Editing Talk:703: Honor Societies

:: How is that not a tautology? For any proposition A, if the proposition is true, then A; if not, then ~A. Logic doesn't allow for a proposition to be both true AND false, nor does it allow for a proposition to be neither true NOR false, so the only remaining possibilities are A and ~A; ergo, A v ~A. [[Special:Contributions/108.162.216.38|108.162.216.38]] 16:44, 11 February 2014 (UTC)

:: How is that not a tautology? For any proposition A, if the proposition is true, then A; if not, then ~A. Logic doesn't allow for a proposition to be both true AND false, nor does it allow for a proposition to be neither true NOR false, so the only remaining possibilities are A and ~A; ergo, A v ~A. [[Special:Contributions/108.162.216.38|108.162.216.38]] 16:44, 11 February 2014 (UTC)

::Not in all forms of logic and mathematics.  Intuitionism, in particular (check Wikipedia) treats "true" as equivalent to "provable" and "false" as equivalent to "disprovable," since math is not an abstract Platonic ideal, but a human construction. Even under conventional math, "The current King of France is bald" is neither true nor is it false, because there is no current King of France.  x \elem S is neither true nor false if x is not well-defined. 02:13, 17 March 2022 (UTC)

::Not in all forms of logic and mathematics.  Intuitionism, in particular (check Wikipedia) treats "true" as equivalent to "provable" and "false" as equivalent to "disprovable," since math is not an abstract Platonic ideal, but a human construction. Even under conventional math, "The current King of France is bald" is neither true nor is it false, because there is no current King of France.  x \elem S is neither true nor false if x is not well-defined. 02:13, 17 March 2022 (UTC)

:Unnecessary nitpick aside, then, there are more serious things. I presume the sentence, "It would also be true if there were somehow 1.000.000 members of the group without 1.000.000 people joining it," should be, "It would also '''not''' be true if there were somehow 1.000.000 members of the group without 1.000.000 people joining it." (Otherwise, the "also" is used incorrectly, and the sentence is useless.) Unfortunately, this would make it wrong; a statement of the form "if A then B" is not false if B is true and A isn't. (This is the difficulty of making formal logic: the traditional conditional leads to bizarre, vacuous truths.) Also, more seriously, you say that "if A then A" is a longer way of saying "A", or, more formally, that "A → A" is logically equivalent to "A." Unfortunately, this is not the case. The statement "if A then A" is always true, and hence a tautology. You also assert that "A = A" (or "A ↔ A") is logically equivalent to "A", where "A" is "The first rule of tautology club." This is even more obviously false. Even if "The first rule of tautology club" yields falsehood, it is still equivalent to itself.

:Unnecessary nitpick aside, then, there are more serious things. I presume the sentence, "It would also be true if there were somehow 1.000.000 members of the group without 1.000.000 people joining it," should be, "It would also '''not''' be true if there were somehow 1.000.000 members of the group without 1.000.000 people joining it." (Otherwise, the "also" is used incorrectly, and the sentence is useless.) Unfortunately, this would make it wrong; a statement of the form "if A then B" is not false if B is true and A isn't. (This is the difficulty of making formal logic: the traditional conditional leads to bizarre, vacuous truths.) Also, more seriously, you say that "if A then A" is a longer way of saying "A", or, more formally, that "A → A" is logically equivalent to "A." Unfortunately, this is not the case. The statement "if A then A" is always true, and hence a tautology. You also assert that "A = A" (or "A ↔ A") is logically equivalent to "A", where "A" is "The first rule of tautology club." This is even more obviously false. Even if "The first rule of tautology club" yields falsehood, it is still equivalent to itself.