Abstract
We study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov's Principle and Weak Konig's Lemma and discuss non-standard semantics admitting assumption-free completeness. We contribute a reusable Coq library for first-order logic containing all results covered in this paper.
| Original language | English |
|---|---|
| Pages (from-to) | 112-151 |
| Number of pages | 40 |
| Journal | Journal of Logic and Computation |
| Volume | 31 |
| Issue number | 1 |
| DOIs | |
| State | Published - 1 Jan 2021 |
| Externally published | Yes |
Keywords
- Coq
- First-order logic
- Kripke semantics
- Tarksi semantics
- algebraic semantics
- completeness
- constructive logic
- constructive reverse mathematics
- dialogue semantics
- type theory
ASJC Scopus subject areas
- Theoretical Computer Science
- Software
- Arts and Humanities (miscellaneous)
- Hardware and Architecture
- Logic