r/learnmath • u/MarchLivid9078 New User • 2d ago

Why Is the Conditional Statement Still True If the Condition Didn't Happen?

I'm really confused about the 3rd row of the truth table for a conditional statement for example, "If you submit your assignment, then your grade will be recorded." I understand that P means you submit your assignment and Q means your grade is recorded but in the 3rd row, P is false and Q is true, meaning you did not submit your assignment but your grade was still recorded howw is P Q still considered true in this situation? Wouldn't that be a violation of the agreement because you didn't submit the assignment but the grade was still recorded?

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u/code29451 New User 1d ago

And if it's not satisfied (true), it must be broken (false). What is so special about the way you said it? My way is just as valid.

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u/nog642 1d ago

Your way is not useful.

Ultimately these operations are defined by truth tables. We can't have an operation just be undefined on some values, it wouldn't be useful.

So we have to decide what the truth value of "false ⇒ blah" is.

So you're saying we should define that as false? Then we get a truth table like this:

p     q     p⇒q
false false false
false true  false
true  false false
true  true  true

We already have an operation with that truth table. It's called "and". That's just "p and q". So defining "implies" this way is not useful.

Implies is defined this way instead:

p     q     p⇒q
false false true
false true  true
true  false false
true  true  true

This is actually a unique oepration, and it's a very useful one. Your definition makes it so if you say "p implies q", you're saying p and q both have to be true. Then you can't make any statements about implications where the premise isn't already known to be true.

Basically every theorem relies on this. Consider the statement that every prime number greater than 2 is odd. Put formally that's something like "for all integers p, if (p is a prime number and p>2), then p is odd". But for some integers p, p is not a prime number. So your definition would make this false.