Supervaluationism and Classical Logic

Parametric logic is a framework that generalises classical first-order logic. A generalised notion of logical consequence—a form of preferential entailment based on a closed world assumption—is defined as a function of some parameters. A concept of possible knowledge base—the counterpart to the consistent theories of first-order logic—is introduced. The notion of compactness is weakened. The degree of weakening is quantified by a nonnull ordinal—the larger the ordinal, the more significant the weakening. For every possible knowledge base T, a hierarchy of sentences that are generalised logical consequences of T is built. The first layer of the hierarchies corresponds to sentences that can be obtained by a deductive inference, characterised by the compactness property. The second layer of the hierarchies corresponds to sentences that can be obtained by an inductive inference, characterised by the property of weak compactness quantified by 1. Weaker forms of compactness—quantified by nonnull ordinals—determine higher layers in the hierarchies, corresponding to more complex inferences. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory and from topology. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory—classification in the limit, with or without a bounded number of mind changes—and from topology—in reference to the Borel and the difference hierarchies. In this paper, we introduce the key model-theoretic aspects of Parametric logic, justify the concept of the knowledge base, define the hierarchies of generalised logical consequences and illustrate their relevance to Nonmonotonic reasoning. More specifically, we show that the degree of nonmonotonicity that is required to infer a sentence can be characterised by the least nonnull ordinal that quantifies the weakening of compactness used to locate the inferred sentence in the hierarchies.

The purpose of this paper is to examine some passages of Tarski‘s paper ’On the concept of logical consequence’ and to show that some recent readings of those passages are wrong. John Etchemendy has claimed that in those passages Tarski gave an argument purporting to show that the notion of logical consequence defined by him (as opposed to some pretheoretic notion of logical consequence) possesses certain modal properties. Etchemendy further claims that the argument he attributes to Tarski is fallacious. Some of Etchemendy’s critics have granted him that Tarski did give an argument purporting to show that the defined notion possesses certain modal properties ; but they have claimed that Tarski’s argument was not a fallacious one. I will show that both Etchemendy and his critics are wrong; in the relevant passages, Tarski did not offer (nor did he intend to offer) an argument that the defined notion of logical consequence possesses any modal properties

My long-standing interest in the notion of logical consequence became urgent when I was working on my recent book on higher-order logic (1991). The most common complaint is that second-order logic is out of bounds. as logic, because its consequence relation is not effective. Even though higher order logic is squarely within the prevailing model-theoretic tradition, its set of logical truths is not recursively enumerable-it is not even in the Kleene hierarchy.

One of the main functions of legal reasoning is to determine the truth-value of legal statements, i.e., of statements that assert that certain actions are legally required or permitted. In general, the notion of legal system provides a useful framework, which allows us to distinguish between legal norms and other socially relevant norms. Two concepts of legal system are often employed in legal discourse. On the one hand, it is usually claimed that the material provided by a certain authority has systematic structure, i.e. the norms are structured by means of specific relationships. e.g. hierarchical relationships. In this sense, law is an institutional system. On the other hand, it is frequently asserted that legal norms can be reconstructed systematically starting from the notion of logical consequence. In this sense, law can be regarded as a deductive system. Legal reasoning, then, would use in an explicit or implicit way both notions of normative system in the process of establishing the truth conditions of legal statements. However, this traditional conception of legal reasoning and normative systems has been object of important challenges. In this paper. I consider two groups of arguments against the systematic reconstruction of law:

Formal logic can be used as a tool for representing complex and heterogeneous data such as beliefs, knowledge and preferences. This study proposes an approach for defining clustering methods that deal with bases of propositional formulas in classical logic, i.e., methods for dividing formula bases into meaningful groups. We first use a postulate-based approach for introducing an intuitive framework for formula clustering. Then, in order to characterize interesting clustering forms, we introduce additional properties that take into consideration different notions, such us logical consequence, overlapping, and consistent partition. Finally, we describe our approach that shows how the inconsistency measures can be involved in improving the task of formula clustering. The main idea consists in using the measures for quantifying the quality of the inconsistent clusters. In this context, we propose further properties that allow characterizing interesting aspects related to the amount of inconsistency.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

An important class of logics, typified by classical logic (Chapter 2), is based primarily on a concept of truth. The logical consequence functions of such logics are derived from the interpretation of a language by structures of truth-values. These logics could therefore be called semantic logics.

Alfred Tarski called a language semantically closed if it contains its own truth predicate and technical means for (self-) reference. We consider the semantically closed non-Fregean logic ∈I, presented in (Lewitzka, 2009, Notre Dame Journal of Formal Logic, 50, 275–301), and introduce a predicate for validity by combining syntactical constructions with a modeltheoretic semantics. The resulting logic, called ∈I+ (Epsilon-I plus), is able to express its own finitary consequence relation. We show that every intuitionistic (or classical) abstract logic extends to a logic which has the semantic features of ∈I (of ∈I+), respectively, namely a truth predicate that satisfies the Tarski biconditionals, a predicate for falsity (as intuitionistic negation), means for propositional (self-) reference, and in the case of ∈I+ also predicates for validity and logical consequence. Applications of our construction to other non-classical abstract logics remain to be further investigated.

