It is often said that stronger acids have weak conjugate bases and vice versa. Yet also in organic chemistry if an acid is proven to have a stable conjugate base it is said to dissociate more, which means it is stronger. Aren't these contradictory statements? What exactly should a strong acid have- a strong or a weak conjugate base?
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Since $\mathrm{pK_a+pK_b=14}$ and strong acids are defined as dissociating completely in water, then a strong acid must be stronger than hydronium, meaning its $\mathrm{pK_a<-1.74}$. For its conjugate base then, this means $\mathrm{pK_b>15.74}$. So yes, a strong acid and a weak conjugate base are two sides of the same coin.

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if an acid is proven to have a stable conjugate base it is said to dissociate more, which means it is stronger
This is wrong. – Avyansh Katiyar Jun 26 '18 at 15:18