Exploring the Limits of Logic: Discussing its Origins and Effects

[XxFREEofFILTHxX]

I Know That Logic Is Limited And That We Are Enslaved In Its Confined Boundaries.

I Would Like To Hear Your Opinions,
My Question's Are:
1. What Is Logic And What Is It A Product Of?
2. Does It Define Things Or Is It Being Defined By Other Higher Things?

 

[dekoi]

1.) Logic is a root of reason. Reason is either 1: a product of human/intellectual development |or| 2: a Divine gift.

2.) Depends on whether you believe in a central authority or not (God). If you believe in God, and that God created all, then God defines reason. One can refute this, by saying humans have free will and thus, they use their will to define reason. However, reason is based on an objective truth (if in fact, you believe in God). If you are a humanist, then you clearly define your own logic (because you believe you are your own authority).

 

[Goongyae]

1a. "What Is Logic"

Logic is a reliable method for taking one set of truths and manufacturing new truths from them. Using a finite number of axioms, plus logic, one has access to many truths (though not all). Seeing as how it's impossible to prove the consistency of simple systems like arithmetic, I admit that it's possible logic can also yield false results. The theorem about cutting up a grapefruit into a finite number of pieces and then fashioning a solid sphere the size of the sun comes to mind.

1b. "What Is It A Product Of?"

Logic, like science, is something people invented because it proves useful empirically. Most people who reject logic have died out by now. In particular, the bunga bunga tribe was horribly impaled by their foes when they refused to accept the parabolic trajectories of their javelins.

2. "Does It Define Things Or Is It Being Defined By Other Higher Things?"

It's merely a tool that has proven useful to people. So, the answer is that people are the higher things that have defined their logic. I think suggesting much beyond that would take a lot of hubris.

 

[Humbucker]

Logic is weighing your behavior against the consequences.The choice is yours.

 

[wuliheron]

I Know That Logic Is Limited And That We Are Enslaved In Its Confined Boundaries.

I Would Like To Hear Your Opinions,
My Question's Are:
1. What Is Logic And What Is It A Product Of?
2. Does It Define Things Or Is It Being Defined By Other Higher Things?

The term "logic" refers to the category of analytical tools based on the principle of Reductio ad absurdium (reduction to the absurd). In turn, this principle is derived from our demonstrably emotive ability to give or find meaning in the world around us.

Logic provides merely one type of description of things among many. For example, natural language is repleat with vague terms such as love, pile, bald, etc. which logic has little to say about.

 

[XxFREEofFILTHxX]

1. but how did you conclude these statements?
2. Is logic wrong and full of errors?
3. By what do we conclude what is logic, and what is not?
4. On what is Logic based on?
5. What is proof?

 

[XxFREEofFILTHxX]

Dekoi, The opinion base in the existence god should not make a difference...what defines logic?

 

[Prometheus]

I Know That Logic Is Limited And That We Are Enslaved In Its Confined Boundaries.

I Would Like To Hear Your Opinions,
My Question's Are:
1. What Is Logic And What Is It A Product Of?

Logic is an abstract form of reasoning that is based on the model of the Indo-European language grammar. Non I-E languages do not consider, and never developed, logic in the way that it developed in the west.

Logic is a method of formulating thought. It is not a way to prove anthing, except in a theoretical manner. For example, logic could never be used to provide a proof that god exists, because such proof requires axioms, which are beyond the scope of logic.

 

[dekoi]

Prometheus:
Logic can very much prove something. Proving God does not require axioms; the proofs themselves use logic/reason to build up into a general law (an axiom).

XxFREEofFILTHxX: If God exists, he has defined logic for us. Logic and/or reason, in this case, would be founded on the natural law.

 

[hypnagogue]

Prometheus:
Logic can very much prove something. Proving God does not require axioms; the proofs themselves use logic/reason to build up into a general law (an axiom).

