How does the implies connective work in logic?

[guguma]

The "Implies" connective

I have a big problem understanding the logic behind the implies connective.

P \Longrightarrow Q

The truth table for this is the same as the truth table for Q OR NOT P.

The Problem is that I cannot wrap my mind around the fact that the implies statement is equivalent to Q OR NOT P. Writing truth tables for the AND OR NOT connectives is intuitive. But for the implies statement I think the truth value of the statement when P is False is accepted by convention.

What I understand from IMPLIES is that when I say

P \Longrightarrow Q

It means Q follows from P, it means that the Truth of P makes Q also True, and When P is True and Q is False then My implies statement is False. That is OK up to this point.

Now when P is False I immediately assume P \Longrightarrow Q is True! That is the problem. Shouldn't it be indeterminate?

And for the equivalence of Q OR NOT P, this statement looks at two unrelated statements Q and P I am looking for the truth of Q or the truth of NOT P for my statement is correct, but
this P \Longrightarrow Q talks about two statements which are related. Truth of Q should follow from the truth of P.

Example:

S = T \Longrightarrow \left( S\cap T = S \cup T \right)

Assume P is the left hand statement and Q is the right hand statement

Now if S=T is true, this statement is true unless the right hand side is false. I understand that. But when S \neq T this statement is still true.

When we are following the logic here we are using the definitions of the equality, intersection and the union of sets. From that I can only conclude the first two results assuming S = T It follows that Q is True so this statement is True. But if I do not assume S = T then Q is also False so this makes the statement still True due to truth table convention of the implies statement. But the second result does not show that Q followed from P. Take this:

S = T \Longrightarrow \left( S\cap T \neq S \cup T \right)

When P is False Q is True, so this statement is True too. Now how come both statements in the two examples are true?

Please help me with this, I am sure that I am overlooking something and I feel very stupid because I cannot wrap my mind around this.

 

[mXSCNT]

You seem to have a fairly good handle on it, you're familiar with how it is strange. That's just the way it is. It turns out that making more "intuitive" versions of implication is a good deal more complicated, so mathematicians usually don't.

In practice, logical implication is usually enough to express what mathematicians want to express. There's rarely a need to prove statements of the form P => Q when P and Q are not related. So when a mathematician uses implication, there usually is some relationship between P and Q beyond what the logical implication alone would mean.

What you are wishing for is more like causation than implication. One way to express it is to define a new symbol --?-->, where P --?--> Q means that it is somehow "easier" to prove Q starting with the premise P. But that is highly subjective and informal. You could attempt to formalize it by saying that P --?--> Q means either that there is no proof of Q except with the premise P, or that the shortest proof of Q with the additional premise P is shorter than the shortest proof of Q without the premise P. However, in most cases that would be impossible to verify, and in any case wouldn't always capture how "easy" it is to prove Q starting with P. The problem with any formalization of P --?--> Q is that the ease with which Q can be proved is subjective, highly depending on the mathematician.

You may be interested in learning about modal logic. Judea Pearl's book, Causality, also looks very interesting on this subject, but does require knowledge of probability.

 

[honestrosewater]

Perhaps the thing to realize is that connectives in this type of logic are required to be truth-functional, meaning that the truth-values that they assign must depend solely on the truth-values of the arguments. This logic doesn't recognize that two things are related in any other way, e.g., physical causation or some kind of taxonomy. It only cares about how their truth-values are related. You might want something like http://en.wikipedia.org/wiki/Relevance_logic" .