Natural Deduction Research Articles

Absurdity as the impossible command in natural deduction

AbstractIn this paper, we propose a new approach to absurdity in the context of natural deduction for intuitionistic and classical logic. It combines the aspects of both the logical approach, which treats absurdity as a propositional constant, and the structural approach, which treats absurdity as a structural punctuation mark signalling the dead end of derivations. In particular, we will treat absurdity as an impossible command, that is, a speech act composed of an imperative force indicator, and the false propositional constant, that is, a proposition that cannot be true by definition. In return, we obtain a framework that constitutes a middle ground between the logical and structural approaches. For example, it allows us to consider the ex falso quodlibet rule as a kind of structural rule, specifically, a semi‐structural rule, and at the same time also maintain that negation can be reduced to implication of absurdity, specifically, to its propositional content.

A normal paradox

Abstract: For the past 40 years, Neil Tennant has defended a proof-theoretic criterion of self-referential paradoxicality. According to this criterion, the defining characteristic of paradoxes is that, when formulated within a natural deduction system, they produce derivations that cannot be normalized. This paper raises doubts about Tennant’s approach. Recently, Tennant has suggested that Russell’s paradox might not truly fit his criterion. I will argue that the reasoning that rules out Russell’s paradox can similarly be applied to some semantic paradoxes. Therefore, if Tennant’s assessment of Russell’s paradox holds, few cases may genuinely qualify as paradoxes by his standards.

Grokking the Sequent Calculus (Functional Pearl)

The sequent calculus is a proof system which was designed as a more symmetric alternative to natural deduction. The 𝜆𝜇𝜇-calculus is a term assignment system for the sequent calculus and a great foundation for compiler intermediate languages due to its first-class representation of evaluation contexts. Unfortunately, only experts of the sequent calculus can appreciate its beauty. To remedy this, we present the first introduction to the 𝜆𝜇𝜇-calculus which is not directed at type theorists or logicians but at compiler hackers and programming-language enthusiasts. We do this by writing a compiler from a small but interesting surface language to the 𝜆𝜇𝜇-calculus as a compiler intermediate language.

Wright’s First-Order Logic of Strict Finitism

AbstractA classical reconstruction of Wright’s first-order logic of strict finitism is presented. Strict finitism is a constructive standpoint of mathematics that is more restrictive than intuitionism. Wright sketched the semantics of said logic in Wright (Realism, Meaning and Truth, chap 4, 2nd edition in 1993. Blackwell Publishers, Oxford, Cambridge, pp.107–75, 1982), in his strict finitistic metatheory. Yamada (J Philos Log. https://doi.org/10.1007/s10992-022-09698-w, 2023) proposed, as its classical reconstruction, a propositional logic of strict finitism under an auxiliary condition that makes the logic correspond with intuitionistic propositional logic. In this paper, we extend the propositional logic to a first-order logic that does not assume the condition. We will provide a sound and complete pair of a Kripke-style semantics and a natural deduction system, and show that if the condition is imposed, then the logic exhibits natural extensions of Yamada (2023)’s results.

Towards determinism in PDL: relations and proof theory

Abstract Guarded Kleene Algebra with Tests (GKAT) was presented as a fragment of KAT to abstract imperative programming languages, where only if-then-else and while-do statements are allowed in the language. The loss of expressiveness is, nevertheless, compensated by a clear advantage over KAT: it allows almost linear decidability of program equivalence. In this work, we give the first step to optimizing the complexity of dynamic logic equivalence, which is EXPTIME-complete in propositional dynamic logic (PDL). First, and based on strict deterministic PDL, we present guarded propositional dynamic logic (GPDL), a fragment of PDL where programs correspond to GKAT terms. It comes embedded, as expected, with a semantics over relational models and a sound axiomatisation. Then, we present a Natural Deduction system for GPDL, proving its soundness and completeness, concerning the axiomatisation. Based on Smolka et al. (2019, Proceedings of the ACM Programming Language 4), we obtain the main result of this work—the equivalence of GPDL programs can be established in almost linear time.

Meaning is Use: the Case of Propositional Identity

We study natural deduction systems for a fragment of intuitionistic logic with propositional identity from the point of view of proof-theoretic semantics. We argue that the identity connective is a natural operator to be treated under the elimination rules as basic approach.

