Wednesday, August 18, 2010

Wednesday Math, Vol. 125: Logical quantifiers

For a statement to pass muster as "logical" in the modern sense, it must be verifiable as either being true or false. For example, if I type "Birds are black", this statement is too vague to be logical. The most common interpretation of the sentence would be "All birds are black", which would count as being usable in a system of logic, even though it is false. If someone states "The panther is the only creature that is black." and I then type "Birds are black", the most common interpretation would be that I was saying "Some birds are black" in a somewhat lazy way, and that is a logical statement and true as well.

To be precise, logical statements need one of the two quantifiers, the existential or the universal. The backwards capital E is the symbol for the existential quantifier. In regular English usage, this would be "Some birds are black." or "Some x has the property P." Being precise, the existential quantifier says "There exists at least one ___ that is ___." The universal quantifier is pretty much the same in standard speech or in precise statement. the upside down capital A means "All ___ are ___."

The quantifiers are not difficult ideas in themselves, but a lot of logical errors occur because people do not understand how to state the logical opposite of a quantified statement. If you want to prove "All birds are black." is false, this is the same as proving "There exists a bird that is not black." is true.

With the existential quantifier, the contradiction will negate the meaning of the statement and switch to universal. The opposite of "There exists a black bird." is "All birds are not black."

In the biggest so-called "controversy" of the moment, many people are taking the true statement "There exist Muslims that wish us harm" and changing it into the false statement "All Muslims wish us harm". Substitute "Muslims" with whatever the popular scapegoat of the era is and I daresay no mental flaw can match this incorrect switching of quantifiers when it comes to bringing evil action into this world.

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