Logic II: introduction to non-classical logics syllabus

The Computer Journal, 1992

No doubt every reader of this journal is aware that computer science is becoming infiltrated by a strange breed of people called logicians, who try to convince computer people that their arcane symbolism and obscure terminology are just what is needed to solve the software crisis, the hardware crisis, and any other difficulties that the computer world finds itself facing. Unfortunately the symbolism and the jargon can be very off-putting to anyone who has not already become immersed in formal logic; I have often met people who work with computers and are aware of how important logic is claimed to be by its devotees, and who feel that they really ought one day to make an effort to penetrate its mysteries, but who have not known how to set about doing so. This article and its sequel ('Logic as a Formal Method') are intended as a fairly gentle initiation into what logic is about and what it has to offer computer scientists. They are inevitably very sketchy and incomplete -more like the brochures that can be picked up at a travel agent's than a proper guide-book -but it is to be hoped that some, at least, of my readers will come away with a clearer picture of what lies in store for them if they decide to follow up the more detailed references.

2010

The first edition of this book appeared in 2001. It comprised about 250 pages and covered only propositional logic. It was a good but limited introduction to non-classical logic. Its most distinctive features were its concise and clear coverage of many important systems of non-classical propositional logic,

2012

Clarke's book titled Logic printed by Longman Green (London) in the year 1909 has been extensively used. Logical arguments can be divided into two major parts: Hypothetical Syllogism and Categorical Syllogism. The word 'syllogism' means an argument or form of reasoning in which two statements or premises are made and a logical conclusion is drawn from them. Here we shall see the first division i.e., hypothetical syllogism and when you are confident you can move to categorical syllogism. The word 'hypothetical' is derived from 'hypothesis' which means a statement that begins with a hypothetical clause like 'if' which shows a possibility or probability of an action. The word 'categorical' means without qualifications or conditions; absolute; positive; directed; explicit; said of a statement or theory.

Brief summaries of selected sections of Graham Priest’s Introduction to Non-Classical Logic: From If to Is, chosen for their relevance to a study of the logic of Gilles Deleuze

We give a proof-theoretic as well as a semantic characterization of a logic in the signature with conjunction, disjunction, negation, and the universal and existential quantifiers that we suggest has a certain fundamental status. We present a Fitch-style natural deduction system for the logic that contains only the introduction and elimination rules for the logical constants. From this starting point, if one adds the rule that Fitch called Reiteration, one obtains a proof system for intuitionistic logic in the given signature; if instead of adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a proof system for orthologic; by adding both Reiteration and Reductio, one obtains a proof system for classical logic. Arguably neither Reiteration nor Reductio is as intimately related to the meaning of the connectives as the introduction and elimination rules are, so the base logic we identify serves as a more fundamental starting point and common ground between proponents of intuitionistic logic, orthologic, and classical logic. The algebraic semantics for the logic we motivate proof-theoretically is based on bounded lattices equipped with what has been called a weak pseudocomplementation. We show that such lattice expansions are representable using a set together with a reflexive binary relation satisfying a simple first-order condition, which yields an elegant relational semantics for the logic. This builds on our previous study of representations of lattices with negations, which we extend and specialize for several types of negation in addition to weak pseudocomplementation; in an appendix, we further extend this representation to lattices with implications. Finally, we discuss adding to our logic a conditional obeying only introduction and elimination rules, interpreted as a modality using a family of accessibility relations.

Classical and Nonclassical Logics, 2020

Typically, a logic consists of a formal or informal language together with a deductive system and/or a modeltheoretic semantics. The language has components that correspond to a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record arguments that are valid for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions for at least part of the language. The following sections provide the basics of a typical logic, sometimes called "classical elementary logic" or "classical first-order logic". Section 2 develops a formal language, with a rigorous syntax and grammar. The formal language is a recursively defined collection of strings on a fixed alphabet. As such, it has no meaning, or perhaps better, the meaning of its formulas is given by the deductive system and the semantics. Some of the symbols have counterparts in ordinary language. We define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion. The other sentences (if any) in an argument are its premises. Section 3 sets up a deductive system for the language, in the spirit of natural deduction. An argument is derivable if there is a deduction from some or all of its premises to its conclusion. Section 4 provides a model-theoretic semantics. An argument is valid if there is no interpretation (in the semantics) in which its premises are all true and its conclusion false. This reflects the longstanding view that a valid argument is truth-preserving. In Section 5, we turn to relationships between the deductive system and the semantics, and in particular, the relationship between derivability and validity. We show that an argument is derivable only if it is valid. This pleasant feature, called soundness, entails that no deduction takes one from true premises to a false conclusion. Thus, deductions preserve truth. Then we establish a converse, called completeness, that an argument is valid only if it is derivable. This establishes that the deductive system is rich enough to provide a deduction for every valid argument. So there are enough deductions: all and only valid arguments are derivable. We briefly indicate other features of the logic, some of which are corollaries to soundness and completeness.

Logica Universalis, 2023

The paper presents a contra-classical dialectic logic, inspired and motivated by Hegels dialectics. Its axiom schemes are (H1) ∶ ⊢ϕ→¬ϕ (H2) ∶ ¬⊢ϕ→ϕ (H3) ∶ ⊢(ϕ→ψ)→(ϕ→¬ψ) (H4) ∶ ⊢(ϕ→¬ψ)→(ϕ→ψ) (0.1) Thus, in a sense, this dialectic logic is a kind of "mirror image" of connexive logic. The informal interpretation of '→' emerging from the above four axiom schemes is not of a conditional (or implication); rather, it is the relation of determination in the presence of truth-value gaps: ϕ→ψ is read as ϕ determines ψ, namely, necessarily, if ϕ is true, then ψ is either true or false, not gappy. As far as I know, such a connective has not been considered before in the literature.

Logic Journal of IGPL, 2003

The purpose of this paper is to take some of the mystery out of what is known as nonmonotonic logic, by showing that it is not as unfamiliar as may at first sight appear. In fact, it is easily accessible to anybody with a background in classical propositional logic, provided that certain misunderstandings are avoided and a tenacious habit is put aside. In effect, there are logics that act as natural bridges between classical consequence and the principal kinds of nonmonotonic logic to be found in the literature. Like classical logic, they are perfectly monotonic, but they already display some of the distinctive features of the nonmonotonic systems. As well as providing easy conceptual passage to the nonmonotonic case these logics, which we call paraclassical, have an interest of their own.

Essay on the Principles of Logic: A Defense of Logical Monism, 2023

Michael Wolff's new preface to my translation of his Essay on the Principles of Logic: A Defense of Logical Monism.

BRAIN. Broad Research in Artificial Intelligence and Neuroscience , 2014

Logic is a set of well-formed formulae, along with an inference relation. But the Classical Logic is bivalent; for this reason, very limited to solve problems with uncertainty on the data. It is well-known that Artificial Intelligence requires Logic. Because its Classical version shows too many insufficiencies, it is very necessary to introduce more sophisticated tools, as may be Non-Classical Logics; amongst them, Fuzzy Logic, Modal Logic, Non-Monotonic Logic, Para-consistent Logic, and so on. All them in the same line: against the dogmatism and the dualistic vision of the world: absolutely true vs. absolutely false, black vs. white, good or bad by nature, Yes vs. No, 0 vs. 1, Full vs. Empty, etc. We attempt to analyze here some of these very interesting Classical and modern Non-Classical Logics.