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In classical logic, why is (p -> q) True if both p and q are False?

The Logic table for If P then Q is as follows:

P Q      If P then Q
T T       T
T F       F 
F T       T
F F       T

What I don't understand is, How can there be a truth table for this?

As far as I understand, If p then Q means "if P is true, Q has to be true. Any other case, I don't know"

So, from what I understand, the first 2 rows of the truth table state that "If P is true and Q is true, the outcome is correct and If P is true and Q is false, the outcome is incorrect (F)"

What about the last 2 rows?


marked as duplicate by GEdgar, Henning Makholm, t.b., Zev Chonoles Jul 28 '12 at 2:42

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Think of the truth table as describing when the statement "If $P$ then $Q$" is true. If $P$ is false, then the statement "If $P$ then $Q$" doesn't claim anything, so how could it be false? Since it doesn't claim anything, we make the convention that "If $P$ then $Q$" should be true.

One could argue that if "If $P$ then $Q$" doesn't claim anything, then how could it be true either? Well, we accept a basic axiom of logic that tell us that every statement is either true or false, so we have to pick one. In mathematics, we find it more useful to take it to be true, but this is not necessary. Often times in Philosophy one takes the opposite convention. This may be confusing as far as notation goes, but it does not actually cause any problems.

  • $\begingroup$ If it Rains > I carry an umbrella. What about a day it doesn't rain? Both I carry an umbrella and I don't carry an umbrella are logically correct? $\endgroup$ – Inquest Jul 8 '12 at 16:26
  • 4
    $\begingroup$ No, on a day it doesn't rain nothing is said about I don't carry an umbrella. It could be one, or the other, we don't know. But the implication is true. The statement If it rains then I carry an umbrella would still be true on a day it isn't raining. The only way you would object to the implication is if you could show me a day where you were not carrying an umbrella and it was raining. $\endgroup$ – nullUser Jul 8 '12 at 16:32
  • $\begingroup$ I'm having trouble accepting what you are saying, @nullUser. Like OP, I think that a truth table is not even applicable for if-then. Here's why: Let's say P = "The earth is flat" and Q = "I'm 24 years old". We know that P is false. I happen to be 24 years old. So, according to the truth table, the fact that the earth is flat somehow implies that I'm 24 years old. This sounds ridiculous. $\endgroup$ – Sandi Aug 4 '17 at 17:26
  • $\begingroup$ Now let's say I'm 23 years old. Then as well, according to the truth table, the earth being flat implies that I'm 24 years old. This seems equally ridiculous because the earth being flat doesn't imply anything about my age! $\endgroup$ – Sandi Aug 4 '17 at 17:26
  • $\begingroup$ Basically, I feel like the truth value of an if-then statement is partially independent of the truth values of P and Q. They cannot determine the truth value of if P then Q on their own, except on row two, because if P is true and Q is false, of course P cannot imply Q. But in any other case, we cannot be sure of whether or not P implies Q by solely looking at the truth values of P and Q. If the earth is round and I'm a human, does the roundness of the earth imply my humanity? I would say no. So the truth table doesn't hold. $\endgroup$ – Sandi Aug 4 '17 at 17:31

You are saying that the truth table for $A\longrightarrow B$ does not fully capture the ordinary language meaning of "if $A$ then $B$." In ordinary language, often some causal connection is understood. And in ordinary language, one would probably not assert that "if $A$ then $B$" when $A$ is clearly false. These assertions about ordinary language are correct.

However, the language of formal logic is not ordinary language. The standard truth-functional interpretation of $A\longrightarrow B$ does make sense if we insist that $A\longrightarrow B$ be assigned a truth value that depends only on the truth values of $A$ and $B$. Certainly alternative truth-functional interpretations of $A\longrightarrow B$ would be intuitively less appropriate than the standard one.


You might find this hand-out for beginning logic students helpful too, http://www.logicmatters.net/resources/pdfs/Conditionals.pdf


The material conditional is a common source of confusion to new logic students, and their worries have also been raised by philosophers of logic and language. The Stanford Encyclopedia of Philosophy has an extensive article on conditionals which addresses the main issues, including the so-called paradoxes of material implication.


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