What's it called when the logical consequent is indifferent to the antecedent?
I'm looking for the logic term for when
P ⇒ Q
¬P ⇒ Q.
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Conclusion: Whether you live modestly or extravagantly, you can afford charity.
What you have in that case is a consequent Q that holds whether or not the antecedent P is true. In English we often tag a conditional of this kind by using the word 'even' or 'still' or both. E.g. "Even if it rains, the match will go ahead as planned." or "If it rains, the match will still go ahead." The match will happen whether it rains or not, so we have
Rains → Match
¬Rains → Match
Such conditionals are often simply called 'even ifs'. They are closely related to what linguists refer to as concessive conditionals. They are not necessarily the same as relevant conditionals, in the sense of relevance logic. In many cases the point of the antecedent is that it is contextually relevant. Imagine this conversation:
A: There's going to be a football match on Saturday. You're welcome to join us. B: What if it rains? A: If it rains, there will still be a football match. B: What if it snows? A: Even if it snows, there will be a football match. B: What if there is a huge volcanic eruption in Japan? A: Even if there is a huge volcanic eruption in Japan, there will still be a football match.
In each case, A's conditional statements have an antecedent that is relevant in context. It would be weird if A had started the conversation with, "If there is a huge volcanic eruption in Japan, there will be a football match on Saturday." Without context, the antecedent would be irrelevant and it would make the conditional misleading, though arguably not actually false. Using Grice's theory of conversational implicature, we would say that such a statement violates the cooperative principle by breaking the rule: be relevant.
Some writers about conditionals consider that 'even ifs' straightforwardly entail the corresponding 'if', e.g. Jonathan Bennett, "A Philosophical Guide to Conditionals", (Oxford, 2003), chapter 17. Others consider 'even ifs' to be different because the antecedent does not provide evidential support for the consequent, e.g. Igor Douven, "The Epistemology of Indicative Conditionals" (Cambridge, 2016), p119.