I think what Prometheus meant to say is that the formal rules of logic alone are not enough to prove anything; we must start with certain premises that we assume to be true (axioms) and manipulate these axioms using the rules of logic to derive further premises. In this regard, Prometheus is quite right. Given no axioms to work with, logic cannot do much of anything, somewhat like a baker cannot do much of anything if he has a recipe but no ingredients.

XxFREEofFILTHxX: If God exists, he has defined logic for us.

This depends on one's definition of God. If the minimal requirements for a God be that it is omnipotent and omniscient, then it's not hard to suppose a kind of omnipotent, omniscient God that has nonetheless not defined logic for us.

Remember that we are talking about an abstract philosophical concept when we use the word 'God.' Perhaps some religions suppose a God who defines logic (whatever that might mean exactly), and in the context of such a religious definition of God your statement would hold. But please keep in mind that PF does not support discussions about God in the context of religion.

 

[hypnagogue]

XxFREEofFILTHxX, a good introduction to the topic of logic is available on wikipedia.org: http://en.wikipedia.org/wiki/Logic

Please read over this article. It may clear up some lingering confusions you have and help shape any future questions you might have.

 

[Prometheus]

I think what Prometheus meant to say is that the formal rules of logic alone are not enough to prove anything; ...

Well-phrased.

Prometheus:
Logic can very much prove something. Proving God does not require axioms; the proofs themselves use logic/reason to build up into a general law (an axiom).

Please prove something using logic. You might try proving that god exists, or something simpler. Please indicate how a proof using the rules of logic can prove that anything exists with no axiomatic requirements.

 

[dekoi]

Prometheus, i must have misunderstood you as Hypnagogue stated.

However, why could axioms not be produced by logic as well? Why do you state they are outside the scope of reason? I can answer this question myself (likely the same answer you will reply with), although it is very vague.


Hypnagogue: imo, i find it very difficult (as well as a professor i know) to express philosophical thoughts regarding an omnipotent power without -- at a latter point -- making a connection with theology. However, i will try

 

[cogito]

I think what Prometheus meant to say is that the formal rules of logic alone are not enough to prove anything; we must start with certain premises that we assume to be true (axioms) and manipulate these axioms using the rules of logic to derive further premises. In this regard, Prometheus is quite right. Given no axioms to work with, logic cannot do much of anything, somewhat like a baker cannot do much of anything if he has a recipe but no ingredients.

Not quite. Formal Logic is sufficient for proving all theorems, which by definition can be proven from the empty set of premises. So, it is false that we must start with certain premises merely assumed to be true. We can start with no premises, and derive necessary truths. What is the case is that these theorems are true in any possible world, and hence quite uninformative about the particularities of the actual world (i.e., you can't derive any contingent truths about the actual world).

 

[quantumdude]

Formal Logic is sufficient for proving all theorems, which by definition can be proven from the empty set of premises.

What brand of logic is it that can do this?

Logic, as it is normally understood, consists of rules of valid inferences from one statement to another. How do you reason something from nothing?

 

[Goongyae]

"Formal Logic is sufficient for proving all theorems, which by definition can be proven from the empty set of premises."

Incorrect, have you not heard of Goedel's theorem? Even in the simple mathematical system of arithmetic it can be demonstrated that there are truths that are impossible to prove.

 

[cogito]

"Formal Logic is sufficient for proving all theorems, which by definition can be proven from the empty set of premises."

Incorrect, have you not heard of Goedel's theorem? Even in the simple mathematical system of arithmetic it can be demonstrated that there are truths that are impossible to prove.

I'm familiar with both Goedel's incompleteness proof, which proves that all consistent formalizations of number theory will include undecidable propositions (propositions for which the axioms of the system will not allow one to determine their truth-value), and his completeness proof, which proves that first-order predicate calculus is complete and sound. When I used the term 'formal logic', in my original quote, I was referring to first-order predicate calculus (which is what the term 'logic' [in the deductive sense, rather than in, say, a Bayesian inductive sense] almost invariably refers to) and not to extensions of it that require set-theory. I should have been more clear about this, but I'm used to talking about logic with philosophers and not mathematicians. Thanks for keeping me honest!

