We present the Natural Deduction Assistant (NaDeA) and discuss its advantages and disadvantages as a tool for teaching logic. NaDeA is available online and is based on a formalization of natural deduction in the Isabelle proof assistant. We first provide concise formulations of the main formalization results. We then elaborate on the prerequisites for NaDeA, in particular we describe a formalization in Isabelle of "Hilbert's Axioms" that we use as a starting point in our bachelor course on mathematical logic. We discuss a recent evaluation of NaDeA and also give an overview of the exercises in NaDeA.

Classical first-order logic is in many ways central to work in mathematics, linguistics, computer science and artificial intelligence, so it is worthwhile to define it in full detail. We present soundness and completeness proofs of a sequent calculus for first-order logic, formalized in the interactive proof assistant Isabelle/HOL. Our formalization is based on work by Stefan Berghofer, which we have since updated to use Isabelle's declarative proof style Isar (Archive of Formal Proofs, Entry FOL-Fitting, August 2007 / July 2018). We represent variables with de Bruijn indices; this makes substitution under quantifiers less intuitive for a human reader. However, the nature of natural numbers yields an elegant solution when compared to implementations of substitution using variables represented by strings. The sequent calculus considered has the special property of an always empty antecedent and a list of formulas in the succedent. We obtain the proofs of soundness and completeness for the sequent calculus as a derived result of the inverse duality of its tableau counterpart. We strive to not only present the results of the proofs of soundness and completeness, but also to provide a deep dive into a programming-like approach to the formalization of first-order logic syntax, semantics and the sequent calculus. We use the formalization in a bachelor course on logic for computer science and discuss our experiences. * Corresponding author: jovi@dtu.dk Teaching a Formalized Logical CalculusThe development described in this paper is available online: https://bitbucket.org/isafol/isafol/src/master/FOL_Berghofer/ 4253 lines FOL_Berghofer.thy 867 lines FOL_Tableau.thy A contribution of this paper 217 lines FOL_Sequent.thy A contribution of this paper 132 lines FOL_Appendix.thy A contribution of this paperThese numbers include blank lines and a few comments. All in all it takes around 5 seconds in real time to verify on a fairly standard computer. The entire formalization is based on the standard theory Main (the standard library of Isabelle/HOL which comes with many useful functions and facts about e.g. natural numbers and lists).We have recently named our system based on the formalization in Isabelle/HOL: SeCa  Sequent Calculus Verifier SeCaV formalizes first-order logic with constants and functionsSeCaV verifies one-sided sequent calculus proofs SeCaV uses the Isabelle proof assistantSeCaV is a tool for teaching logic Jørgen Villadsen Asta Halkjaer FromAlexander Birch Jensen Anders SchlichtkrullFor our bachelor course we focus on the fragment (499 lines) of the formalization available here:https://bitbucket.org/isafol/isafol/src/master/Sequent_Calculus/ Here only the soundness proof is included but in addition a small and a large proof in the sequent calculus is formalized. The structure of the paper is as follows. Section 2 explains the formalization of the syntax and the semantics of classical first-order logic. Section 3 describes the sequent calculus. Section 4 outlines the formalized soundness and completeness proof...

We describe our Natural Deduction Assistant (NaDeA) and the interfaces between the Isabelle proof assistant and NaDeA. In particular, we explain how NaDeA, using a generated prover that has been verified in Isabelle, provides feedback to the student, and also how NaDeA, for each formula proved by the student, provides a generated theorem that can be verified in Isabelle.

The Students' Proof Assistant (SPA) aims to both teach how to use a proof assistant like Isabelle and also to teach how reliable proof assistants are built. Technically it is a miniature proof assistant inside the Isabelle proof assistant. In addition we conjecture that a good way to teach structured proving is with a concrete prover where the connection between semantics, proof system, and prover is clear. The proofs in Lamport's TLAPS proof assistant have a very similar structure to those in the declarative prover SPA. To illustrate this we compare a proof of Pelletier's problem 43 in TLAPS, Isabelle/Isar and SPA. We also consider Pelletier's problem 34, also known as Andrews's Challenge, where students are encouraged to develop their own justification function and thus obtain a lot of insight into the proof assistant. Although SPA is fully functional we have so far only used it in a few educational scenarios.

Proof assistants are important tools for teaching logic. We support this claim by discussing three formalizations in Isabelle/HOL used in a recent course on automated reasoning. The first is a formalization of System W (a system of classical propositional logic with only two primitive symbols), the second is the Natural Deduction Assistant (NaDeA), and the third is a one-sided sequent calculus that uses our Sequent Calculus Verifier (SeCaV). We describe each formalization in turn, concentrating on how we used them in our teaching, and commenting on features that are interesting or useful from a logic education perspective. In the conclusion, we reflect on the lessons learned and where they might lead us next.