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No, I do not mean simply yelling "you lie" and I am not addressing special phraseology  such as "Jury Negation". Neither am I advocating simply calling a lack of rose colored glasses "negativity".

Instead I am concerned with what just happened above.  Just as a narrative may have active or passive voice, a statement (or proposition) may be cast in the affirmative or the negative. One can communicate as well and in some cases better using the negative, but it can, at times, require care on both the part of the writer and the reader.

With the election safely over I hope that we may soon see attempts at substantive discussion concerning goals, plans, proposals and legislation here on Daily Kos.  I anticipate that there will be a moderate amount of negation employed, if only by me. Reading some of the exchanges here over the past couple of years has convinced me that we might all benefit from a review of its uses, forms, mode of meaning, quirks and pitfalls.

I'll get more specific below the orange ruckus symbol.

What it is

One way to characterize propositions or statements is by whether they are affirmative (Socrates is a man.) or negative (Boiled turkey loaf is not red.). Negation can be achieved in most cases by simply prepending the phrase "It is not the case that" in front of an affirmative statement. The phrase "It is not true that " usually works equally well as does "It is false that". Also, though grammarians decry the dreaded double negative ("I don't got no ..."), it can be quite useful in analysis, and it would not be false to say that jurists and attorneys seem to love it.

Contradiction

The obvious use is simple direct negation of a pre-existing affirmative statement. Somebody says "Mitt would be a great President", and ye olde loyal Democrat says "No, Mitt would not be a great president". One needn't always wait for the affirmative statement, however, but can leap right out there asserting up front that "Mitt would not be a great President." Negation isn't limited to simple contradiction, however as shown by the preceding sentence. The meaning of that sentence would've been less easy to convey had I not started it with negation: "One needn't wait".

Specification

Things are often specified as ranges: "not more than x nor less than y". Special cases are "anything but", as in: "Cheese? Anything but Limburger" and everything but as in: "No onion". These types of usages avoid the time and effort otherwise involved in specifying everything allowed or included.  By excluding what isn't acceptable, identified or known, they delimit what is acceptable, identified or known.  This is often the quickest way to an answer. It can also enhance certainty: "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth." (Sherlock's dictum)

Quirks

Negation not only changes the quality of a proposition from affirmative to negative, but it changes the quantity from universal to specific.

"All fish are purple" is a universal affirmative.
"No fish are purple" is the corresponding universal negative, but it is not obtained by negating the universal affirmative. Instead, if you negate "all fish are purple" you get "it is false that all fish are purple" which actually means "at most some fish are purple", or "some fish are not purple". We have moved from the universal, "all" ,to the particular, "some".

To clarify (I hope):

     In classic form, there are four standard categorical propositions: universal and particular affirmatives and universal and particular negatives, normally identified as A, E, I and O.

A - Universal affirmative, "All S is P"
E - Universal negative, "No S is P"
I - Particular affirmation, "Some S is P"
O - Particular Negative, "Some S is not P"

A and O are contradictories, as are E and I. It is important to remember that contradicting a statement changes it both from affirmative to negative and from universal to particular. All too often you see somebody deny the truth of a universal affirmative, and subsequently, one or both parties to the discussion will take that to be equivalent to assertion of a universal negative, or the denial of a particular affirmative will be interpreted as a particular negative instead of as a universal negative.

Negation similarly changes the nature of the conjunctions in a compound statement

The assertion that Fred murdered Sam in the basement on Friday night with a five iron is a string of "and"s. Direct negation of that statement results in a series of "or"s.

Fred murdered Sam and did it in the basement and did it on Friday night and did it with a five iron is true only if every single "and" is true, which requires each clause to be true. It is false, however, if any clause is false, rendering any "and" false. Thus, the negation, "It is not the case that Fred murdered Sam in the basement Friday night with a five iron." is equivalent to "It is false that Fred murdered Sam or that he did it in the basement or that he did it on Friday night or that he used a five iron."

Syllogisms

Most have at some point heard some form of one of the the basic syllogisms, and very likely:

      All men are mortal
      Socrates is a man
      Therefore Socrates is mortal.

This works because 2 of the three terms are linked in each premise. What if we negate the first premise?

     No men are mortal
     Socrates is a man
     Therefore Socrates is not mortal.

This is because the first premise equates to "All men are non-mortal". The negative premise results in a negative conclusion.  So, now let's negate the second premise -

     All men are mortal
     Socrates is not a man
     Therefore ?????

Therefore nothing.  Socrates has been removed from the syllogism, the first premise doesn't apply to him and hence the syllogism conveys no information about him. The required linkage between the terms is broken.

I'm going to skip the other syllogism forms, but negation in syllogisms, especially those with "some" instead of "all" in one or more premises can be very tricky and are best avoided when writing, and meticulously analyzed when read.

The Hypothetical (or Conditional or Implication)

This refers to statements of the form "If A then B". Such statements do not say anything about the truth of A or B independently, but only that A implies B.  This is interpreted to mean that it is impossible for A to be true while B is false.

If A and B are both true, the statement "If A then B" is true. If A is true and B is false, the statement "If A then B" is false. When A is false however, formal logic says that the statement "If A then B" is true. This is because when A is false, it is not possible for A to be True while B is false. This odd technicality needn't rule discussion and analysis of reality, however, for the simple reality is that when A is false, all statements of the form "If A then ..." are irrelevant. They provide us no information, so we ignore them and move on.

If we negate the entire statement: "It is not the case that if A is true, then B is true", however, we do convey information. We are stating that B may be false some or all of the time when A is true, or, conversely, that it is possible for A to be true while B is false.

Mind your Grammar

It seems like almost once a month that somebody asserts that "you can't yell fire in a crowded theater". This is derived from a judge's statement to the effect that the first Amendment doesn't protect yelling fire in a crowded theater. Let's look at that derivation.

What is the subject of the Judge's statement? The first amendment. The statement is about the first amendment, it is not about yelling fire.What about it said Amendment? It does not protect. The verb is to protect, and is used in the negative.  Is there anything in there about prohibiting or outlawing anything? No.

We can recast the statement to make yelling fire the subject - "Yelling fire is not protected speech". The subject is now yelling fire, but the sentence is still about protection, not about legality or illegality. You don't need a degree in logic to simply follow the grammar.

It is important to understand what is being negated.

Fred ate clams. Simple; subject + verb + object, declarative, affirmative.

"It is not the case that Fred ate clams." The entire statement that Fred ate clams is false but we don't know why.

(Not Fred) ate clams - clumsy, but somebody ate clams, somebody other than Fred.

Fred (not ate) clams - Fred either didn't eat or at least didn't have clams.

Fred ate (not clams) - Fred ate something, but not clams.

Make sure to apply the negative to the right part of the sentence when writing, and make sure to reason out the effect of its placement when reading.

Don't forget to take extra care in deriving affirmative statements from negative ones, and in analyzing instances where others have done such. It is easy to read things into negative assertions that are not there.

12:38 PM PT: Thank you community spotlight people.

6:06 PM PT: Belated thankx to J Town & SciTech for the republications

Originally posted to enhydra lutris on Thu Dec 13, 2012 at 07:52 AM PST.

Also republished by J Town, SciTech, Logic and Rhetoric at Daily Kos, and Community Spotlight.

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