 

[cogito]

What brand of logic is it that can do this?

Logic, as it is normally understood, consists of rules of valid inferences from one statement to another. How do you reason something from nothing?

First-order predicate calculus. You reason from nothing by temporarily assuming some proposition, and seeing what follows from it. You discharge this assumption in various ways. If an assumption leads to a contradiction, then you know that the negation of the assumption is a theorem. If your assumption leads to some conclusion, then you know that the conditional comprised of the assumption as its antecedent and the conclusion as its consequent is a theorem. There are other ways of deriving theorems, but these two methods are very common.

 

[wuliheron]

1. but how did you conclude these statements?
2. Is logic wrong and full of errors?
3. By what do we conclude what is logic, and what is not?
4. On what is Logic based on?
5. What is proof?

1) Demonstrably, words only have meaning according to their use in a given context. Hence the term logic could be used by someone to refer to a love of chickens, but obviously that is not the context used here. The implied context is the normal uses of the word, which includes both formal and informal logic.

Because all types of logic normally refer to the concept of reason, the foundation of logic is obviously the concept of the absurd because reason has no demonstrable meaning outside the concept of the absurd. That is, the idea that some things are just impossible, undesirable, and meaningless. When analyzed, every form of logic thus far investigated has been based on the principle or axiom of Reductio ad Absurdium.

In turn, the concept of meaningful/meaningless demonstrably arises from our ability to emote. My computer can do endless logical functions flawlessly, but it cannot place logic in any kind of meaningful context. It is an idiot savant which cannot tell the difference between the logical and the irrational.

One famous case study involves a man who lost the ability to emote due to a head trauma. He could not hide his condition for any length of time whatsoever. Like a computer with an incredible number of preset responses, all he had left was his memories of how to respond. Any novelty whatsoever always threw him for a loop (Danger Will Robinson, does not compute!)

2) Logic is a specific type of tool and, as such, is limited by definition. Still, I call a wrench "wrong" or "full of errors" when I attempt to do something with it that it was not designed to do. However I can use it occationally as a hammer or whatever, something it was not really designed to be used as. If I use logic for something it simply cannot do, it is my use of the tool that is "wrong" or "full of errors", not the tool itself.

3) See answer #1

4) Ditto

5) Again, demonstrably words only have meaning according to their function in a given context. Logically speaking, logic cannot be used to prove it's own axioms because this would violate the principle of reductio ad absurdium. Hence, the ultimate proof for logic is whether or not it fits our ideas of the absurd. Whether or not we feel it is absurd.

 

[dekoi]

If logic requires some sort of earlier premises, would that not cause us to assume there is an original, absolute, and objective truth or premise in everybody?

 

[hypnagogue]

If logic requires some sort of earlier premises, would that not cause us to assume there is an original, absolute, and objective truth or premise in everybody?

I'm not sure exactly what it would mean for there to be a premise literally residing in a person, let alone an original, absolute, or objective one. That seems to be a poor way to phrase your question.

 

[Prometheus]

We can start with no premises, and derive necessary truths.

Please provide an example where you start with no premises and then derive some necessary proof, however you want to define necessary.

 

[Prometheus]

Prometheus, i must have misunderstood you as Hypnagogue stated.

That happens to us all.

However, why could axioms not be produced by logic as well?

Because that is not the way that logic works. Do you know what an axiom is?

Why do you state they are outside the scope of reason?

Please cite where I said that anything is outside the scope of reason, or where I discussed reason at all.

 

[Prometheus]

If logic requires some sort of earlier premises,

You seem to be unclear as to what logic is and what an axiom is. Nobody, I believe, here said that logic requires earlier premises. As well, that is not a true statement. Why? What do you think that a premise is? It is a word that implies a relationship using logic.

