First-order formalisations are often preferred to propositional ones because they are thought to underwrite the validity of more arguments. We compare and contrast the ability of some well-known logics—these two in particular—to formally capture valid and invalid arguments. We show that there is a precise and important sense in which first-order logic does not improve on propositional logic in this respect. We also prove some generalisations and related results of philosophical interest. The rest of the article investigates the results’ philosophical significance. A first moral is that the correct way to state the oft-cited superiority of first-order logic vis-à-vis propositional logic is more nuanced than often thought. The second moral concerns semantic theory; the third logic’s use as a tool for discovery. A fourth and final moral is that second-order logic’s transcendence of first-order logic is greater than first-order logic’s transcendence of propositional logic.

Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic ${\cal L}$ is associated with a matrix semantics $Mo{d^{\rm{*}}}{\cal L}$. This article is a contribution to the systematic study of the so-called truth sets of the matrices in $Mo{d^{\rm{*}}}{\cal L}$. In particular, we show that the fact that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of ${\cal L}$. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ are implicitly definable if and only if the Leibniz operator is injective on deductive filters of ${\cal L}$ over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of ${\cal L}$ to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in $Mo{d^{\rm{*}}}{\cal L}$ that corresponds to the order-reflection of the Leibniz operator.

This is a companion to a paper by the authors entitled “Gödel’s natural deduction,” which presented and made comments about the natural deduction system in Gödel’s unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to Gödel and general proof theory, one can find a brief summary of Gödel’s notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.

Product configuration is when an artifact from a product family is assembled from a set of predefined components that can only be combined in certain ways. These ways are defined by configuration rules. The product developers inspect the configuration rules when they develop new configuration rules or modify the configuration rules set. The inspection of configuration rules is thereby an important activity to avoid errors in the configuration rules set. Several formulations of constraint satisfaction problems (CSPs) are proposed that facilitate the inspection of configuration rules in propositional logic (IF-THEN, AND, NOT, OR, etc.). Many of the configuration rules are so called production rules; that is, a configuration rule is an IF-THEN expression that fires when the IF condition is met. Several configuration rules build chains that fire during the product configuration. It is therefore important not only to inspect single configuration rules but also to analyze the effect of multiple configuration rules. Formulating the tasks as variations of the CSP can support the inspection activity. More specifically, we address the reformulation of configuration rules, testing of feature variant combinations, and counting of item quantities from an item set. The suggested CSPs are tested on industrial vehicle configuration rules for computational performance. The results show that the time for achieving results from the solving of the CSP is within seconds. Our future work will be to implement the various CSPs into a demonstrator that could be tested by product developers.

This paper presents a Prolog interface to the MiniSat satisfiability solver. Logic programming with satisfiability combines the strengths of the two paradigms: logic programming for encoding search problems into satisfiability on the one hand and efficient SAT solving on the other. This synergy between these two exposes a programming paradigm that we propose here as a logic programming pearl. To illustrate logic programming with SAT solving, we give an example Prolog program that solves instances of Partial MAXSAT.

This paper describes learning in a compiler for algorithms solving classes of the logic minimization problem MINSAT, where the underlying propositional formula is in conjunctive normal form (CNF) and where costs are associated with the True/False values of the variables. Each class consists of all instances that may be derived from a given propositional formula and costs for True/False values by fixing or deleting variables, and by deleting clauses. The learning step begins once the compiler has constructed a solution algorithm for a given class. The step applies that algorithm to comparatively few instances of the class, analyses the performance of the algorithm on these instances, and modifies the underlying propositional formula, with the goal that the algorithm will perform much better on all instances of the class.

In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox (viz., the implication-free fragment of any non-classical normal extension of the relevance-mingle logic). In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule.