Question

What is the logic term for a conditional statement where the consequent is indifferent to the antecedent, such that P implies Q and not P implies Q?

Morton's Fork: Either you live modestly or you live extravagantly. If you live modestly (you saved money), then you can afford charity. If you live extravagantly (you're rich), then you can afford charity. 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 and also ¬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.

A formula that is true regardless of the truth values of any other formulas is known as a "tautology." In computer logic, that is known as a 'don't care condition.' P ⇒ Q and ¬P ⇒ Q. These two implications are self-evidently false. 'Self-evident' means that the layperson could tell intuitively that an instantiation of this sort of implication is false

Answer 1

A conditional statement where the consequent is indifferent to the antecedent is called a tautology or a 'don't care condition' in logic.

Step-by-step explanation:

A conditional statement where the consequent is indifferent to the antecedent, such that P implies Q and not P implies Q, is known as a tautology in logic. It is a statement that is true regardless of the truth values of any other formulas. In computer logic, this is referred to as a 'don't care condition'.

For example, in the case of Morton's Fork, whether you live modestly or extravagantly, you can afford charity. The antecedent of the conditional statement (living modestly or living extravagantly) does not affect the truth of the consequent (being able to afford charity).

Therefore, the logic term for such a conditional statement is a tautology or a 'don't care condition'.