We are talking about axioms. Logic does not require axioms, and has nothing to do with axioms. It is the attempt to apply logic to the real world, to which formal logic does not apply, that requires axioms. As these axioms cannot be applied within the confines of formal logic, logic cannot be applied to the real world as a method of establishing proof.

would that not cause us to assume there is an original, absolute, and objective truth or premise in everybody?

Let me try to be clear about my response to your question. NO.

 

[cogito]

Please provide an example where you start with no premises and then derive some necessary proof, however you want to define necessary.

OK, here you go.

Notation:

P: some arbitrary proposition
~: negation symbol
v: disjunction symbol
#: contradiction symbol

The numbers in the parentheses on the far left of each line refer to those hypothetical suppositions that are operative. The text in brackets on the right refers to that rule of first-order predicate calculus that allows for what is written on each line of the derivation.

(1) 1. ~(P v ~P) [hypothetical supposition]
(1,2) 2. P [hypothetical supposition]
(1,2) 3. P v ~P [from 2, by disjunction introduction]
(1,2) 4. # [from 1 and 3, by contradiction introduction]
(1) 5. ~P [from 2 and 4, by negation introduction]
(1) 6. P v ~P [from 5, by disjunction introduction]
(1) 7. # [from 1 and 6, by contradiction introduction]
8. ~~(P v ~P) [from 1 and 7, by negation introduction]
9. (P v ~P) [from 8, by negation elimination]

Note what happened on steps 5 and 8 of this derivation. On step 4, I discharged that hypothetical supposition that P. Thus, on step 5, the number "2" no longer appears in the parentheses on the left. This means that the hypothetical supposition introduced on line 2 is no longer operative. On step 7, I discharged the hypothetical supposition that ~(P v ~P). Thus, on step 8, the number "1" no longer appears int he parentheses on the left. This means that the hypothetical supposition introduced on line 1 is no longer operative. The conclusion, the Law of the Excluded Middle, is derived from no premises at all. Any conclusion derivable from no premises is a necessary truth. Hence, the conclusion is a necessary truth. This particular conclusion goes by the name "The Law of the Excluded Middle".

 

[Les Sleeth]

I Know That Logic Is Limited And That We Are Enslaved In Its Confined Boundaries.

I Would Like To Hear Your Opinions,
My Question's Are:
1. What Is Logic And What Is It A Product Of?
2. Does It Define Things Or Is It Being Defined By Other Higher Things?

Personally, I can't see how whether or not logic proves propositions or produces truth, etc. has anything to do with your questions (although they might be interesting side issues). In my opinion, there are very definite answers to both of your questions.

It is easier to answer the questions in reverse of the order you asked them:

2. Does It [logic] Define Things Or Is It Being Defined By Other Higher Things? This is an good metaphysical question, but we don't need God to decide it (even if some sort of consciousness might be behind order). To decide the metaphysical question all we need to know is whether order is significantly influential in reality, and it is. That is precisely why math can predict certain situations in advance of observing the situation. Human consciousness might be what defined "logic" for communicating about it, but consciousness was merely giving a name to a way reality operates. So ultimately, it was a higher thing, reality, that gave us the order which the logic operations of consciousnes reflect.

1. What Is Logic And What Is It A Product Of? Logic is order embodied in the conscious process of reason. Logic is a product of our consciousness of reality's order.

 

[quantumdude]

First-order predicate calculus. You reason from nothing by temporarily assuming some proposition, and seeing what follows from it.

Perhaps I'm not understanding you, but these two sentences seem to directly contradict each other. I do not see how you can reason from nothing and at the same time reason from an assumed proposition. Is that assumed proposition nothing? Does it come from the empty set of premises?

 

[Prometheus]

Please provide an example where you start with no premises and then derive some necessary proof, however you want to define necessary.

OK, here you go.

