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1. Is logic limited?

You're diving in the area of complexity theory. Good topics to learn include Gödel's incompleteness theorems - Wikipedia, the free encyclopedia, Church–Turing thesis - Wikipedia, the free encyclopedia and a few others.

These, in essence, handle what can be computed, what can "computing" handle, what can be proven, and what's the relation of provability and computability. It goes a bit to the area of general arithmetics, but it's mostly about logic. But then, there's a one-on-one correspondence on a large area of arithmetics and logic. I.e. Arithmetics and logic can mostly be made to be the other, given a few definitions.

You choose a very intriquing question to study. I wish you a good journey.

2. Originally Posted by Ragingkatsuki
^ I see... Interesting.
This is something I like. Therefore logic must be limited.
Logic is limited by the information availible, and limited in what it can be applied to (ie, emotions). In the same way mathematics cannot be applied to everything, logic is a tool that cannot be used in everything (indeed, mathematics and basic logic rely on a very similar form of intelligence). Although it can often be applied at different levels of things that might otherwise seem outside the realm of logic. For instance, emotions don't follow a logical process PSYCHOLOGICALLY, however, logic can be applied to the process that gives rise to emotions, and to the things that happen physiologically and psychologically with a given emotion or emotional situation. The cognitive process being observed may in itself be an illogical entity, but the surrounding elements still obey predictable laws that logic can be directly applied to. An emotional response to a given situation may be an illogical response and thought process, but you can still use logic to describe why the response occured, why they acted and felt the way they did, or predict how they will respond or think.

3. Originally Posted by Ragingkatsuki
Is human logic limited?
I think all of our faculties are limited, be that emotion, logic, intuition , or something else. I think that is what leads people to have extremists points of views...from religious fundamentalists to staunch materialists/fatalism. An over-reliance on logic creates an imbalance in world view. These things are only limited by us being human. If we use all of them together, equally, we may be able to find more truth than if we were using logic alone.

4. Human logic is not limited, it's a method of coming to logical conclusions.

Humans are limited in their use of logic and can come to wrong conclusions.
Animals use the same logic as humans, but are more limited in it's use.

5. This statement's false.

6. Could we clarify, please? Are we talking about formal logic systems or are we talking about what is more properly called "reason"?

I was going to mention Goedel's incompleteness theorems, but it looks as though Santtu got to it first.

A consistent logic system is one which contains no pairs of valid, yet contradictory formulas.
A complete logic system is one in which all semantically valid statements are provable theorems.

Goedel's first incompleteness theorem states that no logic system can be both consistent and complete. If a system is consistent, there are valid results in that system that cannot be proven within the system.

Goedel proved this at time when Bertrand Russel was trying to prove that the system of mathematics was both consistent and complete. So, in one sense Goedel, I guess, was a big jerk, but he did save Russel and all who might have otherwise attempted to do what Russel was doing a lot of work.

Granted, most of this is consolidated from Wikipedia. I do know a little bit about logic, but am definitely not a logician.

Changing gears now, I will present to you an argument I have been refining for quite a while. I think it's still pretty rough, but hopefully someone will at least understand the idea I am trying to get across.

Before I start with the actual argument, I need to get you to accept a couple axioms. If you need further convincing, or if you want to dispute my axioms outright, then go ahead and comment on them. However, I ask that you at least read the rest of the argument first.

Axiom 1: When we consciously assert that an argument is valid or invalid, we are utilizing logic, although we may be doing so incorrectly.
Axiom 2: If there is no logical justification for a proposition, we cannot determine its truth value with a reasonable degree of certainty. Likewise, if there is a logical justification for a proposition, we can determine its truth value with a reasonable degree of certainty.

The standard justification for logic would probably go something like this:

“Because X, we know that logic works,” with X being held as proof of the proposition “logic works.” Recognize that X does not have to be a single proposition – it can also be an argument or a chain of arguments.

This argument is an enthymeme, meaning it has premises that are not stated explicitly in the argument. Mapped out explicitly, the argument looks something like this:

Argument 1
If there is proof for the proposition “logic works” , then the proposition “logic works” is true.
There is proof (specifically, X) for the proposition “logic works.”
Therefore, the proposition “logic works” is true.

Symbolically, this argument can be represented as follows:

B = “There is proof for the proposition 'logic works.'”
C = “The proposition 'logic works' is true.

B?C
B
-------
C

Convince yourself that this is valid. However, also notice that logic is used to achieve the conclusion “logic works.” This seems to be circular, but we need to prove it.

