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As the title says, is this true?

$$(\lnot P \to \lnot Q) \to (P \to Q)$$

The truth table is

\begin{array}{rrrrrr} P & Q & \lnot P & \lnot Q & \lnot P \to \lnot Q & P \to Q & (\lnot P \to \lnot Q) \to (P \to Q) \\ \hline T & T & F & F & T & T & T \\ T & F & F & T & T & F & F \\ F & T & T & F & F & T & T \\ F & F & T & T & T & T & T \\ \end{array}

It seems like it's true from the table.

If it is true, is it true because $$(\lnot P \to \lnot Q) \to (P \to Q)$$ has the same truth table corresponding to the $\to$ connective which is false only when the antecedent is T but the consequent is F?

Or is it true because the statement is true when the premises of $\lnot P \to \lnot Q$ and $P \to Q$ are true?

If it's not true, why not?

000
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2 Answers2

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$(\lnot P\to\lnot Q)\to(P\to Q)$ is not a tautology because it is not true when $P$ is true but $Q$ is false.   That is shown in the second row of your truth table.

Likewise, it is not a contradiction.   The statement is conditionally true.

The statement is logically equivalent to $\lnot(P\land\lnot Q)$, also to $(\lnot P\lor Q)$.

Now $(\lnot P\to\lnot Q)\to(Q\to P)$ is a tautology in classical logic.   Notice the order of the terms.

Indeed $\lnot P\to \lnot Q$ is the contrapositive of $Q\to P$, and the two are logically equivalent.

Graham Kemp
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  • Is it conditionally true because (¬P→¬Q) is a premise and (P→Q) is a conclusion, and the conclusion is false when the premise is true when P is true and Q is false? In other words whether this statement is true or not doesn't have anything to do with the evaluation of (¬P→¬Q)→(P→Q) itself i.e. I can remove the last column and be able to evaluate the truth table – 000 Nov 20 '18 at 22:27
  • "In other words whether this statement is true or not doesn't have anything to do with the evaluation of $(\neg P\to \neg Q)\to (P\to Q)$ itself", except that truth is defined exactly by the result of valuations. – Git Gud Nov 20 '18 at 22:29
  • @GitGud Ok so the last column is necessary because it gives us the final statement to evaluate it's truthfulness correct? – 000 Nov 20 '18 at 22:32
  • I have a question. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion? – 000 Nov 20 '18 at 22:40
  • @000 We say $\lnot P\to\lnot Q$ is the antecedant, and $P\to Q$ is the consequent, of the statement $(\lnot P\to\lnot Q)\to(P\to Q)$. Premise and conclusion reference parts of a logical argument. – Graham Kemp Nov 20 '18 at 22:42
  • @GrahamKemp I see premise and conclusion being used to describe logical arguments. What is the appropriate context to use those terms? – 000 Nov 20 '18 at 22:43
  • A premise is an undischarged assumption. The conclusion is the final derivation made. The argument is intended to convince that the premises logically entail the conclusion. – Graham Kemp Nov 20 '18 at 22:46
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No, if we have a statement "$P$ then $Q$", then "$\neg P$ then $\neg Q$" is the inverse of the statement. The inverse being true does not imply the statement is true.

For instance consider a class where the cutoff for an $A$ is $90\%$. Consider the statement $$ \text{"If you have above an }80\%\text{, then you will receive an }A\text{."} $$ This statement is not true. However its inverse is true. $$ \text{"If you do not have above an }80\%\text{, then you will not receive an }A\text{."} $$

Joey Kilpatrick
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  • Thank you. Is (¬P→¬Q) the premise and (P→Q) the conclusion? Or are they both premises with (¬P→¬Q)→(P→Q) as the conclusion? – 000 Nov 20 '18 at 22:40
  • ($\neg P\rightarrow \neg Q$) is the premise or hypothesis and ($P\rightarrow Q$) is the conclusion. – Joey Kilpatrick Nov 20 '18 at 22:42