Fast termination of the deductive process in resolution proof systems for non-classical logics

The power of temporal proofs

The main goal of this paper is to provide an abstract framework for constructing proof systems for various many-valued logics. Using the framework it is possible to generate strongly complete proof systems with respect to any finitely valued deterministic and non-deterministic logic. I provide a couple of examples of proof systems for well-known many-valued logics and prove the completeness of proof systems generated by the framework.

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).

Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system $${ G3K} $$ of Negri for the modal logic $$K$$ and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in $${ G3K} $$ corresponds to a bipole—a pair of a positive and a negative phases—in $${ LKF} $$ . Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.

The development of non-classical logics brought to light the question of their philosophical interpretation. Given a non-classical logic L, what is the informal/philosophical interpretation of its vocabulary? We find in the literature arguments defending that logical systems do not have a canonical philosophical interpretation, so the same logical system L has different philosophical interpretations. Although the thesis that a logic can be philosophically interpreted in different ways is well known, the thesis that a specific philosophical interpretation can be captured by different logics has not been widely explored. In this paper, we argue that if Kreisel’s informal rigor method is adequate to show that formal notions of logical consequence capture informal notions of logical consequence, then a specific philosophical interpretation only corresponds to a unique logic.

The generation of too much information prohibits efficient resolution proof search in classical logics. Subsumption is used to discard redundant information and strategies have been developed to guide the proof search avoiding irrelevant information. The extension of the resolution method to temporal logics, further magnifies this problem. Here we develop an algorithm for the removal of irrelevant information during resolution proofs for a particular resolution proof system for propositional linear-time temporal logic. The resolution system operates on formulae in a normal form. Following a phase of classical style resolution, temporal resolution is carried out by detecting sets of formulae that together represent an □-formula to be resolved with a ◊-formula. It is before the application of the temporal resolution rule that the new algorithm is applied to efficiently remove irrelevant information. The completeness of the new algorithm is discussed and its complexity considered. Its practical efficiency is demonstrated by giving timings for a set of examples.

A complete proof system for propositional projection temporal logic

Substitution Frege and extended Frege proof systems in non-classical logics

This paper presents a sound and complete logical system whose atomic sentences are the equalities of recursive terms involving sets. There are two interpretations of this language: one makes use of non-wellfounded sets with finite transitive closure, and the other uses pointed finite graphs modulo bisimulation. Our logical system is a sequent-style deduction system. The main axioms and inference rules come from the $\mbox{{\it FLR}$_0$}$ -proof system from [6], including the Recursion Inference Rule (but an additional axiom is needed), and also axioms corresponding to the extensionality axiom of set theory.

The issue of what consequences to draw from the existence of non-classical logical systems has been the subject of an interesting debate across a diversity of fields. In this paper the matter of alternative logics is considered with reference to a specific belief system and its propositions :the Azande are said to maintain beliefs about witchcraft which, when expressed propositionally, appear to be inconsistent. When the Azande have been presented with such inconsistencies, they either fail to see them as such or else accept them as non-problematical. Is our knowledge of logical truths a relative and culturally determined phenomenon, or is there some (transcendent) criterion that allows us to adjudicate between alternative logical systems? The authors propose an approach for resolving disputes about the status of Azande reasoning which assumes a paraconsistent framework, thus providing a new perspective on this debate

Non-classical logic and ontological non-commitment, avoiding abstract objects through modal operators

The construction of many-valued logical systems is commonly believed to have been one of the major achievements of the Warsaw School, and specifically of Łukasiewicz.1 The first mention of many-valued (three-valued) logic is to be found in Łukasiewicz’s farewell speech at Warsaw University on March 7,1918. He said: In 1910 I published a book on the principle of contradiction in Aristotle’s work, in which I strove to demonstrate that that principle is not so evident as it is believed to be. Even then I strove to construct non-Aristotelian logic, but in vain. Now I believe I have succeeded in this. ... Examination of that issue was the subject-matter of my last lectures. I have proved that in addition to true and false propositions there are possible propositions, to which objective possibility corresponds as a third value in addition to being and non-being. This gave rise to a system of three-valued logic, which I worked out in detail last summer. That system is as coherent and self-consistent as Aristotle’s logic, and is much richer in laws and formulae. That new logic, by introducing the concept of objective possibility, destroyed the former concept of science, based on necessity. Possible phenomena have no cause, although they themselves can be the beginning of a causal sequence. An act of a creative individual can be free and at the same time affect the course of the world (1918, p. 86).

We present a tableau based proof system for νTrTL, a trace based temporal logic with fixpoints. The proof system generalises similar systems for standard interleaving temporal logics with fixpoints. In our case special attention has to be given to the modal rule: First we give a system with an interleaving style modal rule, later we use a technique similar to the sleep set method (known from finite state model checking) to obtain a more efficient proof rule. We briefly highlight the relation of the improved rule with recent advances in tableau systems for classical propositional logic, the tamed cut of the system KE.The treatment of fixpoints leads to possibly infinite tableaux, which however can be finitely represented, yielding treelike structures with back loops: we show this using an automata construction. Indirectly we obtain a (known) decidability result.KeywordsModel CheckTemporal LogicModal RuleProof SystemAtomic PropositionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter deals with fuzzy logic in Zadeh’s ‘narrow sense’ [27], pointing to t-norm based truth functional logics, where the truth values model ‘degrees of truth’, identified with reals from the unit interval [0,1]. In particular, we are interested in Łukasiewicz logic Ł and some of its variants, but also in Godel logic G, and in Product logic P. The literature on formal deduction systems for these and related many-valued logics is vast and even more has been written about their algebraic background. We refer to the monographs [23], [22], and [10] for more information and references. Most authors take the usefulness of these logics in the context of approximate reasoning, i.e., reasoning with vague and imprecise notions for granted. However the corresponding proof systems are hardly ever explicitly related to models of correct reasoning with vague information. In other words: the challenge to derive particular fuzzy logics from first principles about approximate reasoning is not addressed explicitly. The reference to general models of reasoning and to theories of vagueness – a prolific discourse in contemporary analytic philosophy – is only implicit, if not simply missing, in most presentations of inference systems for fuzzy logics. Some notable exceptions, where an explicit semantic foundation for particular fuzzy logics is aimed at, are: Ruspini’s similarity semantics [35]; voting semantics [25]; ‘re-randomizing semantics’ [24]; measurement-theoretic justifications [7]; the Ulam-Renyi game based interpretation of D. Mundici [29]; and ‘acceptability semantics’ of J. Paris [31]. As we have argued elsewhere [12], these formal semantics for various t-norm based logics (in particular Łukasiewicz logic) should be placed in the wider discourse about adequate theories of vagueness, a prolific subfield of analytic philosophy.

Starting with Hoare Logic over 50 years ago, numerous program logics have been devised to reason about the different kinds of programs encountered in the real world. This includes reasoning about computational effects, particularly those effects that cause the program execution to branch into multiple paths due to, e.g., nondeterministic or probabilistic choice. Outcome Logic reimagines Hoare Logic with branching at its core, using an algebraic representation of choice to capture programs that branch into many outcomes. In this article, we give a comprehensive account of the Outcome Logic metatheory. This includes a relatively complete proof system for Outcome Logic with the ability to reason about general purpose looping. We also show that this proof system applies to programs with various types of branching, that it subsumes some well-known logics such as Hoare Logic, and that it facilitates the reuse of proof fragments across different kinds of specifications.