I've been thinking about this qustion quite a long.
It all started reasoning about this sentence "Either I'm very good at foorball or I can't play it".
Now, I know that a truth function is a function that "accepts truth values as input and produces a truth value as output". In propositional logic we use the truth tables of the logic connectives to evaluate the truth-value of complex (or compound ) statements.
Unfortunately(for me), appealing to my intuition Exclusive Disjunction presents a problem. I'll try to analyze the different cases for the above stentence:
- (1,0)=1 : I can imagine a world where I'm indeed very good at playing football, so it makes perfect sense to say that the whole sentence is true.
- (0,1)=1 : I can Imagine a world where I really can't play football, so, again, it makes perfect sense to say that the whole sentence is true.
- (0,0)=0 : I can imagine a world where I'm average at playing footaball, so in this scenario both the statements "I'm very good" and "I can't paly" are false, and so it is evident that the same holds for the whole sentence.
- (1,1)=0 : Now here it is my problem. I can't imagine a world where I'm both good at playing football and unable to play it ( It feels constradictory).Since I can't assign the value truth to both statements at the same time, it doesn't make sense to me to pass them as an argument. Since they can't coexist, they can't be co-assigned as arguments to the function! As soon I choose one to be tru the other turns false so I can say nothing in this scenario.
I think, it turned out a bit philosophical. Hope you got my probelm!