Logical implication and evidence
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Original Question
From what I’ve read on Wikipedia, a valid logical implication is only when P is a sufficient condition for Q. So if something is a cat, it logically implies that it’s an animal. This also means that correlation does not imply causation since it’s not a sufficient condition for causation. So far so good.
But now here’s an issue. People don’t always use the word “imply” in that strict logical sense right? Let me try to elaborate things more. If I know that when my cat is outside it’s usually because my parents are inside the house, does this give me the right to claim that “my cat being outside implies that my parents are inside”? If we analyze the problem, we can see that just because my cat is outside does not mean that my parents are inside, so it’s not a sufficient condition right. This is just a positive correlation. Here’s my main question: can this be counted as “evidence” for my parents being inside even if it isn’t a sufficient condition? And does “evidence” HAVE to be accompanied by the strict logical implication I mentioned?
Answers
1There are two main types of logic, deductive, and inductive. In deductive logic, "P implies Q" means, as you said, that P is a sufficient condition for Q; if P is true then Q is true.
In Inductive Logic, the premises do not "imply" in the strict sense that the conclusion is true; a strong inductive argument is one where if the premises are true, the conclusion is probably true.
So this would be a strong inductive argument:
P1 When my cat is outside, it's usually because my parents are inside.
P2. My cat is outside.
C: Therefore, probably my parents are inside.
Note that if you change " it's usually " in P1 to " it's always the case ", then the Conclusion will be "My parents are inside ( not just probably ), and it becomes a Deductive argument.
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