2001-05-12

tautological

HINT: It is not "instructed in rational thought". Nor is it "relating to the study of tension" (with a tip of the hat to Joe Culhane).

Involving tautology, which is needless repetition of an idea in different words. Sometimes, though, the words don't really change much, as in this tautology: "If you want to change some things in your life, you're going to have to change some things in your life". Another somewhat well-known tautological statement made by Abraham Lincoln in reference to a book: "People who like this sort of thing will find this the sort of thing they like."

In mathematics, a logical statement in which the conclusion is equivalent to the premise (another way of saying that a statement is trivially true).

Consider the logical function FOO(X,Y) = X AND Y, which evaluates to TRUE if and only if X and Y are both TRUE, and evaluates to FALSE otherwise. This is a simple example of a logical function that is not a tautology.

Now, consider the logical function BAR(X,Y) = TRUE. No matter what the values of X and Y, the function will evaluate to TRUE. It is a tautological function: TRUE for every possible input.

In some of my mathematical writings, I actually define the value of TRUE itself to be a tautological function taking any number of arguments: TRUE(...) = TRUE(). I thus consider the name "TRUE" to be a short synonym for "the tautology", since the tautology that defines the value TRUE is the most fundamental tautology there could be.

A related concept is that of contradiction: FALSE for every possible input. I write for FALSE: FALSE(...) = FALSE(), thus considering "FALSE" to be a short synonym for "the contradiction", since the contradiction that defines the value FALSE is the most fundamental contradiction there could be.

One final note, just for fun: The symbol I use for TRUE in my formulae is actually "⊤" (that is UNICODE character 0x22a4 "DOWN TACK", which looks like the letter "T", in case it doesn't look right in the version of this you are looking at). The name I use for this symbol is verum, after the Latin word for true, real, actual or genuine.

The symbol I use for FALSE in my formulae is actually "⊥" (that is UNICODE character 0x22a5 "UP TACK", which looks like an upside-down letter "T", in case it doesn't look right in the version of this you are looking at). The name I use for this symbol is falsum, after the Latin word for an untruth, falsehood, fraud or deceit.

More information at Dictionary.com.

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