(1) 1. ~(P v ~P) [hypothetical supposition]
(1,2) 2. P [hypothetical supposition]
(1,2) 3. P v ~P [from 2, by disjunction introduction]
(1,2) 4. # [from 1 and 3, by contradiction introduction]
(1) 5. ~P [from 2 and 4, by negation introduction]
(1) 6. P v ~P [from 5, by disjunction introduction]
(1) 7. # [from 1 and 6, by contradiction introduction]
8. ~~(P v ~P) [from 1 and 7, by negation introduction]
9. (P v ~P) [from 8, by negation elimination]

You claim that you have derived a necessary proof, whatever necessary means. Please be so kind as to tell me what you have proven, as I have no idea. It seems that you are attempting to demonstrate a structural relationship, rather than dealing with content. You seem not to have attempted to represent content at all.

I notice that although you claimed to be able to start with no premises, you begin with 2 premises, in both 1 and 2.

Let me ask you some questions. You began as follows, did you not?

Notation:

P: some arbitrary proposition
~: negation symbol
v: disjunction symbol
#: contradiction symbol

Therefore, your very first line begins with a premise, which you offered to do without.

More importantly, your subsequent lines use the concepts of negation, disjunction, and contradiction. Did you define these concepts here? If not, then how can you use them? Are you assuming such concepts as given, or as defined elsewhere? This violates your conditions, as you claimed that you could provide a proof where nothing is accepted as given.

Therefore:
1. You have proved nothing of content, but only of structure.
2. You used terms that were not self-evident or defined herein, thereby violating the condition that you accepted of not requiring anyting as given.

 

[cogito]

Perhaps I'm not understanding you, but these two sentences seem to directly contradict each other. I do not see how you can reason from nothing and at the same time reason from an assumed proposition. Is that assumed proposition nothing? Does it come from the empty set of premises?

Do understand what a conditional statement is? It is a statement of the form "if X, then Y". Here's an example:

Suppose, hypothetically, that Tom is 6 feet tall. If so, then Tom is taller than all those shorter then 6 feet. So, if Tom is 6 feet tall, then he is taller than all those shorter than 6 feet. Now, does this little argument actually commit me to claiming that Tom is, in fact, 6 feet tall? No, it doesn't. The argument shows one necessary consequence of an assumption, without taking any position on the truth of that assumption. Go look up the truth-tables for conditional statements if you are still confused.

 

[cogito]

You claim that you have derived a necessary proof, whatever necessary means. Please be so kind as to tell me what you have proven, as I have no idea. It seems that you are attempting to demonstrate a structural relationship, rather than dealing with content. You seem not to have attempted to represent content at all.

I notice that although you claimed to be able to start with no premises, you begin with 2 premises, in both 1 and 2.

Let me ask you some questions. You began as follows, did you not?

Therefore, your very first line begins with a premise, which you offered to do without.

More importantly, your subsequent lines use the concepts of negation, disjunction, and contradiction. Did you define these concepts here? If not, then how can you use them? Are you assuming such concepts as given, or as defined elsewhere? This violates your conditions, as you claimed that you could provide a proof where nothing is accepted as given.

Therefore:
1. You have proved nothing of content, but only of structure.
2. You used terms that were not self-evident or defined herein, thereby violating the condition that you accepted of not requiring anyting as given.

Wow. You've never taken a course in logic, have you? There are no premises in the above argument There are temporary suppositions in sub-proofs that are discharged through the use of the appropriate deductive rules. Here is a tutorial on the subject:

faculty.washington.edu/smcohen/120/Chapter6.pdf

In the argument provided, the truth of the conclusion does not depend on the truth of any of those temporary suppositions. What the proof shows is that if a certain statement is true, then it leads to a contradiction just by virtue of its syntactic structure. Hence, the negation of that statement is true just in virtue of its syntactic structure. Hence, any statement that has that syntactic structure is true. Hence the conclusion of the argument is necessarily true.

