Natural Deduction Research Articles

The Polarized λ-calculus

A natural deduction system isomorphic to the focused sequent calculus for polarized intuitionistic logic is proposed. The system comes with a language of proof-terms, named polarized λ-calculus, whose reduction rules express simultaneously a normalization procedure and the isomorphic copy of the cut-elimination procedure pertaining to the focused sequent calculus. Noteworthy features of this natural deduction system are: how the polarity of a connective determines the style of its elimination rule; the existence of a proof-search strategy which is equivalent to focusing in the sequent calculus; the highly-disciplined organization of the syntax – even atoms have introduction, elimination and normalization rules. The polarized λ-calculus is a programming formalism close to call-by-push-value, but justified by its proof-theoretical pedigree.

Natural Deduction for Fitting’s Four-Valued Generalizations of Kleene’s Logics

In this paper, we present sound and complete natural deduction systems for Fitting’s four-valued generalizations of Kleene’s three-valued regular logics.

A Precise Definition of an Inference (by the Example of Natural Deduction Systems for Logics $I_{\langle \alpha,\beta \rangle}$

In the paper, we reconsider a precise definition of a natural deduction inference given by V. Smirnov. In refining the definition, we argue that all the other indirect rules of inference in a system can be considered as special cases of the implication introduction rule in a sense that if one of those rules can be applied then the implication introduction rule can be applied, either, but not vice versa. As an example, we use logics $I_{\langle\alpha, \beta\rangle}, \alpha, \beta \in \{0, 1, 2, 3,\dots \omega\}$, such that $I_{\langle 0, 0\rangle}$is propositional classical logic, presented by V. Popov. He uses these logics, in particular, a Hilbert-style calculus $HI_{\langle\alpha, \beta\rangle}, \alpha, \beta \in \{0, 1, 2, 3,\dots \omega\}$, for each logic in question, in order to construct examples of effects of Glivenko theorem’s generalization. Here we, first, propose a subordinated natural deduction system $NI_{\langle\alpha, \beta\rangle}, \alpha, \beta \in \{0, 1, 2, 3,\dots \omega\}$, for each logic in question, with a precise definition of a $NI_{\langle\alpha, \beta\rangle}$-inference. Moreover, we, comparatively, analyze precise and traditional definitions. Second, we prove that, for each $\alpha, \beta \in \{0, 1, 2, 3,\dots \omega\}$, a Hilbert-style calculus $HI_{\langle\alpha, \beta\rangle}$and a natural deduction system $NI_{\langle\alpha, \beta\rangle}$are equipollent, that is, a formula $A$ is provable in $HI_{\langle\alpha, \beta\rangle}$iff $A$ is provable in $NI_{\langle\alpha, \beta\rangle}$. DOI: 10.21146/2074-1472-2017-23-1-83-104

Natural deduction system for three-valued Heyting’s logic

A Gentzen-style natural deduction system for the propositional fragment of three-valued Heyting’s logic is presented.

On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

Dummett's logic LC is intuitionistic logic extended with Dummett's axiom: for every two statements the first implies the second or the second implies the first. We present a natural deduction and a Curry-Howard correspondence for first-order and second-order Dummett's logic. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for representing parallel computations and communication between them, and from the viewpoint of logic, Dummett's axiom. We prove that our typed calculus is normalizing and show that proof terms for existentially quantified formulas reduce to a list of individual terms forming an Herbrand disjunction.

Simulation of Natural Deduction and Gentzen Sequent Calculus

We consider four natural deduction systems: Fitch-style systems, Gentzen-style systems (in the form of dags), general deduction Frege systems and nested deduction Frege systems, as well as dag-like Gentzen-style sequent calculi. All these calculi soundly and completely formalize classical propositional logic. We show that general deduction Frege systems and Gentzen-style natural calculi provide at most quadratic speedup over nested deduction Frege systems and Fitch-style natural calculi and at most cubic speedup over Gentzen-style sequent calculi.

