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Truth Functional (TF): Has a true/false value which can be completely determined by the truthfulness/falsefullness (?) of the input's values (got that?).

Question: Show that "It is likely that __" is not a truth functional operator (where the blank, the "input", is filled with a simple sentence with a true/false value).

We can prove "likely" is not TF by finding a single counterexample. How do we find a counterexample? For any simple sentence I create which has a true/false value, I already know if it's likely or not.

For example: $A \equiv$ "The Yankees won the world series", is True. It's also likely. Now I need another sentence, $B$, which is True, but is unlikely. Right? Would like explanation and example (or explanation by way of example). Thx.

Edit: P.S. You cannot answer by saying "likely" is a prediction which is not TF. I want to know why "likely" (or even any prediction) is not TF

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    @Jeff I actually meant to say "I chose the number 1".2012-07-02

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I wouldn't go with finding a concrete counterexample; that seems to get us into a semantic morass that just detracts from the main matter.

Rather, I would say: By definition a "truth-functional operator" is one where the truth of "it is likely that $X$" depends only on whether "$X$" itself is true or not. Thus, if "likely" is a truth functional, then it must have a truth table, and that truth table must explain everything about what "likely" means.

"Likely" has only one argument, so there are only four different possible truth tables it could have:

  X   | X likely       X   | X likely      X   | X likely     X   | X likely   ----+---------       ----+---------      ----+---------     ----+---------   Yes | Yes            Yes | No            Yes | Yes          Yes | No   No  | Yes            No  | Yes           No  | No           No  | No 

But clearly neither of these matches any meaning that the word "likely" can have in English:

  • The first one would mean that everything is likely.
  • The second one would mean "it is likely that $X$" means the same as "$X$ is not the case".
  • The third one would mean that "likely" means nothing at all; "it is likely that $X$" would just be a more verbose way to assert $X$ itself.
  • The fourth one would mean that nothing is ever likely.

Since none of the possible truth tables work, "likely" does not have a truth table, and so it is not a truth-functional operator.

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    @FYI: I am told that she got all of the "prove is not TF" questions correct. Therefore the prof accepted this argument.2012-07-03