Before we go any further, note that the following argument form is invalid:

Form F
If D, then E
If E, then D
Therefore, D (or, alternatively, Therefore, E.)

Some might call this circular reasoning. It seems as though circular reasoning can take on several different formal structures. All that matters here, though, is that an argument of this form is invalid. If we can force Argument 1 into this form, we can determine that it is invalid. To do this, we will have to extract a couple more premises from the argument enthymematically.

Consider the juxtaposition of Argument 1 with an argument (Argument 2, specifically) that lays out a couple more assumed premises underlying it.

Argument 1
If there is proof for the proposition “logic works”, then the proposition “logic works” is true.
There is proof (specifically, X) for the proposition “logic works.”
Therefore, the proposition “logic works” is true.

Argument 2
P1: If the proposition “logic works” is true, then Argument 1 establishes truth (assuming it is sound).
P2: If Argument 1 establishes truth (assuming it is sound), then the proposition “logic works” is true.
Therefore, the proposition “logic works” is true.

Notice that the form of Argument 2 is the same as Form F. (I have named the premises P1 and P2 for convenience.)

It would seem as though we are finished, since we have shown that Argument 1 relies on an argument of Form F. But we are not. In order to prove that Argument 1 relies on an argument of Form F, we need to show that the antecedents (the clauses following “If”) in Argument 2 are unique in implying their respective consequents (the clauses following “then”). In other words, we need to show that Argument 1 establishes truth if and only if the proposition “logic works” is true, and that the proposition “logic works” is true if and only if Argument 1 establishes truth. If the antecedent of either P1 or P2 is not unique – that is, if there could possibly be other antecedents that imply the same respective consequents – either of the original antecedents could be replaced by one of those other possible antecedents, which may or may not preserve the form (Form F) of the argument. For example, try replacing the antecedent in P1, “logic works,” with the antecedent “Joe says Argument 1 establishes truth.” The argument is then as follows:

If Joe says that Argument 1 establishes truth, then Argument 1 establishes truth (assuming it is sound).
If Argument 1 establishes truth (assuming it is sound), then the proposition “logic works” is true.”
Therefore, the proposition “logic works” is true.

Although this is invalid, not in Form F. And, indeed, adding a third premise, “Joe says Argument 1 establishes truth,” makes the argument valid (although not necessarily sound).

Again, we are trying to show that the antecedents in P1 and P2 are unique in implying their respective consequents. How convenient, then, that the axioms stated at the beginning already show this, with Axiom 2 showing the uniqueness of the antecedent of P1, and Axiom 1 showing the uniqueness of the antecedent of P2 (the latter being the case only if our justification, X, is a result of conscious thought. Therefore, Argument 1 is actually invalid because it enthymematically relies on Argument 2. Remember that Argument 1 is the standard justification for logic. So, any justification, X, we can give for the proposition “logic works,” as long as it arises from conscious thought, is not sufficient to establish the conclusion “logic works.”

7. Originally Posted by Shimmy
Human logic is not limited, it's a method of coming to logical conclusions.

Humans are limited in their use of logic and can come to wrong conclusions.
Animals use the same logic as humans, but are more limited in it's use.
Originally Posted by run
This statement's false.
You got me there.

Paradoxes are hole's in logic, they are the limit. Though we can easily solve the 'this statement's false' one by redefining the words.

8. Originally Posted by laughingebony
Changing gears now, I will present to you an argument I have been refining for quite a while. I think it's still pretty rough, but hopefully someone will at least understand the idea I am trying to get across.
I won't claim to understand every detail, maybe a rough outline.

1. We arrive at the conclusion through logic
2. Proposition needs to be justified to be logically valid
3. X as justification or proof of argument

Argument:
A. Proposition: Logic works or not
B. Argument: Logic works or not
C. Proof of Proposition or Argument

Argument justified by Proposition, justified by Proof.
Proof of Proposition or Argument is based on logic, which invalidates justification.
Therefore, the entire thing is moot or unjustifiable.

Is this correct? But wouldn't this mean that we can't logically prove any of our logical arguments?

Edit: Re-posted after rewording, apologies.

9. That is not at all what my argument is saying. The concept pretty simple, so the misunderstanding is probably due to my inability to communicate it clearly.

The gist of my argument is this:

When we consciously work through a justification (that is, an argument, either explicit or implicit) for using logic, we use logic to do so. This is circular.

10. Of course it is. It's only one way of looking at the world. It's necessary, but it's certainly not everything. Anyone who thinks that logic is all they need isn't seeing the big picture.

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