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How can we reason about “if P then Q” or “P only if Q” statements in propositional logic?

In a logic exercise, suppose this argument is given:

P1: If there's a God, then there's a world.
P2: If there's a God.
C : There's a world.

This argument is valid. But if we put the first premise this way:

P1: If there's a world, then there's a God.
P2: If there's a God.
C : There's a world.

Then, it won't be valid because we interchanged the parts of if-then. My question is:

Doesn't "If there's a God, then there's a world" and "If there's a world, then there's a God." mean really the same thing?

If it doesn't mean the same thing in logic, then how do we know which one should we use? Is there a rule to how we should arrange the parts of if-then statement so that we know what we are doing? If we don't be careful with this, our logic can go wrong,

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  • This question came up before. Please visit this and this and this and this, because they definitely answer your question, and be sure to look for a given answer before posting a new one (:
    – iphigenie
    Oct 30, 2012 at 13:25
  • I don't think my question is well understood. All those links that are posted have really nothing to do with my question.
    – user120321
    Oct 30, 2012 at 13:33
  • yes they do. the problem is that you're using a material implication, which is not symmetrical.
    – iphigenie
    Oct 30, 2012 at 13:36
  • @iphigenie it might not be a bad idea at this point to have a subject-matter expert formulate a 'canonical' Q&A on this point, and then link to that from all of these similar ones
    – Joseph Weissman ♦
    Oct 30, 2012 at 13:38
  • @iphigenie: My question is asking something like, if I were to create a if-then argument for "There's a God." and "There's a world", How would you arrange them in if-then statement? Would you say: "If there's a God, then there's a world" or "If there's a world, then there's a God." And How do you know which way you want it?
    – user120321
    Oct 30, 2012 at 13:47

2 Answers 2

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If-Then is not symmetrical. Consider the following cases:

  1. If it is raining, then I am wet.
  2. If I am wet, then it is raining.

We can find a counterexample for each that is not a counterexample of the other.

  1. Suppose it is raining, but because I am inside, I am not getting wet. What this demonstrates is that it is not the case that if it is raining then I am wet. That's because we've identified a case where it is raining (the left hand side of the implication is true) but where I am not wet (the right hand side of the implication is false). But we haven't thereby shown that it's not the case that if I am wet, then it is raining. That's because I'm not wet, so our current example doesn't tell us anything about what happens when I am.
  2. Suppose I'm wet, because I've just fallen into a pool, but it's a sunny day out. This demonstrates the converse - that it is not the case that if I am wet then it is raining - since we've found a case where I am wet (the left hand side of the implication is true) but it's not raining (the right hand side of the implication is false). This, on the other hand, doesn't show that if it is raining then I am wet, because it's not raining, so we aren't usefully informed about what the consequences are of it raining.

You might take away as a lesson from this that if you are interested in chasing up the consequences of a particular proposition or propositional fragment, and seeing whether a particular conclusion follows from that proposition, you should put the proposition on the left hand side.

Can we do better, and demonstrate a counterexample to the symmetricality? Sure. How about:

  1. If you're a Texan, then you're an American.

  2. If you're an American, then you're a Texan.

    1. is easily true, because Texas is in America, and therefore every situation in which someone is a Texan is also a situation in which they're an American. But 2. is clearly false, because California is in America, and because there are Californians that are not Texans. There is a situation in which we have an American (a Californian) that is not a Texan. So 2. fails, even where 1. succeeds, and hence we have an example of the Asymmetricality of If-Then.

That doesn't mean it's Anti-symmetric - we do have some cases where the two implications match up. Consider:

  1. If you're an unmarried man, then you're a bachelor.
  2. If you're a bachelor, then you're an unmarried man.

These implications both go through. In Philosophy, rather than talking about two different propositions here, we sometimes use a shorthand form, which might be what's confusing you. This shorthand form is written as follows:

  • You are an unmarried man if and only if (or sometimes iff) you are a bachelor
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  • Thanks for clearing things up. Perhaps that's what I was really looking for. Can you please comment something on my this doubt with some kind of similar example?
    – user120321
    Oct 30, 2012 at 15:39
  • If this is what you wanted, you should thank Paul by voting up and accepting his answer! (:
    – iphigenie
    Oct 30, 2012 at 15:50
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This question has been asked far too many times lately.

Take "If P then Q"

When P is true, Q MUST be true.

When P is false, Q CAN be true (or false).

When Q is true, P CAN be true (or false).

When Q is false, P MUST be false.

The relationships between P and Q are not symetrical and therefore the original statement can not merely be swapped around without changing the meaning.

The more traditional reading for the statement is "P implies Q". Reduce your statements to something clear to understand, and figure out which fact implies the other, and the order will be clear.

One minor point. If sentence 1 is "If P then Q", then the sentence 2 you're looking for is "P" not "If P", which means nothing on its own.

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  • Lets say if P is false, and Q is true then I don't understand why would the result be true? What does it actually mean to be something that we have "true" in logic? Because If we translate that to English argument that won't make sense. E.g. If there's no God, then there's a world. (P = 0, Q = 1) from my original statement in question which says: If there's a God, then there's a world.
    – user120321
    Oct 30, 2012 at 14:32
  • @user120321, you should stick with more basic cases than worlds and Gods when getting the feel for logical conditionals. Try thinking around the Texan/American case above a bit, perhaps in the form "If Person X is a Texan then Person X is an American". Have you thought about what would happen to the conditional if there were no Texans, but there were some Californians? What if there were neither any Texans nor any other Americans; or if Texans were the only Americans; would that change the conditional?
    – Paul Ross
    Oct 30, 2012 at 20:28
  • @user120321 - The "True" in the "result" in the truth table just means that with these values, the scenario you've created still satisfies the rules you are describing. Not "If Not P, then Q", but "P being False and Q being True still satisfies P->Q" You're modifying one of your hypotheses with the values of your P and Q, which is not an accurate reading of the truth table.
    – Ryno
    Oct 31, 2012 at 10:11

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