We briefly overview some of the historical landmarks on the path leading to the reduction of the number of logical connectives in classical logic. Relying on the duality inherent in Boolean algebras, we introduce a new operator ( Nallor ) that is the dual of Schönfinkel’s operator. We outline the proof that this operator by itself is sufficient to define all the connectives and operators of classical first-order logic ( Fol ). Having scrutinized the proof, we pinpoint the theorems of (...) Fol that are needed in the proof. Using the insights gained from the proof, we show that there are four binary operators that each can serve as the only undefined logical constant for Fol . Finally, we show that from every n -ary connective that yields a functionally complete singleton set of connectives two Schönfinkel-type operators are definable, and all the latter are so definable. (shrink)

We prove several interpolation theorems for many-valued infinitary logic with quantifiers by studying expansions of MV-algebras in the spirit of polyadic and cylindric algebras. We prove for various reducts of polyadic MV-algebras of infinite dimensions that if is the free algebra in the given signature,, is in the subalgebra of generated by, is in the subalgebra of generated by and, then there exists an interpolant in the subalgebra generated by and such that. We call this a varying interpolation property because (...) the integer depends on the inequality. We also address cases where this interpolation property fails, but other weaker ones hold. One such interpolation theorem says that though an interpolant may not be found as above, an interpolant can always be found if finitely many universal quantifiers are applied to making it smaller and the same number of existential quantifiers are applied to making it bigger. This number of quantifiers also varies; it depends on the inequality. Se.. (shrink)

We sketch a general theory of complex objects and their constituents.

An exclusive disjunction is true when exactly one of the disjuncts is true. In the case of the familiar binary exclusive disjunction, we have a formula occurring as the first disjunct and a formula occurring as the second disjunct, so, if what we have is two formula-tokens of the same formula-type—one formula occurring twice over, that is—the question arises as to whether, when that formula is true, to count the case as one in which exactly one of the disjuncts is (...) true, counting by type, or as a case in which two disjuncts are true, counting by token. The latter is the standard answer: counting by tokens. James McCawley once suggested that, when the exclusively disjunctive construction in natural language is at issue, the construction should be treated as involving a multigrade connective whose semantic treatment is sensitive to the set of disjuncts rather than the corresponding multiset. Without any commitment as to whether there actually is such a construction, and conceding that for obvious pragmatic reasons such ‘repeated disjunct’ cases would be at best highly marginal, we note that for the binary case, this requires a nonstandard answer—count by type rather than by token—to the earlier question, and thus, an idempotent exclusive disjunction connective. Section 2 explores that idea and Section 3, a further idempotent variant for which it is the propositions expressed by the disjuncts, rather than the disjuncts themselves, that get counted once only in the case of repetitions. Sections 1 and 4 respectively set the stage for these investigations and conclude the discussion. More detailed considerations of points arising from the discussion but otherwise in danger of interrupting the flow are deferred to a ‘Longer Notes’ appendix at the end. (shrink)

Henry M. Sheffer is well known to logicians for the discovery (or rather, the rediscovery) of the ?Sheffer stroke? of propositional logic. But what else did Sheffer contribute to logic? He published very little, though he is known to have been carrying on a rather mysterious research program in logic; the only substantial result of this research was the unpublished monograph The General Theory of Notational Relativity. The main aim of this paper is to explain, as far as possible (given (...) the scanty evidence), the nature of Sheffer's program, and the reasons for its failure. The paper concludes with a discussion of Sheffer's only true logical descendant, C.H. Langford, and his contributions to model theory. (shrink)

We describe sets of partial Boolean functions being closed under the operations of superposition. For any class A of total functions we define the set ????(A) consisting of all partial classes which contain precisely the functions of A as total functions. The cardinalities of such sets ????(A) can be finite or infinite. We state some general results on ????(A). In particular, we describe all 30 closed sets of partial Boolean functions which contain all monotone and zero-preserving total Boolean functions.

We investigate the application of Courcelle’s theorem and the logspace version of Elberfeld et al. in the context of non-monotonic reasoning. Here we formalize the implication problem for propositional sets of formulas, the extension existence problem for default logic, the expansion existence problem for autoepistemic logic, the circumscriptive inference problem, as well as the abduction problem in monadic second order logic and thereby obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, we exhibit, for each (...) of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu, resp., XLnu) under standard complexity assumptions. (shrink)

A simple propositional operator is introduced which generalizes pairwise equivalence and occurs widely in mathematics. Attention is focused on a replacement theorem for this notion of generalized equivalence and its use in producing further generalized equivalences.

