Question

Difference between first order and second order logic

Answer 1

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Answer 2

Propositional Logic

          Propositional logic consists of a set of atomic propositional symbols (e.g. Socrates, Father, etc), which are often referred to by letters p, q, r etc. (Note that these letters aren't variables as such, as propositional logic has no means of binding variables). These symbols are joined together by logical operators (or connectives) to form sentences. The basic logical operators are:

   Negation: ¬p ("it is not the case that ");

   Conjunction: p ∧ q ("p and q");

   Disjunction: p ∨ q ("p or q");

   Implication: p ⇒ q ("p implies q", or "q if p");

   Equivalence: p ⇔ q ("p if and only if q").

First-order Predicate Logic

First-order Predicate Logic is an extension of propositional logic, which allows quantification over variables. Whereas in propositional logic you can only talk about specifics (e.g. "Socrates is a man"), in predicate logic you can also talk more generally (e.g. "all men are mortal").

First order logic

First-order predicate logic allows variables to range over atomic symbols in the domain. It doesn't allow variables to be bound to predicate symbols, however. A second order logic (such as second order predicate logic) does allow this, and you can write sentences such as:

∀p.p(Socrates).

Higher order logic

A higher order logic allows predicates to accept arguments which are themselves predicates..

Second order logic cannot be reduced to first-order logi.