Simple sentence in intuitionstic logic.

Question

Let's consider the following sentence: $$ A \implies B $$

If $A$ has no a construction / proof, does it mean that sentence is true?

Answer 1

See Brouwer–Heyting–Kolmogorov interpretation :

A proof of $A \to B$ is a function [or procedure] that converts a proof of $A$ into a proof of $B$.

If there is no proof of $A$, we will never have a proof of $B$, but the said function still count as a proof of $A \to B$.

Consider the formula :

$\bot \to P$.

Since there is no proof of $\bot$, any mapping may count as a proof of $\bot \to P$, since it has to be applied to an empty domain.

Thus, the above formula is provable.