THE SUBFORMULA PROPERTY IN CLASSICAL NATURAL DEDUCTION ESTABLISHED CONSTRUCTIVELY

Full classical S5 in natural deduction with weak normalization

There exist valuable methods for theorem proving in non classical logics based on translation from these logics into first-order classical logic (abbreviated henceforth FOL). The key notion in these approaches istranslation from aSource Logic (henceforth abbreviated SL) to aTarget Logic (henceforth abbreviated TL). These methods are concerned with the problem offinding a proof in TL by translating a formula in SL, but they do not address the very important problem ofpresenting proofs in SL via a backward translation. We propose a framework for presenting proofs in SL based on a partial backward translation of proofs obtained in a familiar TL: Order-Sorted Predicate Logic. The proposed backward translation transfers some formulasF TL belonging to the proof in TL into formulasF SL , such that the formulasF SL either (a) belong to a corresponding deduction in SL (in the best case) or, (b) are semantically related in some precise way, to formulas in the corresponding deduction in SL (in the worst case). The formulasF TL andF SL can obviously be considered aslemmas of their respective proofs. Therefore the transfer of lemmas of TL gives at least a skeleton of the corresponding proof in SL. Since the formulas of a proof “keep trace” of the strategy used to obtain the proof, clearly the framework can also help in solving another fundamental and difficult problem:the transfer of strategies from classical to non classical logics. We show how to apply the proposed framework, at least to S5, S4(p), K, T, K4. Two conjectures are stated and we propose sufficient (and in general satisfactory) conditions in order to obtain formulas in the proof in SL. Two particular cases of the conjectures are proved to be theorems. Three examples are treated in full detail. The main lines of future research are given.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

We now discuss the model theory of (classical) first-order logic together with partial interpretations. As in the propositional case, the syntax of partial firstorder logic is essentially the same as in the classical case. The main difference from classical first-order logic is the model theory, and the resulting formal axiomatics. The reader is referred e.g. to Smullyan [94] for a comprehensive discussion of classical first-order logic.

Concerning the three fundamental first-order fuzzy logics, the set of logically valid formulae is \(\varPi _2\)-complete for Łukasiewicz logic, \(\varPi _2\)-hard for Product logic, and \(\varSigma _1\)-complete for Godel logic, as with classical first-order logic. Among these fuzzy logics, only Godel logic is recursively axiomatisable. Hence, it was necessary to provide a hyperresolution-based proof method suitable for automated deduction, as one has done for classical logic. As another step, we can incorporate a countable set of intermediate truth constants of the form \(\bar{c}\), \(c\in (0,1)\), together with the equality, Open image in new window, strict order, Open image in new window, projection, Open image in new window, operators in Godel logic; and modify the hyperresolution calculus, inferring over so-called order clauses. We shall investigate the deduction problem of a formula from a countable theory in this expansion and the so-called canonical standard completeness, where the semantics of Godel logic is given by the standard \(\varvec{G}\)-algebra as well as truth constants are interpreted by ‘themselves’. The hyperresolution calculus is refutation sound and complete for a countable order clausal theory under a certain condition for the set of truth constants occurring in the theory. We get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in this expansion of Godel logic.

Chapter 1 Knowledge Representation and Classical Logic

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

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.

This is a textbook based on an interdisciplinary (advanced undergraduate level) course that was originally taught by the three authors at Duke. It is intended for many different disciplines. Even though it covers mathematical and philosophical topics, it occasionally still feels like a textbook whose origins lie more in computer science than in the other disciplines. This isn’t even surprising, for an interdisciplinary focus on logic easily draws one towards computer science and artificial intelligence. In three parts, each written by one of the authors, a three-chapter overview is given of the topics of proof-theory, computability-theory and philosophical logic. Yet, rather than a juxtaposition of logical, mathematical, computational, and philosophical subjects, this textbook isn’t merely intended as an illustration of how logic is relevant for many different disciplines, but forms the basis for a different kind of introduction to logic. This reviewer surely applauds the intent to teach logic to philosophers more like we would teach it to computer scientists—with an emphasis on what it allows us to achieve, but also to teach it to computer scientists more like we would (or ought to) teach it to philosophers—with more attention to the diversity in logic than to the dominant background role of classical logic. While Part 1 is concerned with proof-theory, with chapters on classical propositional logic, first-order logic, and Prolog, it also double-tasks as an introduction to formal logic. Thus, chapter 1 reviews the language and semantics of propositional logic, and even goes on to prove two basic meta-theoretical results by

A natural deduction formulation is given for the intermediate logic called MH by Gabbay in [4]. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.

We propose methods for solving the satisfiability and validity problems in many-sorted composition-nominative pure predicate logics (without functions and with equality). These logics are algebra-based logics of many-sorted partial predicates constructed in a semantic-syntactic style on the methodological basis that is common with programming; they can be considered as generalizations of traditional many-sorted logics on classes of partial predicates that do not have fixed arity. We show the reduction of the satisfiability problem to the same problem for many-sorted classical first-order pure predicate logic with equality. As validity is dual to satisfiability, the method proposed can be adopted to the validity problem. This enables us to use existent satisfiability and validity checking procedures developed for classical logic also for solving these problems in composition-nominative pure predicate logics with equality.

A normalization procedure is given for classical natural deduction with the standard rule of indirect proof applied to arbitrary formulas. For normal derivability and the subformula property, it is sufficient to permute down instances of indirect proof whenever they have been used for concluding a major premiss of an elimination rule. The result applies even to natural deduction for classical modal logic.

Introducing possibilistic logic in ILP for dealing with exceptions

In this paper, we share our experiences concerning the introduction of the Android-based self-study app AXolotl within the first-semester logic course offered at our university. This course is mandatory for students majoring in Computer Science and Artificial Intelligence. AXolotl was used as part of an optional lab assignment bridging clausal reasoning and SAT solving with classical reasoning, proof construction, and first-order logic. The app provides an intuitive interface for proof construction in various logical calculi and aids the students through rule application. The goal of the lab assignment was to help students make a smoother transition from clausal and decompositional reasoning used earlier in the course to inferential and contextual reasoning required for proof construction and first-order logic. We observed that the lab had a positive influence on students' understanding and end the paper with a discussion of these results.

By combining program logic and static analysis, we present an automatic approach to construct program proofs to be used in Proof-Carrying Code. We use Hoare logic in representing the proofs of program properties, and the abstract interpretation in computing the program properties. This combination automatizes proof construction; an abstract interpretation automatically estimates program properties (approximate invariants) of our interest, and our proof-construction method constructs a Hoare-proof for those approximate invariants. The proof-checking side (code consumer’s side) is insensitive to a specific static analysis; the assertions in the Hoare proofs are always first-order logic formulas for integers, into which we first compile the abstract interpreters’ results. Both the property-compilation and the proof construction refer to the standard safety conditions that are prescribed in the abstract interpretation framework. We demonstrate this approach for a simple imperative language with an example property being the integer ranges of program variables. We prove the correctness of our approach, and analyze the size complexity of the generated proofs.