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Suppose a statement form $\varphi$ always has value T or U. Show $\varphi$ is a classical tautology.

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This question is quite badly formulated. What is T? What is U? Do you mean first-order or just propositional? –  boumol Feb 10 '11 at 0:34
    
See en.wikipedia.org/wiki/Three-valued_logic –  JDH Feb 10 '11 at 0:38
    
Did you write out the truth table? –  Doug Spoonwood Jun 19 '13 at 4:59

1 Answer 1

Suppose that a statement $\varphi$ of propositional logic (using only $\wedge$, $\vee$ and $\to$) is not a propositional tautology in classical logic. It follows that some row of the truth table for $\varphi$ has value $F$.

Now, consider the truth table of $\varphi$ in Łukasiewicz three-valued logic, which is called Kleene logic on the Wikipedia page. The key observation is that Łukasiewicz three-valued logic agrees with classical logic on classical logic input. In other words, if all the propositional variables of $\varphi$ are given classical T/F values, then the truth value of $\varphi$ will agree with the classical logic value. Thus, since $\varphi$ had a classical row with value $F$, the very same row will have value F in the Łukasiewicz truth table of $\varphi$, contrary to the assumption that $\varphi$ had only values T or U.

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The Wikipedia page actually has the correct table for Lukasiewicz three-valued logic. –  Doug Spoonwood Jun 19 '13 at 4:59

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