My complaint is mainly one situation that occurs when a contradiction is found in a logical sequence.

The rules allow almost any type of mischief to be attached to remove the contradiction, whether the attachment has any validity or not. A link is given to a good introductory text on symbolic logic for people with little background in mathematics.

http://www.amazon.com/Understanding-Symbolic-Logic-

Virginia-Klenk/dp/0132051524/ref=sr_1_1?ie=UTF8&s=

books&qid=1275250987&sr=8-1

The example is given on page 129 and 130 of the third edition. When a contradiction occurs in a logic sequence, you can use it to prove anything you wish.

If I offer a contradiction to Hank, something like " the opponents are invited to prove a statement that is not true," then he actually can claim under the written rules of symbolic logic that he is the Pope. Bertrand Russell has done this to us.

There are better choices that didn't make it into that particular rule. For example the rule could have said that when a contradiction occurs, one of the premises must be changed to a conditional.

Then I could have said " the opponents are invited to prove a statement if it is true," Maybe it's more fun the other way, but it certainly looks like Bertrand Russell mischief.

## Comments

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You have to consider that Russell and his followers are all like me and you, circumscribed within a limited human frame, and that logic is not natural to human cognition, it takes effort. I'm with Lakoff personally that I can only think what I can and the way I can and others can only understand what they can according the same human constructional circumscriptions.

In other words the perfect machine is not possible if it originates from us human beings, it will crash as surely as anything in windows does.

In the real world as observed by scientists, a proposition can only be

*probably*true, never

*absolutely*true. In language, especially in English, we have many terms which appear to convey truth values but which cannot do so because of the inherent variability of meaning in every word. Absolute truth cannot co-exist with ambiguity. Fortunately, probability can not only co-exist with ambiguity: it is the flip side of the same coin.

Probability is the tool we use to narrow down the range of ambiguity in every utterance. Ambiguity and probability are thus inherent in every word, every phrase, every utterance. Language, masquerading as logical propositions, can be used to prove or disprove any proposition. But only by completely disregarding the fact that, due to ambiguity, words can only convey

*probable*truth. Common sense is the logic which sorts out the probable from the improbable.

As an example, let us assume that Hank is 'really' in America according to the conventional rules of language and of common-sense logic. I shall now use a form of logic to prove that Hank cannot possibly be in America.

**Given the truth of the following propositions:**

Hank is not in England.

Hank is not in Canada.

Hank is not in Siberia.

**It follows that Hank must be somewhere else.**

If Hank is somewhere else then he cannot possibly be in America.

q.e.d.

You might guess that I, also, am not a Bertrand Russell fan. :-)

**It follows that Hank must be somewhere else**is that

**Hank must be somewhere else other than,**

**England or Canada or Siberia.**

Therefore the exclusion cannot be and is not universal but qualified.

**Eric**.

The expression 'somewhere else' is a relative expression and without a reference to its related term or terms it is hopelessly ambiguous. It is a favorite trick of propagandists to set the mind along a track and to leave unsaid any words which point to alternate routes.

Sometimes there is no need to resolve ambiguity. The command "Be elsewhere!" needs no logical analysis. :-)

But, I would say without a doubt that my training in logic greatly improved my critical thinking skills. It kind of rewires your brain, so that your using the logic even when you are not consciously aware that you are doing so. I personally think it makes for a better citizen.

Otherwise we have all those fighting x-es who use this awkward logic:

- This is clear, what is the most comfortable for a human being.
- This is clear, what stands on the way of most comfortable.
- So lets fight against what stands on the way of it.

And thus, because of lack of logic, we always get those stupid and good fighting against things, which keep them alive - and sometimes, unfortunately, winning.

Mathematical abstractions are usually applied to physical systems in ways that are adequate for a particular purpose, but widely recognized as only an approximation. The mathematics of quantum mechanics is the one that comes closest to disproving Lakoff, and that alone should be a source of embarrassment to Lakoff’s opponents.

Quantum mechanics begins by postulating a Fourier series that inserts infinities and singularities into a physical system that apparently has none and needs none. Then the mathematicians try to arrange the many terms in sets and pairs of functions where the non physical things cancel out. Remember quantum mechanics is mathematics’ best chance for a disproof of Lakoff, a messy ensemble of things that defy logic except as coefficients to probability functions, in a barely rational frame work that no one would accept if the experimental results were not repeatable.

In summary, I believe there is a mathematics in the physical world beyond human biology. It is the same everywhere and emerges from the vacuum of space described in the Zero Point field. It isn’t a mathematics that people want, or like, or easily embrace. It is grudgingly accepted slowly in small parts against formidable opposition, but still accepted because it passes all of the physical test that are supposed to finish it off once and for all.

I make it simple:

Applies everywhere => Applies in my mind => No need for experience.

(the rule of just thinking it out)

It takes some time to experiment physically&it takes almost no time to experiment mentally => Experiment mentally.

(the rule that mathematicians always get it before physicians)

First, a contradiction does not prove anything. To prove something, all the premises have to be true. But extensional systems , such as that of Principia Mathematica, you do find this odd feature that any proposition follows from a contradiction.

Second, this feature can be quite useful. You can prove that a set of premises is not self-contradictory by proving that there is at least one proposition that does not follow from the set of premises. It also is handy in reductio ad absurdum proofs,

Third, this was not just something that Russell or other logicians decided to stick in, for fun. It’s something that follows from some simple and apparently innocent rules of inference.

From ‘p’ you can infer ‘either p or q’. (For example, from “It’s raining” you can infer “Either it’s raining or it’s snowing.”)

This rule, sometimes called “weakening,” is quite useful, for example, allowing us to go from “If it is raining or snowing, the picnic is cancelled” and “It’s raining” to “The picnic is cancelled.”

From ‘Either p or q’ and not-p’ you can infer ‘q’. (From “Either Columbus landed in America in 1942 or Columbus landed in America in 1492” and “Columbus did not land in America in 1942” you can infer “Columbus landed in America in 1492.)

This rule, the “Disjunctive Syllogism,” is, I suspect, familiar to anyone who has taken multiple-choice tests.

Here’s the proof that two contradictory premises imply any proposition whatsoever.

(1) p (premise)

(2) not-p (premise)

(3) Either p or q (from (1), by Weakening)

(4) q (from (3) and (2), Disjunctive Syllogism)

So, to get rid of this “paradox of implication” you have to somehow eliminate or severely modify either one the other of those two rules of inference.

Fourth, I believe Anderson & Belnap did this (in Entailment). But I haven’t followed the literature for about 30 years, so I’m not sure how successful their efforts were.

Oh, one more thing. If I have contradicted myself, here or elsewhere, let me apologize for all those things that I may have implied about your mother. I didn’t really mean to imply them.

it thinks that logic should not be unveiled in high school it will just confuse the hell out of people. "it makes me sad to wake up in the morning for i know while i am still in bed anything is possible but once i start doing i start limiting what is possible"-a jackass who thinks too much