2.7: Beyond First-order Logic

• 2.7.1: Overview

  First-order logic is not the only system of logic of interest: there are many extensions and variations of first-order logic.

• 2.7.2: Many-Sorted Logic

  In first-order logic, variables and quantifiers range over a single domain. But it is often useful to have multiple (disjoint) domains. Many-sorted logic provides this kind of framework.

• 2.7.3: Second-Order logic

  The language of second-order logic allows one to quantify not just over a domain of individuals, but over relations on that domain as well.

• 2.7.4: Higher-Order logic

  Passing from first-order logic to second-order logic enabled us to talk about sets of objects in the first-order domain, within the formal language. Why stop there? For example, third-order logic should enable us to deal with sets of sets of objects, or perhaps even sets which contain both objects and sets of objects. And fourth-order logic will let us talk about sets of objects of that kind. As you may have guessed, one can iterate this idea arbitrarily.

• 2.7.5: Intuitionistic Logic

  In constrast to second-order and higher-order logic, intuitionistic first-order logic represents a restriction of the classical version, intended to model a more “constructive” kind of reasoning.

• 2.7.6: Modal Logics

  One obtains modal propositional logic from ordinary propositional logic by adding a box operator; which is to say, if \(A\) is a formula, so is \(\Box A\). Intuitively, \(\Box A\) asserts that \(A\) is necessarily true, or true in any possible world.

• 2.7.7: Other Logics

  Recent decades have witnessed a veritable explosion of formal logics.