What is a visceral sense?

In most uses, a visceral sense is a "gut-level feeling." It's a unconscious feeling that something is right or wrong, even though we can't explain how we arrived at that conclusion.

A medical dictionary has this definition.

visceral sense, the awareness of sensations that arise from the viscera and stimulate the interoceptors; sensations include pain, pressure or fullness, and organ movement. Called also splanchnesthesia.

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