Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid to which properties of theories result in the presence of which rules of inference, and to restrictions on the sets of formulas to which the rules may be employed, restrictions (...) determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stålmarck. (shrink)

Defines and examines a notion of logical friendliness, a broadening of the familiar notion of classical consequence. Also reviews familiar notions and operations with which friendliness makes contact, providing a new light in which they may be seen.

Section 1 recalls a point noted by A. N. Prior forty years ago: that a certain formula in the language of a purely implicational intermediate logic investigated by R. A. Bull is unprovable in that logic but provable in the extension of the logic by the usual axioms for conjunction, once this connective is added to the language. Section 2 reminds us that every formula is interdeducible with (i.e. added to intuitionistic logic, yields the same intermediate logic as) some conjunction-free (...) formula. Thus it would seem that any detour going via formulas with conjunction can be avoided, which raises a puzzle: how is this consistent with the point from Section 1? Sections 3 and 4 raise and discuss this puzzle. In fact, the puzzle turns out on closer inspection not to be so puzzling after all, but it does serve as a convenient centrepiece around which to organize a discussion of the phenomenon illustrated by the Bull–Prior example. Section 5 notes that Prior's observation can be extended to the case of the result of adding disjunction to Bull's logic, while Section 6 includes some further remarks aimed at diagnosing one source of possible residual puzzlement. A subtext of our discussion—spanning several of the notes—is that this work by Bull and Prior has been overlooked, their results having to be rediscovered, by many algebraists and logicians in more recent years. (shrink)

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers ∀p, ∃p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most ω, the resulting logics are decidable. (...) This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some Gödel-Dummett logics with quantifiers over propositions. (shrink)

We prove that the two-variable fragment of first-order intuitionistic logic is undecidable, even without constants and equality. We also show that the two-variable fragment of a quantified modal logic L with expanding first-order domains is undecidable whenever there is a Kripke frame for L with a point having infinitely many successors (such are, in particular, the first-order extensions of practically all standard modal logics like K, K4, GL, S4, S5, K4.1, S4.2, GL.3, etc.). For many quantified modal logics, including those (...) in the standard nomenclature above, even the monadic two-variable fragments turn out to be undecidable. (shrink)

Only propositional logics are at issue here. Such a logic is contra-classical in a superficial sense if it is not a sublogic of classical logic, and in a deeper sense, if there is no way of translating its connectives, the result of which translation gives a sublogic of classical logic. After some motivating examples, we investigate the incidence of contra-classicality (in the deeper sense) in various logical frameworks. In Sections 3 and 4 we will encounter, originally as an example of (...) what (in Section 2) we call a contra-classical modal logic, an unusual logic boasting a connective (" demi-negation" ) whose double application is equivalent to a single application of the negation connective. Pondering the example points the way to a general characterization of contra-classicality (Theorems 3.3 and 4.6). In an Appendix (Section 5), we look at one alternative to classical logic as the target for such translational assimilation, intuitionistic logic, calling logics which resist the assimilation, in this case, contra- intuitionistic. We will show that one such logic is classical logic itself, thereby strengthening a result of Wojcicki's to the effect that the consequence relation of classical logic cannot be faithfully embedded by any connective-by-connective translation into that of intuitionistic logic. (What the "faithfully" means here is that not only is the translation of anything provable in the 'source' logic.. (shrink)

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

This paper considers the question: what becomes of the notion of a logic as a way of codifying valid arguments when the customary assumption is dropped that the premisses and conclusions of these arguments are statements from some single language? An elegant treatment of the notion of a logic, when this assumption is in force, is that provided by Dana Scott's theory of consequence relations; this treatment is appropriately generalized in the present paper to the case where we do not (...) make this assumption of linguistic homogeneity. Several applications of the resulting concept of a heterogeneous logic are suggested, but the main emphasis is on the formal development. One topic touched on is a certain contrast between the boolean and the intensional sentence-connectives in this more general setting. (shrink)

Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic (...) completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, Łukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates. (shrink)

We explore a relation we call 'anticipation' between formulas, where A anticipates B (according to some logic) just in case B is a consequence (according to that logic, presumed to support some distinguished implicational connective →) of the formula A → B. We are especially interested in the case in which the logic is intuitionistic (propositional) logic and are much concerned with an extension of that logic with a new connective, written as "a", governed by rules which guarantee that for (...) any formula B, aB is the (logically) strongest formula anticipating B. The investigation of this new logic, which we call ILa, will confront us on several occasions with some of the finer points in the theory of rules and with issues in the philosophy of logic arising from the proposed explication of the existence of a connective (with prescribed logical behaviour) in terms of the conservative extension of a favoured logic by the addition of such a connective. Other points of interest include the provision of a Kripke semantics with respect to which ILa is demonstrably sound, deployed to establish certain unprovability results as well as to forge connections with C. Rauszer's logic of dual intuitionistic negation and dual intuitionistic implication, and the isolation of two relations (between formulas), head-implication and head-linkage, which, though trivial in the setting of classical logic, are of considerable significance in the intuitionistic context. (shrink)

If a certain semantic relation (which we call 'local consequence') is allowed to guide expectations about which rules are derivable from other rules, these expectations will not always be fulfilled, as we illustrate. An alternative semantic criterion (based on a relation we call 'global consequence'), suggested by work of J.W. Garson, turns out to provide a much better - indeed a perfectly accurate - guide to derivability.

