No. Formal logic (valid reasoning) guarantees only that from true statements, a false conclusion (statement) cannot be deduced. Valid reasoning preserves the truth of premises. It does not say that from false premisses a true conclusion cannot be reached (by a multiple logically invalid steps).

Philadelphia, PA

Dear Radovan,

You are right when you say the following:

No. Formal logic (valid reasoning) guarantees only that from true statements, a false conclusion (statement) cannot be deduced. Valid reasoning preserves the truth of premises. It does not say that from false premises a true conclusion cannot be reached

---End quotation

However, when you go on to suggest that one can only reach a true conclusion from false premises "by a multiple logically invalid steps," then you go wrong.

To see the point consider the following logically valid argument:

(1) All three-legged and inanimate things are mortal. (False, given that "mortal" means dies at some time.)

(2) All men are three-legged and inanimate. (False)

(3) Therefore, All men are mortal. (True)

This argument has the same valid logical form as the following argument:

(1) All men are animals.

(2) All animals are mortal.

(3) Therefore, All men are mortal.

Logical validity is a matter of logical form alone (what the two arguments above have in common,e.g.) , and it does not depend on the specifics of the non-logical vocabulary. What is usually wanted, by way of arguments, though, is an argument which is both valid and sound. An argument where all the premises (or some) are false, can still be logically valid. But it is unsound and will have little practical use.

Stipulating that the premises are false, we have no grounds for asserting the conclusion on the basis of the argument. The argument may be useless, even if the conclusion is otherwise known to be true.

H.G. Callaway

Well, no amount of pure logic can by itself establish the truth or falsness of 

anything. You have to have empirical contrastation.

In the history of the Calculus I believe that the whole thing was thought to work out

because of a compensation of mistakes.

Callaway says: "However, when you go on to suggest that one can only reach a true conclusion from false premises "by a multiple logically invalid steps," then you go wrong."

You are right: strictly speaking, the final part of my answer is not correct. But I am currently too busy, so that I should not answer anything. This question was offered by RG, and so ... I answered in haste.

Dear Callaway,

In fact, I did not "go wrong". My claim is correct!

I wrote that logic "does not say that from false premises a true conclusion cannot be reached (by a multiple logically invalid steps)."

I meant the claim "by a multiple logically invalid steps" as one possible way; you interpreted my claim as a necessary condition, which is a wrong reading of my claim.

You can reach a true conclusion by making "a multiple logically invalid steps". My claim does not exclude other possibilities.

In sum, your explanation is correct, but it does not contradict (exclude) my claim, nor do my claim contradict (exclude) what you say. I must really leave for the time being.

https://en.wikipedia.org/wiki/Simpson's_paradox

Dear Remi Cornwall,

Yes, thank you.  I think that could very well be the answer to my proplem.

Dear George Stoica,

You say:

Maybe Steve is suggesting that, by taking the good part out of two rather bad things, something good might arise. That might work...

Yes that is the kind possibility I am thinking of.  I'm thinking of two integrals that do not converge, but when one is substituted inted the other, or the other substituted into the one, the result is an integrl that does converge.

I need to check my mathematics very carefully, but the application I'm thinking of is in quantum mechancis where states can seemingly, spontaneously arise from nothing. Or from "the vacuum".

A brief remark:

Juan say: "Well, no amount of pure logic can by itself establish the truth or falsness of of anything."

This is mostly correct; strictly speaking, logic deals with valid forms of reasoning, and with formal consistency of discourse. There is no truth in pure logic (which functions at the level of syntax).

Truth enters with semantics (interpretation of symbols). Interpretation is usually empirical, but it does not need to be (that is why I said "mostly correct"). In sum, logic is a rather complex technical matter, not a matter of impressions.

Philadelphia, PA

Dear Radovan,

Sorry to be correcting you so soon again, but you wrote:

This is mostly correct; strictly speaking, logic deals with valid forms of reasoning, and with formal consistency of discourse. There is no truth in pure logic (which functions at the level of syntax).

---End quote

There are many truths in pure logic, one corresponding to each valid form of argument at the least.

If the argument, A & B, therefore C, is logically valid, then there is no way to assign truth-values to the statements A and B, such that statement C turns out false.

