First-Order Classical Modal Logic

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics.     

The Relation between Formal and Informal Logic

The issue of the relationship between formal and informal logic depends strongly on how one understands these two designations. While there is very little disagreement about the nature of formal logic, the same is not true regarding informal logic, which is understood in various (often incompatible) ways by various thinkers. After reviewing some of the more prominent conceptions of informal logic, I will present my own, defend it and then show how informal logic, so understood, is complementary to formal logic.     

Harmony and Autonomy in Classical Logic

Michael Dummett and Dag Prawitz have argued that a constructivist theory of meaning depends on explicating the meaning of logical constants in terms of the theory of valid inference, imposing a constraint of harmony on acceptable connectives. They argue further that classical logic, in particular, classical negation, breaks these constraints, so that classical negation, if a cogent notion at all, has a meaning going beyond what can be exhibited in its inferential use.I argue that Dummett gives a mistaken elaboration of the notion of harmony, an idea stemming from a remark of Gerhard Gentzen"s. The introduction-rules are autonomous if they are taken fully to specify the meaning of the logical constants, and the rules are harmonious if the elimination-rule draws its conclusion from just the grounds stated in the introduction-rule. The key to harmony in classical logic then lies in strengthening the theory of the conditional so that the positive logic contains the full classical theory of the conditional. This is achieved by allowing parametric formulae in the natural deduction proofs, a form of multiple-conclusion logic.     

IF Modal Logic and Classical Negation

The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as ‘Simple IF modal logic’ (IFML _s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML _s is not equivalent to any formula of basic modal logic (ML). We generalize the notion of bisimulation familiar from ML; the resulting asymmetric simulation concept is used to prove that IFML _s is not closed under complementation. In fact we obtain a much stronger result: the only IFML _s formulas admitting their classical negation to be expressed in IFML _s itself are those whose truth-condition is in fact expressible in ML.     

Formal Logic vs. Philosophical Argument

The wider topic to which the content of this paper belongs is that of the relationship between formal logic and real argumentation. Of particular potential interest in this connection are held to be substantive arguments constructed by philosophers reputed equally as authorities in logical theory. A number of characteristics are tentatively indicated by the author as likely to be encountered in such arguments. The discussion centers afterwards, by way of specification, on a remarkable piece of argument quoted in Cicero’s dialog On Divination and ascribed to Stoic thinkers. The Stoics’ formal theory of inference is summarily referred to in this context, with special emphasis on their basic deductive schemata (‘indemonstrables’), some of them recognizable as links in the overall structure of the quoted argument. The main lines of Cicero’s criticism of the Stoic argument are next commented upon, with emphasis on his implied view as to the requirements of a good argument. Towards the end of the paper, a few considerations are added on the changes in the prevailing style of argumentation conspicuous in the three famous Roman Stoics.     

Nice Embedding in Classical Logic

It is shown that a set of semi-recursive logics, including many fragments of CL (Classical Logic), can be embedded within CL in an interesting way. A logic belongs to the set iff it has a certain type of semantics, called nice semantics. The set includes many logics presented in the literature. The embedding reveals structural properties of the embedded logic. The embedding turns finite premise sets into finite premise sets. The partial decision methods for CL that are goal directed with respect to CL are turned into partial decision methods that are goal directed with respect to the embedded logics.     

Classical Logic and the Strict Tolerant Hierarchy

Journal of Philosophical Logic - In their recent article “A Hierarchy of Classical and Paraconsistent Logics”, Eduardo Barrio, Federico Pailos and Damien Szmuc (BPS hereafter) present...     

Logical Foundations and Kant's Principles of Formal Logic

The abstract status of Kant's account of his ‘general logic’ is explained in comparison with Gödel's general definition of a formal logical system and reflections on ‘abstract’ (‘absolute’) concepts. Thereafter, an informal reconstruction of Kant's general logic is given from the aspect of the principles of contradiction, of sufficient reason, and of excluded middle. It is shown that Kant's composition of logic consists in a gradual strengthening of logical principles, starting from a weak principle of contradiction that tolerates a sort of contradictions in predication, and then proceeding to the (constructive) principle of sufficient reason, and to a classical-like logic, which includes the principle of excluded middle. A first-order formalisation is applied to this reconstruction, which reveals implicit modalities in Kant's account of logic, and confirms the implementability of Kant's logic into a sound and complete formal system.     

Logic and the Classical Theory of Mind

I extract several common assumptions in the Classical Theory of Mind (CTM) – mainly of Locke and Descartes – and work out a partial formalisation of the logic implicit in CTM. I then define the modal (logical) properties and relations of propositions, including the modality of conditional propositions and the validity of argument, according to the principles of CTM: that is, in terms of clear and distinct ideas, and without any reference to either possible worlds, or deducibility in an axiomatic system, or linguistic convention.     

A Proof System for Classical Logic

An operation on inferential rules, called H-operation, is used to minimize the axiom basis for classical logic.     

A Second Pretabular Classical Relevance Logic

Pretabular logics are those that lack finite characteristic matrices, although all of their normal proper extensions do have some finite characteristic matrix. Although for Anderson and Belnap’s relevance logic R, there exists an uncountable set of pretabular extensions (Swirydowicz in J Symb Log 73(4):1249–1270, 2008), for the classical relevance logic \( \hbox {KR} = \hbox {R} + \{(A\,\, \& \sim A)\rightarrow B\}\) there has been known so far a pretabular extension: \({\mathcal L}\) (Galminas and Mersch in Stud Log 100:1211–1221, 2012). In Section 1 of this paper, we introduce some history of pretabularity and some relevance logics and their algebras. In Section 2, we introduce a new pretabular logic, which we shall name \({\mathcal M}\), and which is a neighbor of \({\mathcal L}\), in that it is an extension of KR. Also in this section, an algebraic semantics, ‘\({\mathcal M}\)-algebras’, will be introduced and the characterization of \({\mathcal M}\) to the set of finite \({\mathcal M}\)-algebras will be shown. In Section 3, the pretabularity of \({\mathcal M}\) will be proved.     

Falsification-Aware Semantics and Sequent Calculi for Classical Logic

Journal of Philosophical Logic - In this study, falsification-aware semantics and sequent calculi for first-order classical logic are introduced and investigated. These semantics and sequent...     

Grishin Algebras and Cover Systems for Classical Bilinear Logic

Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (??propositions??) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, by a map that preserves all existing joins and meets. This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions.     

近年来我国辩证逻辑研究述评

近年来,我国辩证逻辑研究继续在探索中不断前进,既取得了一些重要进展,也有问题悬而未决;既对老问题不断追问,也向新领域大胆开拓;既联系现实生活的视域,也推动理论自身的更新.反映出辩证逻辑学科学人孜孜求索的努力和收获.     

An Intuitionistic Completeness Theorem for Classical Predicate Logic

This paper presents an intuitionistic proof of a statement which under a classical reading is logically equivalent to Gödel’s completeness theorem for classical predicate logic.