Proof Theory of Finite-valued Logics

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics

Logical consequence in modal logic II: Some semantic systems for S4

ABSTRACT: This 1974 paper builds on our 1969 paper (Corcoran-Weaver [2]). Here we present three (modal, sentential) logics which may be thought of as partial systematizations of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of these three logics coincide with one another and with those of standard formalizations of Lewis's S5. These logics, when regarded as logistic systems (cf. Corcoran [1], p. 154), are seen to be equivalent; but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be as closely linked as previously thought. This 1974 paper uses the linear notation for natural deduction presented in [2]: each two-dimensional deduction is represented by a unique one-dimensional string of characters. Thus obviating need for two-dimensional trees, tableaux, lists, and the like—thereby facilitating electronic communication of natural deductions. The 1969 paper presents a (modal, sentential) logic which may be thought of as a partial systematization of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of this logic coincides those of standard formalizations of Lewis’s S4. Among the paper's innovations is its treatment of modal logic in the setting of natural deduction systems--as opposed to axiomatic systems. The author’s apologize for the now obsolete terminology. For example, these papers speak of “a proof of a sentence from a set of premises” where today “a deduction of a sentence from a set of premises” would be preferable. 1. Corcoran, John. 1969. Three Logical Theories, Philosophy of Science 36, 153–77. J P R 2. Corcoran, John and George Weaver. 1969. Logical Consequence in Modal Logic: Natural Deduction in S5 Notre Dame Journal of Formal Logic 10, 370–84. MR0249278 (40 #2524). 3. Weaver, George and John Corcoran. 1974. Logical Consequence in Modal Logic: Some Semantic Systems for S4, Notre Dame Journal of Formal Logic 15, 370–78. MR0351765 (50 #4253)

Definitions by Rewriting in the Calculus of Constructions

The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules. We prove the strong normalization of the reduction relation generated by the beta-rule and the user-defined rules under some general syntactic conditions including confluence. As examples, we show that two important systems satisfy these conditions: a sub-system of the Calculus of Inductive Constructions which is the basis of the proof assistant Coq, and the Natural Deduction Modulo a large class of equational theories.Comment: Best student paper (Kleene Award

On natural deduction in fixpoint logics

In the current paper we present a powerful technique of obtaining natural deduction (or, in other words, Gentzen-like) proof systems for first-order fixpoint logics. The term "fixpoint logics" refers collectively to a class of logics consisting of modal logics with modalities definable at meta-level by fixpoint equations on formulas. The class was found very interesting as it contains most logics of programs with e.g. dynamic logic, temporal logic and, of course, mu-calculus among them. Fixpoint logics were intensively studied during the last decade. In this paper we are going to present some results concerning deductive systems for first-order fixpoint logics. In particular we shall present some powerful and general technique for obtaining natural deduction (Gentzen-like) systems for fixpoint logics. As those logics are usually totally undecidable, we show how to obtain complete (but infinitary) proof systems as well as relatively complete (finitistic) ones. More precisely, given fixpoint equations on formulas defining nonclassical connectives of a logic, we automatically derive Gentzen-like proof systems for the logic. The discussion of implementation problems is also provided

forall x: Calgary. An Introduction to Formal Logic

forall x: Calgary is a full-featured textbook on formal logic. It covers key notions of logic such as consequence and validity of arguments, the syntax of truth-functional propositional logic TFL and truth-table semantics, the syntax of first-order (predicate) logic FOL with identity (first-order interpretations), translating (formalizing) English in TFL and FOL, and Fitch-style natural deduction proof systems for both TFL and FOL. It also deals with some advanced topics such as truth-functional completeness and modal logic. Exercises with solutions are available. It is provided in PDF (for screen reading, printing, and a special version for dyslexics) and in LaTeX source code

Natural deduction systems for some non-commutative logics

Varieties of natural deduction systems are introduced for Wansing’s paraconsistent non-commutative substructural logic, called a constructive sequential propositional logic (COSPL), and its fragments. Normalization, strong normalization and Church-Rosser theorems are proved for these systems. These results include some new results on full Lambek logic (FL) and its fragments, because FL is a fragment of COSPL

Are there Hilbert-style Pure Type Systems?

