Coarsening Natural Deduction Proofs I: Finding Perfect Proofs

This paper is the second part of a series exploring how, given a proof, we can inductively transform it into a proof that contains no irrelevancies and is as strong as possible. In the prequel paper, I defined a weaker and a stronger notion of what counts as a proof with no irrelevancies, calling them perfect proofs and gaunt proofs, respectively. There, I showed how proofs in core logic and classical core logic can be transformed into perfect proofs. In this paper I study gaunt proofs. I show how proofs in core logic can be inductively transformed into gaunt core proofs, but that this property fails for the natural deduction system of classical core logic.

We explore the consequences, for logical system-building, of taking seriously (i) the aim of having irredundant rules of inference, and (ii) a preference for proofs of stronger results over proofs of weaker ones. This leads one to reconsider the structural rules of REFLEXIVITY, THINNING, and CUT. REFLEXIVITY survives in the minimally necessary form φ:φ. Proofs have to get started. CUT is subject to a CUT-elimination theorem, to the effect that one can always make do without applications of CUT. So CUT is redundant, and should not be a rule of the system. CUT-elimination, however, in the context of the usual forms of logical rules, requires the presence, in the system, of THINNING. But THINNING, it turns out, is not really necessary. Provided only that one liberalizes the statement of certain logical rules in an appropriate way, one can make do without CUT or THINNING. From the methodological point of view of this study, the logical rules ought to be framed in this newly liberalized form. These liberalized logical rules determine the system of core logic. Given any intuitionistic Gentzen proof of Δ:φ, one can determine from it a Core proof of some subsequent of Δ:φ. Given any classical Gentzen proof of Δ:φ, one can determine from it a classical Core proof of some subsequent of Δ:φ. In both cases the Core proof is of a result at least as strong as that of the Gentzen proof; and the only structural rule used is φ:φ.

The fact that classical mathematical proofs of simply existential statements\ncan be read as programs was established by Goedel and Kreisel half a century\nago. But the possibility of extracting useful computational content from\nclassical proofs was taken seriously only from the 1990s on when it was\ndiscovered that proof interpretations based on Goedel's and Kreisel's ideas can\nprovide new nontrivial algorithms and numerical results, and the Curry-Howard\ncorrespondence can be extended to classical logic via programming concepts such\nas continuations and control operators.\n The workshop series "Classical Logic and Computation" aims to support a\nfruitful exchange of ideas between the various lines of research on\ncomputational aspects of classical logic. This volume contains the abstracts of\nthe invited lectures and the accepted contributed papers of the third CL&C\nworkshop which was held jointly with the workshop "Program Extraction and\nConstructive Mathematics" at the University of Brno in August 21-22, 2010, as a\nsatellite of CSL and MFCS. The workshops were held in honour of Helmut\nSchwichtenberg who became "professor emeritus" in September 2010.\n The topics of the papers include the foundations, optimizations and\napplications of proof interpretations such as Hilbert's epsilon substitution\nmethod, Goedel's functional interpretation, learning based realizability and\nnegative translations as well as special calculi and theories capturing\ncomputational and complexity-theoretic aspects of classical logic such as the\nlambda-mu-calculus, applicative theories, sequent-calculi, resolution and\ncut-elimination\n

Full classical S5 in natural deduction with weak normalization

A structure which generalizes formulas by including substitution terms is used to represent proofs in classical logic. These structures, called expansion trees, can be most easily understood as describing a tautologous substitution instance of a theorem. They also provide a computationally useful representation of classical proofs as first-class values. As values they are compact and can easily be manipulated and transformed. For example, we present an explicit transformations between expansion tree proofs and cut-free sequential proofs. A theorem prover which represents proofs using expansion trees can use this transformation to present its proofs in more human-readable form. Also a very simple computation on expansion trees can transform them into Craig-style linear reasoning and into interpolants when they exist. We have chosen a sublogic of the Simple Theory of Types for our classical logic because it elegantly represents substitutions at all finite types through the use of the typed λ-calculus. Since all the proof-theoretic results we shall study depend heavily on properties of substitutions, using this logic has allowed us to strengthen and extend prior results: we are able to prove a strengthen form of the firstorder interpolation theorem as well as provide a correct description of Skolem functions and the Herbrand Universe. The latter are not straightforward generalization of their first-order definitions.

This paper studies the relationship between Argumentation Logic (AL), a recently defined logic based on the study of argumentation in AI, and classical Propositional Logic (PL). In particular, it shows that AL and PL are logically equivalent in that they have the same entailment relation from any given classically consistent theory. This equivalence follows from a correspondence between the non-acceptability of (arguments for) sentences in AL and Natural Deduction (ND) proofs of the complement of these sentences. The proof of this equivalence uses a restricted form of ND proofs, where hypotheses in the application of the Reductio of Absurdum inference rule are required to be “relevant” to the absurdity derived in the rule. The paper also discusses how the argumentative re-interpretation of PL could help control the application of ex-falso quodlibet in the presence of inconsistencies.

