Logic is Empty

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Reconstructo
 
Reply Sun 6 Jun, 2010 08:43 pm
@ughaibu,
ughaibu;174024 wrote:
An equivalent is a implies b, or, b implies a, do you think this principle is true?


I don't know what to make of it. I haven't looked at this until now. (You know I'm eccentric "intuitionist" who means by that term that the foundation of math/logic is intuited. I'm no expert on other opinions, so any information is appreciated.
Quote:

Intuitionistic logic can be succinctly described as classical logic without the Aristotelian law of excluded middle (LEM): (AAA → (AB)).
What is this B?
 
ughaibu
 
Reply Sun 6 Jun, 2010 08:47 pm
@Reconstructo,
Reconstructo;174034 wrote:
What is this B?
In both cases, a and b are any sentence.
 
Reconstructo
 
Reply Sun 6 Jun, 2010 08:50 pm
@Reconstructo,
So not-a implies that a implies b? I don't see why.
 
ughaibu
 
Reply Sun 6 Jun, 2010 09:00 pm
@Reconstructo,
Reconstructo;174039 wrote:
So not-a implies that a implies b? I don't see why.
Here you go: Principle of explosion - Wikipedia, the free encyclopedia
 
Reconstructo
 
Reply Sun 6 Jun, 2010 09:04 pm
@Reconstructo,
Thanks!
Quote:

This can be read as, "If one claims something is both true (http://upload.wikimedia.org/math/c/d/0/cd014731964c742c274df08d7cc238fb.png) and not true (http://upload.wikimedia.org/math/f/c/7/fc7e302ded40b1247108592c66f61cda.png), one can logically derive any conclusion (ψ)."
For me it makes no sense to claim that something is true and not true. It violates my personal intuition of logic. And I also don't see how this contradiction can imply anything. It does not compute for me. What do you think?
 
ughaibu
 
Reply Sun 6 Jun, 2010 09:09 pm
@Reconstructo,
Reconstructo;174047 wrote:
I also don't see how this contradiction can imply anything.
The article demonstrates the implication, from other logical principles.
 
Reconstructo
 
Reply Sun 6 Jun, 2010 09:13 pm
@ughaibu,
ughaibu;174051 wrote:
The article demonstrates the implication, from other logical principles.


I looked around a bit, but I just can't intuit such a thing. Of course one can create any rules one chooses, and apply them consistently. I don't deny that. But I just don't find it convincing. Of course that doesn't mean it can't be useful to others. Am I to suppose you have a different opinion? I definitely appreciate the links. I do wish you would let more of your personal opinions out. I really appreciate talking to someone who knows this stuff. I have learned from you, and this is a subject dear to me.
 
Owen phil
 
Reply Mon 7 Jun, 2010 02:16 am
@ughaibu,
ughaibu;174006 wrote:
But it's not true in intuitionistic logics.


I have not studied intuitionist logic yet, but...

I believe that intuitionist logic defines 'not' as not proven rather than negation.

If (p v ~p) means, p is provable or p is not provable, then (p v ~p) is not tautologous.

I also believe that Tarski and McKinley (1948) have shown that intuitionist logic is a special case of modal logic (S2).

Do you think we can reduce intuitionist logic to modal logic?

---------- Post added 06-07-2010 at 04:40 AM ----------

Reconstructo;174039 wrote:
So not-a implies that a implies b? I don't see why.


~p -> (p -> q), is a tautology, ie. it is true for all values of the variables p and q.

(~p -> (p -> q)) iff (p v (~p v q)), because (p -> q) =df (~p v q).

(p v (~p v q) iff (p v ~p) v q, because (p v ( q v r)) <-> ((p v q) v r), is a theorem.

We can 'calculate' the truth value of ~p -> (p ->q) by exhausting the possible truth values of p and q.

T=true, F=false, ~T=F, ~F=T.

~T -> (T -> T), is true because (F -> T) is true. ie. (T v T)=T.
~T -> (T -> F), is true because (F -> F) is true. ie. (T v F)=T.
~F -> (F -> T), is true because (T -> T) is true. ie. (F v T)=T.
~F -> (F -> F), is true because (T -> T) is true. ie. (F v T)=T.

explosion?!
((p & ~p) -> q) iff ((~p v p) v q), but (~p v p)=T, that is, (T v q)=T

That is to say, (p & ~p) implies any proposition q, is a tautology.

Also: (~p -> (p ->q)) <-> ((p & ~p) -> q), is true.

This sense of 'logical arithmetic' is prior to 'numerical arithmetic' and might fit your pursuit of fundamental calculations.

I believe that we can extend these 'simple' calculations of propositional logic to include predicate logic, by the definition:
(some x)(Fx) =df (Fa v Fb v Fc ...).

That is to say, we can reduce predicate logic to propositional logic.

What do you think about this possible simplification of predicate logic?
 
xris
 
Reply Mon 7 Jun, 2010 03:14 am
@Owen phil,
Should I ask again, my logic asks me? Should I ask , when can this numerical logic be used when our moral or scientific ability fails us? I would like to see examples of this in use, to be convinced it is more than a mathematical exercise.
 
Reconstructo
 
Reply Mon 7 Jun, 2010 04:12 am
@Owen phil,
Owen;174126 wrote:
I have not studied intuitionist logic yet, but...