For many a natural deduction style logic there is a Hilbert-style logic that is equivalent to it in that it has the same theorems (i.e. valid judgements with empty contexts). For intuitionistic logic, the axioms of the equivalent Hilbert-style logic can be propositions which are also known as the types of the combinators I, K and S. Hilbert-style versions of illative combinatory logic have formulations with axioms that are actual type statements for I, K and S. As pure type systems (PTSs)are, in a sense, equivalent to systems of illative combinatory logic, it might be thought that Hilbert-style PTSs (HPTSs) could be based in a similar way. This paper shows that some PTSs have very trivial equivalent HPTSs, with only the axioms as theorems and that for many PTSs no equivalent HPTS can exist. Most commonly used PTSs belong to these two classes. For some PTSs however, including lambda* and the PTS at the basis of the proof assistant Coq, there is a nontrivial equivalent HPTS, with axioms that are type statements for I, K and S.Comment: Accepted in Logical Methods in Computer Scienc

Proof-functional semantics for relevant implication

In this thesis I provide a theory of implication from within the Gentzen/Curry formalist constructivist tradition. Formal consecution and natural deduction systems, which satisfy the formalist and also the intuitionist desiderata for constructivity (including Lorenzen's principle of inversion), are provided for all implication logics. The similar-but simplified- binary relational ("Kripke-style") semantics are also given. The driving force behind this research has been the desire to provide an explanatory semantics for relevant implication in terms of "use as a subproof in a proof". To this end relevant consecution systems which exploit various precisely characterised notions of use are described. The basis of this work has been the development of a way of describing the shapes of proofs in the "object language". In chapter 2 I motivate and introduce the basic machinery used to describe proofs, and show how thereby to capture use. This involves a more detailed consideration of the internal structure of formal systems than exploited by Curry in his epitheory of formal systems. In chapter 3 the completely general "cloned" consecution systems are described, and it is shown that every logic with an axiomatic formulation is captured by such a system. In chapter 4 the corresponding natural deduction systems are described and it is shown that Lorenzen's principle of inversion holds for them by proving the appropriate reduction theorem. Thus every implication logic has a formulation which satisfies the intuitionist formal criterion for constructivity. In chapter 5 we return to the business of providing explanatory semantics for relevant implication, using the similar style of consecution system as in chapter 3, but with list (proof-description) manipulation rules which capture use. In chapter 6 "cloned" binary relation semantics are described which also capture every logic with an axiomatic formulation. These don't quite correspond to the consecution systems of chapter 3 in that they exploit a dramatic simplification of the list machinery (but do involve other complications). The similar relevant semantics using use rules is also given. The corresponding "simplified" consecution and natural deduction systems are described in appendix B .2. These systems do not satisfy the Lorenzen principle of inversion and so are not constructive. Chapter 7 rounds off and offers some thoughts about possible further developments. Appendix A shows an early attempt to capture relevant implication, and is notable as the most complex formulation of intuitionist implication ever devised

Tackling Incomplete System Specifcations Using Natural Deduction in the Paracomplete Setting

In many modern computer applications the significanceofspecificationbasedverificationiswellaccepted.However, when we deal with such complex processes as the integration of heterogeneous systems, parts of specification may be not known. Therefore it is important to have techniques that are able to cope with such incomplete information. An adequate formal set up is given by so called paracomplete logics, where, contrary to the classical framework, for some statements we do not have evidence to conclude if they are true or false. As a consequence, for example, the law of excluded middle is not valid. In this paper we justify how the automated proof search technique for paracomplete logic PComp can be efficiently applied to the reasoning about systems with incomplete information. Note that for many researchers, one of the core features of natural deduction, the opportunity to introduce arbitrary formulae as assumptions, has been a point of great scepticism regarding the very possibility of the automation of the proof search. Here, not only we show the contrary, but we also turned the assumptions management into an advantage showing the applicability of the proposed technique to assume-guarantee reasoning. Keywords - incomplete information, automated natural deduction, paracomplete logic, requirements engineering, assumeguarantee reasoning, component based system assembly