Two well-studied closure operators for relations are based on existentially quantified conjunctive formulas, primitive positive definitions, and primitive positive formulas without existential quantification, quantifier-free primitive positive definitions definitions. Sets of relations closed under p.p. definitions are known as co-clones and sets of relations closed under q.f.p.p. definitions as weak partial co-clones. The latter do however have limited expressivity, and the corresponding lattice of strong partial clones is of uncountably infinite cardinality even for the Boolean domain. Hence, it is reasonable to (...) consider the expressiveness of p.p. definitions where only a small number of existentially quantified variables are allowed. In this article, we consider p.p. definitions allowing only polynomially many existentially quantified variables, and say that a co-clone closed under such definitions is polynomially closed, and otherwise superpolynomially closed. We investigate properties of polynomially closed co-clones and prove that if the corresponding clone contains a k-ary near-unanimity operation for k amp;gt;= 3, then the co-clone is polynomially closed, and if the clone does not contain a k-edge operation for any k amp;gt;= 2, then the co-clone is superpolynomially closed. For the Boolean domain we strengthen these results and prove a complete dichotomy theorem separating polynomially closed co-clones from superpolynomially closed co-clones. Using these results, we then proceed to investigate properties of strong partial clones corresponding to superpolynomially closed co-clones. We prove that if Gamma is a finite set of relations over an arbitrary finite domain such that the clone corresponding to Gamma is essentially unary, then the strong partial clone corresponding to Gamma is of infinite order and cannot be generated by a finite set of partial functions. (shrink)

Tarski 1968 makes a move in the course of providing an account of ?definitionally equivalent? classes of algebras with a businesslike lack of fanfare and commentary, the significance of which may accordingly be lost on the casual reader. In ?1 we present this move as a response to a certain difficulty in the received account of what it is to define a function symbol (or ?operation symbol?). This difficulty, which presents itself as a minor technicality needing to be got around (...) especially for the case of symbols for zero-place functions (for ?distinguished elements?), has repercussions?not widely recognised?for the account of functional completeness in sentential logic. A similarly stark comment in Church 1956 reveals an appreciation of this difficulty, though not every subsequent author on the topic has taken the point. We fill out this side of the picture in ?2. The discussion of functional completeness in ?2 is supplemented by some remarks on what is involved in defining a connective, which have been included in an Appendix. The emphasis throughout is on conceptual clarification rather than on proving theorems, and the main body of the paper may be regarded as an elaboration on the remarks just mentioned by Tarski and Church. The Appendix (?3) is intended to be similarly clarificatory, though this time with some corrective intent, of remarks made in and about Makinson 1973. (shrink)

Polynomial clone reducibilities are generalizations of the truth-table reducibilities. A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions. For a fixed polynomial clone ${\fancyscript{C}}$ , a sequence ${B\in X^{\omega}}$ is ${\fancyscript{C}}$ -reducible to ${A \in {X}^{\omega}}$ if there is an algorithm that computes B from A using only effectively selected functions from ${\fancyscript{C}}$ . We show that if A is Kurtz random and ${\fancyscript{C}_{1} (...) \nsubseteq \fancyscript{C}_{2}}$ are polynomial clones, then there is a B that is ${\fancyscript{C}_{1}}$ -reducible to A but not ${\fancyscript{C}_{2}}$ -reducible to A. This implies a generalization of a result first proved by Lachlan (Z Math Logik Grundlagen Math 11:17–44, 1965) for the case |X| = 2. We also show that the same result holds if Kurtz random is replaced by Kolmogorov–Loveland stochastic. (shrink)

Many proposals for logic-based formalisations of argumentation consider an argument as a pair (Φ,α), where the support Φ is understood as a minimal consistent subset of a given knowledge base which has to entail the claim α. In case the arguments are given in the full language of classical propositional logic reasoning in such frameworks becomes a computationally costly task. For instance, the problem of deciding whether there exists a support for a given claim has been shown to be -complete. (...) In order to better understand the sources of complexity (and to identify tractable fragments), we focus on arguments given over formulæ in which the allowed connectives are taken from certain sets of Boolean functions. We provide a complexity classification for four different decision problems (existence of a support, checking the validity of an argument, relevance and dispensability) with respect to all possible sets of Boolean functions. Moreover, we make use of a general schema to enumerate all arguments to show that certain restricted fragments permit polynomial delay. Finally, we give a classification also in terms of counting complexity. (shrink)

In 1979, H. Lewis shows that the computational complexity of the Boolean satisfiability problem dichotomizes, depending on the Boolean operations available to formulate instances: intractable (NP-complete) if negation of implication is definable, and tractable (in P) otherwise [21]. Recently, an investigation in the same spirit has been extended to nonclassical propositional logics, modal logics in particular [2, 3]. In this note, we pursue this line in the realm of many-valued propositional logics, and obtain complexity classifications for the parameterized satisfiability problem (...) of two pertinent samples, Kleene and Gödel logics. (shrink)

Although a very recent topic in contemporary logic, the subject of combinations of logics has already shown its deep possibilities. Besides the pure philosophical interest offered by the possibility of defining mixed logic systems in which distinct operators obey logics of different nature, there are also several pragmatical and methodological reasons for considering combined logics. We survey methods for combining logics (integration of several logic systems into a homogeneous environment) as well as methods for decomposing logics, showing their interesting properties (...) and applications. (shrink)