Mathematical Foundation of a Functional Implementation of the CNF Algorithm

The conjunctive normal form (CNF) algorithm is one of the best known and most widely used algorithms in classical logic and its applications. In its algebraic approach, it makes use in a loop of a certain well-defined operation related to the “distributivity” of logical disjunction versus conjunction. For those types of implementations, the loop iteration runs a comparison between formulas to decide when to stop. In this article, we explain how to pre-calculate the exact number of loop iterations, thus avoiding the work involved in the above-mentioned comparison. After that, it is possible to concatenate another loop focused now on the “associativity” of conjunction and disjunction. Also for that loop, we explain how to calculate the optimal number of rounds, so that the decisional comparison phase for stopping can be also avoided

Categoricity and Negation. A Note on Kripke’s Affirmativism

We argue that, if taken seriously, Kripke's view that a language for science can dispense with a negation operator is to be rejected. Part of the argument is a proof that positive logic, i.e., classical propositional logic without negation, is not categorical

From Many-Valued Consequence to Many-Valued Connectives

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.Comment: Updated version [corrections of an incorrect claim in first version; two bib entries added

Some Logical Notations for Pragmatic Assertions

The pragmatic notion of assertion has an important inferential role in logic. There are also many notational forms to express assertions in logical systems. This paper reviews, compares and analyses languages with signs for assertions, including explicit signs such as Frege’s and Dalla Pozza’s logical systems and implicit signs with no specific sign for assertion, such as Peirce’s algebraic and graphical logics and the recent modification of the latter termed Assertive Graphs. We identify and discuss the main ‘points’ of these notations on the logical representation of assertions, and evaluate their systems from the perspective of the philosophy of logical notations. Pragmatic assertions turn out to be useful in providing intended interpretations of a variety of logical systems

On Affine Logic and {\L}ukasiewicz Logic

The multi-valued logic of {\L}ukasiewicz is a substructural logic that has been widely studied and has many interesting properties. It is classical, in the sense that it admits the axiom schema of double negation, [DNE]. However, our understanding of {\L}ukasiewicz logic can be improved by separating its classical and intuitionistic aspects. The intuitionistic aspect of {\L}ukasiewicz logic is captured in an axiom schema, [CWC], which asserts the commutativity of a weak form of conjunction. This is equivalent to a very restricted form of contraction. We show how {\L}ukasiewicz Logic can be viewed both as an extension of classical affine logic with [CWC], or as an extension of what we call \emph{intuitionistic} {\L}ukasiewicz logic with [DNE], intuitionistic {\L}ukasiewicz logic being the extension of intuitionistic affine logic by the schema [CWC]. At first glance, intuitionistic affine logic seems very weak, but, in fact, [CWC] is surprisingly powerful, implying results such as intuitionistic analogues of De Morgan's laws. However the proofs can be very intricate. We present these results using derived connectives to clarify and motivate the proofs and give several applications. We give an analysis of the applicability to these logics of the well-known methods that use negation to translate classical logic into intuitionistic logic. The usual proofs of correctness for these translations make much use of contraction. Nonetheless, we show that all the usual negative translations are already correct for intuitionistic {\L}ukasiewicz logic, where only the limited amount of contraction given by [CWC] is allowed. This is in contrast with affine logic for which we show, by appeal to results on semantics proved in a companion paper, that both the Gentzen and the Glivenko translations fail.Comment: 28 page