Natural Philosophy, Deduction, and Geometry in the Hobbes-Boyle Debate

This paper examines Hobbes’s criticisms of Robert Boyle’s air-pump experiments in light of Hobbes’s account in De Corpore and De Homine of the relationship of natural philosophy to geometry. I argue that Hobbes’s criticisms rely upon his understanding of what counts as “true physics.” Instead of seeing Hobbes as defending natural philosophy as “a causal enterprise … [that] as such, secured total and irrevocable assent,” 1 I argue that, in his disagreement with Boyle, Hobbes relied upon his understanding of natural philosophy as a mixed mathematical science. In a mixed mathematical science one can mix facts from experience (the ‘that’) with causal principles borrowed from geometry (the ‘why’). Hobbes’s harsh criticisms of Boyle’s philosophy, especially in the Dialogus Physicus, sive De natura aeris (1661; hereafter Dialogus Physicus), should thus be understood as Hobbes advancing his view of the proper relationship of natural philosophy to geometry in terms of mixing principles from geometry with facts from experience. Understood in this light, Hobbes need not be taken to reject or diminish the importance of experiment/experience; nor should Hobbes’s criticisms in Dialogus Physicus be understood as rejecting experimenting as ignoble and not befitting a philosopher. Instead, Hobbes’s viewpoint is that experiment/experience must be understood within its proper place – it establishes the ‘that’ for a mixed mathematical science explanation.

Formal Approach for Verifying Galois Field Arithmetic Circuits of Higher Degrees

This paper presents an efficient approach to verifying higher-degree Galois-field (GF) arithmetic circuits. The proposed method describes GF arithmetic circuits using a mathematical graph-based representation and verifies them by a combination of algebraic transformations and a new verification method based on natural deduction for first-order predicate logic with equal sign. The natural deduction method can verify one type of higher-degree GF arithmetic circuit efficiently while the existing methods require an enormous amount of time, if they can verify them at all. In this paper, we first apply the proposed method to the design and verification of various Reed-Solomon (RS) code decoders. We confirm that the proposed method can verify RS decoders with higher-degree functions while the existing method needs a lot of time or fail. In particular, we show that the proposed method can be applied to practical decoders with 8-bit symbols, which are performed with up to 2,040-bit operands. We then demonstrate the design and verification of the Advanced Encryption Standard (AES) encryption and decryption processors. As a result, the proposed method successfully verifies the AES decryption datapath while an existing method fails.

Conflict Resolution: A First-Order Resolution Calculus with Decision Literals and Conflict-Driven Clause Learning

This paper defines the (first-order) conflict resolution calculus: an extension of the resolution calculus inspired by techniques used in modern SAT-solvers. The resolution inference is restricted to (first-order) unit-propagation and the calculus is extended with a mechanism for assuming decision literals and a new inference rule for clause learning, which is a first-order generalization of the propositional conflict-driven clause learning (CDCL) procedure. The calculus is sound (because it can be simulated by natural deduction) and refutationally complete (because it can simulate resolution), and these facts are proven in detail here.

Paraconsistent Logic, Evidence, and Justification

In a forthcoming paper, Walter Carnielli and Abilio Rodrigues propose a Basic Logic of Evidence (BLE) whose natural deduction rules are thought of as preserving evidence instead of truth. BLE turns out to be equivalent to Nelson’s paraconsistent logic N4, resulting from adding strong negation to Intuitionistic logic without Intuitionistic negation. The Carnielli/Rodrigues understanding of evidence is informal. Here we provide a formal alternative, using justification logic. First we introduce a modal logic, KX4, in which \(\square X\) can be read as asserting there is implicit evidence for X, where we understand evidence to permit contradictions. We show BLE embeds into KX4 in the same way that Intuitionistic logic embeds into S4. Then we formulate a new justification logic, JX4, in which the implicit evidence motivating KX4 is made explicit. KX4 embeds into JX4 via a realization theorem. Thus BLE has both implicit and explicit possibly contradictory evidence interpretations in a formal sense.

Commentary and Illocutionary Expressions in Linear Calculi of Natural Deduction

We argue that the need for commentary in commonly used linear calculi of natural deduction is connected to the “deletion” of illocutionary expressions that express the role of propositions as reasons, assumptions, or inferred propositions. We first analyze the formalization of an informal proof in some common calculi which do not formalize natural language illocutionary expressions, and show that in these calculi the formalizations of the example proof rely on commentary devices that have no counterpart in the original proof. We then present a linear natural deduction calculus that makes use of formal illocutionary expressions in such a way that unique readability for derivations is guaranteed – thus showing that formalizing illocutionary expressions can eliminate the need for commentary.

