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A day after he spoke about healthy children "who wound up with profound mental disorders after vaccines," Sen. Rand Paul issued a statement clarifying that he supports vaccines and that he "did not allege causation" between vaccinations and disorders.

"I did not say vaccines caused disorders, just that they were temporally related -- I did not allege causation," Paul said. "I support vaccines, I receive them myself and I had all of my children vaccinated."

The Kentucky Republican, who is mulling a 2016 presidential bid, said that he received a booster shot on Tuesday for vaccinations he got for a trip to Guatemala last year.

And he tweeted a photo of his trip to the doctor:

Paul told CNBC on Monday that vaccines "are a good thing" but that parents should have "some input" when it comes to decisions about vaccinating their kids.

"I have heard of many tragic cases of walking, talking normal children who wound up with profound mental disorders after vaccines," he said in the interview.

On Tuesday, possible Republican contenders Gov. Bobby Jindal of Louisiana and Sen. Marco Rubio of Florida both expressed strong support for vaccinations.

- Carrie Dann