In consequence, the compound statement,

"If (A & B) then C" turns out true on every interpretation of the non-logical vocabulary, which is to say it is a logical truth.

For example, if A= "All animals are mortal"; and B = "All men are animals"; and C = "All men are mortal."

Then,

Corresponding to the valid argument,

(1) All animals ae mortal.

(2) All men are animals.

Therefore, (3) All men are mortal,

We get the logical truth:

"if all animals are mortal and all men are animals, then all men are mortal."

This form of statement is true on every uniform substitution of the non-logical vocabulary; and that is what a logical truth is.

H.G. Callaway

Philadelphia, PA

Dear Cornwall,

A small correction to your earlier note in answer to the present question. You wrote:

Also you can prove anything from a false premise (see diagrams below and third)

---End quotation

You seem to conflate two importantly different points. The first is that any "material implication" with a false antecedent is true. This is a matter of the truth table for the truth-functional connective "-->" (usually its symbol is the horseshoe). If the statement "A" if false, then the statement "If A then B" (A --> B) is true.

A   B   A--> B

T   T       T

T   F       F

F   T      T

F   F      T

However, a proof that "If A then B" is not a proof of B.

The second point of interest is that any statement at all follows logically from a contradiction. Here it is a matter of arguments, while above it was a matter of statements. The argument, A and not-A, therefore B, is logically valid, no matter what statement B may be. 

However, none of this is strictly a matter of "proof," if a proof be a matter of our being able to assert the conclusion of an argument on the basis of the premises offered. Where the premises are known to be false, we cannot assert the conclusion of a logically valid argument, on the basis of that argument. But that is usually what is wanted from a proof. 

H.G. Callaway

I think that false physical models can combine and build together correct predictions. The big problem about this is that sometimes correct statements stay on false proofs. People find errors in the proofs, and then they claim that the statement itself is false. This is not always true. They proved only that this particular proof is false.

The situation is particularly embarassing as the last 150 years compose together the most active time in developping and concatenating physical theories.

Particularly, the statement E = mc^2 stays only on proofs which are literally false, but can still be correct. I don't know it to have been hardly fasified by experiments. Also "dark energy" and "dark matter" are almost sure inconsistent, but they build also some correct predictions together with other theories, which are not completely true. We need a fast and radical clean and clear action, or we run directly in a major crisis of science, faster than we believe.

Thank you Miihai.

Always you have something useful to say.

Dear Callaway, I must tell you that you know about logic slightly less than you think you do. Logic is a highly technical matter, about which most people know nothing beyond trivial tricks.

You say: "If the argument, A & B, therefore C, is logically valid, then there is no way to assign truth-values to the statements A and B, such that statement C turns out false."

Correct; but logic does not need to assign anything to anything. Axiomatic systems do not need any "truths". That is what I said in the previous post, when I wrote that logic functions at the level of syntax. Assignments (meaning & truth) are a matter of semantics, which may be added to the syntax, but does not need to be. And so forth. Let me conclude with a paragraph from the book on which I currently work:

---------

Bertrand Russell wrote that "pure mathematics is the subject in which we do not know what we are talking about, or whether what we are saying is true". This famous claim is essentially correct, but it is actually misleading because it is imprecise. Strictly speaking, pure mathematics does not speak about anything, nor does it tell anything, true or false. Pure mathematics is a game with symbols and with symbolic expressions, which is played in accordance with the adopted rules of shaping the expressions (formulas) and of inference (of new formulas). This game by itself contains no specific meanings and makes no meaningful statements, so that it cannot express either truths or falsities. Meaning and truth enter into this story with the interpretation of a mathematical game (a formal system, pure mathematics). The interpretation includes a projection of the symbols and symbolic relations on the phenomena of the real world. With such a projection, the meaning and truth enter the story; however it is not the pure mathematics that tells ("carries") the meaning and truth, but its interpretation.

----------

What I said about mathematics, holds for logic, which is a part of mathematics.

When you start assigning truth of falsness to  statements, you are out of pure logic, into applied logic.