A possible worlds semantics for trustworthy non-deterministic computations

The notion of trustworthiness, central to many fields of human inquiry, has recently attracted the attention of various researchers in logic, computer science, and artificial intelligence (AI). Both conceptual and formal approaches for modeling trustworthiness as a (desirable) property of AI systems are emerging in the literature. To develop logics fit for this aim means to analyze both the non-deterministic aspect of AI systems and to offer a formalization of the intended meaning of their trustworthiness. In this work we take a semantic perspective on representing such processes, and provide a measure on possible worlds for evaluating them as trustworthy. In particular, we intend trustworthiness as the correspondence within acceptable limits between a model in which the theoretical probability of a process to produce a given output is expressed and a model in which the frequency of showing such output as established during a relevant number of tests is measured. From a technical perspective, we show that our semantics characterizes the probabilistic typed natural deduction calculus introduced in D'Asaro and Primiero (2021)[12] and further extended in D'Asaro et al. (2023) [13]. This contribution connects those results on trustworthy probabilistic processes with the mainstream method in modal logic, thereby facilitating the understanding of this field of research for a larger audience of logicians, as well as setting the stage for an epistemic logic appropriate to the task.

The Pioneering Proving Methods as Applied in the Warsaw School of Logic – Their Historical and Contemporary Significance

Justification of theorems plays a vital role in any rational human activity. It is indispensable in science. The deductive method of justifying theorems is used in all sciences and it is the only method of justifying theorems in deductive disciplines. It is based on the notion of proof, thus it is a method of proving theorems. In the Warsaw School of Logic (WSL) – the famous branch of the Lvov-Warsaw School (LWS) – two types of the method: axiomatic deduction method and natural deduction method were developed and practiced. In this paper, both of these methods are briefly discussed with an emphasis on their historical, groundbreaking significance for logic. The axiomatic method by means of rejection (proposed by Jan Łukasiewicz – a co-creator of the WSL), which is the method of the so-called rejection proof (rejection/refutation method) in logical systems and the proving method of generalized natural deduction, which is a hybrid deduction–refutation method of proving theorems, are also outlined in the paper. The author discusses their historical significance. This paper also contains a brief mention of the most significant results which the application of the discussed methods introduced into contemporary scientific research, not only logical one.

The Problem of Natural Representation of Reasoning in the Lvov-Warsaw School

ABSTRACT The problem of precise characterisation of traditional forms of reasoning applied in mathematics was independently investigated and successfully resolved by Jaśkowski and Gentzen in 1934. However, there are traces of earlier interests in this field exhibited by the members of the Lvov-Warsaw School. We focus on the results obtained by Jaśkowski and Leśniewski. Jaśkowski provided the first formal system of natural deduction in 1926. Leśniewski also demonstrated in some of his papers how to construct proofs in accordance with intuitively correct principles.

Meaning and identity of proofs in a bilateralist setting: A two-sorted typed Lambda-calculus for proofs and refutations

Abstract In this paper, I will develop a $\lambda $-term calculus, $\lambda ^{2Int}$, for a bi-intuitionistic logic and discuss its implications for the notions of sense and denotation of derivations in a bilateralist setting. Thus, I will use the Curry–Howard correspondence, which has been well-established between the simply typed $\lambda $-calculus and natural deduction systems for intuitionistic logic, and apply it to a bilateralist proof system displaying two derivability relations, one for proving and one for refuting. The basis will be the natural deduction system of Wansing’s bi-intuitionistic logic 2Int, which I will turn into a term-annotated form. Therefore, we need a type theory that extends to a two-sorted typed $\lambda $-calculus. I will present such a term-annotated proof system for 2Int and prove a Dualization Theorem relating proofs and refutations in this system. On the basis of these formal results, I will argue that this gives us interesting insights into questions about sense and denotation as well as synonymy and identity of proofs from a bilateralist point of view.

Probability and natural deduction

Abstract We develop a system of basic probability reasoning founded on two great logical concepts, Gentzen’s natural deduction systems and Carnap–Popper probability of sentences. Our system makes it possible to manipulate with probabilized sentences and justify their causal relationships: if probabilities of sentences $A$ and $B$ are in $[r,1]$ and $[s,1]$, respectively, then the probability of sentence $C$ belongs to $[t,1]$, i.e. $A^{r},B^{s}\vdash C^{t}$, for $r,s,t\in [0,1]$. We prove that our system is sound and complete with respect to the traditional Carnap–Popper type probability semantics. This approach opens up a new perspective of proof-theoretic treatment of sentence probability, potentially allowing immediate algorithmic use of the pure syntactic convenience of natural deductions in programming.

Coarsening natural deduction proofs II: finding gaunt proofs

Abstract 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.

