First-order Logic

First-order logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables.

Synonyms: First-order Predicate Calculus , FOL , and Quantificational Logic .

How it works

For propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations.

The term "first-order" distinguishes first-order logic from higher-order logic, in which there are predicates having predicates or functions as arguments, or in which quantification over predicates, functions, or both, are permitted.

Considerations

• Advantages: -- First-order logic is more expressive than propositional logic; one can talk about objects and their properties, relations between objects. -- First-order logic is able to make use of variables and quantifiers (e.g., "for all" and "exists".) -- First-order logic supports power forms of reasoning, such as inferring the properties of an unknown object from the properties of known objects.

• Disadvantages: -- First-order logic is more difficult to learn and use than propositional logic, due to its greater complexity. -- First-order logic is also less tractable than propositional logic in many cases; reasoning about quantifiers and variables adds complexity. -- First-order logic can be difficult to apply in practice, due to the need to find appropriate axioms and rules for each application.

Verification Approach

• Automated theorem provers can assist in formal verification, performing automated reasoning over system modeled in first-order logic and explore a complete space of system behaviors
• First-order logic may be more expressive than necessary for many types of problems and may be more difficult to verify by SMEs.
• Theorem provers based in FOL are capable of use in software verification tasks, but an SMT solver such as Z3 might be more appropriate.
• Defining a set of competency questions (i.e., query use cases for a first-order logic ontology) can help scope the logic required for a complete solution.

Validation Approach

• Domain SMEs should be identified to review the analytics results and compare them to expected results for a given input.
• Where possible, an outside team of SMEs should inspect the formal logic specification of a system against its stated requirements and suitability to address its domain problem sets.
• Defining a set of competency questions and the expected results provides one means of validation.

References

1. First-order logic. (2023, May 26). In Wikipedia. Link
2. Shapiro, S. and Kissel, T. Classical Logic. (2022). Stanford Encyclopedia of Philosophy. Link
3. A.I. For Anyone. First-order Logic (n.d.). Link
4. Smith, P. An Introduction to Formal Logic. (2020). Link
5. Gruninger, M. and Fox, M. (1995). Methodology for the Design and Evaluation of Ontologies. Link
6. Keet, C., Suarez-Figurosa, M., and Poveda-Villalon, M. (2014). Pitfalls in Ontologies and TIPS to Prevent Them. Link
7. Bjorner, N. et al. The inner magic behind the Z3 theorem prover. (2019) Link