P = False
Q = False
If P, then Q = True (?)
What is "if P, then NOT Q," true or false?
Best answer + 5 stars if you can explain as well.
Q = False
If P, then Q = True (?)
What is "if P, then NOT Q," true or false?
Best answer + 5 stars if you can explain as well.
-
P -> Q has truth table:
P=t, Q=t ==> P -> Q = t
P=t, Q=f ==> P -> Q = f
P=f, Q=t ==> P -> Q = t
P=f, Q=f ==> P -> Q = t
Now if Q = f, then ~Q(or, "NOT Q") = t. So if P=f, ~Q=t then we have P -> ~Q = t.
As for an explanation: this specific case in the truth table is known as the "Paradox of Material Implication". At first glance, it seems to make no sense: the formula renders the statement "if unicorns exist, then 2+2=4" true. It's worth noting that logicians have been working to address this particular issue in Sentential Logic; many embrace something called a "Relevance Logic", where P -> Q is only well defined if Q is relevant to P.
However, that doesn't help you much because you're still in school working in Sentential Logic. So I suppose the closest thing to an intuitive explanation is as follows: if I make a statement dependent on an impossible premise, then that statement can never be tested. For example, were I to say "if I can shoot laser beams from my eyes, then Barack Obama isn't president in 2012," you are not in a position to disprove my claim - after all, I can't shoot laser beams from my eyes. Thus, you have no way to demonstrate that it's possible for me to shoot laser beams from my eyes, and for Barack Obama to remain president in 2012. By the same token, you have no way of disproving a different statement that, if I could shoot laser beams from my eyes, then Barack Obama WOULD be president in 2012.
Is it a great explanation? No. That's why the logicians are at work. Hope that helps.
P=t, Q=t ==> P -> Q = t
P=t, Q=f ==> P -> Q = f
P=f, Q=t ==> P -> Q = t
P=f, Q=f ==> P -> Q = t
Now if Q = f, then ~Q(or, "NOT Q") = t. So if P=f, ~Q=t then we have P -> ~Q = t.
As for an explanation: this specific case in the truth table is known as the "Paradox of Material Implication". At first glance, it seems to make no sense: the formula renders the statement "if unicorns exist, then 2+2=4" true. It's worth noting that logicians have been working to address this particular issue in Sentential Logic; many embrace something called a "Relevance Logic", where P -> Q is only well defined if Q is relevant to P.
However, that doesn't help you much because you're still in school working in Sentential Logic. So I suppose the closest thing to an intuitive explanation is as follows: if I make a statement dependent on an impossible premise, then that statement can never be tested. For example, were I to say "if I can shoot laser beams from my eyes, then Barack Obama isn't president in 2012," you are not in a position to disprove my claim - after all, I can't shoot laser beams from my eyes. Thus, you have no way to demonstrate that it's possible for me to shoot laser beams from my eyes, and for Barack Obama to remain president in 2012. By the same token, you have no way of disproving a different statement that, if I could shoot laser beams from my eyes, then Barack Obama WOULD be president in 2012.
Is it a great explanation? No. That's why the logicians are at work. Hope that helps.
-
If P then Q is not true
If False then False means that Q is false only if P is first.Since neither depends on the other, this statement can not be definitive. P being false does not define Q as false (it already is)
Conditionals, such as these, need to have something connecting both conditions to determine the truth or false conclusions.In this case - there is no connection.
If P the Not Q wuld also not be true for the same reason
If False then False means that Q is false only if P is first.Since neither depends on the other, this statement can not be definitive. P being false does not define Q as false (it already is)
Conditionals, such as these, need to have something connecting both conditions to determine the truth or false conclusions.In this case - there is no connection.
If P the Not Q wuld also not be true for the same reason