A Formalization of Core Why3 in Coq

Intermediate verification languages like Why3 and Boogie have made it much easier to build program verifiers, transforming the process into a logic compilation problem rather than a proof automation one. Why3 in particular implements a rich logic for program specification with polymorphism, algebraic data types, recursive functions and predicates, and inductive predicates; it translates this logic to over a dozen solvers and proof assistants. Accordingly, it serves as a backend for many tools, including Frama-C, EasyCrypt, and GNATProve for Ada SPARK. But how can we be sure that these tools are correct? The alternate foundational approach, taken by tools like VST and CakeML, provides strong guarantees by implementing the entire toolchain in a proof assistant, but these tools are harder to build and cannot directly take advantage of SMT solver automation. As a first step toward enabling automated tools with similar foundational guarantees, we give a formal semantics in Coq for the logic fragment of Why3. We show that our semantics are useful by giving a correct-by-construction natural deduction proof system for this logic, using this proof system to verify parts of Why3's standard library, and proving sound two of Why3's transformations used to convert terms and formulas into the simpler logics supported by the backend solvers.

Coarsening Natural Deduction Proofs I: Finding Perfect Proofs

Abstract 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.

Scoping Review on the Interactive Digital Tools Used for the Physical and Cognitive Stimulation of Healthy Older Adults

As more of our lives are spent using electronic devices, it comes as a natural deduction that those digital tools could be used to maintain people’s health. Gamified exercise or exergames are indeed promising means to motivate the population to get physically active and even cognitively active if paired with the appropriate games. Considering the global concern of an aging population which could benefit from both physical and cognitive stimulation, these tools appear to be an encouraging solution to keep the population healthier over time. This scoping review reports on the digital tools used in publications between January 2015 and December 2023 regarding the physical and cognitive stimulation of healthy elderly people. The search was conducted in PubMed, Web of Science, and ScienceDirect databases. Of the 1579 publications retrieved, a total of 68 publications were analyzed in this review. A wide variety of digital tools were used in the corpus for the combined physical and cognitive stimulation of the elderly. These tools can be categorized into six types of hardware: pressure plates, optical motion capture, inertial motion capture, virtual reality, ergometers, and driving simulators. The apparition of publications using virtual reality and an increase in publications using inertial motion capture in 2020 could be an indicator that digital tools used for cognitive and physical stimulation of the elderly are evolving. Another finding is the wide variety in evaluation tools used to monitor the outcomes of each protocol. A standardization of the testing process might be needed in order to improve comparisons between experiments.

On a modification of Visser’s formal logic and its connection with Solovay’s modal logic

We present a new logic called SPL, embedded into Solovay’s provability logic S using a translation that embeds Visser’s formal logic FPL into G¨odel-L¨ob’s provability GL. SPL is formulated in the form of sequent and natural deduction calculi, a relational semantics is proposed.

Parallel Verification of Natural Deduction Proof Graphs

Parallel Verification of Natural Deduction Proof Graphs

Core Type Theory

Neil Tennant’s core logic is a type of bilateralist natural deduction system based on proofs and refutations. We present a proof system for propositional core logic, explain its connections to bilateralism, and explore the possibility of using it as a type theory, in the same kind of way intuitionistic logic is often used as a type theory. Our proof system is not Tennant’s own, but it is very closely related, and determines the same consequence relation. The difference, however, matters for our purposes, and we discuss this. We then turn to the question of strong normalization, showing that although Tennant’s proof system for core logic is not strongly normalizing, our modified system is.

Bilateral Connexive Logic

This paper proposes a bilateral analysis of connexivity, presenting a bilateral natural deduction system for a weak connexive logic. The proposed logic deviates from other connexive logics and other bilateral logics in the following respects: (1) The logic induces a difference in meaning between inner and outer occurrences of negation in the connexive axioms. (2) The logic allows incoherence—assertion and denial of the same formula—while still being non-trivial.

Law and logic: A deductive criterion for determination of truth value of evidence in civil and administrative procedures – Part I

The study examines the possibility of applying a deductive logical criterion for determination of the truth value of evidence in civil and administrative procedures. Part I points out types of evidence and facts with ex lege definite truth value according to current legislation in Bulgaria. It reviews the opportunity evidence without such ex lege definite truth value to be subjected to verification using the proposed deductive criterion. It is also argued that in the criterion’s application as premises evidence and facts with ex lege definite truth value should be involved in order to guarantee the truth value of the conclusions. The study provides a detailed review of the propositional logic apparatus involved. Part II reviews the deductive method applied – the method of natural deduction. The application of the deductive criterion in determining the truth value of evidence in an example from legal practice is also shown. At the end the method of natural deduction is applied to a particular kind of proof – based only on circumstantial evidence with ex lege definite truth value. The study concludes with summary of the advantages and limitations of the application of the deductive criterion, some new areas of application of deductive logic in civil procedure are reviewed and some open research issues are pointed.