context is "assumption". Propositions are subject to variation as they must fundamentally adapt through time...they must adapt fundamentally to an absence of perceivable form determining them.

This may sound like a play on words or rhetoric, and in some degrees it is...but in its totality, not really.

Provide an English example considering you are programming using a meta language that will have to adapt to standard English. If the language you are using cannot transition over to English, or any other language, it is not self contained. It is not self contained because it cannot adapt. Adaption is self containment.

1. (P ---> -P)--->Q
2. (Q ---> -P)---> Q

1. The cat therefore noncat) therefore (dog, cow, tree....)
2. (The dog, cow, tree... therefore noncat) therefore (the dog, cow, tree...)

(+1--->-1)--->0
(0--->-1)--->-1

Skepdick
Posts: 2556
Joined: Fri Jun 14, 2019 11:16 am

### Re: Correcting the definition of VALID inference

PeteOlcott wrote:
Fri Sep 06, 2019 5:16 am
POE and RAA contradict each other forming an inconsistent system.
And what's so terrible about inconsistent systems? Why are you so fixated on avoiding inconsistency?

Ironically, the pursuit of consistency/determinism actually creates systems which suppress volatility.

Perfect is the enemy of good.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Correcting the definition of VALID inference

Eodnhoj7 wrote:
Fri Sep 06, 2019 7:28 am
PeteOlcott wrote:
Fri Sep 06, 2019 5:16 am
Eodnhoj7 wrote:
Fri Sep 06, 2019 4:08 am

Not really.
You did that incorrectly. It was very hard to read the red because I have a blue blocker.
(1) Create a Reductio ad absurdum proof proving that P is false on the basis of a contradiction.
(2) Plug this contradiction into the principle of explosion and derive P.
Now you have just proven that Reductio ad absurdum can be overruled so it does not work.

POE and RAA contradict each other forming an inconsistent system.
If it is that hard to read, then maybe you read it wrong.
My BLUE BLOCKER makes red almost the same color as the background color.

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Correcting the definition of VALID inference

Skepdick wrote:
Fri Sep 06, 2019 8:05 am
PeteOlcott wrote:
Fri Sep 06, 2019 5:16 am
POE and RAA contradict each other forming an inconsistent system.
And what's so terrible about inconsistent systems? Why are you so fixated on avoiding inconsistency?

Ironically, the pursuit of consistency/determinism actually creates systems which suppress volatility.

Perfect is the enemy of good.
Inconsistency indicates errors.

Skepdick
Posts: 2556
Joined: Fri Jun 14, 2019 11:16 am

### Re: Correcting the definition of VALID inference

PeteOlcott wrote:
Fri Sep 06, 2019 3:48 pm
Inconsistency indicates errors.
*sigh* you didn't answer my question. sed s/consistency/errors/g

And what's so terrible about errors? Why are you so fixated on avoiding errors?

Ironically, the pursuit of consistency/determinism actually creates systems which suppress volatility.

Perfect is the enemy of good.

Error-free is not the same thing as resiliency/robustness. Have you heard of Chaos engineering?

PeteOlcott
Posts: 730
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Correcting the definition of VALID inference

Skepdick wrote:
Fri Sep 06, 2019 4:12 pm
PeteOlcott wrote:
Fri Sep 06, 2019 3:48 pm
Inconsistency indicates errors.
*sigh* you didn't answer my question. sed s/consistency/errors/g

And what's so terrible about errors? Why are you so fixated on avoiding errors?

Ironically, the pursuit of consistency/determinism actually creates systems which suppress volatility.

Perfect is the enemy of good.

Error-free is not the same thing as resiliency/robustness. Have you heard of Chaos engineering?
So in other words you are fine if someone short changes you in a transaction
costing you thousands of dollars even though arithmetic shows that they are
wrong they are entitled to their own opinion regarding these math errors that
cheated you?

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