Question

The statement comprising the limitations of FOL is/are ____________

(a) Expressiveness

(b) Formalizing Natural Languages

(c) Many-sorted Logic

(d) All of the mentioned

The question was asked during an interview for a job.

My doubt stems from First-Order Logic topic in division Logical Agents of Artificial Intelligence

Answer 1

Right option is (d) All of the mentioned

Best explanation: The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitely logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a meta-logical cost, however: by Lindström’s theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.

Formalizing Natural Languages : First-order logic is able to formalize many simple quantifier constructions in natural language, such as “every person who lives in Perth lives in Australia”. But there are many more complicated features of natural language that cannot be expressed in (single-sorted) first-order logic.

Many-sorted Logic: Ordinary first-order interpretations have a single domain of discourse over which all quantifiers range. Many-sorted first-order logic allows variables to have different sorts, which have different domains.