Applied math would say for instance t=2, pure math would  only say  t=t

Philadelphia, PA

Dear all,

I quote from the Stanford Encyclopedia of Philosophy's article on "Logical Truth":

On any view, logic has as one of its goals to characterize (and give us practical means to tell apart) a peculiar set of truths, the logical truths, of which the following English sentences are paradigmatic examples:

(1) If death is bad only if life is good, and death is bad, then life is good.

(2) If no desire is voluntary and some beliefs are desires, then some beliefs are not voluntary.

(3) If Drasha is a cat and all cats are mysterious, then Drasha is mysterious.

See:

http://plato.stanford.edu/entries/logical-truth/

It is simply a very standard view of logic.

H.G. Callaway

Dear Callaway, let me try to clarify the situation, if I may.

In brief, I do not think your "corrections" of my claims were needed. They actually confirmed what I said. 

You first "correction" was a result of wrong interpretation of my claim. I said "A", and you said: wrong, it is "B". - I then explained (in the next post) that it is both: "A" and "B". My claim "A" does not mean "not B". So, we are both right

Regarding your "second correction", you wrote:

(1) "If the argument, A & B, therefore C, is logically valid, then there is no way to assign truth-values to the statements A and B, such that statement C turns out false."

"Correct", said I. But let me remind you that in my previous post I wrote:

(2) "... strictly speaking, logic deals with valid forms of reasoning, and with formal consistency of discourse."

Your statement (1) confirms what I said in (2). Namely, if "there is no way to assign ...", then this means that this property (of your formula) is a matter of its "form", as I wrote in (2). In sum, your "correction" confirms my claim. We re saying the same: you with an example, I with the general principle.

Regarding SEP, there are several levels of dealing with logic. Regarding the "level of truth" (that SEP illustrates), let me repeat what I wrote my first post:

"Formal logic (valid reasoning) guarantees only that from true statements, a false conclusion (statement) cannot be deduced. Valid reasoning preserves the truth of premises. It does not say that from false premises a true conclusion cannot be reached ..."

This tells the essence, much better than SEP does.

Anyway, there is no need for us to "quarrel" because we are saying the same things in different ways. You are using examples, and I tried to express the essence behind the examples.

I would now leave this story to new forces.

Philadelphia, PA

Dear Radovan,

No need to carry that particular discussion further, I suppose. Certainly, I hold that logical truths are recognized by their forms. We can equally say that they are all capable of being derived by logical rules from zero premises. That is a more "syntactic" approach. I like your revisions of your earlier claims better.

I suspect, in any case, that the "combine" of the question here,

If two false statements can combine to prove a statement that is true , is there something remarkable about this kind of proof?

may not be best looked at as a matter of of logical addition and deduction, though there is certainly some tendency above to adopt such an interpretation of the question. One alternative is to consider that we have two "false" theories and we are wondering how to get beyond them to some new synthesis? 

H.G. Callaway

There is a hidden factor (so serious that it amounts to an error) in posing the question.  For the question regarding what is deducible, if anything, from false statements, I completely concur with Remi's answer.

But the "explanation" of the question concerns integrals that don't exist, and because the mapping is not stated, it implies that "integral such and such does not exist" always maps to a false statement.

But on close reading, you will see that actually "such an integral doesn't exist" is a true statement in this context.  The implied mapping is in error!

@Steve,

Please see this example:

4 + 5 = 11;

2 + 3 = 7;

Both equalities are false but if we subtract the second one from the first one we get

2 + 2 = 4,

which is true.

So, my advice is as follows: check the two integrals and find what is wrong is each one of them. Then, check if the operation of "inter-substitution" that you do, eliminates the problem in the two integrals.

I think that Sofia offered a very powerful answer. What I can do is to repeat her argument using a different language. Any statement is neither true or false if not referred to the specific environment in which it is embedded.

Let us denote by 1, 2 two statements and write

1=F1+T1

2=F2+T2

where F and T are the false or true contents referring to an environment or to an other 

(in Sofia's example the syntax of the statements is true, the arithmetic is false)

If the intersection 1&2 falls in T then we get a T statement. But the logic has been abused. This is the kind of abuses currently adpted in the theory of renormalization.

See the paper by Del Franco and myself "Euler Legacy to modern science"

The best

Pino

Here is the actual example that prompted my original question

I think this is possible by using an exclusive extraction method, however is there is not any "true", we can not do anything about "false" ones.