Suppose that $X$ and $Y$ are propositional statements. Show that $X \implies Y$ holds if and only if $X \rightarrow Y$ is a tautology.

Asked Sep 24 '21 at 11:04

Active Sep 24 '21 at 11:04

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Suppose that $X$ and $Y$ are propositional statements. Show that $X \implies Y$ holds if and only if $X \rightarrow Y$ is a tautology.

If $X \rightarrow B $ is a tautology, then this means that it’s always true. So the value of the $\rightarrow $ connective is always $1$ in the truth table. Could I have some hints on how is one supposed to use this to show the statement $X \implies Y$?

asked Sep 24 '21 at 11:04

Jiming Le

It means that the case when $X$ is True and $Y$ is False is ruled out. What does it means in terms of entailment? – Mauro ALLEGRANZA Sep 24 '21 at 12:04
See also this post – Mauro ALLEGRANZA Sep 24 '21 at 12:04
See Tautological consequence for an helpful review of the basic definition. – Mauro ALLEGRANZA Sep 24 '21 at 12:18
You're well on your way! Can you add the definition of $\implies$ to your post? – Bram28 Sep 24 '21 at 13:25
Bram's point is pertinent: if $\implies$ is defined more weakly than as entailment (logical implication), then $X\implies Y$ holds if (but not automatically only if) $X\to Y$ is a tautology. – ryang Sep 24 '21 at 15:41

Suppose that $X$ and $Y$ are propositional statements. Show that $X \implies Y$ holds if and only if $X \rightarrow Y$ is a tautology.

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