Natural Deduction for Four-Valued both Regular and Monotonic Logics

The development of recursion theory motivated Kleene to create regular three-valued logics. Remove it taking his inspiration from the computer science, Fitting later continued to investigate regular three-valued logics and defined them as monotonic ones. Afterwards, Komendantskaya proved that there are four regular three-valued logics and in the three-valued case the set of regular logics coincides with the set of monotonic logics. Next, Tomova showed that in the four-valued case regularity and monotonicity do not coincide. She counted that there are 6400 four-valued regular logics, but only six of them are monotonic. The purpose of this paper is to create natural deduction systems for them. We also describe some functional properties of these logics.

Variants of Gödel’s Ontological Proof in a Natural Deduction Calculus

This paper presents detailed formalizations of ontological arguments in a simple modal natural deduction calculus. The first formal proof closely follows the hints in Scott’s manuscript about Godel’s argument and fills in the gaps, thus verifying its correctness. The second formal proof improves the first one, by relying on the weaker modal logic KB instead of S5 and by avoiding the equality relation. The second proof is also technically shorter than the first one, because it eliminates unnecessary detours and uses Axiom 1 for the positivity of properties only once. The third and fourth proofs formalize, respectively, Anderson’s and Bjordal’s variants of the ontological argument, which are known to be immune to modal collapse.

A natural deduction and its corresponding sequent calculus for positive contraction - less relevant logic

Reports on Mathematical Logic » 2017 » Number 52 » A natural deduction and its corresponding sequent calculus for positive contraction - less relevant logic A A A

The normalization theorem for extended natural deduction

The normalization theorem for the system of extended natural deduction will be proved as a consequence of the cut-elimination theorem, by using the connections between the system of extended natural deduction and a standard system of sequents.

On harmony and permuting conversions

The paper exposes the relevance of permuting conversions (in natural-deduction systems) to the role of such systems in the theory of meaning known as proof-theoretic semantics, by relating permuting conversion to harmony, hitherto related to normalisation only. This is achieved by showing the connection of permuting conversion to the general notion of canonicity, once applied to arbitrary derivations from open assumption. In the course of exposing the relationship of permuting conversions to harmony, a general definition of the former is proposed, generalising the specific cases of disjunction and existential quantifiers considered in the literature.

Rasiowa–Harrop Disjunction Property

We show that there is a purely proof-theoretic proof of the Rasiowa–Harrop disjunction property for the full intuitionistic propositional calculus (\(\mathbf {IPC}\)), via natural deduction, in which commuting conversions are not needed. Such proof is based on a sound and faithful embedding of \(\mathbf {IPC}\) into an atomic polymorphic system. This result strengthens a homologous result for the disjunction property of \(\mathbf {IPC}\) (presented in a recent paper co-authored with Fernando Ferreira) and answers a question then posed by Pierluigi Minari.

GÖDEL’S NOTRE DAME COURSE

AbstractThis is a companion to a paper by the authors entitled “Gödel’s natural deduction,” which presented and made comments about the natural deduction system in Gödel’s unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to Gödel and general proof theory, one can find a brief summary of Gödel’s notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.

On constructing a logic for the notion of complete and immediate formal grounding

In [14] we have introduced a rigorous definition of the notion of complete and immediate formal grounding; in the present paper our aim is to construct a logic for the notion of complete and immediate formal grounding based on that definition. Our logic will have the form of a calculus of natural deduction, will be proved to be sound and complete and will allow us to have fine-grained grounding principles.

The logic of the reverse mathematics zoo

Building on previous work by Mummertet al.(2015, The modal logic of Reverse Mathematics.Archive for Mathematical54(3–4) 425–437), we study the logic underlying the web of implications and non-implications which constitute the so called reverse mathematics zoo. We introduce a tableaux system for this logic and natural deduction systems for important fragments of the language.