Logics for “generally” were introduced for handling assertions with vague notions,such as “generally”, “most”, “several”, etc., by generalized quantifiers, ultrafilter logic being an interesting case. Here, we show that ultrafilter logic can be faithfully embedded into a first-order theory of certain functions, called coherent. We also use generic functions (akin to Skolem functions) to enable elimination of the generalized quantifier. These devices permit using methods for classical first-order logic to reason about consequence in ultrafilter logic.

With parametric logic, we propose a unified approach to deduction and induction, both viewed as particular instances of a generalized notion of logical consequence. This generalized notion of logical consequence is Tarskian, in the sense that if a sentence φ is a generalized logical consequence of a set T of premises, then the truth of T implies the truth of φ. So in particular, if φ can be induced from T, then the truth of T implies the truth of φ. The difference between deductive and inductive consequences lies in the process of deriving the truth of φ from the truth of T.

We extend classical Propositional Logic (PL) by adding a new primitive binary connective $\varphi|\psi$, intended to represent the "superposition" of sentences $\varphi$ and $\psi$, an operation motivated by the corresponding notion of quantum mechanics, but not intended to capture all aspects of the latter as they appear in physics. To interpret the new connective, we extend the classical Boolean semantics by employing models of the form $\langle M,f\rangle$, where $M$ is an ordinary two-valued assignment for the sentences of PL and $f$ is a choice function for all pairs of classical sentences. In the new semantics $\varphi|\psi$ is strictly interpolated between $\varphi\wedge\psi$ and $\varphi\vee\psi$. By imposing several constraints on the choice functions we obtain corresponding notions of logical consequence relations and corresponding systems of tautologies, with respect to which $|$ satisfies some natural algebraic properties such as associativity, closedness under logical equivalence and distributivity over its dual connective. Thus various systems of Propositional Superposition Logic (PLS) arise as extensions of PL. Axiomatizations for these systems of tautologies are presented and soundness is shown for all of them. Completeness is proved for the weakest of these systems. For the other systems completeness holds if and only if every consistent set of sentences is extendible to a consistent and complete one, a condition whose truth is closely related to the validity of the deduction theorem.

10 - ALL: Formalizing Access-Limited Reasoning

We presentepistemic multilateral logic, a general logical framework for reasoning involving epistemic modality. Standard bilateral systems use propositional formulae marked with signs for assertion and rejection. Epistemic multilateral logic extends standard bilateral systems with a sign for the speech act of weak assertion (Incurvati & Schlöder, 2019) and an operator for epistemic modality. We prove that epistemic multilateral logic is sound and complete with respect to the modal logic$\mathbf {S5}$modulo an appropriate translation. The logical framework developed provides the basis for a novel, proof-theoretic approach to the study of epistemic modality. To demonstrate the fruitfulness of the approach, we show how the framework allows us to reconcile classical logic with the contradictoriness of so-called Yalcin sentences and to distinguish between various inference patterns on the basis of the epistemic properties they preserve.

A formal language of two-valued logic is developed, whose terms are formulas of the language of Kleene's three-valued logic. The atomic formulas of the former language are pairs of formulas of the latter language joined by “consequence” operators. These operators correspond to the three “sensible” types of consequence (strong-strong, strong-weak and weak-weak) in Kleene's logic in analogous way as the implication connective in the classical logic corresponds to the classical consequence relation. The composed formulas of the considered language are built from the atomic ones by means of the classical connectives and quantifiers. A deduction system for the developed language is given, consisting of a set of decomposition rules for sequences of formulas. It is shown that the deduction system is sound and complete.

We introduce a new approach to probabilistic logic programming in which probabilities are defined over a set of possible worlds. More precisely, classical program clauses are extended by a subinterval of [0,1] that describes a range for the conditional probability of the head of a clause given its body. We then analyze the complexity of selected probabilistic logic programming tasks. It turns out that probabilistic logic programming is computationally more complex than classical logic programming, More precisely, the tractability of special cases of classical logic programming generally does not carry over to the corresponding special cases of probabilistic logic programming. Moreover, we also draw a precise picture of the complexity of deciding and computing tight logical consequences in probabilistic reasoning with conditional constraints in general. We then present linear optimization techniques for deciding satisfiability and computing tight logical consequencesof probabilistic logic programs. These techniques are efficient in the special case in which we have little relevant purely probabilistic knowledge. We finally show that probabilistic logic programming under certain syntactic and semantic restrictions is closely related to van Emden's quantitative deduction, and thus has computational properties similar to calssical logic programming. Based on this result, we present an efficient approximation technique for probabilistic logic programming.