Discuss the limitations of predicate logic. Cite practical example to support your answer.
Basically some of what is possible in other logics (Modal, Belief..etc) is considered a limitation of predicate logic.
Consider this example regarding belief :
- Premise 1: Joe believes Tokyo is the capital of France
- Premise 2: Tokyo is the eastern capital
- Conclusion Joe believes that the eastern capital is the capital of France.
Although in predicate (and propositional, syllogistic...) terms the conclusion follows from the premises, according to this form :
- Joe believes A is B
- A is identical to C
- Therefore, Joe believes C is B
It is obvious that even if A is C, it does not follow that Joe also believes C is B.
Even if A is C, you cannot just replace A by C in the first premise, because it contains the verb "believe".
Because here you need another type of logic : Belief (or doxastic) logic.
Let us see how the argument is valid in predicate logic
Let B(x) denote : Joe believes that x is the capital of France.
This is the predicate argument (t=Tokyo, e=the eastern capital)
- ∴ B(e)
That looks valid in predicate logic, but in fact it is invalid, which means that we cannot use first-order logic with arguments that consist of the verb (believe, know, think...etc).
If you believe that Bob is the killer, and the Bob is the butcher, then it does not follow that you believe that the butcher is the killer (since you may not know that Bob is the butcher).
Doxastic logic can show that the argument is invalid, while first-order logic cannot.