We illustrate, with three examples, the interaction between boolean and modal connectives by looking at the role of truth-functional reasoning in the provision of completeness proofs for normal modal logics. The first example (§ 1) is of a logic (more accurately: range of logics) which is incomplete in the sense of being determined by no class of Kripke frames, where the incompleteness is entirely due to the lack of boolean negation amongst the underlying non-modal connectives. The second example (§ 2) (...) focusses on the breakdown, in the absence of boolean disjunction, of the usual canonical model argument for the logic of dense Kripke frames, though a proof of incompleteness with respect to the Kripke semantics is not offered. An alternative semantic account is developed, in terms of which a completeness proof can be given, and this is used (§ 3) in the discussion of the third example, a bimodal logic which is, as with the first example, provably incomplete in terms of the Kripke semantics, the incompleteness being due to the lack of disjunction (as a primitive or defined boolean connective). (shrink)

In this paper we provide cut-free tableau calculi for the intuitionistic modal logics IK, ID, IT, i.e. the intuitionistic analogues of the classical modal systems K, D and T. Further, we analyse the necessity of duplicating formulas to which rules are applied. In order to develop these calculi we extend to the modal case some ideas presented by Miglioli, Moscato and Ornaghi for intuitionistic logic. Specifically, we enlarge the language with the new signs Fc and CR near to the usual (...) signs T and F. In this work we establish the soundness and completeness theorems for these calculi with respect to the Kripke semantics proposed by Fischer Servi. (shrink)

Modern belief revision theory is based to a large extent on partial meet contraction that was introduced in the seminal article by Carlos Alchourrón, Peter Gärdenfors, and David Makinson that appeared in 1985. In the same year, Alchourrón and Makinson published a significantly different approach to the same problem, called safe contraction. Since then, safe contraction has received much less attention than partial meet contraction. The present paper summarizes the current state of knowledge on safe contraction, provides some new results (...) and offers a list of unsolved problems that are in need of investigation. (shrink)

In this article we firstly present a Kripke semantics for the description logic ALC which is directly inspired by the semantics for Intuitionistic logic. Moreover, we discuss why a direct translation of this kind of semantics is not adequate in the description logic context and propose a constructive semantics that differs from the previous one by the fact that we impose a condition on the partial order. We also present a tableau calculus which is sound and complete with respect to (...) our semantics. As an application of the calculus we prove that this semantics meets the disjunction property, a key criterion used in assessing whether a logic is constructive. (shrink)

Whether assent ("acceptance") and dissent ("rejection") are thought of as speech acts or as propositional attitudes, the leading idea of rejectivism is that a grasp of the distinction between them is prior to our understanding of negation as a sentence operator, this operator then being explicable as applying to A to yield something assent to which is tantamount to dissent from A. Widely thought to have been refuted by an argument of Frege's, rejectivism has undergone something of a revival in (...) recent years, especially in writings by Huw Price and Timothy Smiley. While agreeing that Frege's argument does not refute the position, we shall air some philosophical qualms about it in Section 5, after a thorough examination of the formal issues in Sections 1-4. This discussion draws on - and seeks to draw attention to - some pertinent work of Kent Bendall in the 1970s. (shrink)

We prove that the positive fragment of first-order intuitionistic logic in the language with two individual variables and a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals \ and \, where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser’s basic and formal logics, (...) respectively. We also show that, for most “natural” first-order modal logics, the two-variable fragment with a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz—among them, QK, QT, QKB, QD, QK4, and QS4. (shrink)

We study a class of formulas generalizing the weak law of the excluded middle and provide a characterization of these formulas in terms of Kripke frames and Brouwer algebras. We use these formulas to separate logics corresponding to factors of the Medvedev lattice.

Inquisitive first order logic is an extension of first order classical logic, introducing questions and studying the logical relations between questions and quantifiers. It is not known whether is recursively axiomatizable, even though an axiomatization has been found for fragments of the logic. In this paper we define the \—classical antecedent—fragment, together with an axiomatization and a proof of its strong completeness. This result extends the ones presented in the literature and introduces a new approach to study the axiomatization problem (...) for fragments of the logic. (shrink)

In his 1964 note, ‘Two Additions to Positive Implication’, A. N. Prior showed that standard axioms governing conjunction yield a nonconservative extension of the pure implicational intermediate logic OIC of R. A. Bull. Here, after reviewing the situation with the aid of an adapted form of the Kripke semantics for intuitionistic and intermediate logics, we proceed to illuminate this example by transposing it to the setting of modal logic, and then relate it to the propositional logic of what have been (...) called ‘Hilbert algebras with infimum’. (shrink)

We assemble material from the literature on matrix methodology for sentential logic—without claiming to present any new logical results—in order to show that Gödel once made (or at least, is quoted as having made) an uncharacteristically ill-considered remark in this area.

The volume analyses and develops David Makinson’s efforts to make classical logic useful outside its most obvious application areas. The book contains chapters that analyse, appraise, or reshape Makinson’s work and chapters that develop themes emerging from his contributions. These are grouped into major areas to which Makinsons has made highly influential contributions and the volume in its entirety is divided into four sections, each devoted to a particular area of logic: belief change, uncertain reasoning, normative systems and the resources (...) of classical logic. Among the contributions included in the volume, one chapter focuses on the “inferential preferential method”, i.e. the combined use of classical logic and mechanisms of preference and choice and provides examples from Makinson’s work in non-monotonic and defeasible reasoning and belief revision. One chapter offers a short autobiography by Makinson which details his discovery of modern logic, his travels across continents and reveals his intellectual encounters and inspirations. The chapter also contains an unusually explicit statement on his views on the role of logic in philosophy. (shrink)