Now, the conclusion of the argument, expressed in English, is that any proposition is either true or not true. Hardly informative, I know. This is why I claimed earlier that logic doesn't tell us interesting stuff about the world. By itself, all logic can do is establish truths that hold in every possible world; it cannot establish anything idiosyncratic or contingent about any particular world.

Yes, I've used the rules of deduction in the proof above. Are you claiming that the rules of deduction are premises to the above argument? If so, then you are using 'premise' in a completely different way than anybody who has studied formal logic. If you equate rules with premises, then you will be committed to the claim that proofs are impossible to construct, as they would require an infinite number of lines. For more information on this consequence of your equating rules to premises, please look up "Carroll’s Paradox ".

 

[Prometheus]

Dude, take a course on the subject.

Dude, what a quaint word.

Your proof proves nothing at all in the sense that I am speaking. Your proof proves that within the rules of logic there are relationships of logic. How incredibly wonderful of you. I said that logic can prove nothing in the real world, and you attempt to prove that relationships of logic can be proven.

Take a look at any of my posts in this thread before you came along with this. I said that logic cannot be used to prove anything in the real world. Your example is off that topic.

 

[quantumdude]

Do understand what a conditional statement is?

Yes, and I understand your symbolic proof, and I understand truth tables. What I am not understanding from your previous discussion is how you can derive necessary truths from an empty set of premises.

When you say, "There are no premises in the above argument,", it begs the question: Why don't you consider your "hypothetical suppositions" premises?

 

[cogito]

Dude, what a quaint word.

Your proof proves nothing at all in the sense that I am speaking. Your proof proves that within the rules of logic there are relationships of logic. How incredibly wonderful of you. I said that logic can prove nothing in the real world, and you attempt to prove that relationships of logic can be proven.

Take a look at any of my posts in this thread before you came along with this. I said that logic cannot be used to prove anything in the real world. Your example is off that topic.

Your claim is false. The above proof proves that a proposition must be either true or false. That applies to every world, including the real one.

 

[cogito]

Yes, and I understand your symbolic proof, and I understand truth tables. What I am not understanding from your previous discussion is how you can derive necessary truths from an empty set of premises.

When you say, "There are no premises in the above argument,", it begs the question: Why don't you consider your "hypothetical suppositions" premises?

Hypothetical suppositions are not premises because their actual truth value has nothing to do with the proof. The premises of an argument are such that their actual truth value is essential to the establishment of the argument's conclusion. Again, take the example of a conditional proof. A conditional proof starts by saying "Let's see what would follow if P was true. Now, if P was true, Q would follow. So, we know that the conditional claim if P, then Q is true, regardless of whether P is, in fact, true".

Hypothetical suppositions are agnostic as to the actual truth value of what is supposed, whereas premises are declarations that some statement has a particular truth value. When, in formal logic, a hypothetical supposition is discharged, it is thereby shown that the truth of the conclusion does not rest upon that supposition.

 

[Les Sleeth]

Hypothetical suppositions are not premises because their actual truth value has nothing to do with the proof. The premises of an argument are such that their actual truth value is essential to the establishment of the argument's conclusion. Again, take the example of a conditional proof. A conditional proof starts by saying "Let's see what would follow if P was true. Now, if P was true, Q would follow. So, we know that the conditional claim if P, then Q is true, regardless of whether P is, in fact, true".

Hypothetical suppositions are agnostic as to the actual truth value of what is supposed, whereas premises are declarations that some statement has a particular truth value. When, in formal logic, a hypothetical supposition is discharged, it is thereby shown that the truth of the conclusion does not rest upon that supposition.

I don't want to put words in Prometheus' mouth, but I understood his statement about logic not producing proof as meaning it doesn't produce proof about external reality. Everybody knows one can produce a proof within the system of logic, but all it tells you is if the logic is correct. Logic without empirical data really can prove nothing about external reality, just as Prometheus said.