I recently started reading Logic and read the 1st order logic or predicate logic, where the main problem is to prove the validity or invalidity of an argument.
While proving validity or invalidity it's the first task to have an intuition about the validity or invalidity of the argument,then you can proceed to solve the problem.My question is about that intuition.
e.g consider the problem :
- All tenors are either overweight or effeminate.No overweight tenor is effeminate.Some tenors are effeminate.Therefore some tenors are overweight.
In this problem I thought in such a way : as all tenors are either overweight or effeminate,and as some tenors are effeminate,so all tenors are effeminate,hence the argument is invalid and I looked for a substitution instance in which the conclusion is false while the premises are true,and fortunately I found and the answer was right.
But in the next problem viz.,
- All tenors are either overweight or effeminate.No overweight tenor is effeminate.Some tenors are effeminate.Therefore some tenors are not overweight. I thought in the same way and found that as some tenors are effeminate so there is some tenors which are not overweight as all tenors are either overweight or effeminate.So it is a valid argument but unfortunately it was not true $($ I think as I found a substitution instance in which the conclusion is false but the premises are true. If we write down the problem in symbolic notation then we have : The premises are
($\forall x)(Tx \rightarrow (Ox \lor Ex))$
($\forall x)((Tx \land Ox) \rightarrow \sim Ex)$
($\exists x)(Tx \land Ex)$.
The conclusion is ($\exists x)(Tx \land \sim Ox)$
Now consider a universe consisting of ${x,y,}$ such that $Tx ,Ty , Ey $ are true and others are false.
Then the conclusion becomes false,still the premises are true.Hence the argument is invalid. $)$
So what's wrong in my intuition ? Can anyone please help me to point it out ? Also is there another way to make good intuition in such problems,or only experience can led to right intuition ?
Thanks.