I believe that intuitionist logic defines 'not' as not proven rather than negation.

If (p v ~p) means, p is provable or p is not provable, then (p v ~p) is not tautologous.

I also believe that Tarski and McKinley (1948) have shown that intuitionist logic is a special case of modal logic (S2).

Do you think we can reduce intuitionist logic to modal logic?

---------- Post added 06-07-2010 at 04:40 AM ----------



~p -> (p -> q), is a tautology, ie. it is true for all values of the variables p and q.

(~p -> (p -> q)) iff (p v (~p v q)), because (p -> q) =df (~p v q).

(p v (~p v q) iff (p v ~p) v q, because (p v ( q v r)) <-> ((p v q) v r), is a theorem.

We can 'calculate' the truth value of ~p -> (p ->q) by exhausting the possible truth values of p and q.

T=true, F=false, ~T=F, ~F=T.

~T -> (T -> T), is true because (F -> T) is true. ie. (T v T)=T.
~T -> (T -> F), is true because (F -> F) is true. ie. (T v F)=T.
~F -> (F -> T), is true because (T -> T) is true. ie. (F v T)=T.
~F -> (F -> F), is true because (T -> T) is true. ie. (F v T)=T.

explosion?!
((p & ~p) -> q) iff ((~p v p) v q), but (~p v p)=T, that is, (T v q)=T

That is to say, (p & ~p) implies any proposition q, is a tautology.

Also: (~p -> (p ->q)) <-> ((p & ~p) -> q), is true.

This sense of 'logical arithmetic' is prior to 'numerical arithmetic' and might fit your pursuit of fundamental calculations.

I believe that we can extend these 'simple' calculations of propositional logic to include predicate logic, by the definition:
(some x)(Fx) =df (Fa v Fb v Fc ...).

That is to say, we can reduce predicate logic to propositional logic.

What do you think about this possible simplification of predicate logic?

Oh yes, it makes sense to me know. It's been a while. I'm used to the classic "and, or, not." Computer programming doesn't much use the rest. Of course Nand and Nor are made from Nots.

So that explosion is a tautology. I was thinking of implies in more ordinary temporal terms. Now it clicks. Well, this is good. I like to learn. Tanks.

I haven't messed much with predicate logic. I know the some/all basics but haven't really played w/ it. Sounds like a good idea, though. Reduction and simplicity are good goals here.

I actually feel that the most basic of basics in beneath both math and logic. It would be the concept of abstract unity itself. As far as a variable abstract unity, I think that logic and bits (base 2) are as close as it gets. But what about pure unity? Or the notion of abstract unity? Not variable but basically the concept/number 1. I feel like this is inborn. And the rest is possible because of this. What do you think?Smile
 
ughaibu
 
Reply Mon 7 Jun, 2010 06:54 am
@Owen phil,
Owen;174126 wrote:
I also believe that Tarski and McKinley (1948) have shown that intuitionist logic is a special case of modal logic (S2).
This is all I can get, of what I guess is the relevant article. JSTOR: An Error Occurred Setting Your User Cookie
Owen;174126 wrote:
Do you think we can reduce intuitionist logic to modal logic?
I dont know, in any case, is that Tarski's claim?
 
Owen phil
 
Reply Mon 7 Jun, 2010 08:37 am
@ughaibu,
ughaibu;174181 wrote:
This is all I can get, of what I guess is the relevant article. JSTOR: An Error Occurred Setting Your User CookieI dont know, in any case, is that Tarski's claim?


I don't know either.
I vaguely recall that somewhere Tarski did claim that S2 includes intuitionist logic, but, I have not pursued it as of yet.

I believe that intuitionist logic can be reduced to modal propositional logic.
 
Dr Seuss
 
Reply Mon 7 Jun, 2010 09:19 am
@Owen phil,
I think all of you are correct. There is logic because there is no logic, and there is no logic because there is logic. Very Happy Enjoy!

What exists exists because there is something that doesn't.
 
zoot
 
Reply Sun 7 Apr, 2013 03:06 pm
@Reconstructo,
It's very difficult to comment on your post, because a number of things are unclear: 1) what you're taking "empty" to mean; 2) what you mean when you say that logic tells us nothing; 3) how the Wittgenstein quotes relate to your ideas. It'd be helpful if you could flesh all of these things out a little.

On the idea of logic telling us nothing: it's worth bearing in mind that one of the more controversial elements of standard logic is that it makes existential assumptions. Consider, for example, the theorem: (Ǝx)(Fx v ~Fx). The standard reading of this is: "there is an x such that: either x is an F, or it is not the case that x is an F". Yet, if the universe of discourse is empty, this is obviously false. Standard logic thus comes with the assumption that the universe is non-empty; if we accept standard logic, we must also accept that it's impossible for there to be nothing.

Then there are the controversies regarding non-denoting terms and so on. Free logics were developed precisely to deal with this kind of stuff.

(Also note that existential assumptions aren't always accidental: there are some logics that were explicitly designed to make existential assumptions. For example, one of the stipulated axioms of Routley & Meyer's "minimal dialectical logic" (system DL) is: (p & ~p). This is not an axiom schema (obviously), but is simply designed to guarantee that there is at least one contradiction.)
 
Congruencismdotcom
 
Reply Fri 3 May, 2013 10:25 am
@Reconstructo,
Congruencism is not empty.
 
 

 
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