[math-fun] Nth roots of tautology
Dear all: Negation and tautology are both square-roots of tautology, as: "This statement is true." = "This statement is not not true." Do there exist words (in any language) which act as cube-roots of tautology (excluding the trivial case of tautology itself)? Or, more generally, Nth roots of tautology, for arbitrary natural numbers N? If we did have such words in the English language, what logical effect would they have on the meaning of a sentence? Sincerely, Adam P. Goucher
This doesn't have to do with logic strictly, but I think the term you're looking for is "functional square root" or "half-iterate", and their generalizations. Given a function f, the half iterate of f is the function g such that g.g = f, where the period denotes composition. You can generalize to g^n = f, of course. -Robert See http://en.wikipedia.org/wiki/Functional_square_root On Sun, Mar 6, 2011 at 10:18 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Dear all:
Negation and tautology are both square-roots of tautology, as:
"This statement is true." = "This statement is not not true."
Do there exist words (in any language) which act as cube-roots of tautology (excluding the trivial case of tautology itself)? Or, more generally, Nth roots of tautology, for arbitrary natural numbers N?
If we did have such words in the English language, what logical effect would they have on the meaning of a sentence?
Sincerely,
Adam P. Goucher _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
My twin's twin... My spouse's spouse... My triplet's other triplet... The enemy of my enemy... Shirtsleeves to shirtsleeves in 3 generations... At 08:18 AM 3/6/2011, Adam P. Goucher wrote:
Dear all:
Negation and tautology are both square-roots of tautology, as:
"This statement is true." = "This statement is not not true."
Do there exist words (in any language) which act as cube-roots of tautology (excluding the trivial case of tautology itself)? Or, more generally, Nth roots of tautology, for arbitrary natural numbers N?
If we did have such words in the English language, what logical effect would they have on the meaning of a sentence?
Sincerely,
Adam P. Goucher
That's an interesting one. If I, A and B are the three siblings, and since the interpretation of *My triplet's other triplet* depends on whether you choose A or B as the first referent, *My triplet's other triplet's other triplet* could refer to any of the three. But if the first "triplet" always refers to A and "other" always picks the one that has been mentioned least recently, then both of these: *My triplet is A* *My triplet's other triplet's other triplet's other triplet is A* could be interpreted as true, making "*'s other triplet*" the cube root of a tautology. Presumably by "Shirtsleeves to shirtsleeves in 3 generations" you suggest something like *My work dress is shirtsleeves* *My father's work dress is a dress shirt and tie * *My father's father's work dress was overalls * *My father's father's father's work dress was shirtsleeves* the trouble with this one is you probably cannot add another 3 generations, or 6 or 9, etc. as suggested by the original question (because the square, cube, etc. of the tautology should still be a tautology) On Mon, Mar 7, 2011 at 08:38, Henry Baker <hbaker1@pipeline.com> wrote:
[...] My triplet's other triplet... [...] Shirtsleeves to shirtsleeves in 3 generations... [...] -- Robert Munafo -- mrob.com Follow me at: fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com- youtube.com/user/mrob143 - rilybot.blogspot.com
The quantum computing folks have an operator called "the square root of Not". tautology^.25 ? --Rich ----- Quoting Robert Munafo <mrob27@gmail.com>:
That's an interesting one. If I, A and B are the three siblings, and since the interpretation of
*My triplet's other triplet*
depends on whether you choose A or B as the first referent,
*My triplet's other triplet's other triplet*
could refer to any of the three. But if the first "triplet" always refers to A and "other" always picks the one that has been mentioned least recently, then both of these:
*My triplet is A* *My triplet's other triplet's other triplet's other triplet is A*
could be interpreted as true, making "*'s other triplet*" the cube root of a tautology.
Presumably by "Shirtsleeves to shirtsleeves in 3 generations" you suggest something like
*My work dress is shirtsleeves* *My father's work dress is a dress shirt and tie * *My father's father's work dress was overalls * *My father's father's father's work dress was shirtsleeves*
the trouble with this one is you probably cannot add another 3 generations, or 6 or 9, etc. as suggested by the original question (because the square, cube, etc. of the tautology should still be a tautology)
On Mon, Mar 7, 2011 at 08:38, Henry Baker <hbaker1@pipeline.com> wrote:
[...] My triplet's other triplet... [...] Shirtsleeves to shirtsleeves in 3 generations... [...] -- Robert Munafo -- mrob.com Follow me at: fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com- youtube.com/user/mrob143 - rilybot.blogspot.com
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