The problem or issue here lies with the truth table for the conditional (or material implication) ⊃
F G (F⊃G) True True True True False False False True True False False True
especially with lines 3 and 4. The problem is that if the antecedent (ie F) is false the whole thing is true. So, for example, 'If the moon is made of green cheese, I am a happy man' is true and so too is 'If the moon is made of green cheese, I am not a happy man' and many feel this to be paradoxical.
Much has been said about this (which you can follow up for yourself). But one line of suggesting that the result is reasonable is to argue that 'If F then G' is exactly the same as 'Either not-F or G' and then the 'paradoxical' result follows, for
'If the moon is made of green cheese, I am a happy man' becomes 'Either the moon is not made of green cheese or I am a happy man' which is true (thanks to the moon not being green cheese)
'If the moon is made of green cheese, I am a not a happy man' becomes 'Either the moon is not made of green cheese or I am not a happy man' which is also true (thanks to the moon not being green cheese)
Exclusive and Inclusive 'Or'
This bring up another point 'F or G' can be 'F or G and possibly both' or it can be 'F or G and definitely not both'. As in 'The Wildcats will win or the lady Wildcats will win' and 'You can have honey or jam'. The former is known as the inclusive 'Or' and the latter the exclusive 'Or'.
In Latin there are two different words for 'Or', there is 'vel' which is inclusive and 'aut' which is exclusive.
In introductory propositional logic we use the inclusive 'Or' symbolized by 'v' (from the Latin 'vel').
If we want the exclusive 'Or' we just symbolize it long hand, for example ( FvG)∧∼(F∧G) (or ( FvG).∼(F.G) or ( FvG)&∼(F&G), depending on what logical symbol we are using for 'and').