If neither fire's absence would have prevented the house from burning, the ancestral-of-dependence relation fails to identify either fire as a cause, contradicting causal intuition.

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Ancestral-of-dependence relation(a specific philosophical tool being tested in this statement): A technical framework philosophers use to determine whether one thing actually caused another by tracing chains of dependence backwards—basically asking 'did this thing really depend on that thing happening?'
Causal intuition(what our everyday thinking tells us about cause and effect): Our gut feeling or common sense about what actually causes what—for example, most people intuitively feel that fire causes a house to burn.
Causation/Cause(metaphysics): In philosophy, this means one thing genuinely produces or brings about another thing, rather than just happening before it or being correlated with it.
Counterfactual/Absence (in philosophy)(the 'fire's absence' part of the statement): A way of testing if something is really a cause by imagining what would have happened if it had been absent or different. For instance: 'If there had been no fire, would the house still have burned?'

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