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Negation
Of Simple Statements
Negation is a tricky thing. There are two senses. One is to falsify, the other is to make the statement to say the opposite. Consider the following
Assume the following statement is true. "All crows are black." All it would take to make it false would be for one crow to be something other than black, say brown. So to falsify that statement we can say "Some crows are not black." Now to state the opposite of "All crows are black," one could say "No crows are black." or "Crows are not black," or "Crows are non-black."
Of Quantifiers (subjects and the copula)
If no quantifier is present then all is assumed.
Sometimes the quantifier "no" is replaced by putting "not" with the copula (is/are). The the truth of the alternate statements is the same.
Examples of "no" vs. "are not".
| Standard form | Alternate form | Comments |
| No cows are cats.
No cow animals are cat animals. |
Cows are not cats.
Cow animals are not cat animals. |
Two mutually exclusive groups (cows and cats) in a larger universe (animals). |
| No boys are girls.
No boy persons are girl persons. |
Boys are not girls.
Boys persons are not girl persons. |
The division of a universe (persons) into two regions (boys and girls). |
It is a good practice to transform "are not" statements to "no" statements at the very beginning. This will prevent confusion later on when "are not" is misread as a negative predicate.
| p | ~p | Comments Use this regardless of the quantifier. |
| T | F | |
| F | T |
| Statement | Negation/Falsification | Comments |
| All | Some ... not | The negation of "all" is not "no!" There still be some, just not all the individuals have this attribute. |
| Some | No ... are not |
|
| Some ... not | All | |
| No ... are not |
Some | The negation/falsification of "no" is not "all!" |
Examples
| Group | Statement | Negation/Falsification | Comments |
| All/Some not | All cows are black cows. | Some cows are not black cows. | This allows cows to be black or some other color. |
| All/Some not | All roses are red in color. | Some roses are not red in color. | The statement asserts that roses are only red. The negation allows for roses to be white, or yellow as well as some of them being red. |
| Some/No | Some cows are black cows. | No cows are black cows. | If there is only one black cow in the world they you have to say some cows are black. If it is never possible for a cow to be black then no cows are black. |
| Some ... not/ All | Some cows are not black cows. | All cows are black cows. | If some cows are not black is false then they all have to be black. |
| No/Some | No cows are black cows. | Some cows are black cows. | Since no cows are black is false then some cows must be black, but all cows to have to be black. |
Of Predicates
Predicates are negated by prefixing "non" to the noun or noun phrase that comprises the predicate. When the predicate is an adjective then it must be expanded to a complete noun phrase. Negating a predicate both falsifies the statement and makes it state the exact opposite of the original.
| q | ~q | Comments |
| T | F | |
| F | T |
Examples
| Statement | Restated with only the predicate negated. | Comments |
| All crows are black
All crows are black colored birds |
All crows are non-black.
All crows are non-black colored birds. |
Here the adjective had to be expanded to the implied noun
phrase.
It is common in these cases to negate the quantifier (No crows are black colored birds.) |
| Cows are grass eaters | Cows are non grass eaters. | |
| Cows are non grass eaters | Cows are non non grass eaters Cows are grass eaters |
If a predicate is already negative then negating it make a double negative (non non) and may be removed by the law of double negatives. |
Double Negation
Not not p
~(~p) may be replaced with p in any logical statement.
| p | ~p | ~(~p) |
| T | F | T |
| F | T | F |
Be careful here. If the quantifier is "some" it is easy to get into trouble. Basically if you are not sure leave it alone.
Examples
| Statement with double negative | Double negative eliminated | Comments |
| Crows are not not black | Crows are black. | Double negative in the copula |
| No crows are non-black | Crows are black | Quantifier and predicate were negative. |
| Crows are not non-black | Crows are black | Copula and predicate were negative. |
| No crows are not black | Crows are black | Quantifier and copula were negative. |
Compound Statements
Compound statements do not present the same difficulties with falsification and negation. Here the two terms can be used interchangeably.
In the follow p and q stand for the entire simple statement on each side of the operator.
And (the conjunctive)
Or (the disjunctive)
If - Then (the conditional)
