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What's it called when the logical consequent is indifferent to the antecedent?

I'm looking for the logic term for when

P ⇒ Q
and
¬P ⇒ Q.

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  • 4
    This is classically equivalent to asserting that Q is true.
    – PW_246
    Sep 18 at 2:31
  • @PW_246 Why true and not false or indeterminate?
    – Geremia
    Sep 18 at 3:13
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    Classically, P or not P is theorem, and it is also a theorem that (A->P)&(B->P)&(A or B) implies P.
    – David Gudeman
    Sep 18 at 4:36
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    If your 'indifferent' means the colloquial 'irrelevant', it could be said that in classic logic 'material implication' is the name when the logical consequent is indifferent to its antecedent since the conditional statement P ⇒ Q is considered true as long as either P is false or Q is true (or both), regardless of any particular relationship or dependency between P and Q. And your specific form for the tautological consequence of Q is just a special case with additional constraint...
    – Double Knot
    Sep 18 at 4:44
  • In relevance logic, there is a notion of "irrelevant implication", but that applies to the type of implication as a whole rather than to specific conditionals. In particular, material and strict implications are "irrelevant" regardless of semantic relations between P and Q because they do not track them. Some authors do talk about "irrelevant conditionals" individually, see e.g. Walters, pp.3ff, but it is not a standard term.
    – Conifold
    Sep 18 at 5:23

5 Answers 5

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Morton's Fork

  1. Either you live modestly or you live extravagantly
  2. If you live modestly (you saved money), then you can afford charity
  3. If you live extravagantly (you're rich), then you can afford charity

Conclusion: Whether you live modestly or extravagantly, you can afford charity.

😆

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  • 1
    Is what I'm asking not a "false dilemma"?
    – Geremia
    Sep 19 at 3:03
  • That's a tough question to answer. You might wanna do something about that.
    – Agent Smith
    Sep 19 at 6:11
  • A false dilemma occurs when someone argues that either P or Q must be true, when in fact those are not the only possibilities.
    – Bumble
    Sep 19 at 10:54
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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.

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A formula that is true regardless of the truth values of any other formulas is known as a "tautology".

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In computer logic, that is known as a don't care condition.

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  • Can you please provide a source for this?
    – Geremia
    Sep 19 at 0:21
  • Nope. It's just one of those things you learn on the job from programmers.
    – niels nielsen
    Sep 19 at 2:18
  • Well, real world experience is a source.
    – Geremia
    Sep 20 at 22:32
-2

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 implications is false.

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  • According to modern analytical/symbolic logic.
    – Geremia
    Sep 19 at 23:32
  • @Geremia "According to modern analytical/symbolic logic." According to common sense. This is what "self-evident" means. If you need theoretical expertise to decide, then it is not self-evident.
    – Speakpigeon
    Sep 20 at 5:17

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