This paper surveys the work of the last two decades on the complexity of proofs in classical propositional logic, and its connection with open problems in theoretical computer science. The paper ends with a short list of open problems.

It is argued that the Curry-Howard correspondence for classical logic implies limitations for logical reasoning that can be learned and performed by large language models. The correspondence establishes an isomorphism between proofs in logic and programs in functional typed lambda calculus. While intuitionistic logic maps to a version of lambda calculus that can be carried out in a local way, i.e., considering local parts of the code in isolation, the version of lambda calculus that corresponds to classical logic requires non-local relations – and this non-locality cannot be learned by large language models due to their restriction to investigate a relative short sequence of tokens. A possible way to go beyond this limitation is sketched as well. Implications for other areas are investigated as well.

Debates concerning philosophical grounds for the validity of classical and intuitionistic logics often have the very nature of proofs as a point of controversy. The intuitionist advocates for a strictly constructive notion of proof, while the classical logician advocates for a notion which allows the use of non-constructive principles such as reductio ad absurdum. In this paper we show how to coherently combine logical ecumenism and proof-theoretic semantics ( $$\boldsymbol{\mathrm{PtS}}$$ ) by providing not only a medium in which classical and intuitionistic logics coexist, but also one in which their respective notions of proof coexist. Intuitionistic proofs receive the standard treatment of $$\boldsymbol{\mathrm{PtS}}$$ , whereas classical proofs are given a semantics based on ideas by David Hilbert. Furthermore, we advance the state of the art in $$\boldsymbol{\mathrm{PtS}}$$ by introducing a key contribution: treating the absurdity constant $$\bot$$ as an atomic proposition and requiring all bases to be consistent. This treatment is essential for the obtainment of some ecumenical results, and it can also be used in standard intuitionistic $$\boldsymbol{\mathrm{PtS}}$$ . Additionally, we employ normalization techniques to demonstrate the consistency of simulation bases. These innovations provide fresh technical and conceptual insights into the study of bases in $$\boldsymbol{\mathrm{PtS}}$$ .

By rephrasing quantifier-free axioms as rules of derivation in sequent calculus, we show that the generalized Steiner–Lehmus theorem admits a direct proof in classical logic. This provides a partial answer to a question raised by Sylvester in 1852. We also present some comments on possible intuitionistic approaches.

Parallelized elimination rules in natural deduction correspond to Left rules in the sequent calculus; and introduction rules correspond to Right rules. These rules may be construed as inductive clauses in the inductive definition of the notion of sequent proof. There is a natural isomorphism between natural deductions in Core Logic and the sequent proofs that correspond to them. We examine the relations, between sequents, of concentration and dilution; and describe what it is for one sequent to strengthen another. We examine some possible global restrictions on proof-formation, designed to prevent proofs from proving dilutions of sequents already proved by a subproof. We establish the important result that the sequent rules of Core Logic maintain concentration, and we explain its importance for automated proof-search.

We study a conservative extension of classical propositional logic distinguishing between four modes of statement: a proposition may be affirmed or denied, and it may be strong or classical. Proofs of strong propositions must be constructive in some sense, whereas proofs of classical propositions proceed by contradiction. The system, in natural deduction style, is shown to be sound and complete with respect to a Kripke semantics. We develop the system from the perspective of the propositions-as-types correspondence by deriving a term assignment system with confluent reduction. The proof of strong normalization relies on a translation to System F with Mendler-style recursion.

The HOLF proof assistant supports specification and proof in classical higher order logic. It is the latest in a long line of similar systems. In this short overview, we give an outline of the HOLF system and how it may be applied in formal verification.

Prawitz proved a theorem, formalising ‘harmony’ in Natural Deduction systems, which showed that, corresponding to any deduction there is one to the same effect but in which no formula occurrence is both the consequence of an application of an introduction rule and major premise of an application of the related elimination rule. As Gentzen ordered the rules, certain rules in Classical Logic had to be excepted, but if we see the appropriate rules instead as rules for Contradiction, then we can extend the theorem to the classical case. Properly arranged there is a thoroughgoing ‘harmony’, in the classical rules. Indeed, as we shall see, they are, all together, far more ‘harmonious’ in the general sense than has been commonly observed. As this paper will show, the appearance of disharmony has only arisen because of the illogical way in which natural deduction rules for Classical Logic have been presented.

We study some structural and numerical properties of varieties for determinative disjunctive normal forms, introduced here for given propositional formula. We consider for classical and non-classical propositional logics some proof systems, which are constructed on the base of determinative disjunctive normal forms. We investigate also the relation between the proof complexities in some well-known classical and non-classical proof systems (Resolution, Cut-free sequent, Gentzen refutation, Cutting planes etc.) and numerical properties of varieties for determinative disjunctive normal forms for classical and non-classical tautologies.