Coarsening natural deduction proofs II: finding gaunt proofs

This paper explores how, given a proof, we can systematically transform it into a proof that contains no irrelevancies and which is as strong as possible. I define a weaker and stronger notion of what counts as a proof with no irrelevancies, calling them perfect proofs and gaunt proofs, respectively. Using classical core logic to study classical validities and core logic to study intuitionistic validities, I show that every core proof or classical core proof can be transformed into a perfect proof. In a sequel paper, I show how proofs in core logic can also be transformed into gaunt proofs and I observe that this property fails for classical core logic.

Full classical S5 in natural deduction with weak normalization

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

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.

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.

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.

System SKS is a set of rules for classical propositional logic presented in the calculus of structures. Like sequent systems and unlike natural deduction systems, it has an explicit cut rule, which is admissible. In contrast to sequent systems, the cut rule can easily be reduced to atomic form. This allows for a very simple cut elimination procedure based on plugging in parts of a proof, like normalisation in natural deduction and unlike cut elimination in the sequent calculus. It should thus be a good common starting point for investigations into both proof search as computation and proof normalisation as computation.

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.

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.

Prawitz suggested expanding a natural deduction system for intuitionistic logic to include rules for classical logic constructors, allowing both intuitionistic and classical elements to coexist without losing their inherent characteristics. Looking at the added rules from the point of view of the Gödel-Gentzen translation, led us to propose a general method for the coexistent combination of two logics when a conservative translation exists from one logic (the source) to another (the host). Then we prove that the combined logic is a conservative extension of the original logics, thereby preserving the unique characteristics of each component logic. In this way there is no collapse of one logic into the other in the combination. We also demonstrate that a Gentzen calculus for the combined logic can be induced from a Gentzen calculus for the host logic by considering the translation. This approach applies to semantics as well. We then establish a general sufficient condition for ensuring that the combined logic is both sound and complete. We apply these principles by combining classical and intuitionistic logics capitalizing on the Gödel-Gentzen conservative translation, intuitionistic and S4 modal logics relying on the Gödel-McKinsey-Tarski conservative translation, and classical and Jaśkowski’s paraconsistent logics taking into account the existence of a conservative translation.

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.

Our terminology and notation is the same as that of [1], of which this note is a sequel.We wish to show that if we take the natural deduction system (N) described in [1], and delete the rules for the quantifiers, we obtain a complete system for propositional logic. [Of course we now construe “X”, “Y”, “Z” as syntactic variables ranging over formulas of propositional logic, rather than sentences of quantification theory.] Moreover the system serves as a neat decision procedure.

This is a companion to a paper by the authors entitled “Godel on deduction”, which examined the links between some philosophical views ascribed to Godel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Godel presented with the help of Gentzen’s sequents, which amounts to Jaśkowski’s natural deduction system of 1934, and which may be found in Godel’s unpublished notes for the elementary logic course he gave in 1939 at the University of Notre Dame. Here one finds a presentation of this system of Godel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Godel’s Notre Dame course, and is followed by comments concerning Godel’s natural deduction system.

Intuitionistic logic encompasses the principles of reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [Bro07]. Intuitionistic logic can be succinctly described as classical logic without the law ϕ ∨ ¬ϕ of excluded middle. Brouwer observed that this law was abstracted from finite situations and its application to statements about infinite collections is not justified. One of the consequences of the rejection of the law of excluded middle is that every intuitionistic proof of an existential sentence can be effectively transformed into an intuitionistic proof of an instance of that sentence. More precisely, if a formula of the form ∃xϕ(x) without free variables is provable in the intuitionistic predicate logic, then there is a term t without free variables such that ϕ(t) is provable. In particular, if ϕ ∨ ψ is provable, then either ϕ or ψ is provable. In that sense intuitionistic logic may provide a logical basis for constructive reasoning. A formal system of intuitionistic logic was proposed in [Hey30]. A relationship between the classical propositional logic PC and intuitionistic propositional logic INT was proved by Glivenko in [Gli29], namely a PC-formula ϕ is PC-provable if and only if ¬ ¬ϕ is INT-provable. Kripke semantics for intuitionistic logic was developed in [Kri65].